pi.math.cornell.edupi.math.cornell.edu/~srubin/teaching/esslli06/reader.pdf · logic and...

Logic and computation

in finitely presentable infinite structures

Valentin Goranko1, Sasha Rubin2

1School of Mathematics, University. of the Witwatersrand, Johannesburg2Department of Computer Science, University of Auckland

E-mails: [email protected], [email protected]

Course outline: June 26, 2006

This is a tentative outline of the course, which will comprise ten 40-min lectures, delivered2 per day, with a 5 min break in between.

More course material, including updated notes, exercises, slides, and references

will be placed on the course website:

http://www.math.wisc.edu/∼rubin/teaching/teaching.html

Background. This is an advanced course and the participants are expected to have somebackground on first-order logic, finite and pushdown automata, and modal logic. Basicknowledge of Buchi automata, MSO, and extended modal logics such as LTL and CTLwould be an advantage. Some background notes are included in this reader.

Attendance of the ESSLLI introductory course on ‘Verification of infinite state systems’presented by Montanari and Puppis is a strongly recommended supplement to the course.

This course reader includes, with the kind permission of the authors, the following publica-tions:

1) W. Thomas. A short introduction to infinite automata. In Proc. of DLT 2002, LNCS2295, pp. 130-144. Springer, 2002.

2) W. Thomas. Constructing infinite graphs with a decidable MSO-theory. In Proc. ofMFCS 2003, LNCS 2747. Springer, 2003.

3) A. Carayol and S. Wohrle. The Caucal hierarchy of infinite graphs in terms of logic andhigher-order pushdown automata. In Proc. of FSTTCS 2003, LNCS 2914, pp. 112-123.Springer, 2003.

4) A. Blumensath and E. Gradel, Finite presentations of infinite structures: Automata andinterpretations, Theory of Computing Systems, vol. 37, pp. 641 - 674, 2004.

1

Day 1

Lecture 1: Introduction

Infinite structures are used to model theoretically unbounded systems or data-structures,such as transition systems of models of computation with infinite or unbounded memory(stacks, queues, FIFO channels) or state space (parameterized network systems, countersystems, Petri nets) etc.

Model checking is the procedure of deciding whether a ‘system’ (program, hardware, proto-col, computer network, etc) satisfies a formal specification; in other words, whether it doeswhat it is supposed to do.

Logical languages are most natural choice of a medium for formal specification and verifica-tion of properties of structures, because they are syntactically unambiguous, closed undernatural operations (Booleans, projections, etc.), and have explicit and precise formal se-mantics. The most popular natural logical languages used for specification and verificationare: first-order logic FO (suitably extended with reachability operators), (weak) monadicsecond-order logic (W)MSO, modal and temporal logics such as LTL, CTL, etc.

However, model-checking of a formal specification, just like any other computational task,is generally undecidable (if algorithmically definable at all) on infinite structures.

Thus, the central problem arising in this area is to identify and study classes of finitelypresentable infinite structures and respective logical languages where the model checkingproblems can be solved algorithmically.

Such infinite structures should be representable in a finite way and should be algorithmicallyaccessible, i.e., operations performed on them for the model checking procedure should becomputable.

Natural examples of finitely presentable infinite structures include integers, infinite trees,grids, etc.; more generally, every recursive structure. A natural construction that producesinfinite structures from finite ones is tree-unfolding of finite transition systems; in terms ofit, every finite transition system is a finitary presentation of its (generally infinite) unfolding.

Other types of finitary representations and abstractions include: configuration graphs ofcomputing devices, e.g., pushdown automata; presentations by means of effective rewritesystems, (e.g., prefix-recognizable graphs), presentations recognizable by computing devises(automatic structures, recursive structures, etc.).

The classes of structures and the logical languages for which model checking is decidable onthese classes form two opposite hierarchies: the larger the class of structures C, the smallerthe language L which can be effectively model checked on every structure from L, and viceversa. Thus, there is a trade-off between effectiveness and expressiveness. In this course wewill focus on some cases where this trade-off is particularly successful and fruitful.

References: [8],[53],[54],[55].

2

Day 1

Lecture 2: Background

The course centers around the interplay between formal languages and logic. This lecturewill consist of definitions and illustrative examples of the basic concepts.

The primary models of computation will be resource bounded Turing machines; specif-ically, finite-state automata, rational-transductions, and pushdown-automata [35]. Also,ω-automata, tree-automata and ω-tree automata [53].

The main logical concepts include: signatures, structures, logical languages (FO, MSO,WMSO), terms, formulas, theories, and most importantly definability and interpretability[29, 33].

Words. The following operations on sets of words are basic: Boolean operations union,intersection, and negation; logical operations projection, co-projection, and permutationof co-ordinates; automata theoretic operations product UV or UL, finite power U?, andω-power Lω; and the following language theoretic operations: (i) for each a ∈ Σ?, thea-successor function suca : w 7→ wa, (ii) the prefix-relation u �p v or u �p α, (iii) theequal-length relation el.

Trees. We consider vertex-labelled trees with directions. So, given a labelling alphabetΣ, and a direction alphabet ∆ of size k, a k-ary branching Σ-labelled tree T is a functionT : D → Σ from a prefix-closed D ⊂ ∆?, where ∆ = {1, · · · , k} is the set of directions.The tree is called finite if D is finite. The set of k-branching Σ-labelled trees is denoted bytreesk(Σ) if the trees are finite, and treesω

k (Σ) if they are infinite. Basic operations ontrees include the usual Boolean and logical ones, as well as the following language theoreticoperations:

i) sucda(T ) extends every leaf of T in direction d and labels each new vertex with the

symbol a,

ii) T �ext T ′ if T ′ extends T , and

iii) T ≡dom T ′ if they have the same domain.

Structures. The following structures are central to this course.

- (N, S) where S : n 7→ n + 1.

- ({0, 1}?, suc0, suc1).

- W(Σ) = (Σ?; (suca)a∈Σ,�p, el).

- Wω(Σ) = (Σ? ∪ Σω; (suca)a∈Σ,�p, el).

- Tk(Σ) = (Treesk(Σ), (sucda)d≤k,a∈Σ,�ext,≡dom).

- T ωk (Σ) = (Treesk(Σ) ∪ Treesω

k (Σ), (sucda)d≤k,a∈Σ,�ext,≡dom).

Interpretations of structures. Loosely, a structure A is interpretable in structure Bif one can define an isomorphic copy of A in B. Then all definability questions concerningA can be asked of B instead.

In this lecture, we will describe different types of interpretations, and give immediate con-sequences such as the preservation of decidability.

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Definition [Interpretation] An interpretation of structure A = (A; (RAi )i) in structure

B consists of the following data:

(∆(x), ε(x, y), φi(x)),

where ε(x, y) interprets the equality, so that A is isomorphic to the quotient structure

(∆B ; (φBi )i)/ε

B .

Here ΦB denotes the tuples b such that B |= Φ(b).

The surjective mapping µ : ∆B → A induced by the isomorphism is called the co-ordinatemap. We say A is interpretable in B.

We identify two main types of interpretations, depending on the free variables of the domainformula:

Type 1) If the free variables of ∆ are first-order, then µ sends elements B to elements ofA. We call the interpretation FO, WMSO, or MSO, depending on whether all theformulas are FO, WMSO, or MSO respectively.

Type 2) If the free variables of ∆ are (W)MSO, then µ sends (finite) subsets of B toelements of A. We call it a (finite-)set interpretation.

Examples.

1. W({0, 1}) and (N,+, |2) are each FO-interpretable in the other.1

2. Both (N, S) and (Q,≤) are MSO interpretable in ({0, 1}?, σ0, σ1).

3. (N,+) is finite-set interpretable in (N, S) via the map sending a finite set X ⊂ N tothe natural number Σi∈X2i.

4. W({0, 1}) is finite-set interpretable in (N, S).

Proposition.

1. If A has decidable L-theory and B is L-interpretable in A, then B has decidable Ltheory. Here L is one of FO, WMSO, or MSO.

2. If A has decidable (W)MSO theory, and B is (finite-)set interpretable in A, then Bhas decidable FO-theory.

References. Useful background references are: [35],[53], [29],[33].

Also, the following very useful lecture notes, available online, can be consulted:

http://www-i7.informatik.rwth-aachen.de/download/papers/thomas/thomas97b.ps,

http://www.labri.fr/perso/igw/Papers/igw-eefss01.ps.

1Here x|2y if x divides y and x is a power of 2.

4

Day 2

Lecture 3: Automata and MSO

In this lecture, we show how to use automata to prove decidability of certain theories. Theidea is that, under a suitable coding, definable relations are computable by finite automata.Although this already gives decidability, the remarkable fact is that the converse holdstoo: the regular relations are, via the coding, definable. Thus, there is a tight connectionbetween definability and regularity. We illustrate the technique with MSO(N, S), explainthe link with word-models and the usual presentation of this result, and then state thecorresponding theorem for two-successors.

Buchi’s Theorem.

Coding: A set A ⊂ N can be viewed as an infinite string str(A) over alphabet {0, 1}, witha 1 in the ith position if and only if i ∈ A. Similarly, a tuple of sets (A1, · · · , Ar) can beviewed as an infinite string over alphabet {0, 1}r . Moreover, write str(R) for the set ofstrings str(A1, · · · , Ar) for which (A1, · · · , Ar) ∈ R.

For instance, the set R = {A ⊂ N | A is infinite} corresponds to the set of strings str(R) ={w ∈ {0, 1}? | w has infinitely many 1s}.

For every formula φ(X1, · · · ,Xr) of the MSO theory of (N, S), we associate the language ofstrings str(A1, · · · , Ar) for which (N, S) |= φ(A1, · · · , Ar).

Theorem. Let R be a set of tuples (A1, · · · , Ar), where Ai ⊂ N. The following areequivalent:

1. R is MSO-definable in (N, S).

2. str(R) is ω-regular.

3. str(R) is FO-definable in Wω({0, 1})

Rabin’s Theorem.

Coding: A set A ⊂ {1, · · · , k}? can be viewed as an infinite k-branching tree whose verticesare labelled with symbols from {0, 1}. Namely, define tree(A) as mapping w ∈ {1, · · · , k}?

to 1 if and only if w ∈ A. Similarly, a tuple of sets (A1, · · · , Ar) corresponds to a tree overalphabet {0, 1}k ; and a set R of tuples corresponds to the set tree(R) of trees.

Rabin proved that ω-tree automata are closed under complementation. This is the hardingredient in the following theorem.

Theorem: Let R be a set of tuples (A1, · · · , Ar), where Ai ⊂ {1, · · · , k}?. The followingare equivalent:

1. R is MSO-definable in ({1, · · · , k}?, σ1, · · · , σk).

2. tree(R) is recognised by a tree-automaton.

3. tree(R) is first-order definable in T ωk ({1, · · · , k}).

Summary.

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The following correspondences imply the decidability of the corresponding theory.

WMSO(N, S) ≡ automata on finite strings

MSO(N, S) ≡ ω-automata on strings

WMSO({1, · · · , k}?, σ1, · · · , σk) ≡ automata on k-branching finite-trees

MSO({1, · · · , k}?, σ1, · · · , σk) ≡ ω-automata on k-branching infinite-trees

References: [28], [54],[49],[53],[48].

Day 2

Lecture 4: Tree-interpretable structures and MSO

Following Rabin’s theorem [48] on decidability of MSO of the infinite binary tree T2, therehas been a unceasing quest for identifying even larger classes of structures with decidableMSO. One way to obtain finitely-presentable infinite structures with decidable theories isvia interpretation in a fixed structure with decidable theory. This idea, permeating thecourse, will be illustrated in this lecture, using as a point of departure Rabin’s theorem.

A structure A is tree-interpretable if there is a (one-dimensional) MSO-interpretation of Ain T2.

Theorem: Every tree-interpretable structure has decidable MSO-theory.

Here are some important cases of tree-interpretable structures that will be introduced anddiscussed in the course:

1. Concrete structures, such as: the k-branching trees for any k, ω-branching trees, thelinear order of the rationals, etc.

2. Pushdown graphs, studied by Muller and Schupp [45, 46, 47]. These are the transition(configuration) graphs of pushdown automata [35, 1], defined as follows: given apushdown automata A, a configuration in A is a tuple, consisting of the current stateand the content of the stack; the vertices in the configuration graph are all (infinitelymany, because of the unbounded stack) configurations, and the edges are the possibleone-step transitions between configurations.

3. Prefix-recognizable graphs, introduced and studied by Caucal [17, 20]. These graphsare defined by prefix-rewriting systems on words as follows. A transition graph G (withpossibly many types of transitions) is prefix-recognizable if the set of its vertices canbe labelled by a regular language W of words in a finite alphabet, in such a waythat all transitions in G can be determined by a finite set of rewrite rules of the typeU1 →a U2 where U1, U2 are regular subsets of W . The rules work as follows: there isa transition of type a from a vertex v1 to a vertex v2 iff there is a rule U1 →a U2 andwords v ∈ W , u1 ∈ U1, u2 ∈ U2 such that v1 = u1v and v2 = u2v; i.e., if the transitioncan be recognized by a rewrite rule by looking only at the prefixes of the labels ofthe vertices. The pushdown graphs are particular case of prefix-recognizable graphs,where U1 and U2 in all rules are singletons.In fact, tree-interpretable structures were introduced in [7] as a generalization of thenotion of a prefix-recognizable graph to arbitrary relational structures.

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4. A hierarchy of infinite graphs, studied by Courcelle [23, 22, 24]. These are definedin terms of graph-rewriting systems acting on vertices or hyper-edges (VR- and HR-recognizable graphs) or algebraically, by way of recursive equations (VR- and HR-equational graphs). The VR-equational graphs are equivalent to prefix-recognizablegraphs.

In this lecture we will discuss some examples, properties, and different characterizations ofthe family of tree-interpretable structures.

References: [48], [47],[23, 22],[18], [6], [55], [7],[57].

Day 3

Lecture 5: Caucal hierarchy and MSO

The family of tree-interpretable structures can be further extended by applying other oper-ations preserving the decidability of MSO. For instance, such operation is the tree-unfoldingof a graph. That operation, applied to a single vertex with two labelled loops producesthe infinite binary tree; whence Rabin’s theorem follows. The unfolding operation was ex-tended in the 1980’s by Muchnik (see [52, 55]) to a construction, now known as Muchnik’stree iteration, which preserves the decidability of the MSO, too.

In [19] Caucal builds up a hierarchy of infinite structures with decidable MSO, by startingfrom the class of finite graphs and alternatively applying two operations preserving thedecidability of the MSO-theory, viz., unfolding and inverse rational mappings (a specialcase of MSO-interpretations).

In [15] Cachat shows, using parity games, that the graphs in Caucal’s hierarchy are equiv-alent to so called higher-order pushdown systems (arising from ‘higher-order pushdownautomata’ involving a hierarchy of stack of stacks).

Furthermore, in [16], Carayol and Wohrle show that Caucal’s hierarchy coincides withthe hierarchy obtained by alternating (a version of) Muchnik’s tree-iteration constructionand MSO-transductions (MSO interpretations in a disjoint union of copies of the targetstructure), as well as with the family of graphs obtained as the ε-closure of configurationgraphs of higher-order pushdown automata.

Important parameters of finitely presentable infinite structures are their tree-width andclique-width. In pursuit of ultimate characterization of the class of structures with decidableMSO, Seese [51] raised conjecture that every (set of) infinite graph(s) having a decidableMSO-theory is the image of a (set of) trees under an MSO-transduction; equivalently (asshown by Courcelle and others), iff it has a bounded clique-width. This conjecture hasrecently been partly confirmed by Courcelle and Sang-il Oum.

In this lecture we will build up Caucal’s hierarchy, and will discuss its different character-izations and the boundaries of the class of structures with decidable MSO, in particularSeese’s conjecture.

References: [52], [18], [17], [16], [51],[25].

7

Day 3

Lecture 6: Automatic structures 1

A natural and direct way to present structures in a finite way, is to have the structuresthemselves be computable. Here is a general definition: Fix a model of computation M. Astructure is M-computable if its domain and atomic operations are computable by machinesin the class M.

Taking M to be the class of all Turing machines, we get the computable structures [30]. How-ever, even basic operations on computable relations, like projection, are not computable. Ingeneral, it is only the quantifier-free fragment of first-order logic that is effectively decidableon computable structures.

On the opposite end of the spectrum, if we take M to be a class of automata, we get whatmay be called automatic structures [34, 39, 5]. Many problems on automata are decidable.In particular, the closure properties of the automata presenting an automatic structure A,imply that every definable relation on A is also computable by an automaton.

Each notion of automaton ♦ ∈ {word, ω-word, tree, ω-tree} yields a notion of ♦-automaticstructure. In this lecture, we will establish basic properties of automatic structures.

As an illustrative example, the structure ({0, 1}?,+), where + is the usual addition on thebinary-representation of natural numbers, is word-automatic. This is because the domainand addition operation are computable by word-automata.

The exact relationship between automatic structures, which have in general only decidableFO-theory, and say tree-interpretable structures which have decidable MSO-theory, is viaset interpretations. For instance, the word-automatic structures are exactly those that arefinite-set interpretable in (N, S).

Definition [Automatic Structure]. A relational structure with equality2 (A; (Ri)i) is♦-automatic over alphabet Σ if it satisfies the following conditions:

1. the domain A consists of ♦s over Σ,

2. the domain A is ♦-regular;

3. every atomic relation (including equality) R ⊂ Ari in A is ♦-regular.

Theorem [Definability on Automatic Structures.] Let A be a ♦-automatic struc-ture. Every relation R ⊂ Ak that is first-order definable in A is computable by an ♦-automaton. Moreover, the translation from FO-formula to ♦-automaton is effective.

Corollary. The first-order theory of every automatic structure is decidable.

The main focus of the study of automatic structures has been with regard to their iso-morphism types. Call a structure (♦-)automatically presentable if it is isomorphic to some(♦-)automatic structure.

Write W-AutStr, T-AutStr, Wω-AutStr, and Tω-AutStr for the classes of word-, tree-,ω-word-, and ω-tree–automatically presentable structures. The following basic relation-ships hold: W-AutStr ⊂ T-AutStr, W-AutStr ⊂ Wω-AutStr, T-AutStr ⊂ Tω-AutStr, and

2This is one whose signature only contains relation symbols, and contains a symbol = for equality.

8

Wω-AutStr ⊂ Tω-AutStr. However, the exact relationship amongst all these classes is notyet known, though we will have something to say about this in the following lectures.

We have seen that one can generate a collection of structures as those interpretable in afixed structure. We have previously established a tight correspondence between being com-putable by some flavour of automaton and (W)MSO definability theory of a certain corre-sponding structure. This allows us to characterise the automatically presentable structuresvia (finite)-set interpretability.

Also, we can achieve a similar characterisation using first-order logic.

Definition [Universal automatic structures]. A structure U is called universal forthe class of ♦-automatic structures, if

i) U is ♦-automatic, and

ii) every ♦-automatic structure is FO-interpretable in U .

Theorem. Suppose |Σ| ≥ 2.

1. The structure W(Σ) is universal for the class of word automatic structures.

2. The structure Wω(Σ) is universal for the class of ω-word automatic structures.

3. The structure T2(Σ) is universal for the class of tree automatic structures.

4. The structure T ω2 (Σ) is universal for the class of ω-tree automatic structures.

References: [4], [34], [39],[40],[36],[5], [49].

Day 4


Extending the Definability Theorem.

The fundamental theorem of automatic structures says that every first-order definable re-lation in an automatic structure is computable by an automaton. We are interested inextending this theorem beyond first-order. However, there are some obvious constraintsthat should guide us. In particular, the extension must still be decidable; so for instance,MSO (or even WMSO) is too strong.

We consider generalised quantifiers (see for instance [32, Section 3]). These include quanti-fiers such as ’there exists at least κ many x such that φ(x, y)’ (for a fixed cardinal κ), ’thereare the same number of x satisfying φ1(x, y) as z satisfying φ2(z, y)’, ’there exists k modulom many x such that φ(x, y)’, etc.

Of course, the quantifiers we are interested in are the ones that, on automatic structures,only define regular relations. A generalised quantifier ∃C preserves ♦-regularity, if for every♦-automatic structure A = (A; (Ri)i), and every atomic relation Ri ⊂ An+1, the quantifiedrelation ∃CxR(x, y1, · · · , yn) is ♦-regular. Moreover, if this translation is effective, thensay that ∃C preserves ♦-regularity effectively. Write Q

reg♦

for the collection of generalisedquantifiers that preserve ♦-regularity. Immediately from the definitions, the DefinabilityTheorem extends to FO + Q

reg♦ .

9

On the positive side, we will show that ’there exists infinitely many’ and ’there exists kmodulo m many’ preserve regularity effectively on Wω-AutStr.

Problem. Classify the quantifiers in Qreg♦ .

Classicaly, there is a natural hierarchy amongst the generalised quantifiers. The simplestof these, the unary monadic ones, are of the form ∃C for C ⊂ N ∪ {∞}, and express that’the number of x satisfying φ(x, y) is in C’. We classify those there preserve regularity inthe W-AutStr-case. Write ∃mod for the collection of quantifiers of the form ’there exists kmodulo m many’, for 0 ≤ k < m.

Theorem. A unary monadic quantifier preserves word-regularity if and only if it is definablein FO plus ∃mod.

References: [41],[49].

Day 4


Question. How do we prove that a given structure is not automatically presentable ? Howis this different from showing that a given structure is not automatic ?

In this lecture, we survey some answers to this question. Suppose we are given a structure Awhose domain consists of a set of words, say. Then to show that A is not automatic, we needto exhibit that either the domain, or one of the atomic relations, is not word-recognisable.The standard techniques of automata theory (pumping lemma, Myhill-Nerode congruence,etc) can be used to achieve this.

Exercise. Show that the following codings of (Q,+) are not automatic.

1. The binary-representation of rationals as ultimately periodic sequences.

2. The representation of rationals as pairs of integers.

However, to show that A is not automatically presentable, requires proving that there doesnot exist a coding of A that gives an automatic structure. This requires more subtlety.We will discuss some techniques that show the following structures are not in W-AutStr:full arithmetic (N; +,×), skolem arithmetic (N;×), the free monoid on two generators({0, 1}?; · ), the ordinal (ωω;≤), and the random graph. Here are some of the availabletechniques.

1. If A ∈ W-AutStr, and contains functions f1, · · · , fl, and a D ⊂ A is listed length-lexicographically as d0, d1, · · · , then the generated sets G0(D) = {d0}, Gn+1(D) =Gn(D) ∪ {dn+1} ∪

⋃

i{fi(x) | x ⊂ Gn(D)}, satisfy that |Gn(D)| = 2O(n).

2. Say that a structure B is a sum-augmentation of a set of structures S (each havingthe same signature as B) if there is a finite partition of B = B1 ∪ · · · ∪ Bn such thatfor each i, the substructure B � Bi is isomorphic to some structure in S.If A ∈ W-AutStr, then for every formula φ(x, y) on A, there is a finite set of structuresS, so that for every tuple of elements b from A, the substructure A � φA(·, b) is a sum-augmentation of S.

References: [4],[5],[26],[49].

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Day 5

Lecture 9: Rational structures and modal logic

Modal and temporal logics are widely recognized as particularly simple, natural, and ver-satile frameworks to specify reachability, liveness, safety, fairness, etc. properties in infinitestate systems. Infinite structures emerge naturally in various areas of applications of modallogic, including: modal logics over transition systems of models of computation with infiniteor unbounded memory; temporal logics over infinite time flows; description logics over in-finite domains; etc. In [38] Kesten et al formulate minimal requirements for an ‘assertionallanguage L to be adequate for symbolic model checking’, according to which the basic modallogic is the minimal natural logical language satisfying these requirements.

In this lecture we will introduce rational Kripke models as a very general framework fordecidable model checking of modal logic on infinite models, and will indicate some potentialapplications to e.g., regular and bounded model checking.

Rational transducers, studied by Eilenberg [27], Elgot and Mezei [28], Nivat, Berstel [2],etc., are asynchronous automata on pairs of words. Intuitively, these are finite automatawith two autonomous heads that read the input pair of words asynchronously, i.e. each ofthem can read arbitrarily farther ahead of the other. The transitions are determined by afinite set of pairs of (possibly empty) words; alternatively, a transition can be labelled eitherby a pair of letters (when both heads make a move on their words) or by 〈a, ε〉 or 〈ε, a〉,where a is a letter, and ε is the empty word (when one of the heads reads on, while the otheris waiting). Formally, a rational transducer is a tuple 〈Q,Σ, qi, F, ρ〉 , where Q is a finiteset of states, Σ is a finite alphabet, qi ∈ Q is an initial state, F is a set of accepting states,and ρ ⊆ Q× (Σ∪ {ε})× (Σ∪ {ε})×Q is the transition relation, consisting of finitely manytuples, each containing the current state, the pair of letters (or ε) triggering the transition,and the new state. Alternatively, one can take ρ ⊆ Q × Σ∗ × Σ∗ × Q.

Rational relations are relations, recognizable by rational transducers. Equivalently, givenan alphabet Σ, a (binary) rational relation over Σ∗ is rational subset of Σ∗ × Σ∗, i.e.,a relation generated by a rational expression (built up using union, concatenation, anditeration) from a finite subset of Σ∗×Σ∗. Besides the references above, these have also beenstudied, inter alia, by Johnson [37], Frougny and Sakarovich [31], and recently Morvan [44].

A Kripke frame (W,R) is rational if W ⊆ Σ∗ is a rational language in a finite alphabetΣ, and R ⊆ Σ∗ × Σ∗ is a rational relation in Σ. A Kripke model (W,R, V ) is rational if(W,R) is a rational frame, and V is a rational valuation, i.e., every V (p) is a rational set inW . Rational frames include automatic graphs, counter systems, transition graphs of Petrinets, etc.

A key property of rational Kripke models is the following. If X ⊆ Σ∗ is a rational set andR ⊆ Σ∗×Σ∗ is a rational relation, then the set

[R]X = {w ∈ Σ∗ | R(w) ⊆ X}

is an effectively computable (from the automata for X and R) rational set.

Theorem: For every rational Kripke model (W,R, V ), and every modal formula φ in themodal language Lrat with modalities [R] , [−R] ,

[

R−1]

,[

−R−1]

:

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[[φ]]M is an effectively computable rational subset of W .

As a corollary, local and global model checking, and satisfiability checking of formulae fromLrat in any rational Kripke model are decidable.

However, the first-order theory of a rational graph, as well as many important queriesoutside first-order logic, such as reachability, on rational models are generally undecidable.

References: [2],[1],[37], [31],[44], [38], [12], [11].

Day 5

Lecture 10: Infinite state model checking of expressive modal logics

While basic modal logic is good enough for model checking of pre-conditions and post-conditions specified over regular sets of states, it fails to capture many important for ver-ification in infinite systems properties, such as reachability, safety, liveness, fairness, etc.along some or all computation paths in a given transition system.

In this lecture we will discuss extensions of basic modal logic, such as LTL, CTL, CTL*,modal µ-calculus, and fragments of these, that can express these properties, and somespecial cases of classes of infinite state transition systems in which model checking of suchextended modal logics is effective.

Given a transition system 〈W,R〉, and a set of ‘initial states’ I ⊆ W , the forward reachabilityproblem is about computing (symbolically) the set of states succ∗(I) which can be reachedby a finite R-path starting from a state in I. Likewise, given a set (of ‘bad’, or ‘desirable’)states B ⊆ W , the backward reachability problem is about computing the set pred∗(B) ⊆ Wof states from which a state in B can be reached by a finite R-path. Repeated reachability isabout computing the set of states, from which a given state, or a set of states, is reachableinfinitely often by an R-path.

Most of the important verification problems, such as safety, liveness, and fairness, are ex-pressible in terms of versions of (repeated) reachability. In general, however, these problemsare undecidable on infinite state transition systems. Some particular decidable cases, thatwill be discussed in the lecture, include:

• Transition systems with decidable MSO, since all these properties are expressible inMSO.

• In particular, LTL and CTL over pushdown systems, see [10],[50], [56].

• Transition systems (in particular, rational Kripke models) with finite bisimulationindex, i.e., those bisimilar to finite transition systems.

• A class of configuration graphs of counter systems, where the CTL* can be encodedand model-checked within Presburger arithmetic.

• Regular ground tree rewriting graphs [42, 43].

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Time permitting, we will also discuss briefly the cases of regular model checking [12] andbounded model checking [3].

The lecture will end with a concluding discussion on the course.

Additional references: [11], [58], [21], [13],[9], [14].

References

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The Caucal Hierarchy of Infinite Graphs inTerms of Logic and Higher-order Pushdown

Automata

Arnaud Carayol1 and Stefan Wohrle2

1 IRISA Rennes, [email protected]

2 Lehrstuhl fur Informatik 7RWTH Aachen, Germany

[email protected]

Abstract. In this paper we give two equivalent characterizations of theCaucal hierarchy, a hierarchy of infinite graphs with a decidable monadicsecond-order (MSO) theory. It is obtained by iterating the graph trans-formations of unfolding and inverse rational mapping. The first charac-terization sticks to this hierarchical approach, replacing the language-theoretic operation of a rational mapping by an MSO-transduction andthe unfolding by the treegraph operation. The second characterizationis non-iterative. We show that the family of graphs of the Caucal hier-archy coincides with the family of graphs obtained as the ε-closure ofconfiguration graphs of higher-order pushdown automata.While the different characterizations of the graph family show their ro-bustness and thus also their importance, the characterization in termsof higher-order pushdown automata additionally yields that the graphhierarchy is indeed strict.

1 Introduction

Classes of finitely generated infinite graphs enjoying a decidable theory are astrong subject of current research. Interest arises from applications in modelchecking of infinite structures (e.g. transition systems, unfoldings of Kripke struc-tures) as well as from a theoretical point of view since the border to undecid-ability is very close and even for very regular structures many properties becomeundecidable.

We are interested in a hierarchy of infinite graphs with a decidable monadicsecond-order (MSO) theory which was introduced by D. Caucal in [7]. Startingfrom the class of finite graphs two operations preserving the decidability of theMSO-theory are applied, the unfolding [9] and inverse rational mappings [6]. It-erating these operations we obtain the hierarchy (Graph(n))n∈N where Graph(n)is the class of all graphs which can be obtained from some finite graph by ann-fold iteration of unfolding followed by an inverse rational mapping. This hi-erarchy of infinite graphs contains several interesting families of graphs (see [7,17]) and has already been subject to further studies [2].

The first level contains exactly the prefix-recognizable graphs [6], which canin turn be characterized as the graphs definable in ∆2 (the infinite binary tree) byan MSO-transduction [1], or as the ε-closure of configuration graphs of pushdownautomata [16] (see [1] for an overview). We extend these characterizations tohigher levels.

In Sect. 3 we show that the iteration of the treegraph operation, a variantof the tree-iteration [18], and MSO-transductions generates exactly the Cau-cal hierarchy. These two operations are to our knowledge the strongest graphtransformations which preserve the decidability of the MSO-theory. In [17] thehierarchy was defined starting from the infinite binary tree by iterating MSO-interpretations (particular MSO-transductions) and unfolding. Since the unfold-ing is definable inside the graph obtained by the treegraph operation, it followsfrom our result these definitions are indeed equivalent.

Pushdown automata can also be seen as the first level of a hierarchy of higher-order pushdown automata, whose stack entries are not only single letters (as forlevel 1), but words (level 2), words of words (level 3) . . . . Similar hierarchieshave already been considered in [14, 11, 15].

In Sect. 4 we show that a graph is on level n of the Caucal hierarchy iff it is theε-closure of a configuration graph of a higher-order pushdown automaton of leveln. This result is incorrectly attributed to [2, 7] in [17]. In [2], in the context ofgame simulation, only the easier direction from higher-order pushdown graphs tographs in the hierarchy is shown. All the proofs in Sections 3 and 4 are effective.

In Sect. 5 we use the characterization of the hierarchy in terms of higher-orderpushdown automata to show that it is strict. Moreover we exhibit a generatorfor every level, i.e. every graph on this level can be obtained from the generatorby applying a rational marking and an inverse rational mapping, or an MSO-interpretation. Finally we give an example of graph with a decidable MSO-theorywhich is not in the hierarchy.

2 Preliminaries

2.1 Operations on Graphs and the Caucal Hierarchy

We fix a countable set A, also called alphabet. Let Σ ⊆ A be a finite subset ofedge labels. A Σ-labeled graph G is a tuple (V G, (EG

a )a∈Σ) where V G is a set ofvertices and for a ∈ Σ we denote by EG

a ⊆ V G × V G the set of a-labeled edgesof G. We assume that V G is at most countable, and that there are no isolatedvertices in G, i.e. for every v ∈ V G there exists an w ∈ V G such that (v, w) ∈ EG

a

or (w, v) ∈ EGa for some a ∈ Σ. If the graph G and the set of edge labels Σ

is clear from the context we drop the superscript G and speak just of a labeledgraph. A graph is called deterministic if (v, w) ∈ Ea and (v, w′) ∈ Ea impliesw = w′ for all v, w,w′ ∈ V and a ∈ Σ.

A path from a vertex u to a vertex v labeled by w = a1 . . . an−1 is a sequencev1a1 . . . an−1vn ∈ V (ΣV )∗ such that v1 = u, vn = v and (vi, vi+1) ∈ Eai forevery i ∈ {1, . . . , n − 1}. In this case we will also write u

w→ v. A tree T is a

graph containing a vertex r called the root such that for any vertex v ∈ V T thereexists a unique path from r to v.

The unfolding Unf(G, r) of a graph G = (V G, (EGa )a∈Σ) from a node r ∈ V G

is the tree T = (V T , (ETa )a∈Σ) where V T is the set of all paths starting from r

in G and for all a ∈ Σ, (w,w′) ∈ ETa iff w′ = w · a · v for some v ∈ V G.

The treegraph Treegraph(G, ]) of a graph G = (V, (Ea)a∈Σ) by a symbol ] 6∈Σ is the graph G′ = (V +, (E′

a)a∈Σ∪{]}) where V + designates the set of all finitenon-empty sequences of elements of V , for all a ∈ Σ and all w ∈ V ∗, (wu, wv) ∈E′

a iff (u, v) ∈ Ea, and E′] = {(wu, wuu) | w ∈ V ∗, u ∈ V }. The tree-iteration as

defined in [18] also contains a son-relation given by {(w, wu) | w ∈ V ∗ and u ∈ V }.If G is connected then the son-relation can be defined in the treegraph.

Let Σ be a set of symbols disjoint from but in bijection with Σ. We extendevery Σ-labeled graph G to a (Σ ∪ Σ)-labeled graph G by adding reverse edgesEa := {(u, v) | (v, u) ∈ Ea}. Let Γ ⊆ A be a set of edge labels. A rationalmapping is a mapping h : Γ → P(Σ ∪ Σ)∗ which associates to every symbolfrom Γ a regular subset of (Σ ∪ Σ)∗. If h(a) is finite for every a ∈ Γ we alsospeak of a finite mapping. We apply a rational mapping h to a Σ-labeled graphG by the inverse to obtain a Γ -labeled graph h−1(G) with (u, v) ∈ Eb iff there isa path from u to v in G labeled by a word in h(b). The set of vertices of h−1(G)is given implicitly by the edge relations. We also speak of h−1(G) as the graphobtained from G by the inverse rational mapping h−1.

The marking M](G,X) of a graph G = (V, (Ea)a∈Σ) on a set of verticesX ⊆ V by a symbol ] 6∈ Σ is the graph G′ = (V ′, (E′

a)a∈Σ∪{]}) where V ′ ={(x, 0) | x ∈ V } ∪ {(x, 1) | x ∈ X}, E′

a = {((x, 0), (y, 0)) | (x, y) ∈ Ea} fora ∈ Σ, and E] = {((x, 0), (x, 1)) | x ∈ X}. A rational marking of a graphG = (V, (Ea)a∈Σ) by a symbol ] 6∈ Σ with a rational subset R over Σ ∪ Σ froma vertex r ∈ V is the graph M]

(G, {x ∈ V | r

w→G x, w ∈ R}). An MSO-marking of a graph G by a symbol ] with an MSO-formula ϕ(x) is the graphM]

(G, {v ∈ V G | G |= ϕ(v)}).

Following [7], we define Graph(0) to be the class containing for every finitesubset Σ ⊆ A all finite Σ-labeled graphs, and for all n ≥ 0

Tree(n + 1) :={Unf(G, r) | G ∈ Graph(n), r ∈ V G

},

Graph(n + 1) :={h−1(T ) | T ∈ Tree(n + 1), h−1 an inverse rational mapping

},

where we do not distinguish between isomorphic graphs.

2.2 Monadic Second-order Logic and Transductions

We define the monadic second-order logic over Σ-labeled graphs as usual, (seee.g. [13]), i.e. we view a graph as a relational structure over the signature con-sisting of the binary relation symbols (Ea)a∈Σ .

A formula ϕ(X1, . . . , Xk) containing at most the free variables X1, . . . , Xk

is evaluated in (G,V) where G = (V, (Ea)a∈Σ) is a Σ-labeled graph and V :V → P({1, . . . , k}) is a function which assigns to every vertex v of G a set V(v)such that v ∈ Xi iff i ∈ V(v). We write (G,V) |= ϕ(X1, . . . , Xk), or equivalently

G |= ϕ[V1, . . . , Vk] where Vi := {v ∈ V | i ∈ V(v)}, if ϕ holds in G under thegiven valuation V.

An MSO-interpretation of Γ in Σ is a family I = (ϕa(x, y))a∈Γ of MSO-formulas over Σ. Applying an MSO-interpretation I = (ϕa(x, y))a∈Γ of Γ inΣ to a Σ-labeled graph G we obtain a Γ -labeled graph I(G) where the edgerelation E

I(G)a is given by the pairs of vertices for which ϕa(x, y) is satisfied in G,

and V I(G) is given implicitly as the set of all vertices occurring in the relationsEI(G)b . Note that the addition of an MSO-formula δ(x) to I defining the vertex

set explicitly does not increase the power of an interpretation if we require thatthere are no isolated vertices in the resulting graph.

Interpretations cannot increase the size of a structure. To overcome this weak-ness the notion of a transduction was introduced, cf. [8]. Let G = (V, (Ea)a∈Σ)be a Σ-labeled graph and K be a finite subset of A disjoint from Σ. A K-copyingoperation for Σ associates to G a (Σ ∪K)-labeled graph G′ = (V ′, (E′

a)a∈Σ∪K)where V ′ = V ∪ (V ×K), E′

a := Ea for a ∈ Σ, and E′b := {(v, (v, b)) | v ∈ V } for

b ∈ K. An MSO-transduction T = (K, I) from Σ to Γ is a K-copying operationfor Σ followed by an MSO-interpretation I of Γ in Σ ∪K.

Note that an inverse rational mapping is a special case of an MSO-interpre-tation and an MSO-marking is a special case of an MSO-transduction.

2.3 Higher-order Pushdown Automata

We follow the definition of [15]. Let Γ be a finite set of stack symbols. A level 1pushdown stack over Γ is a word w ∈ Γ ∗ in reversed order, i.e. if w = a1 . . . am

the corresponding stack is denoted by [am, . . . , a1]. For n ≥ 2 a level n push-down stack over Γ is inductively defined as a sequence [sr, . . . , s1] of level n− 1pushdown stacks si for 1 ≤ i ≤ r. [ε] denotes the empty level 1 stack, the emptylevel n stack, denoted by [ε]n, is a stack which contains for 1 ≤ i < n only asingle empty level i stack.

The following instructions can be executed on a level 1 stack [am, . . . , a1]:

pusha1([am, . . . , a1]) := [a, am, . . . , a1] for every a ∈ Γ

pop1([am, am−1 . . . , a1]) := [am−1, . . . , a1]

Furthermore we define the following function which does not change the contentof a stack:

top([ε]) := ε and top([am, . . . , a1]) := am for m ≥ 1.

For a stack [sr, . . . , s1] of level n ≥ 2 we define the following instructions

pusha1([sr, . . . , s1]) := [pusha

1(sr), sr−1, . . . s1] for every a ∈ Γ

pushn([sr, . . . , s1]) := [sr, sr, . . . , s1]pushk([sr, . . . , s1]) := [pushk(sr), sr−1, . . . , s1] for 2 ≤ k < n

popn([sr, . . . , s1]) := [sr−1, . . . , s1]popk([sr, . . . , s1]) := [popk(sr), sr−1, . . . , s1] for 1 ≤ k < n

and extend top to a level n stack by setting top([sr, . . . , s1]) := top(sr).We denote by Instrn the set of instructions that can be applied to a level n

stack (without the top function). For the sake of easiness we also add an identityfunction denoted by − which does not change the stack at all.

The instruction pusha1 adds the symbol a to the topmost level 1 stack, while

pushk duplicates the topmost level k−1 stack completely. Similarly pop1 removesthe top symbol of the topmost level 1 stack, while popk for 1 < k ≤ n removesthe corresponding level k − 1 stack completely. Note that the instruction popk

for 2 ≤ k ≤ n can only be applied if the resulting stack is again a level n stack,i.e. it does not remove a bottom level k − 1 stack.

A higher-order pushdown automaton of level n is a tuple A = (Q, Σ, Γ, q0,∆)where Q is a finite set of states, Σ is an input alphabet, Γ is a stack alphabet,q0 ∈ Q is an initial state, and ∆ ⊆ Q × (Σ ∪ {ε}) × (Γ ∪ {ε}) × Q × Instrn isa transition relation. A configuration of A is a pair (q, [sr, . . . , s1]) where q is astate of A and [sr, . . . , s1] is a stack of level n. The initial configuration (q0, [ε]n)consists of the initial state q0 and the empty level n stack [ε]n. A can reacha configuration (q′, [s′r′ , . . . , s

′1]) from (q, [sr, . . . , s1]) by reading a ∈ Σ ∪ {ε} if

there is a transition (q, a, top([sr, . . . , s1]), q′, i) ∈ ∆ such that i([sr, . . . , s1]) =[s′r′ , . . . , s

′1]. The automaton A accepts a word w ∈ Σ∗ if A reaches from the

initial configuration the empty level n stack after reading w. We denote byHOPDA(n) the class of all higher-order pushdown automata of level n.

3 Closure Properties

In this part, we prove that the hierarchy is closed under MSO-transductions andthe treegraph operation. We first consider the case of deterministic trees.

3.1 The Deterministic Case

We consider a sub-hierarchy obtained by unfolding only deterministic graphs.Graphd(0) is equal to Graph(0). Treed(n + 1) contains the unfoldings of everydeterministic graph G ∈ Graphd(n) from a vertex in V G. Graphd(n) is defined inthe same way as Graph(n). Note that Graphd(n) also contains non-deterministicgraphs.

Closure under MSO-transductions Using results from [3], we prove thatfor all n ∈ N, Graphd(n) is closed under MSO-transductions. This result wasobtained for the first level by A. Blumensath in [1]. Obviously, Treed(n) is notclosed under MSO-transductions but if we consider only MSO-markings, weobtain also a closure property for Treed(n).

Proposition 1. For all n ≥ 0, all tree T ∈ Treed(n) and all graph G ∈Graphd(n), we have that:

1. M(T ) also belongs to Treed(n), for any MSO-marking M,

2. T (G) also belongs to Graphd(n), for any MSO-transduction T .

Proof (sketch): These results are proved by induction on the level using partialcommutation results of MSO-transductions and unfolding obtained in [3].

1. For every deterministic graph G and every MSO-marking M, there ex-ists an MSO-transduction T ′ and a vertex r′ such that M(Unf(G, r)) ≈Unf(T ′(G), r′).

2. For every deterministic graph G and every MSO-transduction T , there existsan MSO-transduction T ′, a rational mapping h and a vertex r′ such thatT (Unf(G, r)) ≈ h−1(Unf(T ′(G), r′)).

Note that in both cases T ′ preserves determinism. ¤

Closure Under the Treegraph Operation The unfolding is a particular caseof the treegraph operation in the sense that for any graph G the unfolding froma definable vertex r, Unf(G, r), can be obtained by an MSO-interpretation fromTreegraph(G, ]) (see [9]). In the case of deterministic trees, we show a converseresult: how to obtain treegraph using MSO-interpretations and unfolding. Thisconstruction is due to T. Colcombet.

Lemma 1. For any finite set of labels Σ, there exist two finite mappings h1,h2

and a rational marking M such that for any deterministic tree T with root r:

Treegraph(T, ]) ≈ h−12

(M (Unf

(h−1

1 (T ), r)))

.

Proof (sketch): The finite mapping h1 adds backward edges labeled by elementsof Σ and a loop labeled by ] to every vertex. Thus, for all a ∈ Σ, h1 is definedby h1(a) = {a}, h1(a) = {a} and h1(]) = {ε}.

Let H be the deterministic tree equal to Unf(h−1

1 (T ), r), every node x of

H is uniquely characterized by a word in (Σ ∪ Σ ∪ {]})∗. The rational markingM$ marks all the vertices corresponding to a word which does not containxx or xx for x ∈ Σ. Finally, h2 is used to erase unmarked vertices and toreverse the remaining edges with labels in Σ. h2 is given by h2(]) = {]} andh2(a) =

{$$a$$, $$¯a$$

}for a ∈ Σ. ¤

Figure 1 illustrates the construction above on the semi-infinite line. The filleddots represent the vertices marked by M$. The closure of the deterministichierarchy under the treegraph operation is obtained from Lem. 1 and Prop. 1,using the fact that for all trees T and all rational mappings h which do notcontain ], Treegraph(h−1(T ), ]) = h−1

] (Treegraph(T, ])) where h] designates therational mapping that extends h with h](]) = {]}.

Proposition 2. For all n ≥ 0, if G ∈ Graphd(n) then Treegraph(G, ]) ∈Graphd(n + 1).

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3.2 Deterministic Trees are Enough

We now prove that for all n, Graph(n) is equal to Graphd(n). From the technicalpoint of view, this means that even if the hierarchy contains non-deterministicgraphs and even graphs of infinite out-degree, we can always work with an ”un-derlying” deterministic tree.

Lemma 2. For all n > 0, if G ∈ Graph(n), then there exists a deterministictree T ∈ Treed(n) and a rational mapping h such that G = h−1(T ).

Proof (sketch): The proof proceeds by induction on the level n. Let T ∈ Tree(n+1), we want to prove that T belongs to Graphd(n+1). By definition of Tree(n+1) and by induction hypothesis, we have T ≈ Unf(h−1(Td), s) for some de-terministic tree Td ∈ Treed(n) and some rational mapping h. Using the factthat the unfolding can be defined in the treegraph operation (see [9]), we haveT ≈ T (Treegraph(h−1(Td), ])) for some MSO-transduction T . If h] denotes therational mapping obtained by extending h with h](]) = {]}, we have T =T (h−1

] (Treegraph(Td, ])). Applying Lem. 1, we have T = T ′(Unf(h−11 (Td), r))

where T ′ = M◦ h−12 ◦ h−1

# ◦ T . It is easy to check that Unf(h−1

1 (Td), r)

belongsto Treed(n + 1). Using Prop. 1, we prove that T belongs to Graphd(n + 1). Thecase of G ∈ Graph(n + 1) is easily derived from this. ¤

We can now prove that every graph of the hierarchy has a decidable MSO-theory. Note that this does not follow directly from the definition because un-folding from an arbitrary (i.e. not necessarily MSO-definable) vertex does notpreserve the decidability of MSO-logic. However, using Lem. 2 we can always

come back to the case where we unfold from a MSO-definable vertex (see [4] formore details).

Theorem 1. Each graph of the hierarchy has decidable MSO-theory and thisremains true if we add to MSO-logic the predicates |X| < ∞ , |X| = k mod pfor all k and p ∈ N which are interpreted as X is finite respectively X has sizeequal to k modulo p for k, p ∈ N.

Combining Prop. 1, Prop. 2 and Lem. 2, we now have two equivalent char-acterizations of the hierarchy: one “minimal” in terms of unfolding and inverserational mappings and one “maximal” in terms of the treegraph operation andMSO-transductions. The maximal characterization shows the robustness of thehierarchy and its interest because it combines the two, to our knowledge, mostpowerful MSO-preserving operations. On the other side, the minimal charac-terization allows us to make the link between the hierarchy and the graphs ofhigher-order pushdown automata.

Theorem 2. The Caucal hierarchy is equal to the hierarchy obtained by iteratingthe treegraph operation and MSO-transductions.

4 Higher-order Pushdown Graphs vs. Caucal Graphs

In this section we give an automata-theoretic characterization of the classes ofthe Caucal hierarchy. This characterization provides us with a “flat” model fordescribing a graph of any level, i.e. we do not have to refer to a sequence ofoperations. Furthermore it extends the characterization of the first level of thehierarchy as the ε-closure of configuration graphs of pushdown automata givenin [16] to any level. We recall some definitions.

The configuration graph C(A) of A ∈ HOPDA(n) is the graph of all config-urations of A reachable from the initial configuration, with an edge labeled bya ∈ Σ ∪ {ε} from (q, s) to (q′, s′) iff there is a transition (q, a, top(s), q′, i) ∈ ∆such that i(s) = s′.

Let C(A) be the configuration graph of A ∈ HOPDA(n). We will assume forthe remainder of the paper that for every pair (q, α) of state q and top stacksymbol α only ε-transitions or only non-ε-transitions are possible. The ε-closureof C(A) is the graph G obtained from C(A) by removing all vertices with onlyoutgoing ε-transitions and adding an a-labeled edge between v and w if there isan a-labeled path from v to w in C(A).

A higher-order pushdown graph G of level n is the ε-closure of the configu-ration graph of some A ∈ HOPDA(n). We call G the higher-order pushdowngraph generated by A and denote by HOPDG(n) the class of all higher-orderpushdown graphs of level n (up to isomorphism).

This notion of ε-closure was used in [16] to show that the class HOPDG(1)coincides with the class of prefix recognizable graphs, i.e. with the the graphson the first level of the hierarchy. We extend this result to every level of thehierarchy.

The easier part of the equivalence is to show that every HOPDG of leveln is a graph on the same level of the hierarchy. The main idea is to find agraph in Graph(n) such that every node of this graph can be identified witha configuration of a higher-order pushdown automaton, and to construct aninverse rational mapping which generates the edges of the configuration graphof the automaton. Such a construction is already contained in [2] in a slightlydifferent setting. We propose here to use the family ∆n

m of graphs obtained byan (n− 1)-fold application of the treegraph operation to the infinite m-ary tree∆m. This has the advantage that there is almost a one-to-one correspondencebetween configurations of the higher-order pushdown automaton and the verticesof the graph. Using the fact that ∆n

m ∈ Graph(n) we obtain:

Lemma 3. If G ∈ HOPDG(n) then G ∈ Graph(n).

We now turn to the converse direction: every graph G on level n of the hier-archy is indeed the ε-closure of a configuration graph of a higher-order pushdownautomaton of level n. We show this using the following two Lemmas.

Lemma 4. If G ∈ HOPDG(n) and r ∈ V G, then Unf(G, r) ∈ HOPDG(n + 1).

Lemma 5. If G ∈ HOPDG(n), r ∈ V G and h is a rational mapping, thenh−1(Unf(G, r)) ∈ HOPDG(n + 1).

While the proof of Lem. 4 consists of a straightforward modification of theHOPDA for G, the proof of Lem. 5 requires some technical preparation. Weneed to show that for an automaton as constructed in the proof of Lem. 4 thereexists a higher-order pushdown automaton which generates exactly the graphUnf(G, r) extended by reverse edges, i.e. for all v, w ∈ Unf(G, r), v

a−→ w inUnf(G, r) iff w

a−→ v in the extended graph.To show that such an automaton exists we introduce the notion of a weak

popping higher-order pushdown automaton. A weak popping automaton is onlyallowed to execute a pop instruction of level j ≥ 2 if the two top level j stackscoincide. We skip a formal definition of a weak popping higher-order pushdownautomaton and just mention that even though this automaton model is equippedwith a built-in test on the equality of two stacks of the same level, it is equivalentto the usual model. All proofs are given in the full version of this article [4].

Theorem 3. For every n ∈ N, G ∈ HOPDG(n) iff G ∈ Graph(n).

5 More Properties of the Caucal Hierarchy

We give a generator for each level of the hierarchy. Then we use the traces ofthe graphs of the hierarchy to prove its strictness and to exhibit a graph havinga decidable MSO-theory which is not in the hierarchy.

5.1 Generators

For the first level of the hierarchy, the infinite binary tree ∆2 is a generator forrational markings (without backward edges) from the root and inverse rationalmappings. As hinted by the proof of Lem. 3, a similar result can be obtainedat any level. Recall that ∆n

2 is the graph obtained from ∆2 by an (n − 1)-foldapplication of the treegraph operation.

Proposition 3. Every graph G ∈ Graph(n) can be obtained from ∆n2 by ap-

plying a rational marking (with backward edges) from its source and an inverserational mapping.

5.2 On Traces — The Strictness of the Hierarchy

A direct consequence of Theo. 3 is that the traces of the graphs of level n arerecognized by a higher-order pushdown automaton of level n. These families oflanguages have been studied by W. Damm and form the OI-hierarchy [11]. Theequivalence between the OI-hierarchy and the traces of higher-order pushdownautomata is proved in [12]. In [10, 14], this hierarchy is proved to be strict ateach level.

Theorem 4. For all n ≥ 1,

(a) for all T ∈ Tree(n) the branch language of T (i.e. the set of all words labelinga path from the root to a leaf) is recognized by a HOPDA of level n− 1.

(b) for all G ∈ Graph(n) and u, v ∈ VG, L(u, v, G) = {w ∈ Γ ∗ | uw→ v} is

recognized by a HOPDA of level n.

According to Theo. 4, the strictness level-by-level of the OI-hierarchy impliesthe strictness of the Caucal hierarchy. An obvious example of a graph which is atlevel n but not at level n−1 is the generator ∆n

2 . To obtain more natural graphsthat separate the hierarchy, we consider the trees associated to monotonicallyincreasing mappings f : N → N. The tree Tf associated to f is defined by thefollowing set of edges: Ea = {((i, 0), (i + 1, 0)) | i ∈ N} and Eb = {((i, j), (i, j +1)) | i ∈ N and j + 1 ≤ f(i)}. The branch language of Tf is {anbf(n) | n ≥ 0}.

Using a property of rational indexes of k-OI languages (see [10]), we obtainthe following proposition.

Proposition 4. If {anbf(n)|n ∈ N} is recognized by a higher-order pushdownautomaton of level k then f ∈ O(2↑k−1 (p(n))) for some polynomial p where2↑0(n) = n and 2 ↑k+1 (n) = 22↑k(n).

Let us consider the mapping expk(n) = 2↑k(n). It has been proved in [7]that Texpk

belongs to Graph(k + 1). Note that using Theo. 3 the constructiongiven in [7] can be avoided by providing a deterministic higher-order pushdownautomaton of level k + 1 that recognizes {anbexpk(n)|n ∈ N}.

It is natural to consider the “diagonal” mapping expω(n) = expn(1). Figure 2shows an initial segment of the tree associated to expω. By Prop. 4, the associatedtree Texpω is not in the hierarchy. However, using techniques from [5], we canprove that Texpω has a decidable MSO-theory.

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b

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b

b

b

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Proposition 5. There exists a graph with a decidable MSO-theory which is notin the Caucal hierarchy.

6 Conclusion

We have given two characterizations of the Caucal hierarchy. We have shown thatit coincides with the hierarchy obtained by alternating the treegraph operationand MSO-transductions, and thus have partly answered a question posed in[17]. It remains open whether one can extend this result to structures other thangraphs, i.e. with symbols of higher arity.

We have also characterized the Caucal hierarchy as the ε-closure of configu-ration graphs of higher-order pushdown automata and have used this result toobtain that the hierarchy is indeed strict, but does not contain all graphs witha decidable MSO-theory.

Despite these characterization results we know surprisingly few about thegraphs obtained on level n ≥ 2. This deserves further study. Also a thoroughcomparison with other methods to generate infinite graphs with a decidabletheory misses (see [17] for a more precise account on this).

Futhermore we like to mention that neither the constructions used to buildthe hierarchy nor Proposition 5 contradicts Seese’s conjecture that every infinitegraph (or every set of finite graphs) having a decidable MSO-theory is the imageof a tree (or a set of trees) under an MSO-transduction.

Finally, many of the questions posed in [7] on the corresponding hierarchy oftrees remained unsolved so far.

Acknowledgment

We thank D. Caucal and W. Thomas for stimulating discussions while workingon this paper and for financial support enabling our visits in Aachen respectivelyRennes. We also thank T. Colcombet for pointing to a solution for Lem. 1, T.

Cachat for reading a first draft of Sect. 4, and the anonymous referees for manyuseful remarks.

References

1. A. Blumensath. Prefix-recognisable graphs and monadic second-order logic. Tech-nical Report AIB-2001-06, RWTH Aachen, 2001.

2. T. Cachat. Higher order pushdown automata, the Caucal hierarchy of graphs andparity games. In Proceedings of the 30th International Colloquium on Automata,Languages, and Programming, volume 2719 of LNCS, pages 556–569. Springer,2003.

3. A. Carayol and T. Colcombet. On equivalent representations of infinite structures.In Proceedings of the 30th International Colloquium on Automata, Languages, andProgramming, volume 2719 of LNCS, pages 599–610. Springer, 2003.

4. A. Carayol and S. Wohrle. The Caucal hierarchy of infinite graphs in terms of logicand higher-order pushdown automata. Technical report, RWTH Aachen, 2003.

5. O. Carton and W. Thomas. The monadic theory of morphic infinite words andgeneralizations. In Proceedings of the 25th International Symposium on Mathe-matical Foundations of Computer Science, volume 1893 of LNCS, pages 275–284.Springer, 2000.

6. D. Caucal. On infinite transition graphs having a decidable monadic theory. InProceedings of the 23rd International Colloquium on Automata, Languages andProgramming, volume 1099 of LNCS, pages 194–205, 1996.

7. D. Caucal. On infinite terms having a decidable monadic theory. In Proceedingsof the 27th International Symnposium on Mathematical Foundations of ComputerScience, volume 2420 of LNCS, pages 165–176. Springer, 2002.

8. B. Courcelle. Monadic second-order definable graph transductions: A survey. The-oretical Computer Science, 126:53–75, 1994.

9. B. Courcelle and I. Walukiewicz. Monadic second-order logic, graph coverings andunfoldings of transition systems. Annals of Pure and Applied Logic, 92:35–62, 1998.

10. W. Damm. An algebraic extension of the Chomsky-hierarchy. In Proceedings of the8th International Symposium on Mathematical Foundations of Computer Science,volume 74 of LNCS, pages 266–276. Springer, 1979.

11. W. Damm. The IO and OI hierarchies. Theoretical Computer Science, 20:95–208,1982.

12. W. Damm and A. Goerdt. An automata-theoretical characterization of the OI-hierarchy. Information and Control, 71:1–32, 1986.

13. H.D. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 1995.14. J. Engelfriet. Iterated stack automata and complexity classes. Information and

Computation, 95:21–75, 1991.15. T. Knapik, D. Niwinski, and P. Urzyczyn. Higher-order pusdown trees are easy. In

Proceedings of the 5th International Conference on Foundations of Software Scienceand Computation Structures, volume 2303 of LNCS, pages 205–222. Springer, 2002.

16. C. Stirling. Decidability of bisimulation equivalence for pushdown processes. Sub-mitted.

17. W. Thomas. Constructing infinite graphs with a decidable MSO-theory. In Pro-ceedings of the 28th International Symposium on Mathematical Foundations ofComputer Science, volume 2747 of LNCS. Springer, 2003.

18. I. Walukiewicz. Monadic second-order logic on tree-like structures. TheoreticalComputer Science, 275:311–346, 2002.

DOI: 10.1007/s00224-004-1133-y

Theory Comput. Systems OF1–OF34 (2004) Theory ofComputing

Systems© 2004 Springer Science+Business Media, Inc.

Finite Presentations of Infinite Structures:Automata and Interpretations

Achim Blumensath and Erich Gradel

Mathematical Foundations of Computer Science,Aachen University of Technology,D-52065 Aachen, Germany{blume,graedel}@informatik.rwth-aachen.dewww-mgi.informatik.rwth-aachen.de

Abstract. We study definability problems and algorithmic issues for infinite struc-tures that are finitely presented. After a brief overview over different classes offinitely presentable structures, we focus on structures presented by automata or bymodel-theoretic interpretations. These two ways of presenting a structure are re-lated. Indeed, a structure is automatic if, and only if, it is first-order interpretablein an appropriate expansion of Presburger arithmetic or, equivalently, in the infinitebinary tree with prefix order and equal length predicate. Similar results hold forω-automatic structures and appropriate expansions of the real ordered group. Wealso discuss the relationship to automatic groups.

The model checking problem for FO(∃ω), first-order logic extended by thequantifier “there are infinitely many”, is proved to be decidable for automatic andω-automatic structures. Further, the complexity for various fragments of first-orderlogic is determined. On the other hand, several important properties not express-ible in FO, such as isomorphism or connectedness, turn out to be undecidable forautomatic structures.

Finally, we investigate methods for proving that a structure does not admit anautomatic presentation, and we establish that the class of automatic structures isclosed under Feferman–Vaught-like products.

1. Computational Model Theory

The relationship between logical definability and computational complexity is an im-portant issue in a number of different fields including finite model theory, databases,

OF2 A. Blumensath and E. Gradel

knowledge representation, and computer-aided verification. So far most of the researchhas been devoted to finite structures where the relationship between definability andcomplexity is by now fairly well understood (see, e.g., [19] and [34]) and has many ap-plications in particular to database theory [1]. However, in many cases the limitation tofinite structures is too restrictive. Therefore, in most of the fields mentioned above, therehave been considerable efforts to extend the methodology from finite structures to suitableclasses of infinite ones. In particular, this is the case for databases and computer-aidedverification where infinite structures (like constraint databases or systems with infinitestate spaces) are of increasing importance.

Computational model theory extends the research programme, the general approach,and the methods of finite model theory to interesting domains of infinite structures. Froma general theoretical point of view, one may ask what domains of infinite structures aresuitable for such an extension. More specifically, what conditions must be satisfied bya domain D of not necessarily finite structures such that the approach and methods offinite model theory make sense. There are two obvious and fundamental conditions:

Finite representations. Every structure A ∈ D should be representable in a finiteway (e.g. by a binary string, by an algorithm, by a collection of automata, by anaxiomatisation in some logic, by an interpretation, . . . ).

Effective semantics. For the relevant logics to be considered (e.g., first-order logic),the model-checking problem onD should be decidable. That is, given a sentenceψ ∈ L and a representation of a structure A ∈ D, it should be decidable whetherA |= ψ .

These are just minimal requirements that may need to be refined according to thecontext and the questions to be considered. We may for instance also require:

Closure. For every structure A ∈ D and every formula ψ(x), also (A, ψA), theexpansion of A with the relation defined by ψ belongs to D.

Effective query evaluation. Suppose that we have fixed a way of representing struc-tures. Given a representation of A ∈ D and a formula ψ(x) we should be ableto compute a representation of ψA (or of the expanded structure (A, ψA)).

Note that contrary to the case of finite structures, query evaluation does not necessarilyreduce to model checking. Further, instead of just effectiveness of these tasks, it may berequired that they can be performed within some complexity bounds.

After a brief survey on different classes of finitely presented structures in the nextsection, we focus on domains where structures are presented by two closely relatedmethods, namely by finite automata or by model-theoretic interpretations. While auto-matic groups have been studied rather intensively in computational group theory (see[22] and [24]) a general notion of automatic structures has only been defined in [36] andits theory has been developed in [6] and [8]. These structures are defined in Section 3.Informally, a relational structure A = (A, R1, . . . , Rm) is automatic if it is universe andits relations can be recognised by finite automata reading their inputs synchronously.We believe that automatic structures are very promising for the approach of compu-tational model theory. Not only do automatic structures admit finite presentations, butthere are also numerous interesting examples and a large body of methods that havebeen developed in five decades of automata theory. Further, automatic structures admit

Finite Presentations of Infinite Structures OF3

effective evaluation of all first-order queries and possess many other pleasant algorithmicproperties.

Automatic structures can also be defined via interpretations. As we show in Section 4a structure is automatic if, and only if, it is first-order interpretable in an appropriateexpansion of Presburger arithmetic or, equivalently, in the infinite binary tree with prefixorder and equal length predicate. Similar results hold for ω-automatic structures andappropriate expansions of the real ordered group.

Such results suggest a very general way for obtaining other interesting classesof infinite structures suitable for the approach of computational model theory: Fix astructure A (or a class of such structures) with “nice” algorithmic and/or model-theoreticproperties, and consider the class of all structures that are interpretable in A, for instance,via first-order or monadic second-order logic. Obviously each structure in this class isfinitely presentable (by an interpretation). Further, since many “nice” properties arepreserved under interpretations, every structure in the class inherits them from A. Inparticular, every class of queries that is effective on A and closed under first-orderoperations is effective on the interpretation-closure of A.

In Section 5 we turn to decidability and complexity issues. It is shown that themodel-checking problem for FO(∃ω), first-order logic extended by the quantifier “thereare infinitely many”, is decidable for automatic and ω-automatic structures, and thecomplexity for various fragments of first-order logic is investigated. On the other hand,we prove that several properties not expressible in FO, such as isomorphism of automaticstructures or connectivity of automatic graphs, are undecidable.

While it is usually rather easy to show that a structure is automatic (by constructingan automatic presentation), it is often difficult to prove a structure does not admit anyautomatic presentation. In Section 6 we present some methods to achieve this goal.

In the final section, Feferman–Vaught-like products are introduced, and it is shownthat every domain of structures that can be characterised via interpretations of a cer-tain kind is closed under such products. In particular, this applies to automatic andω-automatic structures.

2. Finitely Presentable Structures

We briefly survey some domains of infinite but finitely presentable structures which maybe relevant for computational model theory.

Recursive structures are countable structures whose functions and relations are com-putable and therefore finitely presentable. They have been studied quite intensively inmodel theory since the 1960s (see, e.g., [2] and [23]). Although recursive model theoryis very different from finite model theory, there have been some papers studying classicalissues of finite model theory on recursive structures and recursive databases [28], [31],[32], [46]. However, for most applications, the domain of recursive structures is far toolarge. In general, only quantifier-free formulae admit effective evaluation algorithms.

Constraint databases are a modern database model admitting infinite relations that arefinitely presented by quantifier-free formulae (constraints) over some fixed background


structure. For example, to store geometrical data, it is useful to have not just a finiteset as the universe of the database, but to include all real numbers ‘in the background’.Also the presence of interpreted functions, like addition and multiplication, is desirable.The constraint database framework introduced by Kanellakis et al. [35] meets bothrequirements. Formally, a constraint database consists of a context structure A, like(R, <,+, ·), and a set {ϕ1, . . . , ϕm} of quantifier-free formulae defining the databaserelations. Constraint databases are treated in detail in [38].

Metafinite structures are two-sorted structures consisting of a finite structure A, abackground structure R (which is usually infinite but fixed), and a class of weightfunctions from the finite part to the infinite one. Simple examples are finite graphs whoseedges are weighted by real numbers. For any fixed infinite structure R, the metafinitestructures with background R are finitely presentable and admit effective evaluation oflogics that make use of arithmetic operations on R, but do not admit full quantificationover its elements. Metafinite model theory has been developed in [27] and has been putto use for studying issues in database theory, optimisation, and descriptive complexity.In particular, metafinite structures have provided the basis for logical characterisationsof complexity classes over real numbers [29].

Automatic structures are structures whose functions and relations are represented byfinite automata. Informally, a relational structure A = (A, R1, . . . , Rm) is automatic ifwe can find a regular language Lδ ⊆ �∗ (which provides names for the elements of A)and a function ν : Lδ → A mapping every word w ∈ Lδ to the element of A that itrepresents. The function ν must be surjective (every element of A must be named) butneed not be injective (elements can have more than one name). In addition it must berecognisable by finite automata (reading their input words synchronously) whether twowords in Lδ name the same elements, and, for each relation Ri of A, whether a given tupleof words in Lδ names a tuple in Ri . Automatic structures provide many examples of highrelevance for computer science. There are also interesting connections to computationalgroup theory, where automatic groups have already been studied quite intensively [22],[24]. The general notion of structures presentable by automata has been proposed in [36]and their theory has been developed in [6] and [8]. Recently, results on automatic linearorders were obtained in [18] and [37]. The notion of an automatic structure can bemodified and generalised in many directions. By using automata over infinite words, weobtain the notion of ω-automatic structures (which, contrary to automatic structures,may have uncountable cardinality).

One of the main reasons for the importance of automatic andω-automatic structuresis that they admit effective (in fact, automatic) evaluation of all first-order queries. Thisfollows immediately from the closure properties of regular and ω-regular relations andfrom the decidability of emptiness problems of finite automata. Indeed, we establish amore general result.

Theorem 2.1. The model-checking problem for FO(∃ω), first-order logic extended bythe quantifier “there are infinitely many”, is decidable on the domain of ω-automaticstructures.

The proof is given in Section 5.


Tree-automatic structures, which are defined by automata on finite or infinite trees,are further natural generalisations of automatic structures. They also admit effectiveevaluation of first-order formulae. The theory of tree-automatic structures has beendeveloped in [6]. On the other hand, first-order logic is not effective on another popularextension of automatic graphs, the so-called rational graphs [41], which are defined byasynchronous multihead automata.

Tree-interpretable structures are structures that are interpretable in the infinite binarytreeT 2 = ({0, 1}∗, σ0, σ1) via a one-dimensional monadic second-order (MSO) interpre-tation (see Section 4 for details on interpretations). By Rabin’s theorem, (MSO) formu-lae can be effectively evaluated on T 2, and since MSO is closed under one-dimensionalinterpretations, the same holds for all tree-interpretable structures. Tree-interpretablestructures form a proper subclass of the automatic structures that generalises various no-tions of infinite graphs that have been studied in logic, automata theory, and verification.Examples are the context-free graphs [42], [43], which are the configuration graphs ofpushdown automata; the HR-equational and VR-equational graphs [15], which aredefined via graph grammars; and the prefix-recognisable graphs [13], which can for in-stance be defined as graphs of form (V, (Ea)a∈A)where V is a regular language and eachedge relation Ea is a finite union of sets X (Y × Z) := {(xy, xz)|x ∈ X, y ∈ Y, z ∈ Z},for regular languages X , Y , Z .

It has been established in a series of papers that some of these classes coincide withtree-interpretable graphs (see [3], [7], and [13]).

Theorem 2.2. For any graph G = (V, (Ea)a∈A) the following are equivalent:

(i) G is tree-interpretable.(ii) G is VR-equational.

(iii) G is prefix-recognisable.(iv) G is the restriction to a regular set of the configuration graph of a pushdown

automaton with ε-transitions.

On the other hand, the classes of context-free graphs and of HR-equational graphsare strictly contained in the class of tree-interpretable graphs.

Tree-constructible structures: the Caucal hierarchy. The question arises whetherthere are even more powerful domains than the tree-interpretable structures on which(MSO) logic is effective. An interesting way to obtain such domains are tree construc-tions that associate a kind of tree unravelling with any structure. A simple variant is theunfolding of a labelled graph G from a given node v to the tree T (G, v). Courcelle andWalukiewicz [16], [17] show that the MSO theory of T (G, v) can be effectively com-puted from the MSO theory of (G, v). A more general operation, applicable to relationalstructures of any kind, has been invented by Muchnik [44]. Given a relational structureA = (A, R1, . . . , Rm), let its iteration A∗ = (A∗, R∗1 , . . . , R∗m, suc, clone) be the struc-ture with universe A∗, relations R∗i = {(wa1, . . . , war )|w ∈ A∗, (a1, . . . , ar ) ∈ Ri }, thesuccessor relation suc = {(w,wa)|w ∈ A∗, a ∈ A}, and the predicate clone consistingof all elements of form waa. It is not difficult to see that unfoldings of graphs are first-


order interpretable in their iterations. Muchnik’s theorem states that the monadic theoryof A∗ is decidable if the monadic theory of A is (for proofs, see [5] and [49]). Definethe domain of tree-constructible structures to be the closure of the domain of finitestructures under (one-dimensional) MSO interpretations and iterations. By Muchnik’stheorem, and since effective MSO model checking is preserved under interpretations,tree-constructible structures are finitely presentable and admit effective evaluation ofMSO formulae.

Tree-constructible graphs form the Caucal hierarchy, which was defined in [14] ina slighly different way. The definition easily extends to arbitrary structures: Let C0 be theclass of finite structures, and let Cn+1 be the class of structures that are interpretable in theiteration A∗ of a structure A ∈ Cn . There are a number of different, but equivalent, waysto define the levels of the Caucal hierarchy. For instance, one can use the inverse rationalmappings from [13] rather than monadic interpretations, and simple unfoldings ratherthan iterations without changing the hierarchy [12]. Equivalently, the hierarchy can bedefined via higher-order pushdown automata. It is known that the Caucal hierarchy isstrict, and that it does not exhaust the class of all structures with decidable MSO theory.We refer to [48] and [12] for details and further information.

Ground tree rewriting graphs are defined by tree rewriting [39]. Vertices are repre-sented by finite trees and edges are generated by ground rewriting rules. In this way onecan obtain graphs that are not tree-interpretable (for instance, the infinite two-dimensionalgrid), but for which, in addition to first-order theory, the reachability problem also remainsdecidable. While universal reachability and universal recurrence (and hence general MSOformulae) are undecidable on ground tree rewriting graphs, Loding [39] exhibits a frag-ment of CTL (permitting EF and EGF operations, but not EG, EFG, or until operations)that can be effectively evaluated on this class.

3. Automatic Structures and Automatic Groups

As usual in logic, we consider structures A = (A, R1, R2, . . . , f1, f2, . . .) where A isa non-empty set, called the universe of A, where each Ri ⊆ Ari is a relation on A, andevery f j : Asj → A is a function on A. The names of the relations and functions of A,together with their arities, form the vocabulary of A. We consider constants as functionsof arity 0. A relational structure is a structure without functions. We can associate withevery structure A its relational variant which is obtained by replacing each functionf : As → A by its graph G f := {(a, b) ∈ As+1| f (a) = b}.

For a structure A and a formula ϕ(x), let ϕA := {a|A |= ϕ(a)} be the relation(or query) defined by ϕ on A.

We assume that the reader is familiar with the basic notions of automata theoryand regular languages. One slightly non-standard aspect is that, in order to present astructure by a list of finite automata, we need a notion of regularity not just for languagesL ⊆ �∗ but also k-ary relations of words, for k > 1. Instead of introducing synchronousmultihead automata that take tuples w = (w1, . . . , wk) of words as inputs and worksynchronously on all k components of w, we reduce the case of higher arities to the unaryone by encoding tuples w ∈ (�∗)k by a single word w1 ⊗ · · · ⊗ wk over the alphabet


(� ∪ {�})k , called the convolution of w1, . . . , wk . Here � is a padding symbol notbelonging to�. It is appended to some of the wordswi to make sure that all componentshave the same length. More formally, for w1, . . . , wk ∈ �∗, with wi = wi1 · · ·wi�i and� = max{|w1|, . . . , |wk |},

w1 ⊗ . . .⊗ wk :=

w′11...

w′k1

. . .

w′1�...

w′k�

∈ ((� ∪ {�})k)∗,

where w′i j = wi j for j ≤ |wi | and w′i j = � otherwise. Now, a relation R ⊆ (�∗)k iscalled regular if {w1 ⊗ · · · ⊗ wk |(w1, . . . , wk) ∈ R} is a regular language. Below we donot distinguish between a relation on words and its encoding as a language.

Definition 3.1. A relational structure A is automatic if there exist a regular languageLδ ⊆ �∗ and a surjective function ν : Lδ → A such that the relation

Lε := {(w,w′) ∈ Lδ × Lδ|νw = νw′} ⊆ �∗ ×�∗

and, for all predicates R ⊆ Ar of A, the relations

L R := {w ∈ (Lδ)r |(νw1, . . . , νwr ) ∈ R} ⊆ (�∗)r

are regular. An arbitrary (not necessarily relational) structure is automatic if and only ifits relational variant is.

We write AutStr[τ ] for the class of all automatic structures of vocabulary τ . Eachstructure A ∈ AutStr[τ ] can be represented, up to isomorphism, by a list d =〈Mδ, Mε, (MR)R∈τ 〉 of finite automata that recognise Lδ , Lε, and L R for all rela-tions R of A. When speaking of an automatic presentation of A we either mean thefunction ν : Lδ → A or such a list d. An automatic presentation d is called deterministicif all its automata are, and it is called injective if Lε = {(u, u)|u ∈ Lδ} (which impliesthat ν : Lδ → A is injective).

Examples. (1) All finite structures are automatic.(2) Important examples of automatic structures are Presburger arithmetic (N,+)

and its expansion Np := (N,+, |p) by the relation

x |p y : iff x is a power of p dividing y.

Using p-ary encodings (starting with the least significant digit) it is not difficult toconstruct automata recognising equality, addition, and |p.

(3) Natural candidates for automatic structures are those consisting of words. (How-ever, note that free monoids with at least two generators do not have automatic presenta-tions.) Fix some alphabet� and consider the structure Tree(�) := (�∗, (σa)a∈�,�, el)


where

σa(x) := xa, x � y : iff ∃z(xz = y), and el(x, y) : iff |x | = |y|.

Obviously, this structure is automatic as well.

The following two observations are simple, but useful:

(1) Every automatic structure admits an automatic presentation with alphabet{0, 1} [6].

(2) Every automatic structure admits an injective automatic presentation [36].

Automatic Groups. The class of automatic structures that have been studied most in-tensively are automatic groups. Let (G, ·) be a group and let S = {s1, . . . , sm} ⊆ G be aset of semigroup generators of G. This means that each g ∈ G can be written as a productsi1· · · sir of elements of S and hence the canonical homomorphism ν : S∗ → G is surjec-tive. The Cayley graph�(G, S) of G with respect to S is the graph (G, S1, . . . , Sm)whosevertices are the group elements and where Si is the set of pairs (g, h) such that gsi = h.By definition (G, ·) is automatic if there is a finite set S of semigroup generators and aregular language Lδ ⊆ S∗ such that the restriction of ν to Lδ is surjective and providesan automatic presentation of �(G, S). (In other words, the inverse image of equality,

Lε = {(w,w′) ∈ Lδ × Lδ|νw = νw′},

and ν−1(Si ), for i = 1, . . . ,m, are regular.)Note that it is not the group structure (G, ·) itself that is automatic in the sense of Def-

inition 3.1, but the Cayley graph. There are many natural examples of automatic groups(see [22] and [24]). The importance of this notion in computational group theory comesfrom the fact that an automatic presentation of a group yields (efficient) algorithmicsolutions for computational problems that are undecidable in the general case.

ω-Automatic structures. The notion of an automatic structure can be modified andgeneralised in a number of different directions (see [6] and [36]). In particular, we obtainthe interesting class ω-AutStr of ω-automatic structures. The definition is analogous tothe one for automatic structures except that the elements of an ω-automatic structure arenamed by infinite words from some regular ω-language and the relations of the structureare recognisable by Buchi automata.

Examples. (1) All automatic structures are ω-automatic.(2) The real numbers with addition, (R,+), and indeed the expanded structure

Rp := (R,+,≤, |p, 1) are ω-automatic, where p ≥ 2 is an integer and

x |p y : iff ∃n, k ∈ Z: x = pn and y = kx .

(3) The tree automatic structures Tree(�) extend in a natural way to the (uncount-able) ω-automatic structures Treeω(�) := (�≤ω, (σa)a∈σ ,�, el).


4. Characterising Automatic Structures via Interpretations

Interpretations constitute an important tool in mathematical logic. They are used to de-fine a copy of a structure inside another one, and thus permit the transfer of definability,decidability, and complexity results among theories.

Definition 4.1. Let L be a logic, and let A = (A, R0, . . . , Rn) and B be relationalstructures. A (k-dimensional) L-interpretation of A in B is a sequence

I = ⟨δ(x), ε(x, y), ϕR0(x1, . . . , xr ), . . . , ϕRn (x1, . . . , xs)⟩

of L-formulae of the vocabulary of B (where each tuple x , y, xi consists of k variables)such that

A ∼= I(B) := (δB, ϕBR0, . . . , ϕB

Rn

)/εB.

To make this expression well-defined we require that εB is a congruence relation on thestructure

(δB, ϕB

R0, . . . , ϕB

Rn

). We denote the fact that I is an L-interpretation of A in B

by I : A ≤L B. If A ≤L B and B ≤L A we say A and B are mutually L-interpretable.The epimorphism

(δB, ϕB

R0, . . . , ϕB

Rn

)→ A is called a coordinate map and is alsodenoted by I. If it is the identity function, i.e., A = I(B), we say that A is L-definablein B. An interpretation I is injective if its coordinate is injective, i.e., if ε(x, y) ≡ x = y.

Examples. (1) Recall that we write a|pb to denote that a is a power of p dividing b. LetVp : N→ N be the function that maps each number to the largest power of p dividingit. It is very easy to see that the structures (N,+, |p ) and (N,+, Vp) are mutually first-order interpretable. Indeed, we can define the statement x = Vp(y) in (N,+, |p ) by theformula x |p y∧∀z(z|p y → z|px). In the other direction, Vp(x) = x∧∃z(x+ z = Vp(y))is a definition of x |p y.

(2) For every p ∈ Nwe write Tree(p) for the tree structure Tree({0, . . . , p−1}). Thestructures Np and Tree(p) are mutually interpretable, for each p ≥ 2 (see [6] and [26]).

If I : A ≤FO B, then every first-order formula ϕ over the vocabulary of A can betranslated to a formula ϕI over the vocabulary of B by replacing every relation symbol Rby its definition ϕR , by relativising every quantifier to δ, and by replacing equalities by ε.

Lemma 4.2 (Interpretation Lemma). If I : A ≤FO B, then

A |= ϕ(I(b)) iff B |= ϕI(b) for all ϕ ∈ FO and b ⊆ δB.

This lemma states the most important property of interpretations. For any logic L , anotion of interpretation is considered suitable if a similar statement holds, and if thelogic is closed under the operation ϕ �→ ϕI . Note that in the case of MSO, arbitraryk-dimensional MSO interpretations are too strong since they translate sets to relations


of arity k which takes us out of MSO. On the other hand, the Interpretation Lemma doeshold for one-dimensional MSO interpretations.

Interpretations provide a general and powerful method to obtain classes of finitelypresented structures with a set of desired properties. One fixes some structure B hav-ing these properties and chooses a kind of interpretation that preserves them. Then oneconsiders the class of all structures which can be interpreted in B. Each structure A

of this class can be represented by an interpretation I : A ≤FO B which is a finiteobject, and model checking and query evaluation for such structures can be reduced tothe corresponding problem for B. If I : A ≤FO B, then Lemma 4.2 implies that

ϕA = {a |A |= ϕ(a)} = {I(b) |B |= ϕI(b)}.

Hence, the desired representation of ϕA can be constructed by extending the interpreta-tion I to 〈I, ϕI〉 : (A, ϕA) ≤FO B.

Automatic structures are closed under first-order interpretations.

Proposition 4.3. If A ≤FO B and B is (ω-) automatic, then so is A.

Proof. Since B is automatic there are regular languages Lδ for the universe, Lε forequality, and L R for each relation R of B. By the closure of regular languages underboolean operations and projections it follows that, for each first-order formula ϕ, thelanguage encoding the relation ϕB is also regular.

Corollary 4.4. The classes of automatic, resp. ω-automatic, structures are closed un-der (i) extensions by definable relations, (ii) factorisations by definable congruences,(iii) substructures with definable universe, and (iv) finite powers.

As stated above the class of automatic structures can be characterised via first-orderinterpretations.

Theorem 4.5. For every structure A, the following are equivalent:

(i) A is automatic.(ii) A ≤FO Np for some (and hence all) p ≥ 2.

(iii) A ≤FO Tree(p) for some (and hence all) p ≥ 2.

Proof. The facts that (ii) and (iii) are equivalent and that they imply (i) follow immedi-ately from the mutual interpretability of Np and Tree(p), from the fact that these struc-tures are automatic, and from the closure of automatic structures under interpretation.

It remains to show that every automatic structure is interpretable in Np (or Tree(p)).Suppose that d is an automatic presentation of A with alphabet [p] := {0, . . . , p−1} forsome p ≥ 2 (without loss of generality, we could take p = 2). For every wordw ∈ [p]∗,let val(w) be the natural number whose p-ary encoding isw, i.e., val(w) :=∑i<|w|wi pi .By a classical result, sometimes called the Buchi–Bruyere Theorem, a relation R ⊆ Nk


is first-order definable in (N,+, Vp) if and only if

{(val−1(x1), . . . , val−1(xk))|(x1, . . . , xk) ∈ R}

is regular. (See [10] for a proof of this fact and for more information on the relationshipbetween automata and definability in expansions of Presburger arithmetic.) The formulaethat define in this sense the regular language and the regular relations in an automaticpresentation of A provide an interpretation of A in (N,+, Vp). Hence also A ≤FO Np.

For automatic groups we are not free to change the coordinate map. Indeed, thedefinition of an automatic group requires that the function ν : Lδ → G be the restrictionof the canonical homomorphism from S∗ to G. Hence the arguments used above give usa characterisation of automatic groups in terms of definability rather than interpretability.

Theorem 4.6. (G, ·) is an automatic group if and only if there exists a finite set S ⊆ Gof semigroup generators such that �(G, S) is first-order definable in Tree(S).

By definition, if G is an automatic group, then for some set S of semigroup genera-tors, the Cayley graph �(G, S) is an automatic structure. Contrary to a claim in [36] theconverse does not hold. A counterexample, which has been pointed out by Senizergues,is the Heisenberg group H which is the group of affine transformations of Z3 generatedby the maps

α : (x, y, z) �→ (x + 1, y, z + y),

β : (x, y, z) �→ (x, y + 1, z),

γ : (x, y, z) �→ (x, y, z + 1).

Using this matrix representation of H, it is not difficult to construct a (three-dimensional)interpretation of �(H, S) in (N,+), which implies that �(H, S) ∈ AutStr. However, in[22] it is shown that H is not automatic.

Proposition 4.7. There exist groups G with a set of semigroup generators S such thatthe Cayley graph�(G, S) is an automatic structure without G being an automatic group.

We now turn toω-automatic structures. To provide a similar characterisation we canuse an equivalent of the Buchi–Bruyere Theorem for encodings of ω-regular relations.One such result has been obtained by Boigelot et al. [9]. Using natural translations be-tween ω-words over [p] and real numbers, they prove that a relation over [p]ω can berecognised by a Buchi automaton if and only if its translation is first-order definable inthe structure (R,+, <,Z, X p) where X p ⊆ R

3 is a relation that explicitly representsthe translation between [p]ω and R. X p(x, y, z) holds iff there exists a representationof x by a word in [p]ω such that the digit at the position specified by y is z. A some-what unsatisfactory aspect of this result is the assumption that the encoding relation X p

must be given as a basic relation of the structure. It would be preferable if more naturalexpansions of the additive real group (R,+) could be used instead.


We show here that this is indeed possible if, as in the case of Np, we use a re-stricted variant of the divisibility relation. Recall that the structures Rp and Treeω(p)(introduced at the end of Section 3) are ω-automatic. As a first step we show that thebehaviour of Buchi automata recognising regular relations over [p]ω can be simulated byfirst-order formulae in Treeω(p). Secondly we show that Treeω(p) and Rp are mutuallyinterpretable. As a result we obtain the following model-theoretic characterisation ofω-automatic structures.

Theorem 4.8. For every structure A, the following are equivalent:

(i) A is ω-automatic.(ii) A ≤FO Rp for some (and hence all) p ≥ 2.

(iii) A ≤FO Treeω(p) for some (and hence all) p ≥ 2.

Proof. In order to construct interpretations of Treeω(p) in Rp and vice versa we defineformulae which allow us to access the digits of, respectively, some number in Rp andsome word in Treeω(p). In the latter case we set

digk(x, y) := ∃z(el(z, y) ∧ σk z � x)

which states that the digit of x at position |y| is k. For Rp the situation is more compli-cated as some real numbers admit two encodings. The following formula describes thatthere is one encoding of x such that the digit at position y is k (this corresponds to thepredicate X of [9]):

digk(x, y) := ∃s∃t (|x | = s + k · y + t ∧ p · y|ps ∧ 0 ≤ s ∧ 0 ≤ t < y).

For Rp ≤FO Treeω(p) we represent each number as a pair of words. The first oneis finite and encodes the integer part, the other one is infinite and contains the fractionalpart. In the other direction we map finite words a1 · · · ar ∈ [p]∗ to the interval [2, 3] via

p−r+1 +r∑

i=1

ai p−i + 2 ∈ [2, 3].

Infinite words a1a2 · · · ∈ [p]ω are mapped to two intervals [−1, 0] and [0, 1] via

±∑

i

ai p−i ∈ [−1, 1].

This is necessary because some words, e.g., 0(p − 1)ω and 10ω, would be mapped tothe same number otherwise. Now the desired interpretations can be constructed easilyusing the formulae digk defined above.

It remains to prove that if R ⊆ ([p]ω)n is ω-regular, then it is definable in Treeω(p).Let M = (Q, [p]n,�, q0, F) be a Buchi–automaton for R. Without loss of generalityassume Q = [p]m for some m and q0 = (0, . . . , 0). We prove the claim by constructinga formula ψM(x) ∈ FO stating that there is a successful run of M on x1 ⊗ . . .⊗ xn . The


run is encoded by a tuple (q1, . . . , qm) ∈ ([p]ω)m of ω-words such that the symbols ofq1, . . . , qm at some position equal k1, . . . , km iff the automaton is in state (k1, . . . , km)

when scanning the input symbol at that position. ψM(x) has the form

∃q1 · · · ∃qm[ADM(q, x) ∧ START(q, x) ∧ RUN(q, x) ∧ ACC(q, x)],

where the admissibility condition ADM(x, q) states that all components of x and q areinfinite, START(x, q) says that the first state is 0, ACC(x, q) says that some final stateappears infinitely often, and RUN(x, q) ensures that all transitions are correct.

Define the following auxiliary formulae. To access the digits of a tuple of words atsome position we define Syma(x, z) :=∧i digai

(xi , z), and to characterise the ω-wordsof [p]≤ω we set

Inf(x) := ∀y(x � y → x = y).

ADM and START are defined as

ADM(q, x) :=m∧

i=1

Inf(qi ) ∧n∧

i=1

Inf(xi ),

START(q, x) := Sym0(q, ε),

RUN states that at every position a valid transition is used,

RUN(q, x) := ∀z∨

(k,a,k ′)∈�

(Symk(q, z) ∧ Syma(x, z) ∧ Symk ′(q, σ0z)

),

and ACC says that there is one final state which appears infinitely often in q:

ACC(q, x) :=∨k∈F

∀z∃z′(|z′| > |z| ∧ Symk(q, z′)

).

5. Model Checking and Query Evaluation

In this section we study decidability and complexity issues for automatic structures. Twofundamental algorithmic problems are:

Model checking. Given a (presentation of a) structure A, a formula ϕ(x), and a tupleof parameters a in A, decide whether A |= ϕ(a).

Query evaluation. Given a presentation of a structure A and some formula ϕ(x),compute a presentation of (A, ϕA). That is, given automata for the relations of A,construct an automaton that recognises ϕA.

Decidability. We first observe that all first-order queries on (ω-)automatic structuresare effectively computable since the construction in Proposition 4.3 is effective. In fact,this is the case not only for first-order logic but also for formulae containing the quan-tifier ∃ω meaning “there are infinitely many”. To prove this result for the case of ω-automatic structures we require some preparations.


Lemma 5.1. Let R ⊆ �ω⊗�ω be regular. There is a regular relation R′ ⊆ R such that

(i) if (x, y) ∈ R, then there is some y′ such that (x, y′) ∈ R′, and(ii) for all x there is at most one y with (x, y) ∈ R′.

Proof. Without loss of generality, assume that R �= ∅. First we introduce some notation.By v[i, k) we denote the factor vi · · · vk−1 of v = v0v1 · · · ∈ �ω. Similarly, v[i, ω) isequal to vivi+1 · · ·, and v[i] := v[i, i + 1).

Let A = (Q, � × �,�, q0, F) be a Buchi automaton recognising R. Fix an order-ing � of Q such that all final states are less than non-final ones. For a run (sequence ofstates) � ∈ Qω define

Inf(�) := {q ∈ Q|there are infinitely many i such that �[i] = q},µ(�) := min Inf(�).

Let fi (�) be the greatest number such that �[k, fi (�)) contains exactly i times thestate µ(�) where k is the least position such that all states appearing only finitely oftenare contained in �[0, k).

Denote the lexicographically least run of A on x ⊗ y by �(x, y). Fix x ∈ �ω andset µ(y) := µ(�(x, y)), fi (y) := fi (�(x, y)). We define an order on �ω by

y ≤ y′ iff µ(y) < µ(y′),or µ(y) = µ(y′) and there is some n such that

fn(y) < fn(y′) and fi (y) = fi (y

′) for all i < n,

or µ(y) = µ(y′), fi (y) = fi (y′) for all i , and

y is lexicographically less than or equal to y′.

It should be obvious that the relation ≤ is regular. Finally, R′ is defined by

R′ := {(x, y) ∈ R | there is no y′ < y such that (x, y′) ∈ R}.

Clearly, R′ is regular, is contained in R, and satisfies (ii). Hence, it remains to prove(i). We directly construct the minimal element of V := {y | (x, y) ∈ R} as follows. LetY−1 ⊆ V be the subset of those y with minimal µ(y). We define a sequence of setsY−1 ⊇ Y0 ⊇ Y1 ⊇ · · · by

Yi := { y ∈ Yi−1 | fi (y) ≤ fi (y′) for all y′ ∈ Yi−1 }.

For i ≥ 0, fix some element yi ∈ Yi such that yi [0, fi ) is lexicographically minimal.Hence, y0 ≥ y1 ≥ · · ·. Define y by

y[n] := limk yk[n].

We claim that y is the minimal element of V .


(a) y exists. Set fn := limk fn(yk) = fn(yn). The pointwise limit of yk exists sinceyn+1[0, fn) = yn[0, fn) for all n. For, otherwise, there is some k < fn such that

yn+1[0, k) = yn[0, k) and yn+1[k] �= yn[k].

Since yn , yn+1 ∈ Yn it follows that yn[k] < yn+1[k]. On the other hand, y′ :=yn[0, fn)yn+1[ fn, ω) is in V and thus in Yn+1, as �(x, yn)[ fn] = �(x, yn+1)[ fn]. Thus,yn+1[k] ≤ y′[k] = yn[k] by choice of yn+1. Contradiction.

(b) y ∈ V . An accepting run of A on x ⊗ y is given by � where

�[ fn−1, fn) = �(x, yn)[ fn−1, fn)

since �[ fn] = µ(y0) ∈ F for all n.(c) y is minimal. Suppose y′ ≤ y for some y′ ∈ V . Then µ(y′) ≤ µ(y) and, by

induction on n, one can show that y′ ∈ Yn since fn(y′) ≤ fn(y). Thus by constructionµ(y′) = µ(y) and fn(y′) = fn(y). Suppose y′ < y. Then y′ must be lexicographicallyless than y and there exists some k such that y′[0, k) = y[0, k) and y′[k] < y[k]. Choosen such that fn−1 ≤ k < fn . Then yn ≤ y′ by construction. However, yn[0, fn) =y[0, fn) and hence yn[0, k) = y′[0, k) which implies that y′[k] ≥ yn[k] = y[k].Contradiction.

Proposition 5.2. Every (ω-)automatic structure has an injective presentation.

Proof. For automatic structures this result is due to Khoussainov and Nerode [36].Let ν : D → A be a presentation of an ω-automatic structure A. By the preceding

lemma applied to the relation {(x, y)|νx = νy} there is a function e such that

(i) νx = νex for all x ∈ �ω, and(ii) νx = νy implies ex = ey.

Thus we obtain a regular subset D′ ⊆ D containing exactly one representation for eachelement of A by defining D′ := {x ∈ D|ex = x}.

We say that a logic L effectively collapses to L0 ⊆ L on a structure A if, given aformula ϕ(x) ∈ L , one can compute a formula ϕ0(x) ∈ L0 such that ϕA

0 = ϕA.

Proposition 5.3.

(1) FO(∃ω) effectively collapses to FO on Tree(p).(2) FO(∃ω) effectively collapses to FO on Treeω(p).

Proof. (1) In the case of automatic structures the quantifier ∃ω can be handled using apumping argument. Consider for simplicity the formula ∃ωxψ(x, y). By induction theformulaψ is equivalent to some first-order formula and, hence, there is some automatonrecognising the relation defined by ψ . There are infinitely many x satisfying ψ iff forany m there are infinitely many such elements whose encoding is at least m symbolslonger than that of y. If we take m to be the number of states of the automaton for ψ


then, by the Pumping Lemma, the last condition is equivalent to the existence of at leastone such x . Thus ∃ωxψ(x, y) ≡ ∃x(ψ(x, y) ∧ “x is long enough”) for which we canobviously construct an automaton.

(2) For ω-automatic structures the proof is more involved.Let M be a deterministic Muller automaton with s states recognising the language

L(M) ⊆ �ω ⊗�ω. For w ∈ �ω let V (w) := {v ∈ �ω|w ⊗ v ∈ L(M)}.Let v, w ∈ �ω and define v ≈∗ w iff v[n, ω) = w[n, ω) for some n. Let

[v]∗ := {v′ ∈ V (w)|v′ ≈∗ v} be the ≈∗-class of v in V (w).

Claim. V (w) is infinite if and only if there is some v ∈ �ω such that [v]∗ ∈ V (w)/≈∗is infinite.

Proof. (⇐) is trivial and (⇒) is proved by showing that V/≈∗ contains at most s finite≈∗-classes.

Assume there are words v0, . . . , vs ∈ V (w) belonging to different finite ≈∗-classes. Denote the run of M on w ⊗ vi by �i . Define Ii j := {k < ω|�i [k] = �j [k]}.Since there are only s states, for each k < ω there have to be indices i , j such thatk ∈ Ii j , i.e.,

⋃i, j Ii j = ω. Thus, at least one Ii j is infinite. For each [vi ]∗ there is a

position ni such that v[ni , ω) = v′[ni , ω) for all v, v′ ∈ [vi ]∗. Let m be the maxi-mum of n0, . . . , ns . Fix i, j such that Ii j is infinite. Since vi �≈∗ vj there is a positionm ′ > m such that vi [m,m ′) �= vj [m,m ′). Choose some m ′′ ∈ Ii j with m ′′ ≥ m ′. Letu := vi [0,m)vj [m,m ′′)vi [m ′′, ω). Thenw⊗vi ∈ L(M) iffw⊗u ∈ L(M)which impliesthat u ∈ [vi ]∗. However, u[m, ω) �= vi [m, ω) in contradiction to the choice of m.

To finish the proof let ϕ(x) := ∃ωyψ(x, y) and A be ω-automatic. One can expressthat [v]∗ is finite by

finite(x, v) := ∃n∀v′[ψ(x, v′) ∧ v ≈∗ v′ → equal(v, v′, n)],

where

equal(v, v′, n) := n = 1i 0ω ∧ v[i, ω) = v′[i, ω).

Clearly, ≈∗ and equal can be recognised by ω-automata. By the claim above,

ϕ(x) ≡ ∃v(ψ(x, v) ∧ ¬finite(x, v)).

Hence, we can construct an automaton recognising ϕA.

We are now ready to prove Theorem 2.1, saying that the model checking problemfor FO(∃ω) is decidable on ω-automatic structures.

Proof of Theorem 2.1. Given a presentation of an ω-automatic structure A, and a for-mula ϕ ∈ FO(∃ω) we compute an injective interpretation I : A ≤FO Treeω(p). SinceFO(∃ω) is closed under injective interpretations, we have that ϕI ∈ FO(∃ω) and A |= ϕ


iff Treeω(p) |= ϕI . By Proposition 5.3 we can effectively compute a formula ψ ∈ FOthat is equivalent to ϕI on Treeω(p). Since the first-order theory of any ω-automaticstructure is decidable, the result follows.

Corollary 5.4. The FO(∃ω) theory of any (ω)-automatic structure is decidable.

As an immediate consequence we conclude that full arithmetic (N,+, ·) is neitherautomatic nor ω-automatic. For most of the common extensions of first-order logic usedin finite model theory, such as transitive closure logics, fixed point logics, MSO logic,or first-order logic with counting, the model-checking problem on automatic structuresbecomes undecidable.

Complexity. The complexity of model checking can be measured in three differentways. First, one can fix the formula and ask how the complexity depends on the inputstructure. This measure is called structure complexity. The expression complexity on theother hand is defined relative to a fixed structure in terms of the formula. Finally, onecan look at the combined complexity where both parts may vary.

Of course, the complexity of these problems may very much depend on how au-tomatic structures are presented. Since the decision methods for Np and Tree(p) areautomaton based, a presentation d consisting of a list of automata, is more suitablefor practical purposes than using an interpretation. Here, we focus on presentations bydeterministic automata because these admit boolean operations to be performed in poly-nomial time, whereas for non-deterministic automata, complementation may cause anexponential blow-up.

In the following we always assume that the vocabulary of the given automatic struc-tures and the alphabet of the automata we deal with are fixed. Furthermore, the vocabularyis assumed to be relational when not stated otherwise. For a (deterministic) presentation d

of an automatic structure A, we denote by |d| the maximal size of the automata in d, and,fixing some surjective function ν : L(Mδ)→ A, we define λ : A → N to be the function

λ(a) := min{|x ||ν(x) = a}

mapping each element ofA to the length of its shortest encoding. Finally, letλ(a1, . . . , ar )

be an abbreviation for max{λ(ai )|i = 1, . . . , r}.While we have seen above that query evaluation and model checking for first-order

formulae are effective on AutStr, the complexity of these problems is non-elementary,i.e., it exceeds any fixed number of iterations of the exponential function. This followsimmediately from the fact that the complexity of Th(Np) is non-elementary (see [26]).

Proposition 5.5. There exist automatic structures such that the expression complexityof the model-checking problem is non-elementary.

It turns out that model checking and query evaluation for quantifier-free and exis-tential formulae are still—to some extent—tractable. As usual, let �0 and �1 denote,respectively, the class of quantifier-free and the class of existential first-order formulae.


Structure complexity Expression complexity

Model checking�0 LOGSPACE-complete ALOGTIME-complete�0 + fun NLOGSPACE PTIME-complete�1 NPTIME-complete PSPACE-complete

Query Evaluation�0 LOGSPACE PSPACE

�1 PSPACE EXPSPACE

Theorem 5.6.

(i) Given a presentation d of a relational structure A ∈ AutStr, a tuple a in A, anda quantifier-free formula ϕ(x) ∈ FO, the model-checking problem for (A, a, ϕ)is in

DTIME[O(|ϕ|λ(a)|d| log|d|)] and

DSPACE[O(log|ϕ| + log|d| + log λ(a)

)].

(ii) The structure complexity of model checking for quantifier-free formulae isLOGSPACE-complete with respect to first-order reductions.

(iii) The expression complexity is ALOGTIME-complete with regard to deterministiclog-time reductions.

Proof. (i) To decide whether A |= ϕ(a) holds, we need to know the truth value of eachatom appearing in ϕ. Then all what remains is to evaluate a boolean formula which can bedone in DTIME

[O(|ϕ|)] and ATIME

[O(log|ϕ|)] ⊆ DSPACE

[O(log|ϕ|)] (see [11]). The

value of an atom Rx can be calculated by simulating the corresponding automaton onthose components of a which belong to the variables appearing in x . The naıve algorithmto do so uses time O

(λ(a)|d| log|d|)) and space O

(log|d| + log λ(a)

).

For the time-complexity bound we perform this simulation for every atom, store theoutcome, and evaluate the formula. Since there are at most |ϕ| atoms the claim follows.

To obtain the space bound we cannot store the value of each atom. Therefore we usethe LOGSPACE-algorithm to evaluate ϕ and, every time the value of an atom is needed,we simulate the run of the corresponding automaton on a separate set of tapes.

(ii) We present a reduction of the LOGSPACE-complete problem DETREACH, reach-ability by deterministic paths (see, e.g., [34]), to the model-checking problem. Given agraph G = (V, E, s, t) we construct the automaton M = (V, {0},�, s, {t}) with

� := {(u, 0, v)|u �= t, (u, v) ∈ E and there is no v′ �= v with (u, v′) ∈ E}∪ {(t, 0, t)}.

That is, we remove all edges originating at vertices with out-degree greater than 1 andadd a loop at t . Then there is a deterministic path from s to t in G iff M accepts someword 0n iff 0|V | ∈ L(M). Thus,

(V, E, s, t) ∈ DETREACH iff A |= P0|V |,


where A = (B, P) is the structure with the presentation ({0}∗, L(M)). A closer inspec-tion reveals that the above transformation can be defined in first-order logic.

(iii) Evaluation of boolean formulae is ALOGTIME-complete (see [11]).

For most questions we can restrict attention to relational vocabularies and replacefunctions by their graphs at the expense of introducing additional quantifiers. Whenstudying quantifier-free formulae we will not want to do this and hence need to con-sider the case of quantifier-free formulae with function symbols separately. This class isdenoted �0 + fun.

Lemma 5.7. Given a tuplew of words over�, and an automaton A = (Q, �, δ, q0, F)recognising the graph of a function f , the calculation of f (w) is in

DTIME[O(|Q2| log|Q|(|Q| + |w|))] and DSPACE

[O(|Q| log|Q| + log|w|)] .

Proof. The following algorithm simulates A on input w0 ⊗ . . .⊗wn−1 ⊗ x where x isthe result that we want to calculate. For every position i of the input, the set Qi of stateswhich can be reached for various values of x is determined. At the same time the setsQi and Qi+1 are connected by edges Ei labelled by the symbol of x by which the secondstate could be reached. When a final state is found, x can be read off the graph.

We use the function Step(Q, a) depicted in Figure 1 to compute Qi+1 and Ei from Qi

and the input symbol a. If E is realised as an array containing, for every q ∈ Q, the valuesq ′ and c such that (q ′, c, q) ∈ E , then this function needs spaceO

(|Q| log|Q|) and time

O(|Q|(|Q| log|Q| + |Q| log|Q|)) = O(|Q2| log|Q|).

Two slightly different algorithms are used to obtain the time- and space-complexitybounds (see Figure 2). The first one simply computes all sets Qi and Ei and determines x .The second one reuses space and keeps only one set Qi and Ei in memory. Therefore

Step(Q, a)Q′ := ∅E := ∅forall q ∈ Q

forall c ∈ �q ′ := δ(q, ac)if q ′ /∈ Q′ then

E := E ∪ {(q, c, q ′)}Q′ := Q′ ∪ {q ′}

endend

return (Q′, E)

Fig. 1. Computing Qi+1 and Ei from Qi .


Input: A = (Q, �, δ, q0, F), w Input: A = (Q, �, δ, q0, F), wQ0 := {q0} Q := {q0}i := 0 i := 0whileQi ∩ F = ∅ whileQ ∩ F = ∅

if i < |w| then if i < |w| thena := w[i] a := w[i]

else elsea := ✷ a := ✷

(Qi+1, Ei ) := Step(Qi , a) (Q, E) := Step(Q, a)i := i + 1 i := i + 1

end endlet q ∈ Qi ∩ F let q ∈ Q ∩ Fwhilei > 0 whilei > 0

i := i − 1 i := i − 1Q := {q0}fork = 0, . . . , i − 1if k < |w| then

a := w[k]else

a := ✷

(Q, E) := Step(Q, a)end

let (q ′, c, q) ∈ Ei let (q ′, c, q) ∈ Ex[i] := c x[i] := cq := q ′ q := q ′

end endreturn x return x

Fig. 2. Two algorithms to compute f (w).

it has to start the computation from the beginning in order to access old values of Ei inthe second part.

In the first version the function Step is invoked |x | times, and the second part isexecuted in time O

(|x ||Q| log|Q|).The space needed by the second version consists of storage for Q, E , and the counters

i and k. Hence, O(|Q| + |Q| log|Q| + log|x |) bits are used.Since A recognises a function, the length of x can be at most |Q| + |w| (see Propo-

sition 6.1 for a detailed proof). This yields the given bounds.

Theorem 5.8.

(i) Let τ be a vocabulary which may contain functions. Given the presentation d

of a structure A in AutStr[τ ], a tuple a in A, and a quantifier-free formulaϕ(x) ∈ FO[τ ], the model-checking problem for (A, a, ϕ) is in

DTIME[O(|ϕ||d2| log|d|(|ϕ||d| + λ(a)))] and

DSPACE[O(|ϕ|(|ϕ||d| + λ(a))+ |d| log|d|)].


(ii) The structure complexity of the model-checking problem for quantifier-freeformulae with functions is in NLOGSPACE.

(iii) The expression complexity is PTIME-complete with regard to ≤logm -reductions.

Proof. (i) Our algorithm proceeds in two steps. First the values of all functions ap-pearing in ϕ are calculated starting with the innermost one. Then all functions can bereplaced by their values and a formula containing only relations remains which can beevaluated as above. We need to evaluate at most |ϕ| functions. If they are nested theresult can be of length |ϕ||d| + λ(a). This yields the bounds given above.

(ii) It is sufficient to present a non-deterministic log-space algorithm for evaluatinga single fixed atom containing functions. The algorithm simultaneously simulates theautomata of the relation and of all functions on the given input. Components of theinput corresponding to values of functions are guessed non-deterministically. Each sim-ulation only needs counters for the current state and the input position which both uselogarithmic space.

(iii) Let M be a p(n) time-bounded deterministic Turing machine for some polyno-mial p. A configuration (q, w, p) of M can be coded as word w0qw1 with w = w0w1

and |w0| = p. Using this encoding both the function f mapping one configuration toits successor and the predicate P for configurations containing accepting states can berecognised by automata. We assume that f (c) = c for accepting configurations c. Letq0 be the starting state of M . Then M accepts some word w if and only if the configura-tion f p(|w|)(q0w) is accepting if and only if A |= P f p(|w|)(q0w) where A = (A, P, f )is automatic. Hence, the mapping taking w to the pair q0w and P f p(|w|)x is the desiredreduction which can clearly be computed in logarithmic space.

Theorem 5.8 says that, on any fixed automatic structure, quantifier-free formulaecan be evaluated in quadratic time. This extends a well-known result on automaticgroups [22].

Corollary 5.9. The word problem for every automatic group is solvable in quadratictime.

Proof. Let G be an automatic group with semigroup generators s1, . . . , sm . We presentthe Cayley graph of G in a functional way, by the structure (G, e, g �→ gs1, . . . , g �→gsm) which is therefore automatic. Each instance of the word problem is described bya quantifier-free sentence (a term equation) on this structure, which can be checked inquadratic time by Theorem 5.8.

Theorem 5.10.

(i) Given a presentation d of a structure A in AutStr, a tuple a in A, and a formulaϕ(x) ∈ �1, the model-checking problem for (A, a, ϕ) is in

NTIME[O(|ϕ||d|λ(a)+ |dO|(|ϕ|))] and

NSPACE[O(|ϕ|(|d| + log|ϕ|)+ log λ(a)

)].


(ii) The structure complexity of model checking for�1-formulae is NPTIME-completewith respect to ≤p

tt-reductions.(iii) The expression complexity is PSPACE-complete with regard to ≤log

m -reductions.

Proof. (i) As above we can run the corresponding automaton for every atom appearingin ϕ on the encoding of a. However, now there are some elements of the input missingwhich we have to guess. Since we have to ensure that the guessed inputs are the samefor all automata, the simulation is performed simultaneously.

The algorithm determines which atoms appear in ϕ and simulates the product au-tomaton constructed from the automata for those relations. At each step the symbolfor the quantified variables is guessed non-deterministically. Note that the values ofthose variables may be longer than the input so we have to continue the simulationafter reaching its end for at most the cardinality of the state-space number of steps.Since this cardinality is O

(|d||ϕ|) a closer inspection of the algorithm yields the givenbounds.

(ii) We reduce the NPTIME-complete non-universality problem for non-deterministicautomata over a unary alphabet (see [40] and [33]), given such an automaton, checkwhether it does not recognise the language 0∗, to the given problem. This reduction isperformed in two steps. First the automaton must be simplified and transformed into adeterministic one, then we construct an automatic structure and a formula ϕ(x) such thatϕ(a) holds for several values of a if and only if the original automaton recognises 0∗.As model checking has to be performed for more than one parameter this yields not amany-to-one but a truth-table reduction.

Let M = (Q, {0},�, q0, F) be a non-deterministic finite automaton over the al-phabet {0}. We construct an automaton M ′ such that there are at most two transitionsoutgoing at every state. This is done be replacing all transitions from some given stateby a binary tree of transitions with new states as internal nodes. Of course, this changesthe language of the automaton. Since in M every state has at most |Q| successors, wecan take trees of fixed height k := 'log|Q|(. Thus, L(M ′) = h(L(M)) where h is thehomomorphism taking 0 to 0k . Note that the size of M ′ is polynomial in that of M .

M ′ still is non-deterministic. To make it deterministic we add a second componentto the labels of each transition which is either 0 or 1. This yields an automaton M ′′ suchthat M accepts the word 0n iff there is some y ∈ {0, 1}kn such that M ′′ accepts 0kn ⊗ y.

M ′′ can be used in a presentation d := ({0, 1}∗, L(M ′′)) of some {R}-structure B.Then

B |= ∃y R0kn y iff 0kn ⊗ y ∈ L(M ′′) iff 0n ∈ L(M).

It follows that

L(M) = 0∗ iff B |= ∃y R0kn y for all n < 2|Q|.

The part (⇒) is trivial. To show (⇐) let n be the least number such that 0n /∈ L(M). Byassumption n ≥ 2|Q|. However, then we can apply the Pumping Lemma and find somenumber n′ < n with 0n′ /∈ L(M). Contradiction.


(iii) Let M be a p(n) space-bounded Turing machine for some polynomial p. Weencode configurations as words appending enough spaces to increase their length top(n)+ 1. Let L) := { c0⊗ c1|c0 ) c1 } be the transition relation of M . The run of M oninput w is encoded as sequence of configurations separated by some marker #. L) canbe used to check whether some word x represents a run of M . Let y be the suffix of xobtained by removing the first configuration. The word x ⊗ y has the form

c0 # c1 #c1 # c2 #

· · · # cs−1 # cs

# cs #.

Thus x encodes a valid run iff x ⊗ y ∈ LT where

LT :=(

L)##

)∗(�∗ ⊗ ε).

Clearly, the language L F of all runs whose last configuration is accepting is regular.Finally, we need two additional relations. Both the prefix relation � and the shift sare regular where s(ax) := x for a ∈ � and x ∈ �∗. Therefore, the structureA := (A, T, F, s,�) is automatic, and it should be clear that

w ∈ L(M) iff A |= ϕw(q0w�k−|w|#

),

where k := p(|w|) and

ϕw(x) := ∃y0 · · · ∃yk+1

(∧i≤k

syi = yi+1 ∧ x � y0 ∧ T y0 yk+1 ∧ Fy0

).

ϕw(x) states that there is an accepting run y0 of M starting with configuration x .y1, . . . , yk+1 are used to remove the first configuration from y0, so we can use T tocheck whether y0 is valid.

Clearly, the mapping of w to ϕw and q0w�k−|w|# can be computed in logarithmicspace.

We now turn to the query-evaluation problem for these formula classes.

Theorem 5.11. Given a presentation d of a structure A in AutStr and a formula ϕ(x),an automaton representing ϕA can be computed

(i) in timeO(|dO|(|ϕ|)) and spaceO

(|ϕ| log|d|) in the case of quantifier-free ϕ(x),and

(ii) in time O(2|dO|(|ϕ|))

and space O(|dO|(|ϕ|)) in the case of existential formulae

ϕ(x).

In particular, the structure complexity of query evaluation is in LOGSPACE for quantifier-free formulae and in PSPACE for existential formulae. The expression complexity is inPSPACE for quantifier-free formulae and in EXPSPACE for existential formulae.


Proof. Enumerate the state-space of the product automaton and output the transitionfunction.

Undecidability. In the remainder of this section we present some undecidability re-sults for automatic structures. When we say that a property P of automatic structures isundecidable, we mean that there is no algorithm that, given an automatic presentationof a structure, decides whether the structure has property P .

Lemma 5.12. The configuration graph of any Turing machine is automatic.

Proof. We encode a configuration of a Turing machine M with state q, tape contentsw,and head position p by the word w0qw1 where w = w0w1 and |w0| = p. At every tran-sition w0qw1 )M w′0q ′w′1 only the symbols around the state are changed. This can bechecked by an automaton.

Corollary 5.13. REACHABILITY is undecidable for automatic structures.

This is an immediate consequence of Lemma 5.12 and the undecidability of thehalting problem. For further results, we make use of a normal form for Turing machines.

Lemma 5.14. For any deterministic 1-tape Turing machine M , one can effectivelyconstruct a deterministic 2-tape Turing machine M ′ such that the configuration graphof M ′ consists of a disjoint union of

(a) a countably infinite number of infinite acyclic paths with a first but without alast element, and

(b) for each word x ∈ L(M), one path starting at an initial configuration andending in a loop.

Proof. It is well known that any Turing machine M can be translated into an equivalentreversible Turing machine M ′ (see [4]). We slightly modify this construction. Whilesimulating M on its first tape the machine M ′ appends to the second tape the transitionsperformed at each step. Hence, the storage content of M ′ at each step is a pair (y, z)wherey describes a configuration of M , and z is a sequence of transitions of M . Further, wedefine M ′ such that if M terminates without accepting, then M ′ diverges, i.e., its compu-tation is infinite and not periodic (for instance, it moves the head to the right at every step).Thus, if x /∈ L(M), then the run of M ′ on input x consists of an infinite path of type (a).

The construction as presented so far does not suffice since there may exist configura-tions that cannot be reached from any initial configuration. We have to ensure that everypath containing such a configuration is of type (a). Clearly, every such path must have afirst element since z never decreases. We modify M ′ such that whenever M reaches anaccepting configuration, M ′ reverses its computation, using the transitions stored on thesecond tape, but without changing the content of the second tape. That is, from config-urations with storage content (y, z) where y is accepting for M , M ′ will either reach aconfiguration with storage content (y0, z) where y0 is an initial configuration of M andz represents the computation of M from y0 to z, or M ′ will detect that no such y0 exists.


In the second case, M ′ diverges as above. In the first case, M ′ enters a cycle as follows:it restarts the simulation of M from y0 (leaving z unchanged and checking at everystep that the sequence of transitions on the second tape is correct). When the acceptingconfiguration (y, z) is reached again, the computation is again reversed completing thecycle.

Note that with this modification the run of M ′ on inputs x with x ∈ L(M) is oftype (b). Further, the nodes of the configuration graphs with two incoming edges (thestarting points of cycles) are precisely the accepting configurations that are reachablefrom an initial configuration.

Theorem 5.15. It is undecidable whether two automatic structures are isomorphic.

Proof. Let M be a deterministic 1-tape Turing machine, and let M ′ be the associ-ated Turing machine as described in the preceding lemma. Given M , we can effectivelyconstruct an automatic presentation of the configuration graph G of M ′. Further, we con-struct an automatic presentation of the graph H consisting ofℵ0 copies of (ω, suc). ThenG ∼= H iff L(M) = ∅. Since the emptiness problem for Turing machines is undecidable,so is the isomorphism problem for automatic structures.

We recall that a directed graph is called connected if its underlying undirected graphis. Further, it is called strongly connected if every node is reachable from every othernode via a directed path.

Theorem 5.16. It is undecidable whether an automatic graph is (i) connected or (ii)strongly connected.

Proof. Given a Turing machine M , we again construct an automatic presentation ofthe configuration graph G of the associated machine M ′ from Lemma 5.14. Let I bethe set of initial configurations of M ′; obviously, I is automatic. Further, let X2 be theset of nodes of G with two predecessors and let X0 be the set of nodes without pre-decessors. Finally, let G ′ be the graph obtained from G by connecting all nodes fromX2 ∪ (X0 − I ) with each other. Since X2 and X0 are first-order definable it followsthat the resulting graph G ′ is still automatic. Further, G ′ is connected if all compo-nents of G correspond to accepting computations of M or to computations from un-reachable configurations. Hence, G ′ is connected iff M accepts all inputs, which isundecidable.

Finally, note that the inverse of the graph G ′′ obtained from G ′ by adding to eachedge is also first-order definable from G ′ and is strongly connected if and only if G ′ isconnected. Hence, strong connectedness of automatic graphs is also undecidable.

6. Structures that Are Not Automatic

To prove that a structure is automatic, we just have to find a suitable presentation. How-ever, how can we prove that a structure is not automatic? The main difficulty is that


a priori nothing is known about how elements of an automatic structure are named bywords of the regular language.1

Besides the two obvious criteria, namely, that automatic structures are countableand that their first-order theory is decidable, not much is known. The only non-trivialcriterion that is available at present for general structures uses growth rates for the lengthof the encodings of elements of definable sets. For recent progress on the classificationof automatic linear orders see [18] and [37].

Proposition 6.1 [21]. LetAbe an automatic structure with injective presentation (ν, d),and let f : An → A be a function of A. Then there is a constant m such thatλ( f (a)) ≤ λ(a)+ m for all a ∈ An .

The same is true if we replace f by a relation R where for all a there are only finitelymany values b such that Rab holds.

This result deals with a single application of a function or relation. In the remainingpart of this section we study the effect of applying functions iteratively, i.e., we considersome definable subset of the universe and calculate upper bounds on the length of theencodings of elements in the substructure generated by it. First we need bounds for the(encodings of) elements of some definable subsets. The following lemma follows easilyfrom classical results in automata theory (see, e.g., Proposition V.1.1 of [20]).

Lemma 6.2. Let A be a structure in AutStr with presentation d, and let B be anFO(∃ω)-definable subset of A. Then λ(B) is a finite union of arithmetical progressions.

In the process of generating a substructure we have to count the number of applica-tions of functions.

Definition 6.3. Let A ∈ AutStr with presentation d, let R1, . . . , Rs be finitely manyrelations of A with arities r1 + 1, . . . , rs + 1, respectively, and let E = {e1, e2, . . .} besome subset of A with λ(e1) ≤ λ(e2) ≤ · · ·. Then Gn(E), the nth generation of E , isdefined inductively by

G1(E) := {e1},Gn(E) := {en} ∪ Gn−1(E) ∪ {b | (a, b) ∈ Ri , a ∈ Gri

n−1(E), 1 ≤ i ≤ s}.

Putting everything together we obtain the following result. The case of finitelygenerated substructures already appeared in [36].

Proposition 6.4. Let d an injective presentation of an automatic structureA, let f1, . . . , fr be finitely many definable operations on A, and let E be a defin-able subset of A. Then there is a constant m such that λ(a) ≤ mn for all a ∈ Gn(E). Inparticular, |Gn(E)| ≤ |�mn+1| where � is the alphabet of d.

1 In the case of automatic groups, where the naming function is fixed, more techniques are available suchas the k-fellow traveller property, see [22].


The proof consists of a simple induction on n.

Theorem 6.5. None of the following structures has an automatic presentation.

(i) Any trace monoid M = (M, ·) with at least two non-commuting generatorsa and b.

(ii) Any structure A in which a pairing function f can be defined.(iii) The divisibility poset (N, |).(iv) Skolem arithmetic (N, ·).

Proof. (i) We show that {a, b}≤2n ⊆ Gn+1(a, b) by induction on n. We have {a, b} ⊆{a, aa, b} = G2(a, b) for n = 1, and, for n > 1,

Gn+1(a, b) = {uv | u, v ∈ Gn(a, b)}⊇ {uv | u, v ∈ {a, b}≤2n−1}= {a, b}≤2n

.

Therefore, |Gn(a, b)| ≥ 22nand the claim follows.

(ii) is analogous to (i), and (iv) immediately follows from (iii) as the divisibilityrelation is definable in (N, ·).

(iii) Suppose (N, |) ∈ AutStr. We define the set of primes

Px iff x �= 1 ∧ ∀y(y|x → y = 1 ∨ y = x),

the set of powers of some prime

Qx iff ∃y(Py ∧ ∀z(z|x ∧ z �= 1 → y|z)),

and a relation containing all pairs (n, pn), where p is a prime divisor of n,

Sxy iff x |y ∧ ∃=1z(Qz ∧ ¬Pz ∧ z|y ∧ ¬z|x).

The least common multiple of two numbers is

lcm(x, y) = z iff x |z ∧ y|z ∧ ¬∃u(u �= z ∧ x |u ∧ y|u ∧ u|z).

For every n ∈ N there are only finitely many m with Snm. Therefore S satisfiesthe conditions of Proposition 6.1. Consider the set generated by P via S and lcm,and let γ (n) := |Gn(P)| be the cardinality of Gn(P). If (N, |) is in AutStr then(N, |, P, Q, S) ∈ AutStr, and γ (n) ∈ 2O(n) by Proposition 6.4. Let P = {p1, p2, . . .}.For n = 1 we have G1(P) = {p1}. Generally, Gn(P) consists of

(1) numbers of the form pk11 ,

(2) numbers of the form pk22 · · · pkn

n , and(3) numbers of a mixed form.


In n steps we can create

(1) p1, . . . , pn1 (via S),

(2) γ (n − 1) numbers with k1 = 0, and(3) for every 0 < k1 < n, γ (n − 2)− 1 numbers of a mixed form (via lcm).

All in all we obtain

γ (n) ≥ n + γ (n − 1)+ (n − 1)(γ (n − 2)− 1)

= γ (n − 1)+ (n − 1)γ (n − 2)+ 1

≥ nγ (n − 2) (as γ (n − 1) > γ (n − 2))

≥ n(n − 2) · · · 3γ (1) (without loss of generality assume that n is odd)

= n(n − 2) · · · 3≥ ((n + 1)/2)!

∈ 2�(n log n).

Contradiction.

Remark. (1) Since it is easy to construct a tree-automatic presentation of Skolemarithmetic this result implies that the class of structures with tree-automatic presentationstrictly includes the class of automatic structures (see [6]).

(2) The structure (N,⊥), where ⊥ stands for having no common divisor, is au-tomatic. To see this, we represent each number n ∈ N by a pair (w, k) where w =w0w1 · · · ∈ {0, 1}∗ such that wi = 1 iff the i th prime divides n, and k is the number ofelements m < n with the same set of prime divisors as n. Then (w, k) ⊥ (w′, k ′) iff wand w′ represent disjoint sets which can obviously be checked by an automaton.

7. Composition of Structures

The composition method developed by Feferman and Vaught [25] and by Shelah [45](see also [30] and [47]) considers compositions (products and sums) of structures ac-cording to some index structure and allows one to compute—depending on the type ofcomposition—the first-order or MSO theory of the whole structure from the respectivetheories of its components and the monadic theory of the index structure.

The characterisation given in Section 4 can be used to prove closure of automaticstructures under such compositions of finitely many structures. A generalised product—as it is defined below—is a generalisation of a direct product, a disjoint union, and anordered sum. We will prove that given a finite sequence (Ai )i of structures first-orderinterpretable in some structure C, all their generalised products are also first-order inter-pretable in C.

The definition of such a product is a bit technical. Its relations are defined in termsof the types of the components of its elements. The atomic n-type atpA(a) of a tu-ple (a0, . . . , an−1) in a structure A is the conjunction of all atomic and negated atomicformulae ϕ(x) such that ϕ(a) holds in A.


We first look at how a direct product and an ordered sum can be defined using types.

Example. (1) Let A := A0×A1 where Ai = (Ai , Ri ), for i ∈ {0, 1}, and R is a binaryrelation. The universe of A is A0 × A1. Some pair (a, b) belongs to R iff (a0, b0) ∈ R0

and (a1, b1) ∈ R1. This is equivalent to the condition that the atomic types of a0b0 andof a1b1 both include the formula Rx0x1.

(2) Let A := A0 + A1 where Ai = (Ai , <i ), for i ∈ {0, 1}, and <0, <1 are partialorders. The universe of A is A0∪A1

∼= A0 × {✸} ∪ {✸} × A1, and we have

a < b iff a = (a0,✸), b = (b0,✸), and a0 <0 b0,

or a = (✸, a1), b = (✸, b1), and a1 <1 b1,

or a = (a0,✸), b = (✸, b1).

Again, the condition ai <i bi can be expressed using types.

Definition 7.1. Let τ = {R0, . . . , Rs} be a finite relational vocabulary, let rj be thearity of Rj , and let r := max{r0, . . . , rs}. Let (Ai )i∈I be a sequence of τ -structures, andlet I be an arbitrary relational σ -structure with universe I .

Fix for each k ≤ r an enumeration {t k0 , . . . , t k

n(k)} of the atomic k-types and set

σk := σ ∪ {D0, . . . , Dk−1} ∪ {T ml |m ≤ k, l ≤ n(m)}.

The σk-expansion I(b) of I belonging to a sequence b ∈ (∏i∈I (Ai ∪{✸}))

k is given by

DIl (b) := {i ∈ I |(bl)i �= ✸},

(T ml )

I(b) := {i ∈ I |atpA((bj0)i · · · (bjm−1)i ) = tml and

{ j |(bj )i �= ✸} = { j0, . . . , jm−1} }.

For D ⊆ BI and βj ∈ FO[σrj ], C := (I, D, β0, . . . , βs) defines the generalised

product C(Ai )i∈I := (A, R0, . . . , Rs) of (Ai )i∈I where

A :=⋃d∈D

∏i∈I

χdi

({✸}, Ai),

Ri := {b ∈ Ari |I(b) |= βi },

and χb(a0, a1) := ab.

Example (continued).(1) For the direct product of A0 × A1 we would set I := (I ) with I = {0, 1},

D := {(1, 1)}, and

β :=∨l∈L

T 2l 0 ∧

∨l∈L

T 2l 1,

where L is the set of atomic types containing the formula Rx0x1.


(2) In this case we would set I := (I ) with I = {0, 1}, D := {(1, 0), (0, 1)}, and

β :=(

D00 ∧ D10 ∧∨l∈L

T 2l 0

)∨(

D01 ∧ D11 ∧∨l∈L

T 2l 1

)∨ (D00 ∧ D11),

where L is the set of atomic types containing the formula x0 < x1.

Theorem 7.2. Let τ be a finite relational vocabulary, let K be a class of τ -structurescontaining all finite τ -structures, and let there be a structure C such that K ⊆{A|A ≤FO C}. Let I be a finite relational σ -structure, let (Ai )i∈I be a sequence of struc-tures in K, and let C = (I, D, β) be a generalised product. Then C(Ai )i∈I ∈ K, andan interpretation C(Ai )i∈I ≤FO C can be constructed effectively from the interpretationsAi ≤FO C and I ≤FO C.

Proof. Let τ = {R0, . . . , Rs}. Without loss of generality assume that I = {0, . . . , |I |−1} and that C contains constants 0 and 1. We have to construct an interpretation ofA := C(Ai )i∈I in C. Let rj be the arity of Rj . Consider ni -dimensional interpretations

I i := ⟨hi , δi (x i ), εi (x i , yi ), ϕi0(x

i0, . . . , x i

r0−1), . . . , ϕis(x

i0, . . . , x i

rs−1)⟩

of Ai in C. We represent an element a of A by a tuple of (|I |+n0+ . . .+n|I |−1) elements

x := (d, x0, . . . , x |I |−1),

where d ∈ D determines which components are empty and x i encodes the i th componentof a. The desired interpretation is constructed as follows:

I := ⟨h, δ(x), ε(x, y), ϕ0(x0, . . . , xr0−1), . . . , ϕs(x0, . . . , xrs−1)⟩,

where

h(d, x0, . . . , x |I |−1) := (χd0

(✸, h0(x0)

), . . . , χd|I |−1

(✸, h|I |−1(x |I |−1)

)),

δ(d, x0, . . . , x |I |−1) :=∨c∈D

(d = c ∧

∧i :ci=1

δi (x i )

),

and

ε(d, x0, . . . , x |I |−1, e, y0, . . . , y|I |−1) := d = e ∧∧

i<|I |

(di = 1 → εi (x i , yi )

).

In order to define ϕj we consider an interpretation I I of I in C. Since I is finite suchan interpretation exists. Let αj := βI I

j be the formula defining Rj . Note that βj contains


additional relations Dl and T ml which are not in σ . Thus αj is a sentence over the vocab-

ulary τ extended by the symbols Dl and T ml for appropriate l and m. We have to replace

them in order to obtain a definition of ϕj . Let x0, . . . , xrj−1 be the parameters of ϕj where

xk = (dk, x0k , . . . , x |I |−1

k )

for k < rj . Dl and T ml can be defined by

Dli := (dl)i = 1 and T ml i := (tm

l )Ii(x i

0, . . . , x irj−1).

Note that those definitions are only valid because i ranges over a finite set. ϕj can nowbe defined as αj with Dl and T m

l replaced by the above definitions.Obviously, all steps in the construction above are effective.

Corollary 7.3. Both AutStr and ω-AutStr are effectively closed under finitary gener-alised products.

As promised we immediately obtain closure under several types of compositions.

Corollary 7.4. Let A0, . . . ,An−1 ∈ AutStr. Then there exists automatic presentationsof

(i) the direct product∏

i<n Ai ,

(ii) the disjoint union⋃

i<nAi , and(iii) the ω-fold disjoint union ω · A0 of A0.

Corollary 7.5. Let A0, . . . ,An−1 ∈ AutStr be ordered structures. There exist auto-matic presentations of

(i) the ordered sum∑

i<n Ai and(ii) the ω-fold ordered sum

∑i<ω A0 of A0.

8. Conclusion

We have seen different methods of how to represent infinite structures in a finite wayeffectively. Among the most interesting classes of finitely presentable structures are au-tomatic and ω-automatic structures. Besides the original presentations based on finiteautomata these classes also admit characterisations in terms of first-order interpretationsinto expansions of Presburger arithmetic or the additive group of reals. We have shownthat these representations allow for effective model checking and algorithms for opera-tions like unions, products, and the evaluation of queries. We have also given complexitybounds for some of these problems. Finally, we have started to investigate algebraicproperties of automatic structures. In particular, we have developed methods to provethat certain structures are not automatic.


Most of the techniques that we have used can be generalised to other classes ofinfinite structures. For instance, we can modify the automaton model to obtain tree-automatic or rational structures. The approach based on interpretations is even moregeneral. Fixing any structure A with good algorithmic or model-theoretic properties,we can consider the class of all structures interpretable in A (for any suitable notion ofinterpretation). Since many relevant properties of structures are preserved by interpre-tations, every structure in such a class inherits them from A. This is a very broad andversatile approach that encompasses many of the classes of finitely presentable structuresappearing in the literature.

So far we have mostly studied rather powerful classes that contain as many inter-esting structures as possible, while preserving decidability of first-order or MSO logic.An obvious drawback of this generality is complexity. For instance, even rather sim-ple formulae are quite difficult to evaluate on automatic structures or tree-interpretablestructures. For future studies it might be interesting to search for classes for which thealgorithmic problems are somewhat more manageable. For classes defined by interpre-tations one might choose structures with relatively modest complexity and notions ofinterpretations that do not blow up the complexity too much. We believe that this is apromising line of future research that may also be relevant for practical applications.

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Received October 2, 2002. Online publication September 13, 2004.

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