a logic for metric and topology

A Logic for Metric and TopologyAuthor(s): Frank Wolter and Michael ZakharyaschevSource: The Journal of Symbolic Logic, Vol. 70, No. 3 (Sep., 2005), pp. 795-828Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/27588395 .

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The Journal of Symbolic Logic

Volume 70. Number 3. Sept. 2005

A LOGIC FOR METRIC AND TOPOLOGY

FRANK WOLTER AND MICHAEL ZAKHARYASCHEV

Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It com

bines the 'qualitative' interior and closure operators with 'quantitative' operators 'somewhere in the sphere of radius r.' including or excluding the boundary. We supply the logic with both the intended metric

space semantics and a natural relational semantics, and show that the latter (i) provides finite partial

representations of (in general) infinite metric models and (ii) reduces the standard 'e-definitions' of closure

and interior to simple constraints on relations. These features of the relational semantics suggest a finite

axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational

numerical parameters are coded in binary). An extension with metric variables satisfying linear rational

(in)equalities is proved to be decidable as well. Our logic can be regarded as a 'well-behaved' common

denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and

qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is

a natural fragment of decidable temporal logics for specification and verification of real-time systems. On

the real plane, it is closely related to quantitative and qualitative formalisms for spatial representation and

reasoning, but this time the logic becomes undecidable.

?1. Introduction. The concept of a metric space with the induced topology is

obviously fundamental for any serious course in mathematics or computer science.

Standard exercises an undergraduate student is confronted with look as follows.

Let (V,d) be a metric space, X ? V and a > 0.

(1) Is the set A = {y G V \ 3x G X d(x,y) < a} open?

(2) If y G V belongs to the topological closure of the set A above, does it mean

that there exists x G X such that d(y, x) < al

A proof of (1) usually involves an arbitrarily small s > 0. A counterexample for (2) can only be found in the class of infinite metric spaces.

The main aim of this paper is to find out to which extent this kind of reasoning can be mechanised?e.g., is it decidable? what is its computational complexity?? and whether reasoning with the 'arbitrarily small s > 0' and infinite spaces can be

performed usingfinitely many logical rules and finite models. In this perspective, our

direction of research can be regarded as a 'metric extension' of Tarski's programme of the algebraisation of topology ("of creating an algebraic apparatus adequate for the treatment of portions of point-set topology." to be more precise) [29]; see also

[34,25,6.30,1]. There is another idea underpinning this paper. As is well-known, reasoning

about metric and topology is fundamental in various areas of computer science.

For example:

Received April 8. 2004; revised January 22. 2005.

? 2005, Association for Symbolic Logic 0022-4812/05/7003-0007/S4.40

795

796 FRANK WOLTER AND MICHAEL ZAKHARYASCHEV

Temporal logics for specification and verification of real-time systems deal with

the real line R and the standard Euclidean metric (see. e.g.. [4. 23]). Using their qualitative temporal operators such as 'since' and 'until' interpreted by the metric of R. one can define the topological closure and interior over R.

For example, a point x is in the topological closure of a set X if and only if

there is no point y > x until which 'not X" and there is no point y < x since

which 'not X.'

Spatial representation and reasoning uses various topological and metric rela

tions between regions [29. 16.21.7.31. 1]. The intended models are based on?

among others and in the decreasing order of abstractness?arbitrary topo

logical spaces, metric spaces with their topologies, and the two-dimensional

Euclidean space R2 (see. e.g., [36. 14]).

Similarity measures that are used to classify various sets of objects (e.g., pro teins or viruses in bio-informatics) give rise to reasoning about metric spaces not related at all to the standard Euclidean spaces [13]. [18. 19] suggest a com

bination of topological relations between regions in spaces with similarity measures for classifying and identifying objects. Logics for reasoning about

similarity have been developed in the field of approximate reasoning [15, 17].

In this respect, we are looking for a logic which can be regarded as a sort of'common

denominator' of the formalisms constructed in these fields and which reveals most

important expressivity and complexity issues that arise in special purpose temporal,

spatial, and similarity logics. (In modal logic, such a position is occupied by the

minimal logic K which is a 'common denominator' of various 'qualitative' temporal,

description, spatial, epistemic, dynamic, etc. logics.)

The logic we construct here is a natural combination of the logic of metric spaces from [35] which comes equipped with the operators

3<a for 'somewhere in the sphere of radius a excluding its boundary.' where a G Q+.

3-a for 'somewhere in the sphere of radius a including the boundary.' where a GQ+.

and the well established logic S4U of topological spaces which has the operators of

the standard modal logic S4. namely.

G for topological interior and O for topological closure.

as well as the 'universal modalities'

V for 'everywhere in the space' and

3 for 'somewhere in the space.'

Besides the intended metric space models, we supply the logic with a natural rela

tional semantics and show that it (i) provides finite partial representations of (in

general) infinite metric models and (ii) reduces the standard 'e-definitions' of clo sure and interior to simple constraints on relations. Using these features of the

relational semantics, we give a finite axiomatisation of the logic and prove its

EXPTIME-completeness (even if the numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well.

A LOGIC FOR METRIC AND TOPOLOGY 797

?2. Semantics and syntax.

Intended models. A metric model is a structure of the form

m = (v.d.p

.p ....).

where the Pf31 are subsets of a nonempty set V and d is a function from V x V

into the set R+ of non-negative real numbers such that ( V.d) is a metric space?i.e..

it satisfies the standard metric axioms

d(x.y)=0 iff x = y.

d(x.z) <d(x.y)+d(y.z).

d(x.y) =

d(y.x)

for all x.y.z G V. Remember that each metric space (V.d) induces the interior

operator ld on V: for all X ? V.

ld(X) =

{u e X \3e>0Vv (d(u.v) <e -> v G X)}.

(Vld) is called the topo logical space induced by the metric space (V.d). The closure

operator Cd in this space is dual to ld. that is. Cd(X) = V -

ld(V -

X): in other

words.

Cd(X) =

{u eV \Ve>03v e X d(u.v) <e}.

Thus, the intended models of the logic we are looking for are of the form

m = (v.d.i(i.p

.p ....). (i)

where (V.d. Pf1, P _) is a metric model and ld is the interior operator induced

by (V.d). These structures will be called top?me trie models.

Language I. There are many approaches to devising languages that are capable of speaking about topometric models. The first natural choice would be the appro

priate (fragment of) two-sorted first-order logic, with one sort for the elements of the metric space (V.d). another one for the distances in R+. and countably many unary predicates interpreted over the domain V. ternary predicates d(x.y) < z.

where x. y G V. z G R+. and additional constants and operators like 0 and + with their standard interpretation over R. However, as was shown in [26], even

the two-variable fragment of monadic predicate logic with only one sort for the

domain of the metric space and binary predicates d(x. y) < a. a a natural number

between 0 and 80. is undecidable. It was also proved in [26] that propositional operators 'somewhere in the open sphere of radius a but not in its centre' give rise to an undecidable logic. On the other hand, we know from [35] that the (proposi tional) logic of metric spaces with operators 'somewhere in the open/closed sphere of radius a' is decidable and finitely axiomatisable. To speak about the topological component we take the standard language of S4 [29] extended with the universal and existential modalities over the space (which turned out to be very useful in

spatial representation and reasoning [7. 1]). Thus, the basic language J?ET we propose to use for speaking about topometric

models is built from atomic terms P?. i < co. with the help of the following rule:

t ::= Pi | -t | TiriT2 I 3<aT I 3^az I Dr | 3r.

where a is an arbitrary positive rational number, i.e., a e Q+. Wecalli an^ST-term

or simply a term. We remind the reader of the definitions of standard 'topological

fragments' of JtZT. The language J?3? of S4 is defined by

t ::= P, | -it | ti n T2 | Gr

and the language M2\ o? S4U is

t ::= P, | -it | ti n t2 | Gt I 3x.

For a topometric model ?JI of the form (1). the extension xm of a term t is

computed inductively as follows:

\X\ n T2) =

t1 ht2 .

Ki)9* = K-Tj*,

(B^nT = {u e V I 3v e z?ld{u,v) < a},

{3^'xl)m =

{w G F I 3v e x d{u,v) < a},

( nr = iA*n

(3ti) an _ J

^ u ci V ifrP1 ̂ 0,

otherwise.

Say that a term t is satisfied in 9JI if t3^ ^ 0; t is ?rwe in Tl if t911 ? F. We denote

by V<?/, V-a, O. and V the operators dual to 3<a, 3-a, G, and 3, respectively. For

instance.

(V<gt)^ =

{?eK|VveK (?(w,t;) < a -> v e xm)}

and O is interpreted by the closure operator C?. The following examples illustrate the expressive capabilities of MET.

Example 1. (i) Set tj U i2 = ^(-n n ^t2). Then the term V(-<n U t2) is

true in a topometric model 9Jt if and only if xf1 ? xf1 holds in Tl. So we use the

subsumption n G t2 as an abbreviation for V(-ti U t2), and write n = t2 for the

conjunction of n G t2 and t2 G x\.

(ii) The terms t = Gt,t = Ot and x ? OGt mean then that (the extension of) t is open, closed and regular closed, respectively. The exercises mentioned in the

introduction can be formalised as follows:

U3<aX = 3<aX and 03<flX g 3^aX.

(iii) The topology induced by a finite metric space is trivial (every set is both open and closed). Thus, even the term P n O-^P (which is clearly satisfiable, say, in every Euclidean space) is not satisfiable in any finite topometric model.

(iv) The Hausdorff distance between two closed sets is bounded by a if every point of one set is within distance a from some point in the other set. It is used, e.g., to

measure the approximate matching between images [24]. We write <i#(n, t2) < a

to abbreviate the conjunction

(On g 3^0t2) n (Ot2 g a^flOn)

(which formalises the definition of Hausdorff distance); dn(T\.z2) = a is an abbre

viation for

(dH(ri, r2) < a) n ((On g 3<aOz2) U (O^g 3<aOzx)).

Note that, in contrast to the standard definition via infimum, du(?ir T2) = a implies

that there is a point in one set which is within distance > a from any point in the other set.

(v) The min-distance between two sets is bounded by a if there exists a point in one set which is within distance < a from a point in the other set. This definition can be formalised as follows

dmm(i\^2) < a iff i\ n 3-az2 ^ JL,

where 1 stands for the empty set, say, P n -P.

(vi) The MCC-% (aka Egenhofer-Franzosa) relations [16, 21] between nonempty regular closed sets can be expressed in the language Jt3?u of S4U [1, 31]. So they are expressible in JIS" as well. The Hausdorff and min-distances introduced above

suggest natural metric extensions of these relations.

(vii) JtET is not compact in the sense that there is an infinite set Y of terms such

that, for every finite Y' ? Y, there exists a model DJ? with f\er' zm ^ 0, but there

exists no model Wl for which f]rer zm ^- 0. An example is given by the set of terms

{^OP}U{3<L?P \ n GN+}. (viii) The term (Oti

= z2) n 3z2 n -<3?ti says that t\ is dense in a nonempty z2,

but has no interior.

Logics in language I. Given a class Ji of topometric models, let L(ji). the logic of Jt. be the set of those JtZT-terms that are true in all models frornJt. By MT we

denote the logic of the class of all topometric models. In this paper we investigate in detail the minimal logic MT as well as the logics of the real line and plane, L(R) and L(R2) (that is the logics of the classes of topometric models based on R and

R2. respectively).

Note that the topological fragments of these logics are well known. The fragments of MT. L(R) and L(R2) in the language Jt^ coincide with the modal logic $4 [29], which is PSPACE-complete [27]. The fragment of MT in the language J?3?u is the modal logic S4U. which is also PSPACE-complete [5]. The fragments of L(R) and

L(R2) in JtS?u coincide with the logic of all connected topological spaces (induced by metric spaces): it can be obtained from S4U by adding the 'connectivitv axiom' of [32]

3BPln3BP2ny(oplu\DP2) \z 3(nplnnp2). (2)

Language II. In the language J?J~ we can formalise metric relations, but we cannot compare two distances without specifying their absolute values. To en

able reasoning about relations between distances, we extend J?ET with numerical variables. More precisely, let J?ST[Y] be the language that is defined similarly to

J?ET. with the only difference being that instead of the parameters from Q+ we use

variables from the list W = {xo, x\,...}. Given an assignment a : W ?>

Q+ and

an JtZr\V\-\&xm z, we denote by za the ̂ ?T-term that results from z by replacing each x,- with a(x?).

Fet T be a set of constraints for the variables in "V. say. a set of rational linear in

equalities over *V or a polynomial equation. An ^^[^"J-term x is called satisfiable relative to Y if there exists an assignment a such that it solves Y and xa is satisfiable in a topometric model. Here are two simple examples of constraint systems Y:

T consists of equalities x? = a?. where a? G Q+. In this case an jf^p^J-term t is satisfiable relative to Y iff xa is satisfiable for a : x? i?? a?. Y consists of strict inequalities xo < x\. x\ < x2. etc. In this case no absolute

value for variables is fixed.

Example 2. (i) The terms ?///(n. t2) = x. ?///fo, T4)

? y under the constraint

x > 2 v say that the Hausdorff distance between n and t2 is at least two times larger than the Hausdorff distance between T3 and T4.

(ii) We can express incomplete knowledge about distances by using interval con

straints a\ < x < ?2.

(iii) Comparative distance statements can be used within quantifiers, e.g., V<xt

under the constraint x = ?/#(n, T2).

In the remaining three sections of the paper we consider in turn the logic for

malisms defined above.

?3. The logic MT. The purpose of this section is a comprehensive analysis of the

logic MT. With every ./CT'-term x we associate a finite set of axiom schemata from

which t is derivable iff x G MT. We also introduce a sound and complete relational

semantics for JfET with respect to which MT has the finite model property (remember that, according to Example 1 (iii). MT does not enjoy the finite model property relative to the intended topometric models). Using this relational semantics, we

prove EXPTIME-completeness of MT (for binary coding of the parameters). Fet

us begin by introducing the notation we need.

Parameter set. Suppose that M ? Q+ is such that

(+) if a, b G M and a + b < y m , then a + b G M.

{-) if a.b G M and a - b > 0. then a - b G M.

where y m = sup M if M is bounded and y m = 00 otherwise. Then M is called a

parameter set. We also let sm ? inf M.

Given an jf^-term t, we definite ^finite parameter set M[x] ? Q+ containing all

parameters from x. Suppose that the numerical parameters occurring in x comprise the set

*?>-{*.?}?Q+ where a\,b\ are mutually prime for 1 < / < n. Fet yr =

^ be the maximal number

in N{x) and let

sz =

l/(6i x x bn).

?T =

Qj x b\ x x

bj-\ x

bj+\ x x bn.

M[x] =

{n x eT \ n G N+. 1 < n < ?T}.

It is an easy exercise to check that M[x] is closed under (+) and (-). and so is

a finite parameter set. Note that the closure under (+) and (?) of a finite set of real

numbers can be infinite.

Length of term. Let us agree on how to measure the length ? {x ) of a term x. Given a positive rational number c =

a/b. where a and b are relatively prime integers, let

?(c) denote the smallest natural number exceeding 1 H- log2(a + 1) + log2(6 + 1). Then the length ?(x) of a term x is defined inductively in the usual way (say. as the

number of subterms of t) with the only exception:

?(3<cx) = ?(3^cx) = ?(x) + 1 + ?(c).

In other words, the parameters in x are assumed to be represented in binary. It is

not hard to see that

log2(\M[z]\)<log2([[aixbi)<e(T), i=\

where the cn/bi are elements of N(x). It follows that \M[x]\ < 2^T\

Axiom system. Given a parameter set M, denote by Axmt[M] the following axiomatic system. First, it contains some set of axiom schemata and inference rules

(say, modus ponens) of classical propositional logic. Second, it has the standard

axiom schemata and rules of the minimal multi-modal logic K with the 'necessity

operators' op of the form G, V, \/<a, a G M, and \/-a, a e M, namely,

i

op (x ?? p) ?? (op x ?> op p) and -, op x

where x ?> p is an abbreviation for -i(-it n p). Third, to ensure that G is an

^/-operator and V is an 55-operator quantifying over the whole metric space, we

include the well-known axiom schemata

Gt -

t, Gt -+ GGt, Vt^t, Vt^Wt, t-^V3t, Vt -> V-"t,

for a G M. Finally, the following axiom schemata govern the metric operators and

their interaction with topology (here a and b range over M):

T^vâ^n (3) x^\/<a3<ax, (4)

V^t - T. (5)

V<flT -> T, (6)

3<ax - 3âx, (1)

3âx-^3<hx, fora<b, (8) 3<?3<bT ̂ 3â+bx, if a + ^ G M, (9) 3<*3<6t -> 3<a+?T, if a + b G M, (10) 3<fl3<?T -> 3<a+\, if ? + b G M, (11) Ot^3<?t, (12)

3<?Ot -

3<flT. (13)

Axioms (3) and (4) ensure symmetry of metric, (5) and (6) guarantee the condition

d(x,x) = 0, and (9)?(11) the triangle inequality. Axioms (7) and (8) reflect the

relation between open and closed spheres. (12) and (13) are the only axioms we

need to grasp the interaction between metric and topology.

Relational semantics. An J??T[M\-frame, for a parameter set M, is a structure

of the form

d=(W.R,(D^)aeM.(D<)aeM). (14) where W is a nonempty set and R, Df, D< (a G M) are binary relations on W

satisfying the following properties (qoR)-(D<R) for all u, v. w G W and a,b G M:

(qoR) R is reflexive and transitive (a quasi-order). D< and Df are symmetric and reflexive, (rcZ))

(D<D^) U>*<) (trD^) (trD^D<

(trD<D^

(RD<) (D<R)

KnJ?,T\

if uD^v and a < b, then uD^v. if uDfvD^w and #

if uDfvD^w and a - -b e M, then uD^+hw, b G M, then uD<i,w,

) if uD^vD~w and a +b e M, then w?><+/? if w7^>, then wZ><i; for all a > 0,

if uD<vRw, then uD<w.

M]-model is a structure

? = ^f-P

? D? (15) where 5 is an ^#5^[Af ]-frame of the form (14) and the Pf are subsets of W. The

va/we xA of an ^#iT[M]-term r in ? is defined inductively: the values of the P? are

given by the model, the Boolean cases are standard, and

(3<?t)? =

{ueW\3versi ??>>}.

(3^t)? =

{? e ^ 3u G rs ?/)?%},

(nT)? =

{u e W | V? e W Mv -

jV ifr<V0 I 0 otherwise.

? G t*)},

Or)'

We say that z is satisfied in an JtZT\M\-model & if t^ /- 0; t is true in i^ if r^ = W.

Throughout this paper we shall often use a number of simple properties of

J?<T[M]-frames that can be easily derived from conditions (qoR)-(D<R). We

formulate them in the following lemma, where s = xoRDxn is an arbitrary finite

sequence of the form xqSoX\S\ ... Sn-\xn such that each S? is one of R. D< or Df,

for some a G M, and the sum as of the numerical parameters occurring in s belongs to Af:

Lemma 3. (a) ifuDfv and a < b. then uD^v. (b) ifuRvD-iu anda < b. then uD^w, (c) ifuD<vD?w and a + b G M, then uD<+bw, (?) ifuD^v anduRw, then wD<v,

(e) ifuD^v (or vD<u), s ? vRDw and a + as G M. then uD<+a w.

(f) if s ? uRDv and a = em + a s ? M. then uD<v,

(g) if s ? vRDw, sf = vRDw'. and a =

em + cis + as< G M, /?e? wD<w'.

Proof. We only show how to prove (b) and (g), and leave the remaining (equally

easy) cases to the reader.

(b) Suppose that uRvDfw and a < b. By (RD<), we then have uDfMvDfw. Since M is closed under (-) and (+). em < b - a and em + a G M. It follows by

(trD<D-) that uD<?+aw. If sm + a = b then we are done. And if sm + a < b*

then we use (D<D-) and (D-<) which also give uD^w.

(g) Suppose that s = vRDw, s' = vRDw'. and a =

eM + as + av, g M. By

(f), vD<M+aw, and so, by symmetry, wDfM+av, from which, by (e), we obtain

wD?w'M S S

H

Remark 4. Intuitively, uDfv (uD<v) in J says that the distance between u and v

is < a (respectively. < a)\ uRv means that u belongs to the topological closure of

{v}. In fact, the truth-condition for G means that G is interpreted by the interior

operator 1? of the topological space T? = (W,I<&) induced by the quasi-order

0 = (W,R)? i.e., by

Iv(X) =

{je G X | Vy G "PF (xPy -> j G X)}.

Such spaces are known as Aleksandrov spaces. Alternatively they can be defined as

topological spaces where arbitrary (not only finite) intersections of open sets are

open: for details see [2, 11]. The constraints on the relations in J?!T\M\-frames reflect the connection be

tween metric and topology: (qoR) corresponds to the 5?-axioms for G, and (rsD)

(D<R) to axioms (3)?(13). In fact, it is an easy exercise to check that the axioms

of Axmt[M] are true in all Jt^\M\-n\o?t\s and that these models preserve the

inference rules for every parameter set M. Note, however, that the topological in

terpretation of R differs drastically from the topology induced by metric spaces: for

example, in a metric space every singleton {x} is closed, while every .#y~[M]-frame with a non-trivial R contains a non-closed singleton.

Given a term t, we write AxMj\A f?r AxMt[M[x]] and J^^f[x] for y?T[M[x]]. We are now in a position to formulate our main result about the basic logic MT

of metric and topology: Theorem 5. (i) The following conditions are equivalent for every JtET-term x:

1. t eMT.

2. x is derivable in Axmt[Q+] 3. t is derivable in Axmt[t] 4. t is true in all y??/[x]-models. 5. x is true in all finite J?ff\x\-models.

( ii ) The decision problem fo r MT is EXP TIME- comp le te.

Before going into details of the proof, we separate the conceptually interesting part from the 'folklore' or trivial observations. Clearly, (3) => (2) => (1). By Sahlqvist's

completeness theorem (see, e.g., [9, Theorem 4.42]), we also have (4) <^> (3). Indeed,

Axmt[c] can be regarded as a normal multi-modal logic with Sahlqvist axioms. So

Axmt[A is determined by its Kripke frames which clearly coincide with ,?^[x] frames. In other words, a term x is derivable in Axmt[t] iff t is true in all MT[x]

models.

Thus, to complete the proof of (i), it remains to show the implications (5) => (4) and (1) =4> (5). The former one?which means that MT has \h& finite model prop

erty (fmp, for short) with respect to JtZT\M\-models?as well as the EXPTIME

completeness of MT will be proved in Section 3.1. The core part of Theorem 5

is the latter implication to be proved in Section 3.2. It states that every term x

satisfiable in a finite jf^[M]-model (with M[x] ? M) is satisfiable in a topometric

model. Actually, we show that every finite ̂ /'[Mj-model is a p-morphic image (in the natural sense to be defined below) of some (almost always infinite) topometric

model. Thus. ^T[M]-modds can be regarded as a sort of partial descriptions of scenarios which can be realised in topometric models. In the algebraic setting, which is closer to Tarski's programme mentioned in the introduction, this can be

reformulated as follows: every Boolean algebra with operators for . V<". \/-a

(a G M). V. that is induced by a finite ^^[M]-frame, can be embedded into the

Boolean algebra with operators induced by some topometric model. The reader

familiar with algebraic semantics of modal logics and duality theory should not

have problems with reformulating these results in the algebraic manner (consult.

e.g.. [20]).

Remark 6. It is to be noted that the axiomatic system Axmt[t] is not complete if we do not include in it the axioms for all parameters from the closure of N(z) under both (+) and (-). For example. 03-2P ? 3<3P is not derivable from the

axioms formulated for M = {2. 3} only.

3.1. The fmp and EXPTIME-completeness.

Theorem 7. (i) If an JtET-term z is satisfiable in an JtZr\r\-modeL then it is

satisfiable in a finite yOT[z]-model.

(ii) The satisfiability problem for Jt^-terms in J?H -models is decidable in exponen tial time in the length of the term.

Proof. The proof of both (i) and (ii) is based on the standard elimination method

used. e.g.. for PDL [22]. Roughly, it works as follows. Given a term z. take a suitable

closure cl(z) (the Fischer-Ladner closure in case of PDL) of the set of subterms of r. form a set Yr of appropriate types over cl(z). define appropriate relations for D.

V<". V-". a G M[z]. on YT and. finally, eliminate recursively all of those types from

rr that contain terms of the form -iD/?. -N<a p. or -N-ap having no 'witnesses' in

Tr (i.e.. properly located types with ->/?). Then we show that z is satisfiable iff the

elimination procedure terminates with a set containing a type with z. (This is done

by proving that if r is satisfiable then z is satisfiable in a finite model constructed from

the resulting set of types.) Finally, we show that this procedure is 'only' exponential in ?(z) by proving that YT is 'only' exponential in ?(z).

Given the simple first-order form of the constraints on Jt?/[z]-models, it is not

too difficult to implement this scheme for MT?provided that we use the unary

coding of the numerical parameters. The main novelty of the proof is the additional

ingredients required to devise an EXPTIME algorithm in ?(z). i.e.. under binary

coding.

Without loss of generality we may assume that z is constructed using the Booleans.

the metric operators V<w. V-?/. the interior operator D. and the universal quantifier V.

Since M[z] is exponential in ?(z). a na?ve adoption of. say. the Fischer-Ladner

closure would result in our case in 22 types. To avoid this, we add to the

language new operators V-a\ a G M[z]. with the following semantics. Given

an Jf^r]-model ? of the form (15). we have x G (\/-a p)? iff y G p? whenever s =

xRDy and as < a. where s ? xRDy is an arbitrary finite sequence of the form

xSqX\S\ ... Sny such that each 5/ is one of R. D^ or Df. for some b G M[z]. and

the sum as of the numerical parameters occurring in s belongs to M[z].

A LOGIC FOR MEFRIC AND TOPOLOGY 805

Denote by sub(x) the set of all subterms of x and by cl(x) the closure under single

negation of the set

sub(x) U {Bp | \/<hp g sub(x) or V-;> G sub(x)} U

{f<ap.V-ap.V-> | (V<;> G ^w6(t) or V^> G A'w?(r)) and ? G M[t]}.

Say that a subset T of cl(x) is a x-type if it satisfies the following conditions, where

b G M[t]: (t?) -n/? G riff/? 0 7\ for all-/; G ?7(t):

(/2) />i n/;2 G T iff/? 1./72 7\ for all/?i n p2 G c/(r):

(r3) if Dp g rthen/? G T:

(t4) ifV^^/? eiandK ?/. then V^>.V^/? G T:

(;5) ifV^V G TandZ) < a. then V^Y v</> e ^:

(i6) if V<ap e T and ? < a. then V<Y D/> G T:

(i7) if V<?//; G T and ?z - eT > 0 then V^("-?r)/> G T.

(?8) if V/? G Tthen/; G 7\

A typical example of a T-type is the set

T(u) = {Pecl(T)\uep*}

generated by any point u G W. where ? is an Jt^\x\-model of the form (15).

Femma 8. The number of distinct x-types does not exceed 2c"e^~.for some constant

Of).

Proof. Every T-type T is uniquely determined by the set of subterms /; of x

it contains and. for every subterm p of x. by the maximal a G M[x] such that

\/-a* p G T. the maximal a G M[x] such that \/-ap G T. the maximal ?7 G M[x] such that V<?//? G T. and. finally, whether or not Dp g 7\ This follows from the

conditions (t\) and (14)-(te) for T-types. Therefore, the number of distinct T-types does not exceed

2|,?Mr)| x 2\,uh(r)\ x |M[r]|3-!-MD? < 22.^(r) x ^-Hr)\

as required H

Now we are going to show that, if x is satisfiable at all. then a certain subset X of

the set of all T-types can serve as the domain of a model satisfying x. The points u

in this model will coincide with the T-types they generate. The relations R and D?r and D< in this model will be the restrictions to T of the following relations ??#.

<-><? and <-??,. for ?/ G M[t]. defined on the set of all T-types.

Fet T\ ?

# Ti iff conditions (R1)-(R5) are satisfied:

(Rl) if Up g T\ thenD/? g TV

(R2) ifV^V G ri thenV^> G TV

(R3) if V?'/? G T? and 0 > eT then V^'"^/; G TV

(R4) if V<?v; G 71! then D/> g 7V

(R5) if V<"/> G T2 then V<?'/> ? 7V Fet T\ <^><a Ti iff the following four conditions hold for / = 1. 2. where 1 = 2.

2 = 1 '?n? a G M[t]:

(<1) ifV^"/> ? ^/ then/? G 7>.

(<2) if V^/> G 7} and b > a then V^"V G 7).

(<3) if V<V e Ti and 6 > ?/ then \J<h~ap G 7).

(<4) if V^> G Ti and b > a then M^b~a^p g Tt. Let T\ ^<a T2 iff, for i ? 1,2:

(<1) if \J<ap G Ti then Up g Tt,

(<2) if V</> G rz- and ?7 > a then \/^b~a)+p g 7>. Given a set ̂ =

{Vti ..... Vt,-} ? sub(z), denote by S, we have

Vy9 G 71 iff V/> G^.

That is to say, T G g$V if and only if the set of terms of the form \fp in T coincides

with "U. Clearly, the cardinality of the set of distinct S% is exponential in ?(z). Now, to decide whether z is satisfiable, we enumerate all ?% and perform on each

c$V the following elimination procedure which checks whether it induces a model

for t or not. Form the sequence

of sets of T-types as follows. Suppose ^ is defined. Delete all those types T from

STi for which one of the following conditions holds:

there is -A^> G T such that there is no T G ̂ with -./> G T' and r ^<? T', there is -tf<a p G T" such that there is no T' G ?7} with -1/7 G T7 and T ^<a T', there is -iV/? G T7 such that there is no T' G <?/? with -^p e Tf, there is -?D/? g jT such that there is no T' G ̂ with -1/? G T1 and 7" ?># T', there is some -A/-a p <E T for which there is no sequence S = T -^r^ T' of

T-types such that as < a and -1/? G ?T' (where S ? T -^r^> T' and as are

defined similarly to s = xRDy and ?z5 using the relations ??#, <-><fl, <-

<&)

Observe that, for every set % of the form {Vti, ..., Vrr} ? sub(z), this elimination

procedure terminates after at most exponentially many in i(z) steps. As there are

at most exponentially many in?(z) different %. we can find out in exponential time

whether for at least one "U the elimination procedure applied to ?<% terminates with a set of T-types at least one of which contains z.

Lemma 9. Ifzis satisfiable then there exists a set ?I of the form

{Vti....,Vt;..} Csub(z)

such that the elimination procedure applied to ??/ terminates with a set ̂ ofz-types at least one of which contains z.

Proof. Suppose z is satisfied in a model

?= (5, Pf,Pf,...), where

$ = (W,R.(D^)aeM[z],(D<)aeM[T]).

Let

^ - {\fp G sub(z) I p? = W}, ?~& =

{T(u) I u G W}.

We show that the elimination procedure applied to ?<% terminates with some set

^ ?5 ̂r Assume that the elimination procedure generates a sequence

Sv = Fo 2 ^1 2

Then it is sufficient to prove that (i) ?9^ G 5^ and (ii) for all i > 0. if ̂ G ?9^, then

^/+i 2 ?^k- Claim (i) follows from the observation that V/> G 7"(w) if and only if

p? = W, for all u e W and V/? G sub(x). For (ii) we consider the five elimination

rules for all T = T(u) such that u G PF.

Suppose that-V-"/? G r(w). Then there exists v e W such that uDfav and

i? ̂ /??, from which we obtain T(u) ^<a T(v) and -i/? G r(i;).

Suppose -^/<ap G r(w). Then there is i? G PF such that uDfav and v ^ /r*\ Therefore. T(w) ^<fl T(v) and ->/? G T(t;).

Suppose -Np G !T(w). Then there is v G W such that v ^ p?, and so

->/? G 7?.

Suppose -iD/? G r(w). Then there is v G PF such that uRv and v ^ pR, from

which r(w) ?># r(tO and ->p G 7"(?;).

Suppose -iV-fl p g !T(w). Then there exists a sequence uSqU\S\ ... Snun such

that each S? is one of R, Df or Df, for some a G M[x] such that as < a, where

As is the sum of the numerical parameters of this sequence, and un 0 pR. It

follows that there is a sequence S = T(u) -^r^ T(un) such that as < a and

-i/> G T(w?).

This completes the proof of the lemma. H

Femma 10. Suppose that the elimination procedure starts on input S% and termi

nates with EF.Ifxe T,for some T G ??7\ then x is satisfiable.

Proof. Define an MET[-r]-model ? = (#. Pf, P2, ) by taking

3r=<^A,(/)^)fleA/[T]?(/)a<)flG^[T]>?

where

w = sr.

P? =

{T eW\ Pi G T}, for / < co, TRTf iff r -?* r.

TDfT' iff r ?-><? r, r7)<r iff r^<? v.

We show that (i) # is an j?^t]-frame and that (ii) for every /? G c/(t) and every r G PF, T G/^if?> G T.

Fet us start with (i) and check conditions (qoR)-(D<R).

(qoR) That R (i.e., ?>/?) is reflexive and transitive follows immediately from the

definition, (t?) and (t-j).

(rsD) That Df (that is ̂ <a ) is reflexive and symmetric follows from the definition

and (u)-(t6).

(D<D-) follows from the definition and (^3)?(^5).

(D^<) Fet Ti ^<a T2 and a < b. If \J<bp G T{ then, by (<3), y<h~ap G T2

and, by (te). G/? g T2. Thus we have (<1). Now, if \/<c p G T\, for c > b, then by

(<3), y<c~ap G 72. As c - b < c - a. we can use (t-j) (together with (t4), (t5)) and

the definition of M[x] to obtain \J-^'~h) p g T2, which proves (<2).

The properties (trD^), (trD^D<), (trD<D^), (RD<) are easy and left to the

reader.

(D<R) Suppose To ^<a T\ ?^ T2. We need to show that 7o ^<a T2. If

V<V G T0 then Dp g T1; and so, by (RI). D/> g T2. If V</3/? G T0 and b > a, then

we have V^-"}> G Tx. Therefore, by (R2), V^"a)> G T2. Now suppose that

y<ap G T2. Then, by (R5), \/<ap G Tu and so Up g T0. Finally, if V<bp G T2 and

b > a, then, by (R5), \/<bp G Ti, and hence V^~a)> G T0.

Now we show (ii) by induction on the construction of p. If p is an atomic term, then (ii) follows from the definition. The steps for the Boolean connectives can

be proved using conditions (t\) and (?2) for T-types. We now consider the 'modal' connectives.

Let p =

Up0 and T ? (Up0)?. Then there exists T' e W such that TRT' and T' ? p?. By the induction hypothesis, p0 g T'. But then, by (R2), Up0 ? T.

Conversely, suppose Up0 ? T. T is not eliminable. Therefore, by the fourth elimination rule, there exists T' G W such that T ^r T' and po ? T'. By the induction hypothesis, T' ? p?. Hence, T ? (Up0)?.

The case p = V-?/?o can be proved using condition (<1) and the first elimination

rule.

The case p = V<a po can be proved using condition (<1), property (?3) of T-types, and the second elimination rule.

The case p = V/?o can be proved using property (ig) of T-types, the fact that

all T G W contain the same terms of the form V/?o (as ?% D W), and the third elimination rule. H

This proves Theorem 7. H

Our next theorem establishes the corresponding lower bound even without using the interior and closure operators:

Theorem 11. The following problems are EXPTIME-hard :

(i) decide whether an JPET-term built using only the Booleans and the operators V-1 and V belongs to MT,

(ii) decide whether an JtET-term built using only the Booleans and the operators V-",

?eN+, belongs to MT.

Proof. The proofs are by reduction of the global TfT-consequence relation that is known to be EXPTIME-hard [33]. We remind the reader that the language S?k of

modal logic K extends propositional logic (with propositional variables p\, p2,... ) by means of one unary operator O. S?K is interpreted in models of the form

m=(v,s,p*,p?,...),

where F is a nonempty set, S ? V x V and pf1 ? V. The value (p**1 ? F of an

3?k-formula (p in 9? is defined inductively as follows:

{<p A (//)* =

<pmn y/m;

h?)* = V -^:

(Oip)m =

{v G V I 3w (vSw A w G (pm)}. Say that (p\ follows globally from (p2 and write cp2 h <p\ if, for every model ?xl,

(p^1 = V whenever p^1

= V. The problem of deciding whether <p2 h p\ holds is EXPTIME-hard [33].

We define a translation ? from S?K into the set of .MET -terms built from the Booleans and the operator 3-1. For out translation, the important difference between models for 3?k and MET -models for the operator 3-1 is that the O of S?k is interpreted by an arbitrary relation S while the relation Df interpreting 3-1 is

symmetric and reflexive. Thus, the translation has to 'encode' an arbitrary relation

S by means of a symmetric and reflexive relation. We achieve this by interpreting the points of the model for %K by means of the extension t? of a term to and using 'colours' xf* and x^ to encode the relation S. Going from Xq to Tq* by traversing

xf and then xf gives the required arbitrary relation S. More precisely, take atomic

terms F0, Y\ and define t0 =

F0 n Y\, x\ =

Y0 G ^Y\, x2 =

^F0 n ->Fi. Now we

set inductively

( p{ G MT.

The direction from left to right is easy and left to the reader. Conversely, suppose

Lp2 \f ipi We may assume that (p^1 = V and r g cpf1, for a model 9t =

( F 5, pf1,...} such that (KS) is an intransitive irreflexive tree with root r. Now, for any x G

F - {r}, take its predecessor xp and insert two points (x, 1). (x, 2) between x and

x^. In other words, define an ̂ ^[{l}]-frame

%={W,R,Df,Df)

by taking

W=VU((V- {r}) x {1}) U ((F -

{r}) x {2}), 7? and Df to be the identity relation on PF. and Df the reflexive and symmetric closure of the relation containing, for any x G V -

{r}, the pairs (x^, (x, 1)),

((x, 1), (x, 2)), and ((x, 2), x). Define an jfJ~[{l}]-model ? based on ? by taking

Y0*=Vu((V-{r})x{l}).

Yf= V.

It is not difficult to see that r G (t0 A V(t0 ?> pjj))^ ^ut r ^ (^?)^- Therefore,

(t0AV(to-^))^^^MJ. (ii) Call an i^-formula (p valid if (pm

= F. for every model 9F It is proved in

[12] that, for any ip\. Lp2 G j?x\

p2 h Lp\ in (</?2 A Gp2 A A ip2) ?>

(f\

is valid, where ?(ipi) is the number of subformulas of </?,-. Now. it follows immediately from the reduction (i) that

m h Vl iff (to n v^^'^'Vo - p?)) -+ *>? e Mr.

This proves (ii). H

Note that, as follows from (i). the complexity of MT does not depend on whether we use binary or unary coding. This contrasts with real-time logics where, for

example, LTL is PSPACE complete, but being extended with the operators 'after n ticks of the clock,' n coded in binary, it becomes an EXPSPACE-complete logic

(of the same expressive power, of course) [4]. Theorem 11 (ii) is of interest because it states that the universal modalities can be simulated by distance operators with

sufficiently large parameters.

3.2. Representation theorem. Now we show how to prove the implication ( 1 ) =$>

(5). The proof actually extends a representation theorem of McKinsey and Tarski

[29] (see also [32. 8.1] for more recent proofs) which they use to show the implication (1) => (5) for the topological fragment y?^f of our language. To start with, we

formulate their representation theorem for the metric space consisting of the real line R with the standard Euclidean metric. Let IIr denote the standard interior

operator on R. For a quasi-order $ = (W, R) and X ? W, set

UX = {u G X | \fv G W (uRv -> v G X)}. (16)

A map / from R onto W is called a topological morphism from R to # if, for all X CW,

f-l(ux) =

iRf-l(x).

A point r G W is called a root of $ if W = {w \ rRw}.

Theorem 12 (McKinsey and Tarski). If% is a finite quasi-order with root r, then there exists a topological morphism f from R to $ such that f(0)

= r.

This result yields the following topological variant of the implication (1) => (5):

Corollary 13 (McKinsey and Tarski). If an Ji^f-term p is satisfiable in a finite rooted quasi-order, then it is satisfiable in (R, %}

We now introduce the concepts we need to extend this result (with R replaced by arbitrary metric spaces) to our language of metric and topology.

Let 2) = (V,d,Id) be a metric space with the induced interior operator Id. Let

$ ?

(W,R, (Df)a^M, (L>a)aehi) be an ^#5r[M]-frame, for some parameter set M. Define the operator as in (16). We say that a surjective map / : V ?? W is an

M-morphism from 2) to # if the following conditions are satisfied for all x, y G V, alla G M. and all I? W:

(Ml) f^(uX)=ldf-](X); (M2) ifd(x,y) < a then f(x)Dff(y): (M3) if d(x,y) < a then f(x)D<f(y): (M4) if f(x)Dff(y) then there exists a z G V such that d(x, z) < a and f(z)

=

f\y)\ (M5) if f(x)D<f(y) then there exists a z G V such that d(x. z) < a and f(z)

=

/oo. The following proposition is easy and left to the reader (it corresponds to the well-known p-morphism theorem from modal logic; see, e.g., [12, Theorem 3.15] or

[9. Theorem 3.14]):

Proposition 14. (i) Let Wl = (?). Pf1, P

,...) be a topometric model and & =

(g, Pf-. Pj,...) an Jt^\M\-model. If there is an M-morphism f : S) -> $ such

that Pf1 =

f-l(P?)for all i < co. then zm = f~l(zA)for every M ST-term, z

with M[z] ? M.

(ii) Suppose 2). # and z are as above. If z is satisfied in an Jt^\M\-model based on

5 and there is an M -morphism f : 2) ?? $, then z is satisfied in a topometric model based on 2).

A LOGIC FOR METRIC AND TOPOLOGY 8 11

Thus, in view of (ii) above, to achieve our aim and show the implication ( 1 ) => (5), it suffices to prove the following fundamental representation theorem:

Theorem 15. Given a finite JiET\M\-frame

$=(W,R,(Df)aeM,(D<)aeM)

with a finite parameter set M, one can construct a (possibly infinite) metric space D =

(V,d,\ci) and an M -morphism f : D ?> $.

Proof. The metric space D is constructed by means of a sort of 'unravelling' of

5 into a tree-like metric space. However, instead of taking the set PF+ of all finite

nonempty sequences over W as the domain of the new space (which is the standard

construction), we define the domain V of S as

V = (R x PF)+,

i.e., as the set of nonempty sequences (r\,v\)... (rn,vn), n < cd, over Rx W.

Elements of V are denoted by x, /, z, etc., with end(x) being the last element of x.

This generalisation of the standard unravelling construction by means of copies of the real line R allows us to represent the 'local topologies' within #. More precisely, for every w G PF. let Ww =

{u G PF | wRu} and Rw = R \ Ww. The pair

(WW,RW) is known as a subframe of the S4-frame (W,R) generated by w. Note

that, since (W,R) is & quasi-order, it is possible that Wu = Wv for u ^ v.

According to Theorem 12, for every subframe $v = (Wv,Rv) of (W,R), there

exists a surjective map fv : R ?> Wv such that, for all X ? Wv,

f;x(uvx) =

\Rf;x(x) and fv(ti) = v.

Here UVX = {w G Wv | Vw G Wv(wRvu ?> u e X)}. Notice, however, that

UX ? UVX for every X ? Wv because (WV,RV) is generated by v. Fix such a map

fv for each v e W. We are going to introduce a metric on V in such a way that the

map / : V ?> PF defined by taking

/(*(r,i;))=/?(r), (17)

for all x G FU {A}, (r, v) eRx PF, is the required M-morphism. (Here ? denotes

the empty string.) We begin by introducing a notion of ^-distance d$(w,w') between w,w' G W.

Fet M~ ?

{a~ | a G M}. Three cases are possible:

Case 1: if there isa G M such that wDfw' but wD<w' does not hold, then

d$(w, wr) ? a.

Case 2: if there is a G M such that wD^w' but wDfw' does not hold for any b < a, b G M, then d$(w,wf)

= a-.

C?we 3: if wDfw' does not hold for any a G M, then d$(w, wf) = oo.

It is not hard to see that d$ : PFx PF ?> MUM" U {00} is a well-defined

function.

In fact, the ^-distance d$(w, w') is just a convenient way of speaking about the

relations Df and Df. Indeed, d$(w, w') = a G M means that the distance encoded

by these two relations is exactly a, and d$(w,w') = a~ means that this distance is

between a and b, where b is the maximal number in M such that b < a. In the

latter case we cannot yet fix a number for the distance between w and w'.

With every pair (r.v) G R x W we associate a non-negative real number op(r.v) in the following way. Suppose that fv(r)

? w and Ww C pfz, (equivalently. not

wRv). Then

o/7(r.iO=inf{|r-r'||r'g/-1(W',1,)}. As ( Ww. Rw ) is the subframe of ( Wv. Rv ) generated by w. we have

Ww =

uwWw =

uvWw =

UWw.

Note that op(r,v) > 0 since r G f~l(Ww) =

f~{(UWw). and so f~l(Ww) ? M

is open. If Ww = Wv then we set op(r.v) = y m (in particular. op(0.v)

= yM)

Intuitively, we need the number op(r,v) to ensure that points outside the interval

[r ?

op(r. v).r + op(r. v)] are 'sufficiently far away' from points in different copies of R. Take some

e < min{EM- a i + a2

- yM \ ci\.a2 G M. a\Jra2> yM}

and assign to each element x of V a real number dist(x) by taking inductively:

dist((r.v)) =

-mm{op(r.v). e}.

dist(x(r.v)) ?

-nxm{dist(x), op(r.v)}.

We are now in a position to define a metric function d on V. The definition

consists of five cases. The important ones are I and II. whereas the remaining cases

ensure that we obtain a tree-like metric space by taking as distances between points the sum of the already defined distances over the shortest path connecting them.

The maximal distance between any two points in V will be y m + ?m One can easily define a tree metric space without introducing this bound, but having it minimises

the number of case distinctions required.

Case I. Suppose that / =

x(r\. v) and z = x(r2. v) for some r\. r2 G R and some

x G V U {?}, where / is the empty string. Then

d(y.z) =min{yM +em. V\-r2\}.

That is to say. on each copy of the real line R. the metric function d coincides with

the Euclidean distance 'cut' at y m + sm- Notice that neither the behaviour of the

interior operator Ir nor the behaviour of the operators 3<a and 3-a. a G M. are

affected by 'cutting' the distance function at y m + ?m

Case II. Suppose that y = x(r\.v). For each w' G W. we define d(y. y(0. w')).

This is the subtlest case in the definition of d: given the point r\ in a copy of R

corresponding to (WV.RV). we define the distance d between this point and the

Os in the 'successor copies' of R corresponding to (WW>.RW>), for w' G W. Let

w ? fv(r\). So w is the point in Wv to which r\ corresponds. Thus, the distance

d(y* /(O. w')) should depend on the value of d$(w. w'). The definition is clear for

d$(w,w') G M U {oc}: If d$(w,w')

= oo. then d(y.y(0.wf)) = y m + ?m

If d$(w. w') ? a. then d(y. y(0. w'))

= a.

Now suppose that d$(w. w') = a~. Then?/(j7./(O.i?/)) should be between a and

the maximal b G M such that b < a. In other words, it should be between a - eM

and a. Second, we want to make sure that for a = sM the distance d(y. v(0. w')) is

A LOGIC FOR METRIC AND TOPOLOGY 8 1 3

at least \zM because points in other copies of R should not influence the topology on the present copy of R. Finally, we have to ensure that for any r' which is on the same copy of R as r\ and which corresponds to a w" G Wv

- Ww.

|n -

r'\ + d(y,y(0.w')) > a.

We can ensure this by having d(y.y(0.wf) > a ? op(r\.v). Actually, a bit more

is required. We have to ensure that this inequality also holds for sums of such

distances over paths from the root of the constructed tree metric space to y. Taking this into account, we end up with the definition d(y. y(0. w'))

= a ? dlst^

.

We extend d to the remaining pairs x. v G V x F by taking the appropriate sums

of the distances defined so far:

Case III. Fet y = x(r\,v\)(r2,v2)... (rn.vn) and end(x)

= (ro, vo). Then

d(x,y) =

min\yM +?m

n n ? \

Y^\ri\ +

^d(x(ruvy)...(ri.vi),x(ruv\)...(ri,vi)(0,vi+i))y i=\ /=0

Case IV. Suppose now that

x = z(r0.vo)(r\.v\)(r2,v2)... (rn,vn),

y = M.vo)(rl.vl)(r'2,v>)...(r;?.v'J

and either r0 / r'0 or r0 = r'0 and v\ ^ v[. Then set

d(x,y) =min{7M +eM- fo -

r'0\ + d(z(r0. v0).x) + d(z(r,0.vo).y)}.

Case V. Finally, set d(x. y) =

d(y, x) whenever d(y, x) is defined, while d(x, y) is not defined yet. For the remaining undefined d(x. y) set d(x. y)

= yM + ?m

Straightforward yet tedious computations show that the defined function d is

indeed a metric on F (this is left for the reader). Thus we have

Femma 16. (Kd) is a metric space.

Now define a map f : V ?> W by taking

f(x(r,v)) = fv(r) for all x G FU {/}. (r. v) G R x W. To show that / is an M-morphism, we require the following auxiliary lemma.

Femma 17. Suppose that y = x(r\.v\)... (r?.v?) and end (x)

= (ro, vq). Suppose

also that, for 0 < i < n ? 1.

d$(fvj(ri)- Vj+\) G {a?.af} ? M U M-. a= ^

a?. b = a + eM. 0<i<n

Let x' G V be the result of replacing the last element (ro- ̂ o) ofx with (r?. vq). Finally, suppose d(x.y) < yM- Then the following hold true:

(1) ifb < yM. thenf(x)D<f(y) and f(x')D<f(y). (2) d(x.y) > a ?

dist(x).

(3) a < yM

(4) ifds(f(x).vx) = a~. thenf(x)Dff(y).

(5) d(x.y) > a - ^op(ro.vo) andd(x.y) > a ?

\e.

(6) d(x',y) > a + \op(r0,v0) if\r'0 -

r0\ > op(r0,v0),

(7) if\r'0 ?

ro| < op(ro,vo), w G W ande G M, then f(x)Dfw implies f(x')Dfw.

Proof. (1) By the definition of f and d$t we have

v0Rf(x)D^vlRfvfrl)D^v2...Dl_]vnRf(y). (18)

So, by Lemma 3 (f), f(x)D?f(y). And since voRf(x'), by Lemma 3 (g), we

obtain f(x')D<f(y).

(2) We have

0</'<>7 0</<?

where // =

x(î,ri)... (v?,ri), for 0 a-

2^

?-?

0<i<n

and therefore d(x, y) > a ? dist(x).

(3) Suppose a > yM- Then a?E>yu- It follows that a-dist(x) > yM, contrary to (2) and d(x,y) < yM

(4) As f(x)D^v\, by Lemma 3 (e) and (1) we then obtain f(x)D<f(y).

(5) follows from (2), since dist(x) < ôp(ro,vo) and dist(x) <

\e.

(6) We have, by case IV and (5),

d(x',y) =

d(x,y) + \r'0 -

rQ\ > d(x,y) + op(r0,v0)

> a - -op(r0, vq) + op(r0, vo)

>a + -op(r0,v0).

(7) Suppose fVo(r0) = u. As |r0

? r0| < op(r0,v0), we have r'0 G f~l(Wu).

It follows that fVQ(rf0) G Wu. i.e., f(x)Rf(x'). It remains to use Lemma 3 (d). H

We are now in a position to prove that / is an M-morphism from (V,d,\d) onto

?. Obviously, / is surjective.

(Ml) Let X ? W. Then

x(r.v)eldf-{(X) iff

iff 3e > OVjT G F(J(x(r, v), /) < e - f(y) G X) iff 3e G (0,EM/2)\/ye V(d(x(r,v),y) < e -* f(y) G X) iff 3e G (0, EM/2)Wy G {xU v) \ s G R}

(d(x(r,v),y)<e->f(y)eX) iff ?(r,i;)Gl,/-1(^)

iff xfoê/"1^) iff x(r,v) ef~l(uX).

A LOGIC FOR METRIC AND TOPOLOGY 8 15

(Here (0,?M/2) =

{e G I ? 0 < e < eM/2}.) Note that the fourth equivalence follows from the observation that the distance between any two points in different

copies of R is at least \em> This was ensured by the definition of d in case IF

(M4) Suppose that f(x)Dff(y). Fet w = f(y). Then d(x.x(0.w)) < a and

/(x(0, w)) =

fw(0) = w. Thus, z =

x(0, w) is as required.

(M5) Suppose that f(x)Dff(y). Fet w = f(y). Then d((x),x(0.w)) < a and

f(x(0, w)) =

fw(0) = w. Thus, z =

x(0. w) is as required.

(M3) Suppose d(x. y) < a. Consider all cases I-V.

Case I. Suppose that x = z(r\.v) and y =

z(r2.v) for some z G V U {A}, r 1,7*2 G M. Then f(x)

= fv(r\) and /(/)

= fv(r2). Therefore, vRf(x) and

vRf(y). ByFemma3(g)./(x)7)</(/)forallc G M. In particular, f(x)Dff(y). Case IF Supposex

= z(r\. v) and j7

= z(ri,?;)(0, ?/) Then f(x)Dff(y) follows

by definition.

Case III. By induction on ? we show that, for every a G M, if ?(f, j?) < a and

y = x(n, v\)... (rnrvn), then f(x)Dff(y). The case ? = 0 is clear.

Fet y =

x(r\,v\).. .(rn,vn)(rn+\,vn+\), end(x) =

(ro,vo), and d(x,y) < a,

a G M. We find 0/ G M such that, for 0 a1 - eM because

a' ?

5m < a' ? -? < d(x,y) < a.

If a > a' + 8 m then, by Femma 17(1),/' (x)Dff (y), and we are done. So it suffices to consider the case a = a1.

Fet f(x)DfQv\. Then, by Femma 17 (4). f(x)Dff(y), and we are done. Sup

pose now that f(x)DfQv\ does not hold. Then f(x)Dfov\, d(x,x(0,v\))

= ?z0. and therefore d(x(0,v\),y) < a ?

ao- It follows that d(x(r\.v\),y) < a ? ao

and, by Femma 17 (6), |0 -

r\\ < op(r\,v\). By the induction hypothesis.

f(x(ruv{)D<_aJ(y). So. by Femma 17 (7), f(x(0.vi))D<_aJ(y). But then

f(x)DfQvxD<_aJ(y). and by (trD^D<). we obtain f(x)D<f(y). Case IV. Suppose now that

x = z(rQ, v0)(ri.vi)(r2. v2)... (rn,vn),

y = z(rivo)(r[.v[)(r^v^...(r'm,v>J

and either ro ̂ rf0 or ro = rf0 and v\ ^ v[. Fet

d(x.y) =

\r0 -

ro| + d(z(r0, v0). x) + d(z(r'0,v0)* y) < a.

bx = ]r a^ h2 =

J2 a'^ 0<i<n 0<i<m

where

dd(fVi(nlvM) G {cii.ar}. dd(fv;(r?).v?+l) G K-kT}

(v'0 ?

vq). Set b = b\ -f- b2. By Femma 17 (5).

d(z(r0.vo).x) > b[ -

-e. d(z(r'o,vo).y) > b2

- -e.

8 16 FRANK WOLTER AND MICHAEL ZAKHARYASCHEV

So. d(x.y) > b - e > b - em- If # > b + em- then f(x)Saf(y). by Lemma 3 (g), since fVQ(0)Rf(z(r0. v0)) and fVo(0)Rf(z(r'0, v0)).

Thus, it suffices to consider the case a = b.

Let d(z(ro,vo).x) < b\ and |r0 -

r'0\ < op(r0,vo). Then, by (III) above,

f(z(r0.vo))D<J(x) and, by Lemma 17 (7). f(z(r'^vo))D<J(x). Therefore, by

Lemma3(e)J(f)i)</(;). The case d(z(r'Q. vo),y) < b2, |r0

- r'0\ < op(rf0, vo) is dual to the previous one.

Let |ro ?

r'0\ > op(ro.vo) and \ro

? rf0\

> op(rf0,vo). We may assume that

op(ro. vo) > op(rQ, vq). Now, by Lemma 17 (6),

d(z(r'0,v0),x) >b\ + -op(r0,v0)

and, by Lemma 17 (5),

d(z(/0.v0),y) > b2- -op(/0,vo).

Therefore, d(x, y) > b\ + b2, which is a contradiction.

The case d(z(r0,vo),x) > b\, d(z(rf0,vo),y) > b2 leads to a contradiction as

well.

Let d(z(ro,vo),x) > b\ and |ro ?

r'0\ >

op(rf0,vo). Then d(z(ro,vo),y) > b2, by Lemma 17 (6), which is a contradiction.

The case d(z(rf0, vq), x) > b2, |ro ?

r'0\ > op(r^, vq) is dual to the previous one.

Case V is trivial.

(M2) Suppose d(x,y) < a. Again we check cases I-V.

Case I. As we know, x = z(r\, v) and y =

z(r2, v) imply xD<y for all a G M.

So, xDfyfor all a G M.

Case II follows from the definition.

Case III. We show by induction on n that, for every a G M. if d(x, y) < a and

y = x(r\,v\)... (rn,vn), ihzn f(x)Df f(y). Thecase^ = 0is trivial. Supposenow

that y = x(r\,v\)... (rn,vn)(rn+\,vn+\), end(x)

= (ro,^o) and d(x,y) < a G M.

We can find a? G M such that, for 0 . Then,byLemmal7(4),/(x)Z)</(jr). Therefore f(x)Dff(y), and we are done.

Assume now that f(x)D<Qv\ does not hold. Then we have

f(x)DfQv\ and

d(x.x(0.v\)) = ao. Therefore.

d(x(0.v\),y) < ci ? ao.

\fop(r\.v\) < \r\ -

0|, then

d(x(0.v\).y) > a - a0 + -op(ruv\)

(by Lemma 17 (6)) and

d(x.y) >a + -op(rx,v\),

which is a contradiction.

Suppose now that op(r\,v\) > \r\ ?

0|. Fet r\ = 0. Then d(x(0,v\),y)

= d(x(r\,v\)) < a -

ao, and by the induction

hypothesis, f(x(0,vi)) = f(x(rx,vx))Df_aQf(y). Then we have f(x)Dff(y) by

(trD^). Fet r\ ^ 0. As d(x,y) < a, we have d(x(r\,v\),y) < a -

ao. By (M3),

f(x(rx,vx))D<_aJ(y). By Femma 17 (7), f(x, (0,vx))D<a_aJ(y). Therefore, by (trD^D<), f(x)D<f(y), and so f(x)Dff(y).

Case IV. Suppose now that

x = z(r0, vo)(r\, vx)(r2, v2)... (rn,vn),

y = z(rivo)(r[,v[)(r>,v>)...(r'm,v'J

and either r0 ̂ r'0 or r0 = r? and v\ ^ v[. Fet

d(x,y) =

\r0 -

r?| + d(z(r0, v0), x) + d(z(r'0,v0), y) < a,

bx = ^2 a^ b2 =

X] ai> 0<i<n 0<i<m

where

dd(fvi(ri),vi+i) G {a/,aG}, ^CA'WW+i) ? WiA<t?)~}

(v? =

vo). Set ? = ??i + ?2- As in (M3), it suffices to consider the case a = b. Fet d(z(r0,vo),x) < b\ and |r0

- r'0\ < op(r0,vo). By (M3), f(x)Dff(y), and

*of(Z)Dff(y). The case d(z(r'0, vo),y) < b2 and |ro

? r?| < op(r'0, vo) is dual to the previous one.

The case |ro ?

rf0\ > op(ro, vo) and |ro ?

r'0\ > op(rf0, vq) leads to a contradiction

as in (M3). Fet d(z(ro,vo),x) > b\ and d(z(rQ,vo),y) > b2. Then ro =

r?, d(z(ro,vo),x) =

b\ and d(z(rf0,vo),y) = b2. So, by (III), we obtain

f(z(ro,vo)Dff(x) and

f(z(r0, v0))Df2f(y). f(x)Dff(y) follows from the symmetry of Df}

and (trD^). Fet d(z(ro, vo), x) > b\ and |r0

- r'0\ >

op(r'0, vo). Then, by Femma 17 (6),

d(z(r0,v0),y) >b2+ -op(rQ,v0),

which is a contradiction.

The case d(z(rf0, vo), x) > b2 and |r0 -

r'0\ > op(ro, vq) is dual to the previous one.

This completes the proof of Theorem 15. H

Remark 18. It is worth noting that this result does not hold for infinite parameter sets. For example, take the ,/C7"[Q+]-model

art = ({0,1}, R, (Df)aeq+,(Df)aeQ+,Pm),

where R = <, Df =

Df =

{0,1} x {0,1}, Pm = {I}. Then the underlying frame

of Tl is not a Q+-morphic image of any topometric space, since otherwise the set

{- OP} U {3<* P | n G N+} from Example 1 (vi) would be satisfiable.

?4. The logics of the real line and plane.

The logic of R. Let us consider now the satisfiability problem for J?ET-terms in

topometric models based on the real line, i.e., the one-dimensional Euclidean space

(R, d) with d(r,r') =

\r ?

r'\. This problem can be shown to be decidable by a straightforward embedding into the quantitative monadic logic of order QMLO introduced in [23].

The language of QMLO is built from atoms x\ < x2, x\ = x2, P?(x), using the Booleans, first-order quantifiers and the following rule: if p(x) is a formula of

QMLO with only one free (first-order) variable x, then

(3x)>?0+V "= 3x (x0 < x < x0 + 1 A p(x)).

(3x)<^-]p ::= 3x (x0

- 1 < x < xq A p(x)) are also formulas of QMLO. The following result is proved in [23]:

Theorem 19 (Hirshfeld and Rabinovich). Validity of formulas of QMLO in R is

decidable.

Hirshfeld and Rabinovich [23] also prove that .

(3x)|-+Vand(3x)|i:-'V, .

(3x)^o+Vand(3x)^o-'V are expressible by QMLO-formulas, for n G N.

To show that the satisfiability problem for ̂ r-terms over (R, d) is decidable, we observe first that this problem is reducible to satisfiability of ̂ ^[Nj-terms over

(R, d) (of course, this holds for satisfiability in many other classes of topometric

models): given an .^?r[N]-terrn with parameters f-,..., y, replace any f- in z by a? x b\ x x bi-i x 6,-+i x x b?. Then the resulting term is satisfiable if and

only if t is satisfiable.

Now define inductively a translation ? from the set of J?ET[N]-terms into the

language of QMLO by taking

P] = Pi(x),

(3<nzf = (3y)f+"zKy) V

{3y)lx-?xHy),

(3^z)* = (3y)fx+nzKy)v(3y)irn^(y),

(Oz)* =

Vj (y > x -> 3z(x <z<yA z\z))) V

My (y < x - 3z(y < z < x A z*(z))),

(3zf =

3y (x = x At?(>')).

It is readily seen that, for every jf^NJ-term t, we have a G zm for some a G R

and some topometric model Wl over (R, d) iff z$[a] holds in some ?MLO-model. Thus, we obtain from Theorem 19:

Theorem 20. The logic L(R) is decidable.

The computational complexity of L(R) in unknown. It should be clear that the

language of QMLO is more expressive than JtST. For example, the gMLO-formula 3x (x > y A P(x)) cannot be expressed by means of an jfi/'-term (interpreted in

(R, d)). More interestingly, by extending MET with the operators 3<q, a eN, such

that

(3<a0z)m =

{ue V \3v(0 <d(u,v) <aAvezm)},

we obtain an undecidable logic over the class of all topometric models even without

the topological operators [26]. However, when interpreted over (R. d), the extended

language gives rise to a decidable logic because it is clearly embeddable into QMLO. Thus, there are natural metric operators that are 'too expressive' on arbitrary metric

spaces and yet 'harmless' on the real line.

On the other hand, it is known that the 'punctuality' operators 3=a ('at distance =

a') yield undecidable logics on R (both with and without finite variability constraints on the interpretations) [3], but it is an open problem whether the extension of MET

with the punctuality operators is decidable over the class of topometric models.

The logic of R2. The situation becomes quite different if we consider the satis

fiability problem for J?ET -terms in topometric model based on the 2D Euclidean

space. We consider points of R x R, i.e., pairs x = (x\,x2), where x\,x2 G R, as

standard Euclidean vectors of length

11*11 = v^ + 4

A metric space (W,d) is a subspace of the Euclidean space R x R if there is an

injective map / : W ?> R x R such that, for any two x,y G W,

d(x,y) = \\f(x)-f{y)\\.

Theorem 21. The satisfiability problem for MET\{ 1,2}]-terms (even containing no

topological operators U and O) in topometric models over subspaces of'R x R or over

R x R itself is undecidable.

Proof. The proof is by reduction of the undecidable Z x Z tiling problem (see [10] and references therein), which is formulated as follows. Given a finite set ET =

{Tb,..., 7)} of tile types (i.e., 1 x 1-squares T? with colours left(Ti), right(Ti),

up(Ti), and down(T?) on their edges), decide whether the grid Z x Z can be covered

with tiles, each of a type from ET, in such a way that the colours of adjacent

edges on adjacent tiles match, or, more precisely, whether there exists a function

/:ZxZ^J such that, for all n,m G Z, we have

right(f(n,m)) = left(f(n + l,m)),

up(f(n,m)) =

down(f(n,m + 1)).

So suppose that a set ET = {To,..., T?} of tile types is given. Our aim is to

construct an Ji^[{l, 2}]-term z^ which is satisfiable in a subspace of R2 (or in R2

itself) iff F can tile Zx Z.

Define t^ to be the conjunction of the following terms, where Xo,...,Xi,

Yo,..., Yi, Zo,..., Z/ are atomic terms, 0 < i, j < 3, 0 < n, m, k < I, and +4

and ?4 denote addition and subtraction modulo 4:

I0n70/i (19)

XinYJU3^(Xl+4lnYj), (20)

xlnYJu3^(xl-4lnYj), (21)

XlnYJU3^(XlnYJ+4l), (22)

XiUYjU3^(Xlr\Yi^l), (23)

Yj UV<2^YJ+42, (24)

Xi G V<2^-+42, (25)

Xi G Yj G p] -3<1(X,v G Y y), (26)

[J ^GFy = |JZ^ (27)

zj<3 *;</

Zm G V<1-Z? (for w ? m), (28)

i;-nF;nz,?vÛ-+4iU-r;u |J zm), (29)

^ n^nz^vÛ-u^û |J zm). (30) up{Tk) =down ( rm )

The intended meaning of the conjuncts of x^ will become clear from the following consideration. Suppose that x^- is satisfied in a model

X = (W,d,X?,...,X?,Y*,...,Y?,Z?,...,Z?),

where PF ? R x R and d is the restriction of the Euclidean metric to PF. The terms

of the form X? G Yj, for 0 < /, j < 3, will be used to simulate the grid Z x Z in

( PFd) in the following way. Fet Ai = X*, Bj

= Yj8,

for 0 < i,j < 3. By (19), there is some r G Ao n P0 Vector r will represent the point (0,0) of Z x Z. By (20), there is a point xx e Xxn Y0 such that d(r,x\)

= \\r

? x\\\ < 1. But in fact, by (26), d(v,x\)

= 1. So we can

regard jci as representing (1,0). Using again (20) and (26), we can find x2 G X2 n To such that <7(jci, jc2)

= 1. Now, observe that in view of (25), d(r,x2) > 2, while by the triangular inequality, d(r,x2) < 2. Thus, d(r,x2)

? 2 and we can regard x2 as representing (2,0). Using the same kind of argument we can find x^ G Xt> Pi Fo such that d(x2, JC3)

= 1 and d(xx, JC3) = 2. Now, since the metric d is Euclidean, we

conclude that d(r,x^) = 3, and so X3 can represent (3,0). Then we find X4 G XoDYo

for which d(r, X4) ? 4; that will be (4,0). All four points r, xx,x2, x^ lie on the same

straight line. (Observe that if ( W,d) is not assumed to be a subspace of IR2 then we

could have, say, d(v,x^) = 1 and r =

X4.) More generally, it follows from (19), (20), (21), (25), and (26) that there exists

a vector a such that ||?|| = 1 and, for every / < 3,

{r + (An + Z) a \ n G Z} ? Ai n P0

This gives us points of the form (n', 0) from the grid Z x Z.

Similarly, using (19), (22)-(24) and (26), we can find a vector b such that ||A|| = 1

and, for ally < 3,

{v + (Am + 7) b I m G Z} ? Ao H Bj (31)

which gives us points of the form (0, m') from Z x Z. Note that, according to (26), a and b are linearly independent.

We claim now that in general, for all /, j < 3,

{r+(4ra + y) b + (4? + i) a | m,n G Z} ? ?z- nPy, (32)

which gives all points (n', m') from Z x Z. To prove this claim, we first show that,

for j < 3,

{r+(4m+j) b + a \ m G Z} ? A\ HBj. (33)

We know that r + a G A\ (1 Bo. Using the same argument as above, we can show

that there is a vector v such that ||v|| = 1 and

{r + (4m + j) v + a \ m G Z} ? A\ n Bj. (34)

So, we have to prove that v = b. Assume first that v and b are linearly independent. Since ||v||

= ||?||

= 1, for every vector x, there exist n,m G Z such that

\\n b -+- m v ? x\\ < 1.

It follows then from (31) and (34), that there are c G A0 n ?7 and </ G /4i n By, for some 7,7' < 3, such that \\c

- d\\ < 1, contrary to (26). Therefore, b and v are

linearly dependent. So it remains to show b / -v. Suppose otherwise. But then

y + b G Ao n 2?i and r + a + ?e^ini?3, contrary to (24). (Here we again used the

fact that the metric is Euclidean. Without it we could not ensure, for example, the

existence of a point c G A\ n B\ such that d(r + a,c) =

d(v + b,c) =

1.) Thus we have (33) and then obtain (32) by a straightforward induction.

This provides us with an encoding of the grid Z x Z. We use the atomic terms

Zi to encode the tile types Tx : (27) says that every point on the grid is covered with

a tile, (28) ensures that this tile is unique, while (29) and (30) guarantee that the

colours of adjacent edges on adjacent tiles match.

Consider now a map /:ZxZ^J defined by taking, for all n,m G Z and

f(An + i,Am + j) = Tk iff r + (An + i)a + (4m + j)b G Zf.

It is not hard to check that / is a tiling we need.

Conversely, suppose that / : Z x Z ?> ET tiles Z x Z. Consider the model 53

based on the Euclidean metric space R x R in which, for 0 < / < 3 and 0 < k < /,

X?* = {(/ + An, m) I (n, m) G Z x Z},

Y? =

{(m, i + An) \(n,m)eZx Z},

Zf =

{(n,m)\f(n9m) = Tk}.

It is readily checked that (t^)? 7^ 0. H

?5. Reasoning with numerical variables. The main aim of this section is to show

that reasoning with the language Jt3~\V\ (containing numerical variables that

satisfy some rational linear inequalities) is decidable:

Theorem 22. It is decidable whether an J?EF\V\-term z is satisfiable relative to a

finite set of rational linear inequalities.

To formulate the decision procedure, we require the following notation. Sup

pose that an jf^FJ-term z and a set T of rational linear inequalities over 2f =

{x\,..., xn} are given. Without loss of generality we may assume that Y contains

(among others) the constraints

{0 < X\,X\ < x2,x2 < X3,. .. ,Xn-\ < xn}.

(Obviously, the general decidability problem can be reduced to n\ decidability prob lems of this type.) Denote by Sy the set of all assignments a : %? ?> Q+ solving Y.

Our problem is to decide whether there is a E ̂ such that xa is satisfiable. Since

eSy is usually infinite, it is not clear a priori that this can be done effectively. The

main step towards an algorithm solving this problem is to show that if there exists

an a G ?r such that xa is satisfiable, then there exists an assignment which comes

from a finite and effectively computable set of solutions to Y.

Intuitively, the finite set of solutions a we need consists of those solutions that

minimise the number of constraints on MEF[xa]-models. More precisely, given an

assignment a, denote by L(a) the set of all inequalities of the form

/J kXiXj < Xj and Yj kXjXi < x7-,

x,eZ x?eZ

where the kXj are positive natural numbers, Z ? ^, x7 G ̂ , and

]P kXia(xi) < a(xj) or ^ kXja(xi) < a(xy),

x?ez x?ez

respectively. Each L(a) is a finite set of rational linear inequalities. However, there may exist

infinitely many L(a). a G Sr- But it turns out that the set of ?-minimal members

of {L(a) | a G Sy} is finite and provides enough information. Fet

Mr = {L(a) | a G Sr & -3b G ?r L(b) g L(a)}.

Femma 23. My is finite andean be computed effectively from Y.

Proof. Finiteness follows from the observation that

({L(a)\aeSr},Q

is a well partial order (it is easily embeddable into an appropriate (Nm, <), where

(nx,... ,nm) < (n[,... ,nfm) iff ni < n? for all / < m.) Recall that it can be checked

effectively whether, for a given a, Y U L(a) is solvable. If the answer is 'yes' then a

solution can be computed. So, we can check effectively (i) whether L(a) is a member

of My, for a given a, and (ii) whether X = My holds, for a given X ? My. H

Now the algorithm runs as follows:

1. Determine My and take, for every L G My, a b G Sy such that L = L(b).

2. If there exists a b in this list such that xb is satisfiable, then x is satisfiable relative

to F. Otherwise x is not satisfiable relative to Y.

Obviously, all this can be done effectively. The soundness of this algorithm is clear.

Completeness is a consequence of the following lemma:

Femma 24. Suppose a, b G ?>y- Then

(i) if L(a) ? L(b), then xa is satisfiable whenever xb is satisfiable',

(ii) if L(a) ?

L(b), then xa is satisfiable iffx0 is satisfiable.

Proof. We begin by introducing the notions required for the proof Suppose that

x is an ̂ ^-term and, as before, 7V(t) is the set of numerical parameters occurring in t. Suppose also that a frame

1=(WR,(D^aeN{r),(D<)aeNir))

is such that

(W,RU (J (DfuD<)) aeN(r)

is a (possibly infinite) forest of (disjoint) intransitive and irreflexive trees (so here we do not assume that (qoR)-(DKR) hold). Then a structure of the form

6 = (X,if,if,...),

where Pf ? W for i < m, is called a z-skeleton. The value p& of & subterm p of z

in 6 is defined in precisely the same way as for the standard JttT-models. The term t is satisfied in 6 if t6 / 0.

The expansion of a T-skeleton 6 is the JtET\z\-n\odz\

6* = (X*.if, if,...), with (35)

X* = (W,R\(E^)aeM[T],(E<)aeM[T]), (36)

where the X* is the result of closing X under the rules (qoR)-(D<R) for M[z]. Say that a T-skeleton 6 is expandable if, for every subterm p of z, we have

Lemma 25. An Jiff-term z is satisfiable iff it is satisfiable in some expandable z-skeleton.

Proof. The implication (<=) follows from definition.

(=>) Suppose that t? ^ 0 for some finite J^^[z]-model

?= (#, if, if,...), with

$=(W9R, (D^)aeM[T], (D<)aeM[T]).

Consider the reduction

^ = (W,R,(Df)a N{T),(D<)aeN{T))

of # to N(z) and apply to it the standard unravelling procedure (see, e.g., [9]). More

precisely, take some minimal subset roots = {w\,..., Wk} of W such that, for every

w' G W ? roots, there is a path

WiS\U\S2u2Sz.. .Snun, (37)

where un = w'. each Sz- is one of R, Df, D<, a G N(z), and w,- G roots. Now

construct the frame

Z=(wJ?D^)aeN{T)AD:)aeN{T)), where W consists of all finite sequences of the form (w?,u\, ... ,un) for which (37)

holds, and for each relation S G {R, Df,D< \ a G N(z)}, we have

(wi,u\,... ,un)S(wj,v\,... ,vm)

iff m = n + 1. i = j, Ui

= Vi for i =

1,..., n, and unSvm holds in $'. Let

6 = (X,if ,P26,...)

where (wi,u\,..., un) G Pf

iff un G Pf.

It should be clear that 6 is a T-skeleton

such that, for every subterm p of z,

(wi,u\,...,un) G pe iff unep?. (38)

Thus, it remains to show that 6 is expandable. Fet 6* be the expansion of 6 based on the ^J^-frame T* as defined in (35), (36).

We claim that, for every subterm p of x, we have p& =

pe*. The claim is proved by induction on the construction of p. Here we only show that (3<ap)e

= (3<ap)e*.

The inclusion ? is trivial. To prove the converse, suppose (w?,u\, ...

,un) G

(3<ap)e\ Then there is (w?,vx,... ,vm) G p&* such that

(Wi, ui,..., un ) Ef (wj , vi,..., vm ).

Using Femma 3 (e), (g) and the definition of unravelling, it is not hard to see that we then have unD<vm. By the induction hypothesis, (wj,v\,

..., vm) G pe, and so,

by (38), vm G p?. Therefore, un G (3<p)? and, again by (38), (wi,u\,... ,un) G

(3<ap)e. H

Now, clearly, (ii) follows from (i). To prove (i) suppose that L(a) ? L(b) and set Ni =

{a(xi) | 1 < i < n}, N2 = {b(x?) | 1 < i < n}. Denote by Mx and M2

the closure under (+) and (-) of N\ and 7V2, respectively, and set e\ = min Mi, e2

= min M2.

Suppose xb is satisfiable. By Femma 25, we can satisfy xb in an expandable xb-skeleton

62 = (z2,P?2,P?2,.

where

Z2 =

(w9R,(Df)beN2,(D<)beN2 Denote by P*, Ef, Ef, b G M2, the relations in the expansion 62 ?f ?2- Our aim

is to construct an Mi-expandable ra-skeleton satisfying xa.

Fet

61 = (%x,Pf\Pf\.

where P?1 = Pf \ for i<co, and

where, for 1 < i < n,

F- ?

D- F< ?

D<

Denote by P*, G?-, Gf, a G M\, the relations in the expansion &x of 61. We claim that, for all u,v G PF and 1 < i < n,

(1) GK, , CP< ,,

(2) G-( , CEf v Here we only show (1) and leave (2) to the reader. Fet us write s = uRFv if s is a sequence of the form u = xo^oxiSi ... Sn-Xxn such that each Si is one of P, Ff or Fa-, a G Nx. As before, as denotes the sum of the parameters that occur in s.

By s = uRDv we denote sequences that use relations P, Df and Df, for b G N2. Given a sequence s ? uRFv, denote by w (s) the sequence of the form uRDv that

results from s by replacing all F<,s and F3xs

with Df,, and

D^,y respectively.

Suppose (u,v) G G^ y Recall that a */

w,ru\Jf?u\J f<

is an intransitive and irreflexive tree. Therefore, the following cases (which do not

necessarily exclude each other) are possible:

Case 1. If u = v then (u,v) G Ef-, ,, and the claim follows.

Case 2. There exists s = uRFv such that as < a(xj). But then we obtain a s + ?\ <

a(xj). So w(s) = uRDv and, since L(a) ? L(b), we have aw^ < b(xj).

Now, e2 < b(xj)

- aw(s), and, therefore, aw^ + e2 <

b(xj). This implies aw^ <

b(xj), and so we derive (u,v) G E<,y

Case 3. There exists s = ^PFw such that as < a(xj). Then swap v and w and

proceed as in Case 2.

Case A. There exists s = wPi7?; such that s contains an F< and at least one F<

occurs before any R in s and as < a(xj). Then aw^ <

b(xj), since L(a) ? L(b), and so (w, u) G ?5 v

Case 5. As in Case 4 but with w and v swapped. Then proceed as in Case 4.

Case 6. There exist s\ =

uoRFu, s2 ?

uqRFv and ?z5l + aSl < a(xj).

Then

aSx + <zJ2 + ei < a(xj). Again, using L(a) ? L(b), we obtain ow(il) + aw^Sl) + e2 <

b(xj), and so (u,v) G E<,y

Case 1. There exist si =

uoRFu, s2 =

uoRFv such that at least one of s\,s2

contains an F< which occurs before any R and aS] + aS2 < a(xj). Using L(a) ?

L(b), we obtain aw^) + aw(S2) <

b(xj) and, therefore, (u,v) G E<,,.

This finishes the proof of the claim.

Obviously, for any subterm p of z, we have

(/>?)6' =

(/)S2.

It remains to show by induction that, for any subterm p of z,

(pa)e<=(pb)e;-.

The interesting steps are p = V<x,/?o and p

= V-x'/?o- We only consider the former.

Suppose u tfz (pa)ei. Then there is v such that (u,v) G G<,_, and v ^ (po)e*

By the induction hypothesis, v ^ (Po)e^. By the claim above, (u,v) G E<,y

Therefore, u 0 (ph)e*2.

Suppose w 0 (y^)?2*. It follows that ? ? (/>b)@2, and so u i (pa)&l. But then,

by the induction hypothesis, we obtain u ? (pa)&l. H

In general, it is not known whether there is an elementary upper bound for

the complexity of this procedure. However, in some cases it can be considerably

simplified:

Corollary 26. (i) Suppose that Y consists of'pure' strict inequalities

{0 < X\,X\ < X2,X2 < X3, ... ,xn-\ < xn}

and z is an ^?^[V]-term with variables {x\,... ,xn}. Then z is satisfiable relative to

Y iffza is satisfiable, where a(x\) = 1 and, for \ a\X\,x3 > a2x2,... ,xn >

an-\xn^\},

where the a? > 1 are natural numbers, and x is an MET\V\-term with variables

{xi..... xn}. Then x is satisfiable relative to Y iff xa is satisfiable. where a(xi) = 1

and. inductively. a(x/+i) =

a? a(xz).

In both cases the satisfiability problem for x relative to Y is EXPTIME-complete.

Proof. In both cases the set My contains just one set of inequalities L and

L(a) = L. 3

?6. Conclusion. In this paper we have defined and investigated a new framework

for integrating qualitative and quantitative aspects of reasoning about metric spaces and their induced topologies. A number of interesting open problems arise within this framework if we (i) consider satisfiability of M3~ -formulas in various important classes of metric spaces and/or (ii) weaken or strengthen the expressive power of

Msr.

For example, it would be of interest to investigate the logic determined by metric spaces whose induced topological spaces are connected. We conjecture that this logic can be axiomatised by adding the connectivity axiom (2) to MT

and that results similar to those for MT can be obtained.

We have proved that L(R) is decidable. Does there exist a transparent axioma tisation of this logic? What is the computational complexity of L(R)?

Although the logic of M2 (and its subspaces) is undecidable, we do not know

whether it is recursively enumerable. Nor is it known what happens if we omit

the operators 3-a.

What is the computational complexity of satisfiability of MET\V\-izxnys in

metric spaces? In [26], we have investigated the (non-topological) metric language with op erators 3-a and 3>a ('somewhere outside the closed sphere of radius a') and

proved the decidability of satisfiability in arbitrary metric spaces. Is the satis

fiability problem for the extension of this language with topological (interior and closure) operators decidable as well?

The addition of nomin?is (atomic terms interpreted as singleton sets) would

make the language powerful enough for reasoning about concepts, similarities, and prototypes in combinations with, e.g., description logics [18, 28]. It is an

interesting open problem whether the resulting language is still decidable. Note

that the nomin?is would require new axioms like

0{n} =

{n}, 03<a{n} G 3^a{n},

where n is a nominal.

Acknowledgements. The work on this paper was partially supported by the U.K.

EPSRC research grants GR/S61966/01 and GR/S61973/01. Thanks are due to

Ivan Zakharyaschev for his help and comments. We are also grateful to the anony mous referee whose remarks and suggestions helped us to improve the paper.

REFERENCES

[1] M. AiELLO. J. van Benthem. and G. Bezhanishvili, Reasoning about space: the modal way. Journal of Logic and Computation, vol. 6 (2003). pp. 889-920.

[2] P. S. Alexandroff. Diskrete R?ume. Matematicheskii Sbovnik. vol. 2 (44) (1937). pp. 501--518.


[3] R. Alur, T. Feder, and T. Henzinger, The benefits of relaxing punctuality. Journal of the ACM.

vol.43 (1996), pp. 116-146.

[4] R. Alur and T.A. Henzinger, A really temporal logic. Journal of the ACM, vol. 41 (1994).

pp. 181-204.

[5] C. Areces, P. Blackburn, and M. Marx, The computational complexity of hybrid temporal logics,

Logic Journal of the IGPL, vol. 8 (2000), pp. 653-679.

[6] S. Artemov. J. Davoren, and A. Nerode. Modal logics and topological semantics for hybrid

systems, Technical Report MSI 97-05, Cornell University, 1997.

[7] B. Bennett, Modal logics for qualitative spatial reasoning. Logic Journal of the IGPL, vol. 4 ( 1996).

pp. 23-45.

[8] G. Bezhanishvili and M. Gehrke, A new proof of completeness of S4 with respect to the real line,

2002, Preprint PP-2002-06 of ILLC, University of Amsterdam.

[9] P. Blackburn, M. de Rijke, and Y. Venema, Modal logic, Cambridge University Press, 2001.

[10] E. Borger. E. Gr?del, and Yu. Gurevich. The classical decision problem, Perspectives in

Mathematical Logic, Springer. 1997.

[11] N. Bourbaki, General topology, part 1, Hermann, Paris and Addison-Wesley. 1966.

[12] A. Chagrov and M. Zakharyaschev. Modal logic, Oxford Logic Guides, vol. 35, Clarendon

Press, Oxford, 1997.

[13] P. Clote and R. Backofen, Computational molecular biology, John Wiley & Sons. 2000.

[14] A. Cohn and S. Hazarika, Qualitative spatial representation and reasoning: an overview, Funda

menta Informaticae, vol. 46 (2001), pp. 1-29.

[15] D. Dubois, H. Prade, F. Esteva. P. Garcia, and L. Godo, A logical approach to interpolation based on similarity relations, International Journal of Approximate Reasoning, vol. 17 (1997), pp. 1-36.

[16] M. Egenhofer and R. Franzosa, Point-set topological spatial relations. International Journal of

Geographical Information Systems, vol. 5 (1991), pp. 161-174.

[17] F. Esteva. P. Garcia, L. Godo, and R. Rodriguez, A modal account of similarity-based reason

ing, International Journal of Approximate Reasoning, vol. 16 (1997), pp. 235-260.

[18] P. G?rdenfors, Conceptual spaces. The MIT Press, 2000.

[19] P. G?rdenfors and M. A. Williams, Reasoning about categories in conceptual spaces. Proceedings

of the 7th international joint conference on artificial intelligence (IJCAI), Morgan Kaufmann, 2001,

pp. 385-392.

[20] R. Goldblatt, Mathematics of modality. CSLI Lecture Notes, no. 43, CSLI Publications,

Stanford. 1993.

[21] M. Grigni, D. Papadias, and Ch. Papadimitriou, Topological inference. Proceedings of the 14th

international joint conference on artificial intelligence (IJCAI), Morgan Kaufmann. 1995. pp. 901-906.

[22] D. Harel. D. Kozen, and J. Tiuryn, Dynamic logic. MIT Press, 2000.

[23] Y Hirshfeld and A. Rabinovich, Quantitative temporal logic. Proceedings of computer science

logic 1999, Springer. 1999. pp. 172-187.

[24] D. Huttenlocher, G. Klanderman, and W Rucklidge. Comparing images using the Hausdorff distance, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15 (1993). pp. 850-863.

[25] P. Kremer, G. Mints, and V. Rybakov, Axiomatizing the next-interior fragment of dynamic

topological logic, The Bulletin of Symbolic Logic, vol. 3 (1997), pp. 376-377.

[26] O. Kutz, H. Sturm, N.-Y Suzuki, F. Wolter, and M. Zakharyaschev, Logics of metric spaces, ACM Transactions on Computational Logic, vol. 4 (2003). pp. 260-294.

[27] R. Ladner. The computational complexity of provability in systems of modal logic. SI A M Journal

on Computing, vol. 6 (1977), pp. 467-480.

[28] C. Lutz, F. Wolter, and M. Zakharyaschev, A tableau algorithm for reasoning about concepts and similarity, Automated reasoning with analytic tableaux and related methods (M. C. Mayer and F. Pirri,

editors). LNCS. vol. 2796, Springer, 2003.

[29] J.C.C. McKinsey and A. Tarski, The algebra of topology. Annals of Mathematics, vol. 45 (1944).

pp. 141-191.

[30] G. Mints, A completeness proof for propositional S4 in Cantor space. Logic at work (E. Orlowska.

editor), Springer, 1999. pp. 79-88.

[31] J. Renz and B. Nebel, On the complexity of qualitative spatial reasoning. Artificial Intelligence, vol. 108 (1999), pp. 69-123.




[32] V. Shehtman, "Everywhere" and "Here", Journal of Applied Non-classical logic, vol. 9 (1999),

pp. 369-379.

[33] E. Spaan, Complexity of modal logics, Ph.D. thesis, Department of Mathematics and Computer

Science, University of Amsterdam, 1993.

[34] A. Tarski, Der Aussagenkalk?l und die Topologie, Fundamenta Mathematicae, vol. 31 (1938),

pp. 103-134.

[35] F. Wolter and M. Zakharyaschev, Reasoning about distances, Proceedings of the 18th interna

tional joint conference on artificial intelligence (IJCAI), Morgan Kaufmann, 2003, pp. 1275-1280.

[36] M. Worboys, GIS: A computational perspective, Taylor & Francis, 1995.

DEPARTMENT OF COMPUTER SCIENCE

UNIVERSITY OF LIVERPOOL

LIVERPOOL L69 7ZF, UK

E-mail: [email protected] URL: http://www.csc.liv.ac.uk/Trank

DEPARTMENT OF COMPUTER SCIENCE

KING'S COLLEGE LONDON

STRAND, LONDON WC2R 2LS, UK

E-mail: [email protected] URL: http://www.dcs.kcl.ac.uk/staff/mz



a logic for metric and topology

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