a four-valued semantics for terminological logics

ARTIFICIAL INTELLIGENCE 319

A Four-Valued Semantics for Terminological Logics*

P e t e r F . P a t e l - S c h n e i d e r * *

Schlumberger Palo Alto Research, 3340 Hillview Avenue, Palo Alto, CA 94304, U.S.A.

Recommended by Alex Borgida

ABSTRACT

An intuitive four-valued semantics can be used to develop expressively powerful terminological logics which have tractable subsumption. The subsumptions supported by the logic are a type o f "'structural" subsumption, where each structural component of one concept must have an analogue in the other concept. Structural subsumption captures an important set of subsumptions, similar to the subsumptions computed in KL-ONE and NIKL. Thus the trade-off between expressive power and computational tractability which plagues terminological logics based on standard, two-valued semantics can be avoided while still retaining a useful and semantically supported set o f subsumptions.

1. Introduction

Terminological logics formalize the notion of frames--a notion present in many current knowledge representation systems--as structured types, often called concepts. These logics include a set of syntactic constructs that form concepts, and other, related, notions such as roles, and are based on formal model- theoretic semantics which provide firm definitions for the syntactic constructs of the logic. Terminological logics are part of K L - O N E [5], NIKL [14, 20], KRYPTON [3, 4], KANDOR [16], BACK [22], and LOOM [13].

The allowable concepts vary between different terminological logics but concepts generally are the conjunction of a set of more general concepts and a set of restrictions on the attributes of instances of the concept. Such concepts can be loosely rendered as noun phrases such as

* Portions of this paper appear in preliminary form in papers presented at the Fifth National Conference on Artificial Intelligence, Philadelphia, Pennsylvania, August 1986 and the Seventh National Conference on Artificial Intelligence, St. Paul, Minnesota, August 1988.

** Current address: AI Principles Research Department, AT&T Bell Labs, 600 Mountain Ave., Murray Hill, NJ 07974, U.S.A.

Artificial Intelligence 38 (1989) 319-351 0004-3702/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

320 P.F. PATEL-SCHNEIDER

a student and a female whose department is a computer science department, and who has at least three enrolled courses, each of which is a graduate course whose department is an engineering department.

The most important operation in terminological logics is determining if one concept subsumes another.

Informally, one concept subsumes another if the first is more general than the other. A formal definition of subsumption is part of the semantics of every terminological logic. This definition is generally equivalent to a requirement that, in any situation, all instances of the subsumed concept must also be instances of the subsuming concept.

For example, the concept

person with at least two children

subsumes the concept

person with at least three children who are lawyers

in standard terminological logics, because, in the standard semantics for terminological logics, all instances of the second concept must also be instances of the first.

Subsumption is difficult to compute correctly in reasonably expressive terminological logics. This problem first came to light during the formalization of part of KL-ONE where it was discovered that the subsumption algorithm in KL-ONE was incomplete [21]. Levesque and Brachman [12] have shown that computing subsumption in a very simple terminological logic based on a standard two-valued semantics is NP-hard, and Nebel [15] has demonstrated a similar result for a different, expressively limited terminological logic. There- fore determining subsumption in resonably expressive terminological logics is very difficult, at least in the worst case. For more expressive terminological logics, such as the terminological logic of NIKL, subsumption is only semi- decidable [17].

Thus there is an unfortunate trade-off between expressive power and computational tractability in terminological logics based on standard semantics; if subsumption is to be tractable, at least in the worst case, then the logic must be expressively weak- - too weak to be usable in knowledge-based systems) The intractability of subsumption in standard terminological logics is a severe problem when using complete terminological reasoners in knowledge-based systems, since computing subsumption is the most important operation in terminological reasoners and will be performed often 2, and it is not desirable to

~See [12] for a general discussion of the trade-off between expressive power and tractability in knowledge representation.

2In the current design of KRVPXON, new concepts are created as part of resolution steps, leading to a great number of new subsumption questions being asked during deductions.

A FOUR-VALUED SEMANTICS FOR TERMINOLOGICAL LOGICS 321

have components of knowledge-based systems that may take an extremely long time to compute commonly used operations.

There are several ways to partially solve this problem. The first partial solution is to simply ignore the problem. The examples used by Levesque and Brachman to show that computing subsumption in their terminological logic is intractable are not likely to occur in actual knowledge bases. Perhaps computing subsumption will be reasonably fast in actual knowledge bases. This sort of solution occasionally works well but will fail for more expressive terminological logics, such as NIIO~'S, where subsumption is only semi-decidable, and where the most complete method of computing subsumption is equivalent to transla- tion into first-order formulae followed by the employment of a theorem proving program.

The second partial solution is to limit the expressive power of the terminological logic. The problem with this solution is that expressive power must be very severely limited to achieve computational tractability. Nevertheless, this solution is used in the implemented version of KRYPTON [4], which has a very limited terminological logic in which subsumption in easy to compute.

A third solution is to provide only a partial subsumption algorithm, one which does not discover all subsumption relationships, only an easy-to-calcu- late subset of them. This is the solution used in KL-ONE, NIKL, and BACK, where only simple subsumption relationships are computed. In this solution the algorithm for computing subsumption is no longer fully defined by the semantics of the terminological logic. There is little basis for deciding exactly which subsumption relationships to discover, except the reasons of expediency and tractability. The subsumption relationships computed in this solution have little relationship to the semantics of the terminological logic, and it is impossible to semantically characterize which subsumptions are performed by such algorithms.

A combination of all three solutions is used in KANDOR. In KANDOR the terminological logic is more powerful than that of KRYPTON, but still limited expressively. The subsumption algorithm of KANDOR is incomplete [15], but is still ambitious, discovering more subsumptions than the subsumption algorithms of KL-ONE and NIKL, and its worst-case complexity is exponential in the size of numbers appearing in concepts, but otherwise tractable. Moreover, the algorithm is quite fast in normal circumstances, where the numbers appearing in concepts are small. Also, KANDOR, like many other similar systems, keeps track of subsumption relationships in a concept taxonomy so that each subsumption question need only be asked once. In this way, the worst case behavior of the subsumption problem in KANDOR is rendered less harmful .3

3However, ARGON [19], a query language using KANDOR to represent its knowledge, builds concepts for each query, thus leading to a large number of subsumption questions being asked. Therefore, the performance of subsumption in KANDOR is of vital importance to ARGON.


Given that none of these solutions is satisfactory for terminological logics, is there a solution to the problem? Unfortunately there is no solution if the standard semantics for terminological logics is to be strictly followed. However, there is a way to legitimize the third solution, by using a weaker semantics for terminological logics, one that supports fewer subsumption relationships and which has tractable subsumption.

Weaker semantics that provide tractable reasoning operations have also been proposed for assertional knowledge representation systems. (An assertional knowledge representation system is concerned with the representation of assertions or facts instead of concepts and roles.) Levesque [11] suggested that propositional tautological entailment, a weak version of propositional relevance logic, could be used as the basis of a simple assertional knowledge representation system. The advantage of using propositional tautological entailment is that computing inference in it is computationally tractable if formulae are kept in conjunctive normal form. This work was later extended [18] to produce a decidable variant of first-order tautological entailment that could be used as the basis of a knowledge representation system.4 Both of these efforts were based on a four-valued model-theoretic semantics derived from the four truth values of tautological entailment [1,2] where propositions can be assigned not only true or false, but also neither true nor false, and also both true and false.

Of course, there are problems with using different semantics. The standard two-valued Tarskian semantics for logic, which serves as the basis for the standard semantics of terminological logics, has been in existence for quite some time now. It is generally agreed that this semantics captures our intuitions of how the world actually is and that the inferences sanctioned by it are a reasonable set of inferences. Any other semantics is liable to be less intuitive than this standard semantics and perhaps may be so counter-intuitive that it is useless for knowledge representation purposes. The goal of this endeavor is to produce a semantics for terminological logics that is still intuitive but which also has tractable subsumption.

2. Syntax and Intuitive Meaning

The terminological logic developed here has two major syntactic types - - concepts and roles----corresponding to the frames and slots of most frame-based knowledge representation systems. As in these other systems, concepts represent descriptions of related entities and roles describe relations between these entities. The intuitive meanings of the various constructs in the language are derived from the intuitive meanings of the basic constructs in typical frame-

4These developments are along the general line of Frisch [7] who argues that any Artificial Intelligence program--knowledge representation systems in particular--should be the complete implementation of some formalism with model-theoretic semantics.


based knowledge representation systems. The logic here is closely related to the terminological logics of KL-ONE, mKL, KRYPTON, KANDOR, and BACK.

Concepts can be formed in the following ways:

<concept> ::= <atomic concept> ] (and (concept> +) [ (all <role> (concept>) [ (atleast ( minimum > ( role ) ) [ (atmost (maximum> (role>) [ (fillers (role> <concept> +) [

<minimum> ::= (positive integer> (maximum>::= (non-negative integer>.

The construct (and <concept> +) is a conjunction construct. Informally, an entity belongs to (and C, C 2) if it belongs to both C, and C2. The construct (all (role> (concept)) is a role restriction construct. Informally, an entity belongs to (all Iq C) if, for every entity, either it is not related to that entity by Iq, or the entity belongs to C.

The constructs (atleast (minimum) (role)) and (atmost (maximum) (role)) are number restriction constructs. Informally, an entity belongs to (atleast n Iq) if it is related to at least n distinct entities by Iq. If n is 1, this reduces to a role filler existence construct. An entity belongs to (atmost n It) if it is related to at most n distinct entities by lq.

Finally, an entity belongs to (fillers R C , . . . Cn) if, for each of the Ci, the entity is related by F! to at least one entity that belongs to Ci, and, moreover, these entities are all distinct. 5

Roles can be formed in the following ways:

(role) :: = ( atomic role ) [ (and (role) +) [ (restrict ( role ) ( concept ))

The construct (and <role> +) is a conjunction construct, similar to (and ( concept ) +). The construct (restrict ( role ) ( concept )) is a restriction construct. Two entities are related by (restrict FI C) if they are related by Ft and the second is also an instance of (3.

This language is quite expressive--much more expressive than the language for which subsumption was found to be intractable by Brachman and Levesque [12]. Its expressive power is closer to the expressive power of KL-ONE and NIKL, but it is lacking some of their constructs, such as role chains and structural descriptions. The language appears to have sufficient expressive power to be useful in a fairly wide variety of domains.

5This construction would be more useful if objects were used instead of concepts. However, adding objects with reasonable semantics makes reasoning much harder, and probably destroys the computational tractability of subsumption in this logic.


The concepts described in the introduction can be expressed in this language. For example, person with at least two children can be expressed as (and person (atleast 2 child)), person with at least three children who are lawyers can be expressed as (and person (atleast 3 (restrict child lawyer))), and

a student and a female whose department is a computer science department, and who has at least three enrolled courses, each of which is a graduate course whose department is an engineering department.

can be expressed as

(and student female (atmost 1 department) (fillers department computer-science-department) (atleast 3 (restrict enrolled-course

(and graduate-course (fillers department engineering-department))))).

3. Formal Semantics

The formal semantics of the logic is a four-valued model-theoretic semantics with a four-valued identity. The basic ideas behind the semantics are similar to the ideas behind other denotational semantics; it is based on semantic structures or possible worlds, containing a set of entities and a mapping from syntactic constructs--concepts and roles-- into their meaning in the semantic structure. The four truth values in the semantics are the subsets of {t, f } , i.e., {t), { f ) , {}, and { t , f } . The best way of viewing these truth values is as states of knowledge (or belief) of a reasoning system about some proposition. Under this interpretation, if the truth value of a proposition contains t, then the systems knows that---or believes that, or has evidence to the effect tha t - - the proposition is true. Similarly, if the truth value of a proposition contains { f} , then the system knows that the proposition is false. The truth value {} then corresponds to lack of knowledge, and the truth value { t , f } corresponds to inconsistent knowledge. This view of semantic structures makes them into states of knowledge of the entire system, and not models of the world. (For more on this view of the four truth values, see [1].)

Note that this four-valued semantics is different from the treatment of knowledge as a modal operator. In particular, in the four-valued semantics it is possible to have "inconsistent" knowledge about some proposition without being totally inconsistent. This property, which is shared by other relevance logics, is touted as one of the advantages of relevance logics, especially when modeling states of knowledge.

Definition 1. A semantic structure is a triple, (D, V, I ) , where D is a set of


entities, V is a function that takes concepts and roles into their extension, and I is an identity relationship over D. The extension o f a concept is a mapping f rom D to 2 {''f}. (The set 2 {t'f} is the powerset o f {t, f } , containing the four (truth) values {}, {t}, { f} , {t, f } . ) The extension o f a concept is thus a four-valued characteristic function--not a two-valued characteristic function. Similarly, the extension of a role is a mapping f rom D x D to 2 {'d}. Finally, the identity relationship is a mapping f rom D × D to 2 {t'f}, with the properties that I(d, d) = {t} and I(d,e) = I(e ,d) , for all d,e E D.

The extension function can best be understood by viewing the four-valued extensions as two separate two-valued extens ions-- the positive extension and the negative extension. The positive extension of a concept, G, is the set of domain elements that are known to belong to the concept- -def ined as {d ~ D : t ~ V[C](d)}. The negative extension of C is the set of domain elements that are known not to belong to the concept--xlefined as {d ~ D : f E V[C](d)}. Unlike the case in two-valued semantics, these two sets need not be comple- ments of each o the r - - t he re may be domain elements that are members of neither of these sets, and also domain elements that are members of both of these sets. The positive and negative extensions of a role are defined similarly.

Domain elements that are members of neither set then are not known to belong to the concept and are not known not to belong to the concept. This is a perfectly reasonable state for a system that is not a perfect reasoner or does not have complete information. Domain elements which are in both the positive and negative extension of a concept can be thought of as inconsistent with respect to that concept in that there is evidence to indicate that they are in the extension of the concept and, at the same time, not in the extension of the concept (but such domain elements need not be inconsistent with respect to other concepts). This is a slightly harder state to rationalize but can be considered a possibility in the light of inconsistent information.

The identity relationship of the semantics is also four-valued. This relationship is easier to understand if viewed in a manner similar to the positive and negative view of extensions. Under this view, if t E I(d, e) then d and e are known to be identical, and if f E I(d, e) then d and e are known not to be identical, i.e., known to be distinct. As above, it is possible that two domain elements are neither known to be identical nor known not to be identical, and it is also possible that two domain elements are both known to be identical and known not to be identical. These four possibilities make domain elements more like descriptions than individuals, which is consistent with the view of semantic structures as states of knowledge.

This four-valued identity does not have several of the propert ies of equality. First, it is not transitive, that is, t E I(d, e) and t E I(e, f ) and do not imply t ~ I(d, f ) . This is a major depar ture from the propert ies expected even of a four-valued equality. However , requiring that identity be transitive makes reasoning in the logic much harder , and relaxing this requirement is reasonable


in limited reasoning systems. Similarly, if t ~ l(d, e), then d and e can have different properties. Again, adding this property would have severe computational consequences, and relaxing it is reasonable in limited reasoning systems.

Because of the four-valued identity, sets of domain elements do not have unique cardinalities in the semantics. Instead, the semantics defines both a minimum and a maximum cardinality. The minimum cardinality of a set, X, is defined to be the size of its largest subset for which all elements are known to be distinct,

mincard(X) = max{] Y] : Y C_ X /x f E l(d, e), Vd,e E Y, d ~ e}.

(If the minimum cardinality of a set is n, then there is evidence to believe that the set has n pairwise distinct elements.) Similarly the maximum cardinality of a set, X, is defined to be the size of its largest subset for which no two elements are known to be identical,

maxcard(X) = max{I YI: Y c x / x t ~ l ( d , e), Vd,e E Y, d ¢ e}.

(If the maximum cardinality of a set is n, then the set has at most n elements that are all pairwise not known to be identical.)

It is possible for the maximum and minimum cardinality of a set to be different. For example, if no identity relationships, positive or negative, are known, then the maximum cardinality of a non-empty set is its standard cardinality and its minimum cardinality is 1. It is also possible for the maximum cardinality of a set to be less than the minimum cardinality of a set. For example, if the identity relationship is total---everything is both identical and distinct from everything else-- then the maximum cardinality of a set is 1 and its minimum cardinality is its standard cardinality.

The extensions of non-atomic concepts and roles have to meet certain restrictions, designed so that the formal semantics respects the informal meaning of concepts and roles. For example, the positive extension of (and C~ (32) must be the intersection of the positive extension of (31 and (32 and its negative extension must be the union of their negative extensions, thus formalizing the intuitive notion of conjunction.

The conditions for concepts are as follows:

t E V[(and C, . . . C,,)](d) f E V[(and C, . . . C,,)](d) t~E V[(all R C)](d) f E V[(all R C)](d) t ~E V[(atleast m R)](d) f E V[(atleast m R)](d) t E V[(atmost m R)](d)

iff for each i,t ~ V[Ci](d ) iff for some i , f E V[Ci](d) iff Ve f E V[R](d, e) or t ~ V[C](e) iff : l e t E V[R](d, e) and f E V[C](e) iff mincard{e : t E V[R](d, e)} ~>m iff maxcard{e : f ~ V [ R ] ( d , e) } < m iff maxcard{e : f ~ V [ R ] ( d , e)} ~<m


f E V[(atmost m R)](d) iff mincard{e : t E V[R](d, e)} > m t E V[(fillers RC 1 . . . C,)](d) iff

3el . . . e n (Vi ¢ j ei ¢ e j A f E I(ei ,e j ) ) /~ (Vi t E VIRI(d, e,) ^ V[C,](e,))

f ~ V[(fillers R C 1 . . . C,,)](d) iff Vej . . . e,, ( 3 i ¢ j ei = ej v t E l (e i ,e j ) ) v

(3 i f E V[R](d, e,) v f E V[Ci](e~))

The conditions for roles are as follows:

t~E V[(and R, . . . R.)](d, e) f{E V[(and R, . . . R.)](d, e) t E V[(restrict R C)](d. e) f E V[(restrict a C)](d, e)

iff for each i,t E V[Ri](d, e) iff for some i , f E V [ R i ] ( d , e) iff t ~ V[R](d, e) and t E V[C](e) iff f E V[R](d, e) or f E V[C](e)

The final part of the semantics is the definition of subsumption:

Definition 2. One concept or role is subsumed by another, written G ~ C', i f the posit ive extension o f the first is always a subset o f the positive extension o f the second and the negative extension o f the second is always a subset o f the negative extension o f the first.

4. Discussion of the Semantics

The semantics defined here has a close relationship to standard, two-valued semantics for terminological logics. (See [12] for a simple example of this sort of semantics.)

This semantics encompasses a two-valued semantics. Define a mode l as a semantic structure where (1) for every concept C, the positive and negative extensions of C are disjoint and together exhaust the domain of the model; (2) the positive and negative extensions of roles are also disjoint and exhaustive; and (3) the identity relationship is equality. In such semantic structures the above semantics, including the definition of subsumption, reduces to a standard two-valued semantics for terminological logics. Because of this inclusion relationship, all reasoning in this logic is sound with respect to standard terminological logics.

The conditions for concepts and roles, and also the definitions of cardinality and subsumption, are essentially a reinterpretation, in a four-valued setting, of the standard two-valued conditions and definitions. There is nothing changed besides what is needed to get from two truth values to four truth values, apart from the weakening of identity. Thus the semantics is closely related to intuitions about the meanings of concepts and roles.

The changes in the semantics--going from two to four truth values and the weakened, four-valued ident i ty--are reasonable for systems with limited reasoning power. Such systems do not have total information, thus the


presence of truth-value gaps, and also cannot resolve inconsistencies, thus allowing for inconsistent situations. The four truth values of the logic have also been previously used to develop limited assertional reasoners [11, 18].

The set of subsumptions supported by this logic forms an interesting and useful set. Since subsumption is sound with respect to standard terminological logics, if one concept subsumes another in this logic then it will also do so in a standard, two-valued terminological logic. Soundness of subsumption is an important requirement if the semantics is to capture some of the intuitive ideas underlying terminological logics.

The sort of subsumptions that are valid in this logic are the simple ones, such as

(and person (atleast 2 child))

subsuming

(and person (atleast 3 (restrict child lawyer))),

i.e., a person with at least three children who are lawyers must also be a person with at least two children,

(and person (atmost 4 (restrict child doctor)))

subsuming

(and person female (almost 3 child)),

i.e., a female person with at most three children must also be a person with at most four children who are doctors, and

(atleast 2 (restrict child doctor))

subsuming

(fillers child (and doctor Susan) (and doctor lawyer) doctor),

i.e., a person whose children include a doctor named Susan, a different doctor who is also a lawyer, and a third doctor must also be a person with at least two children who are doctors. More complicated valid subsumptions, such as

(atmost 0 (restrict R (and (atleast 3 $1) (atleast 4 $2))))

subsuming

(all (restrict R (atleast 2 $1) ) (atmost 2) $2) )

can be reduced to simpler subsumptions. (Here the latter concept is equivalent to (almost 0 (restrict R (and (atleast 2 $1) (atleast 3 $2)))).) As these examples show, the valid subsumption relationships are not trivial, and include at least some interesting subsumption relationships.

Subsumption relationships involving modus ponens are not valid here.


For example,

(and person (all friend doctor) (all (restrict friend doctor) (atleast 1 speciality)))

is not subsumed by

(and person (all friend (atleast 1 speciality))),

because in four-valued semantic structures it is possible that some friend might both be a doctor and not be a doctor, as well as not specializing.

Also, subsumption relationships that require reasoning from the law of the excluded middle for identity are not valid here. For example 6, in a two-valued terminological logic

(and (atleast 1 (restrict child lawyer)) (atleast 1 (restrict child doctor)))

would be subsumed by

(or (atleast 2 child) (atleast 1 (restrict child (and lawyer doctor)))),

because either the child that is a lawyer is different from the child that is a doctor, in which case there are two children, or they are identical, in which case there is one child which is both a doctor or a lawyer. In the four-valued logic this is not a valid subsumption because it is possible to be uncertain about whether the doctor and the lawyer are identical. The change from two-valued to four-valued identity was made to invalidate this kind of reasoning, which causes severe computat ional problems.

Finally, subsumption relationships that depend on concepts used in (fillers R C 1 . . . C,) constructs mapping into single domain elements are not valid here. For example,

(fillers (and child friend) doctor)

does not subsume

(and (fillers child doctor) (fillers friend doctor))

because the domain elements that satisfy the two conjuncts in the second concept need not be the same. The lack of this sort of subsumption shows how the use of concepts in (fillers S C ] . . . Cn) is less powerful than the use of objects would be.

6This example cannot be expressed in the logic described here because it includes a disjunction operator. However, a more complicated example which embeds this one c a n be expressed in the logic.


5. Computing Subsumption

Subsumption in this logic is weaker than subsumption in logics using the standard semantics, however this does not imply that subsumption is easy to compute here. Fortunately, subsumption is tractable in this logic. A direct subsumption algorithm for the full form of the logic is too long to fit in this paper and too complicated to be easily understood, so an indirect argument has to be used to show its tractability. This is done by converting concepts and roles to a canonical form, giving a subsumption algorithm for concepts and roles in this canonical form, and then showing how this algorithm can be converted into a tractable subsumption algorithm for concepts and roles in arbitrary form.

Concepts and roles in canonical form take the following form:

(concept) ::= (and (primary)*) (primary) ::= (atomic concept) [

~(atomic concept) ] (atleast (minimum) (role)) I -~(atleast (minimum) (role)) [ (fiJlers (and (atomic role)*) (concept) +) [ 7(fillers (and (atomic role) +) (concept) +)

(role) ::= (restrict (and (atomic role) +) (concept)) (minimum) ::= (positive integer)

This canonical form uses the negation operator, -7, which is a classical negation operator with the following semantics:

t E V[~C](d) iff f ~ V[C](d) f E V[~C](d) iff t ~ V[Cl(d)

The conversion to canonical form can be done by using the following equivalences, which do not affect subsumption relationships:

(1) commutativity and associativity for conjunctions of concepts (2) C ---- (and C) (3) (all R (and C I C2))~-(and (all R Cj) (all R C2) ) (4) (all R E)--= 7(atleast 1 (restrict R -~E)) (5) (atmost n R) =-- -q (atleast n + 1 R) (6) (fillers (restrict R C) C ~ . . . C,,) ~(fillers R (and C C 1 ) . . . (and C C,)) (7) commutativity and associativity for conjunctions of roles (8) R--= (and R) (9) R --= (restrict R (and))

(10) (and (restrict R~ C) R2)~ (restrict (and R~ R2) C) (11) (restrict (restrict R Ct) C2)~-(restrict R (and C 1 C2) )

It is often convenient to consider a canonical form concept as a set of primaries and to consider the two parts of a canonical form role as a set of atomic roles and a set of primaries.


Once concepts and roles are in canonical form then the following characterization of subsumption is both sound and complete.

Theorem 1. Let G and C' be canonical form concepts. Then C is subsumed by (3' ((3=:>C') iff for each D' in (3', there exists D in (3 such that

(1) if D' is o f the form A , with A an atomic concept, then D = D', (2) if D' is o f the form --hA, with A an atomic concept, then D = D', (3) i fD' is o f the form (atleast m R'), then either D is o f the form (atleast n R),

with n>~m and R ~ R ' , or D is o f the form (fillers S (31. . .Gn) , with I{J : (restrict S Cj) ~, R'}I 1> m,

(4) if D' is o f the form -7(atleast m R'), then either D is o f the form 7(atleast n R), with m >1 n and R ' ~ R, or D is o f the form 7(fillers S (31 • • • Cn), with m >i n and R' =), (restrict S (3j), for all j,

(5) if D' is o f the form (fillers S' (31 - - - (3"), then either D is o f the form (fillers S (31 • • • (3n), with S' C S and for each i there exists a distinct j such that (3j =z), G'i, or D is of the form (atleast n R), with n >1 m and R :=). (restrict S' (31), for each i,

(6) if D' is o f the form 7(fillers S' (3'1... (3,',,), then either D is of the form -n(fillers $ (31 . . • (3,), with $ C_ 5' and for each j there exists a distinct i such that (3~ ~ C j , or D is o f the form 7(atleast n R), with I{i : (restrict S' C I ) ~ R } I ~>n.

Let R = (restrict S C) and R'= (restrict S' C') be canonical form roles. Then R:::~ R' iffS'C_S and C~,C'.

An algorithm for subsumption for canonical form concepts and roles can easily be derived from this characterization of subsumption. If the effect of determining the existence of matchings in the case of (fillers 9 ' C I . . . Cn)=:), (fillers 5 ' C' 1 . . . (3") and its converse is ignored, this algorithm can easily be determined to take, at worst, time proport ional to the product of the sizes of the two concepts or roles. Determining the existence of matchings takes t ime proport ional to n m V ~ + m (see [6, pp. 120-121]), and thus in- creases the time complexity of the algorithm to the product of the sizes of the two concepts or roles times the square root of the sum of the sizes of the two concepts or roles.

Thus subsumption of canonical form concepts and roles is tractable. For all but the most pathological examples, the extra theoretical complexity added by the necessity of determining matchings will not be noticed, and the algorithm will effectively run in time proport ional to the product of the sizes of the two concepts or roles.

However , the process of converting concepts and roles to canonical form can exponentially increase their size, and thus the tractability of subsumption on arbitrary form concepts and roles has not yet been demonstrated.

Two modifications are needed to produce a tractable algorithm for subsumption. First, the conversion of concepts and roles to canonical form must be


done by means of structure sharing. If this is done the "size" of the canonical form of a concept or ro le - -no t the length of its printed form but the size of the data structure--will be proportional to the size of the original concept or role, and the canonicalization can be done in linear time. Second, the subsumption algorithm has to be changed so as not to redo computations. This can be done by storing previously performed subsumption tests at the appropriate places in the canonical form of the concept or role, and querying these results when applicable. The obvious method of storing and querying the cached subsumption tests results in a subsumption algorithm that runs in the same order as the previous subsumption algorithm for canonical form concepts or roles, but based on the "size" of the canonicalized concepts or roles, and not on their expanded size. Thus the entire subsumption process is of the same order as subsumption for canonical form concepts or roles, essentially the product of the sizes of the two concepts or roles. This makes the subsumption relationship usable as the basis of a tractable, object-based knowledge representation system.

The algorithm for subsumption shows that subsumption in the logic developed here is indeed a sort of "structural" subsumption where each structural component of one concept (or role) must have an analogue in the o the r ) This simple characterization of the subsumption relationship makes it easy to determine by hand which subsumptions are valid and which are not. The subsumption relationship is also very close to the subsumption relationships computed in KL-ONE and N~r<L.

There are several described subsumption algorithms for KL-ONE and r~KL. The best description of a subsumption algorithm for either of these systems is by Schmolze and Israel [21]. Their algorithm is the same as the algorithm developed here, when they are both restricted to the common portion of their respective languages. Recent versions of NIKL have a stronger classification algorithm [10] which performs some chaining between roles, and thus goes beyond structural subsumption. However, the core of this stronger algorithm is still based on structural subsumption. This close correspondence indicates that the subsumption relationships of the logic developed here form a useful set, and, moreover, provides a way of semantically justifying the incomplete subsumption algorithms of KL-ONE and NIKL.

6. Summary

What has been gained from this new semantics for terminological logics? First of all, the semantics is a reasonable semantics, especially when considering systems with limited reasoning capabilities. Second, subsumption in this seman-

7Structural subsumption satisfies the "no-chaining" property of Frisch [7, 8], making structural subsumption a sort of knowledge retrieval.


tics is easy to compute. Also, the valid subsumption relationships form an interesting set---one that includes the easy subsumptions and leaves out the less obvious ones. This set corresponds closely to the set of subsumption relationships computed in KL-ONE and NIKL, lending a degree of credence to that set. Third, the logic used here contains most of the constructs of the languages of KL-ONE and YlKe, and appears to be useful for representing a fairly wide variety of domains, thus showing that tractable subsumption is retained even with expressively powerful languages.

However, there are some problems with the semantics. First, the semantics is not as intuitive as the two-valued semantics. This is a problem with all alternative semantics but the four-valued semantics given here is still a reasonable semantics, especially for limited reasoners. Second, subsumption in this semantics gets only the very easy cases, leaving many that might be important, such as those involving a single application of modus ponens. These seem to be unavoidable problems if a uniform, simple semantics with a fast subsumption algorithm is required.

The most important point about this new semantics is that it forms a principled way to defuse the trade-off between expressive power and computational complexity. It justifies a limited set of subsumption relationships that are easy to compute and, moreover, captures an interesting subset of the standard subsumption relationships. This is not a total solution, because no total solutions are possible; however, this semantics does form an important step towards a principled, computationally tractable yet expressively powerful, knowledge representation system, and thus serves to alleviate the computational problems of terminological logics first reported by Brachman and Levesque.

A. Canonical Form Proofs

This appendix contains the proofs of the soundness and completeness of the subsumption algorithm for concepts and roles.

A.I . Canonicalization of concepts and roles

The first step of this proof is to show that the conversion to canonical form preserves subsumption relationships.

Definition 3. L e t G 1 and C 2 be concepts (or roles). Then C 1 =---C 2 i f f V[G1] = V[C2] in all semantic structures.

Theorem 2. / fC1 ~-D1 and C2 ~=D2, then ClEE~C 2 i f f D l ~ D 2.

Proof. Trivial.


Theorem 3. The following are valid equivalences: (1) (and C l C2)=-(and 0 2 Ci) (2) (and Ct (and C 2 C3)) --- (and (and C 1 C2) C3) (3) C =- (and C) (4) (all R (and C I C2))~-(and (all R Ci) (all R C2) ) (5) (all R E)=---q(atleast 1 (restrict R -qE)) (6) (atmost n R) --=-q(atleast n + 1 R) (7) (fillers (restrict R C) C I . . . C,,)--= (fillers R (and C C I ) . . . (and C C,,)) (8) (and R~ R2)~-(and R 2 R~) (9) (and R l (and R 2 R3) ) =--(and (and R I R2) R3)

(10) R ~ (and R) (11) R ~- (restrict R (and)) (12) (and (restrict R l C) R2)-= (restrict (and R~ R2) C) (13) (restrict (restrict R C1) C2)=-(restrict R (and C 1 C2) )

Proof. (1)

(2)

(3)

(4)

(5)

For any semantic structure and for any d and e in its domain: Trivially, V[(and C 1 02) } = V[(and C 2 C~)].

Trivial ly, V[(and C 1 (and C 2 C~))] = V[(and (and C l C2) C3)].

Trivially, V[C] = V[(and C)].

Here, t E V[(all R (and C, C2))](d ) iff Ve T~V[R](d, e) v tE V[(and C, C2)](e ) iff Ve ( f ~ V[R](d, e) v t E V[C,](e)) ^ ( f ~ V[R](d, e) v t E V[C2](e)) iff (Ve f e e V[R](d, e) v t E V[C,](e))/x

( r e f E V[RI(d, e) v t E V[C2l(e)) iff t@ V[(and (all R C,) (all R C2))](d ). Also, f @ V[(all a (and C, C2))](d ) iff 3e t E V[R](d, e)/x f ~ V[(and C~ C2)](e ) iff 3e ( t E V[R](d, e)/x t ¢ V[C l(e)) v ( t e V[R](d, e) A t e V[C2](e)) iff ( 3 e t E V[RJ(d, e) A t E V[C,](e)) v

(3e t ~ V[R](d, e) A t E VIC2I(e)) i ff t~E V[(and (all R C,) (all R C2))](d ).

Here, t E V[(all R E)](d) iff Ve f¢E V[R](d, e) v t E V[C](e) iff Ve f E V[(restrict R -TC)](d, e) iff maxcard{e : f C V[(restrict R -1C)](d, e)} = 0 i f f f C V[(atleast 1 (restrict R -1E))](d) iff t E V[-7(atleast 1 (restrict R ~E))](d). Also, f E V[(all R E)](d) iff Be t E V[R](d, e)/x f E V[C](e) iff ::let E V[(restrict R -1C)](d, e) iff mincard{e : t C V[(restrict R -1C)](d, e)} > 0 iff t E V[(atleast 1 (restrict R -~E))](d) iff f ¢ V[-7(atleast I (restrict R ~E))](d).


(6) Here, t E V[(atmost n R)](d) iff maxcard{e : f ~ V [ R ] ( d , e)} ~< n iff f ~E V[(atleast n + 1 R)](d) iff t E V[-q(atleast n + 1 R)](d). Also, f E V[(atmost n R)](d) iff mincard{e : t C V[R](d, e)} > n iff t @ V[(atleast n + 1 R)](d) iff f E V[~(atleast n + 1 R)](d).

(7) Here, tEE V[(fillers (restrict R C) C1 . . . Cn)](d ) i f f 3 e ~ . . . e , ( V i • j e i ~ ej ^ f E l(e i, ej)) ^

(Vi t E V[(restrict R C)](d, el) A t E V[Ci](ei) ) i f f : l e ~ . . . e , (Vi # j ei ~: ej ^ f E I(e~, ej)) A

(Vi t E v [a ] (d , ei) A t E V[(and C Ci)](ei)) iff t E V[(fillers R (and C C 1 ) . . . (and C C,))](d). Also, f E V[(fillers (restrict R C) C ~ . . . C,)](d) i f fVe I . . . e n ( 3 i ¢ j e z = e i v t E I ( e i , e j ) ) v

(: l i f E V[(restrict R C)](d, el) v f @ V[Ci](ei) ) i f f V e l . . . e , , ( 3 i ¢ j e i = e j v t E I ( e i , e j ) ) v

(: l i f ~E v[a](d, e~) v f E V[(and C Ci)](ei)) i f f f E V[(fillers R (and C C , ) . . . (and C C,,))](d).

(8) Trivially, V[(and R 1 R2) ] : V[(and R 2 R,)].

(9) Trivially, V[(and R, (and R 2 R3)) ] = V[(and (and R~ R2) R3) ].

(10) Trivially, V[R] = V[(and R)].

(11) Trivially, V[R] = V[(restrict R (and))].

(12) Trivially, V[(and (restrict R 1 C) R2) ] = V[(restrict (and R 1 R~) C)].

(13) Here, t E V[(restrict (restrict R Cl) C2)](d, e) iff t E V[R](d, e) and t ~ V[Cl](e) and t E V[C2](e) iff t E V[R](d, e) and t ~ V[(and C~ C2)](e) iff t E V[(restrict R (and C~ C2))](d, e). Also, f E V[(restrict (restrict R C~) C2)](d, e) iff f E V[R](d, e) or f E V[C~](e) or f E V[C2](e) iff f E V[R](d, e) or T E V[(and C, C2)](e ) iff f E V[(restrict R (and C~ C2))](d , e). [ ]

It is obvious that the above equivalences can be used to convert concepts and roles into an equivalent canonical form.

A.2. Disjoint union

An important part of the completeness proof is the construction of certain semantic structures called the extended disjoint union of a multi-set of semantic structures. A multi-set is similar to a set except that it is possible for an element to occur more than once in a multi-set. In this section, a multi-set will often be


treated as a set of pairs whose first element is a non-negative integer, indicating how many times the second element occurs in the multi-set.

Definition 4. The disjoint union of two (multi-) sets, A and B, is a multi-set, written A (~ B. If A contains n copies of e and B contains m copies of e, then A (~ B contains n + m copies of e. I f D is a multi-set formed from the disjoint union of several copies of a set, say E, then E i is the subset of D which contains the occurrences of the elements of D which derive from the ith copy orE, and if e is an element orE, then ie is the element o l D which derives from the e in the ith copy of E.

Definition 5. Let 5P be a multi-set of semantic structures. An extended disjoint union of 5f is a semantic structure, S', with domain

D s ,={O}(~ ( ~ (D s @ " • " (~)Ds) , ( n , S ) ~ : ! n copies

whose identity relationship satisfies

Is,(d, e) = Is(d, e), for d,e E Dis, ,

and whose extension function satisfies

Vs,[A](d ) : Vs[A](d ) Vs,[P](d, e) : Vs[P](d, e) Vs,[P](d, e) : {f}

for d E D i~, for d,e E D i~, for d @ Dis., e~Dis. ,

where A is an atomic concept, P is an atomic role, and S is an element of 5 ~.

Note that D s, is a multi-set. As far as the semantics is concerned, each occurrence of an element in D s, is t reated as an entirely separate element of the domain of the semantic structure. Also note that the extension function and identity relationship of an extended disjoint union are unconstrained in many places, so there can be many extended disjoint unions of one multi-set of semantic structures.

The fundamental property of extended disjoint unions is that the extensions of structured concepts and roles within the domains of the constituent semantic structures are unaltered.

Theorem 4. Let ~ be a multi-set of semantic structures. Let S' be an extended disjoint union of 5¢. Then Vs,[C](d ) = Vs[C](d ), for d E DI~, and Vs,[R](d, e) = Vs[R](d, e), for d,e E Dis., where C is any concept, R is any role, and S is an element of 5~.


Proof. By structural induction on canonical form concepts and roles. A simple property of S' that is used in the proof is that Vs,[R](d, e) = {f} for

R any role, if d E D s and e JTZD s for S E 0 °. This property can be easily verified by noting that any role includes a conjunction of atomic roles. Another simple property of S' used in the proof is that m i n c a r d s , { d E D s : P ( d ) } = mincards{d ~ Dis. • P(d)} and maxcards,{d E D s • P(d)} = maxcards{d E D s • P(d)}, for any P, since Is ,(d, e) = Is (d , e) for d,e E Dis. and S E oW.

Throughout the proof S will be an arbitrary constituent of 5e, and, unless otherwise specified, d and e will be arbitrary elements of D s. Appeals to the inductive hypothesis occur in almost every piece of the proof, and will not be mentioned.

Case A for A an atomic concept: From the constraints on Vs,, Vs,[A](d ) = Vs[A](d ).

Case -7C: Here, t E Vs,[-qC](d ) iff f ~E Vs,[C](d ) iff f ~_ Vs[C](d ) iff t E Vs[~C](d ). Similarly for f.

Case (and Cj C2): Here, tE Vs,[(and C~ C2)](d ) iff t E Vs,[C,](d ) and t E Vs,[C2l(d ) iff t E Vs[Cl](d ) and t@ Vs[C2](d ) iff t ~ Vs[(and C, C2)](d ). Also, f E Vs,[(and C, C2)](d ) iff f E Vs,[C,](d ) or f U_ Vs,[C2](d ) i f f f E Vs[C,](d ) or f E Vs[C2](d ) i f f f E Vs[(and C, C2)](d ).

Case (atleast n R): Here, t E Vs,[(atleast n R)](d) iff mincards,{e E D s, : t E Vs,[R](d , e)}/> n iff mincards,{e E D s : t E Vs,[R](d , e)}/> n,

since t ~ V s , [ R ] ( d , e) for e ~ D I S , iff mincards,{e ~ DIS : t E Vs[R](d , e)}/> n iff mincards{e E Dis : t@ Vs[R](d , e)}/> n iff t @ Vs[(atleast n R)](d). Also, f E Vs,[(atleast n R)](d) iff maxcards,{e E D s, : f ~ V s , [ R ] ( d , e)} < n iff maxcards,{e E Dis : f ~ V s , [ R ] ( d , e)} < n,

D i since f E Vs,[R](d , e) for e ~ s, iff maxcards,{e E Dis : f ~ V s [ R ] ( d , e)} < n iff maxcards{e E D s : f ~ V s [ R ] ( d , e)} < n iff f E Vs[(atleast n R)](d).


Case (fillers R C1 . . . C,): Here, tE Vs,[(fillers R C l . . . C,)](d) iff =le I . . . e, E D s, (Vi # j e, #= ej A f E ls,(e i, ej)) A

(Vi t E Vs,[R](d, ei) /~ t E Vs,[Ci](ei) ) iff=le 1 . . . e . ~ D i s . (Vi # j ei #= ey A f E ls,(ei, ej)) A

(Vi I E Vs,[R](d, ei) A t ~ Vs,[C,](ei) ), since t~Vs,[R](d, e) for e ~DI~,

i f f 3 e 1 . . . e , , ~ D ! s . (Vi # j ei # ei A f E ls(ei, ei)) A (Vi t ~ Vs[R](d, ei) A t E Vs[Ci](ei) )

iff t E Vs[(fillers R C, . . . C,)](d). Also, f ~E Vs,[(fillers R C, . . . Cn)](d ) i f f V e j . . . e , , E D s, ( 3 i # j e i = e j v t ~ l s , ( e ~ , e j ) ) v

(3i f E Vs,[Rl(d, Ci) V f ~ Vs,[Ci](ei) ) iff Ve~ . . . e , EDis ( 3 i # j e ~ = e j v t ~ I s , ( e , , e i ) ) v

(3i f c V~,[Rl(d, ei) v f ~ V,,[C,l(ei)), since f ¢ Vs,[R](d, e) for e ~E'DIs.,

i f f V e ~ . . . e EDis. ( 3 i ~ j e ~ = e ~ v t E l s ( e ~ , e j ) ) v (3i f ~ Vs[R](d, e,) v f @ Vs[Ci](ei) )

iff f E Vs[(fillers R C1 . . . C,,)](d).

Case P for P an atomic role: From the constraints on Vs,, Vs,[P](d, e) = Vs[P](d, e).

Case (and Rj R~): Here, t E Vs,[(and R~ R~)](d, e) iff t E Vs,[R~](d, e) and t E Vs,[R2](d, e) iff t E Vs[R~](d, e) and t ¢ Vs[R2](d, e) iff t E Vs[(and R, R2)](d, e). Also, f E Vs,[(and R, R1)](d, e) i f f f E Vs,[R,](d, e) o r f E Vs,[Rz](d, e) i f f f ¢ Vs[Rl](d, e) or f ~ Vs[R2](d, e) iff f E Vs[(and R 1 R~)](d, e).

Case (restrict R G): Here, t E Vs,[(restrict R C)](d, e) iff t E Vs,[n](d, e) and t ~ Vs,[C](e ) iff t ~ Vs[R](d, e) and t ~ Vs[C](e ) iff t ¢~ Vs[(restrict R C)](d, e). Also, f ~ Vs,[(restrict R C)](d, e) iff f ~ Vs,[R](d, e) or f ~ Vs,[C](e ) iff f ~ Vs[R](d, e) or f ~ Vs[Cl(e) iff f ~ Vs[(restrict R C)](d, e). []

A.3. Soundness and completeness

Now the soundness and completeness of the algorithmic characterization of subsumption given in Theorem 1 can be shown. In this section, the notation


C=>C' retains the semant ic mean ing of Defini t ion 2, while the no ta t ion ~-C ~ C' refers to the a lgor i thmic charac ter iza t ion , as redef ined here.

The redefini t ion uses the conjunc t ion and dis junct ion of sets of concepts .

Definit ion 6. Let E = {C 1 . . . Cn} be a set o f concepts. Then/~ E is defined to be (and C~ . . . C n) a n d V E is defined to be (or C 1 . . . C n), where

t e V[(or C, . . . C . ) ] ( d ) i f f for some i , tE V[Ci](d), and f E V[(or C1 . . . C , ) ] (d ) iff for each i, f E V[Ci](d ).

7. Let E and E' be sets o f canonical form primaries. Then ~- /~ E Definit ion

~ V E ' iff (1) for some A E E', with A an atomic concept, A E E, (2) for some -TA E E', with A an atomic concept, --7A E E, (3) for some (atleast m R') E E', either

(a) there exists (atleast n R) @ E, with n >t m and [- R =), R', or (b) there exists (fillers S C, . . . Cn) E E,

with [{j • [--(restrict S C / )=>R '} [ /> m, (4) for some --7(atleast m R') E E', either,

(a) there exists --a(atleast n R) @ E, with m >1 n and ~ R' ::~ R, or (b) there exists 7(fillers S C l . . . C,) E E,

with m >t n and ~ R' :ff (restrict S C/), for all j, ! t t (5) for some (fillers S ' C 1 . . . C m) E E , either,

(a) there exists (fillers S C~ . . . C,) E E, with S' C S and for each i there exists a distinct j such that ~-Cj ~ C I, or

(b) there exists (atleast n R) ~ E, with n >1 m and ~ R ~ (restrict S' C'i ), for each i,

o r

for (a)

(6) some ~(fillers S ' C'~ . . . C ' ) E E ' , either there exists -q(fillers S C 1 . . . C , ) E E, with S C S' and for each j there exists a distinct i such that ~-C' i ~ Ci, or there exists -7(atleast n R) E E, with I{ i" ~-(restrict S' CI)~,R} I ~>n.

(b)

Definit ion 8. Let C and C' be canonical form concepts. Then ~ C ~ C' iff for all D' in C', ~ - C ~ V (D'} .

Definit ion 9. Let R = (restrict S C) and R ' = (restrict S' C') be canonical form roles. Then ~ - R ~ R ' iff S' C_S and ~ - C ~ C ' .

It is obvious that the above defini t ions are equivalent to the a lgor i thmic charac te r iza t ion of subsumpt ion in T h e o r e m 1.

G iven these definit ions, the soundness par t of the p roo f is s imple.


Theorem 5. Let C and C' be canonical form concepts or roles. Then ~ C ~ C' implies C ~ C'.

Proof. The proof proceeds by structural induction. Part of the induction will show that if E and E' are sets of canonical form primaries, then ~- A E ~ V E' implies /~ E ~ V E'.

It is convenient to use the notation ~-E ~ E', where E and E' are canonical form primaries, to refer to the relationship between primaries defined in the definition of ~-/~ E ~ V E' above. This extension will be used in the case analysis of the induction.

Throughout the proof, S will be an arbitrary semantic structure and d and e will be arbitrary elements of its domain.

Suppose that C and C' are canonical form concepts and that ~-C ~ C'. Then, for each D' in C', ~- C ~ V {D'}, so, from the induction hypothesis, C ~ V {D'}.

If t E Vs[C](d ), then, from the definition of subsumption, t E Vs[D'](d ) for each conjunct, D', of C', and thus t@ Vs[C'](d ). Also, if f E Vs[C'](d ) then, f E Vs[D'](d ) for some conjunct, D', of C', and thus, from the definition of subsumption, f E Vs[C](d ). Therefore C ~ C'.

Suppose that E and E' are sets of canonical form primaries and that ~-/~ E ~ V E'. Then, for some E E E and for some E' C E', ~- E ~ E', so, from the induction hypothesis, E ~ E'.

If t E Vs[ A E](d) then t E Vs[E](d ) for each E E E. So, from the definition of subsumption, t E Vs[E'](d ) for some E' E E', and thus t E Vs[ V E'](d) . Also, if f E Vs[ V E'](d) then, f E Vs[E'](d ) for each E' ~ E'. So, from the definition of subsumption, f E Vs[E](d ) for some E E E, and t h u s f E Vs[/~ E](d). Therefore

VE'. Suppose that E and E' are canonical form primaries and that ~-E ~ E'.

(1) If E' is of the form A or -TA, where A is a 1-place predicate, then the result is trivial.

(2) If E' is of the form (atleast rn R') then either (a) E is of the form (atleast n R), with n/> m and ~ - R ~ R ' , or (b) E is of the form (fillers S C 1 . . . C,,), with

I{J : I-( restrict S C f l :~R ' } ] /> m. In the first case, R=> R' from the inductive hypothesis, and thus, if t E

Vs[E](d ) then mincard{e : t E Vs[R](d, e)} ~>n. Since t E Vs[R](d, e) implies t C Vs[R'](d, e), thus

mincard{e : t E Vs[Rl(d, e)}/> n i> m,

and thus tEVs[E'](d ). Also, if fEVs[E ' ] (d ) then maxcard{e : f ~ ' Vs[R'](d, e)) < m. Since f ~Vs[R](d, e) implies f ~Vs[R'](d, e), thus


maxcard{e : f ~ V s [ R ] ( d , e)} < m ~< n,

and thus f ~ Vs[E](d ). In the second case, ]{j : (restrict 8 Cj)@R'}}>~rn, from the inductive

hypothesis. If t v= Vs[E](d ) then

3e~ . . . e, (Vi ¢ j ei # ej ^ f ~ I(ei, ej) ) ^ (Vi t ~ V[S](d, e,) ^ t ~ V[C,l(e,)).

Considering those j for which (restrict S C j) ~ R' there are m distinct e i for which t E V[R'](e/), and so mincard{e : t ~ V[R'](d, e)} >~ m. Thus t ~ Vs[E'](d ). Also, if f ~ Vs[E'](d ) then maxcard{e : f y Z V [ R ' ] ( d , e)} < m, i.e.,

V e a . . . e m ( 3 i # j e i = e j v t ~ I ( e i , ej) ) v (3i f E V[R'I(d, e,)).

Consider the j for which (restrict S C j ) ~ R'. For any e l . . . e m with Vi # j e i ~ ei ^ t ~ I ( e ~ , ej) there must be some i such that f ~ V[R'](e/), and thus f E V[(restrict S Cj)](e~). Since n ~> m, thus

V e , . . . e , ( 3 i ¢ j e i = e j v t ~ I ( e i , e j ) ) v (3i f @ V[CA(e~) v f ~ V[S](d, e/)),

and thus f C Vs[El(d). Therefore E ~ E'. (3) If E' is of the form (tillers S' C I . . . Cm') then either,

(a) E is of the form (fillers S C ~ . . . Cn), with S' G S and for each i there exists a distinct j such that ~-Ciz=>C~, or

(b) E is of the form (atleasl n R) E E, with n 1> m and ~- R ~ (restrict S' C~), for each i.

In the first case, from the inductive hypothesis C j ~ C ' i , for the C i that matches each Ci, and thus, if t ~ Vs[E](d ) then

3e , . . . e,, (Vi # j ei C e j A f E I(ei, ej)) A (Vj t ~ V[S](d, e/)/x t E V[Cj](ej)).

Since 8 C S' and for each i there exists a distinct j such that Gj ~ C'i, therefore

Bet . . . e,, (Vi ~ j ei # e j ^ f ~ I(e,, ej)) /x (Vi t ~ V[S'](d, el) A t@ V[C; ](ei)),

and thus t E Vs[E'](d ). Also, if f ~ Vs[E'](d ) then


V e l . . . e m ( 3 i : ~ j e ~ = e j v t E I ( e ~ , e j ) ) v (3 i f E V[S'](d, e~) v f E V[Cl](e,) ).

Since S C_ S' and for each i there exists a distinct j such that C / ~ Gi, therefore

Ve 1 . . . e . ( 3 i # j e i = e i v t ~ l ( e i ,ej)) v (3i f E V[S](d, ei) v f E V[Ci](e~) ),

and thus f E Vs[E](d ). In the second case, R~(restrict S' CI), for each i, from the inductive

hypothesis. If t E Vs[E](d ) then

mincard{e : t E V[R](d, e)}/> n/> m.

(Vi # j ei ~ e j A f E I(ei, ei)) A (Vi t E V[R](ei)).

Since R~(restriet S' C'i), for each i, thus

3e 1 . . . e . , ( V i ¢ j e i ¢ e j A f E I ( e i,ej)) A (Vi t E V[S'](d, ei) A t E V[Ci](ei) ),

and thus t E Vs[E'](d ). Also, if f E Vs[E'](d ) then

V e ~ . . . e m ( 3 i ~ j e i = e j v t E I ( e i , e i ) ) v (3 i f E V[S'](d, el) v f E VIC;I(e~)).

Since Rzff(restrict S' ¢ i ) , for each i, thus

V e ~ . . . e m ( 3 i C j e i = e j v t E I ( e i , ej) ) v (3i f E V[R](ei) ).

Therefore, since n/> m,

V e l . . . e . ( 3 i ~ j e i = e i v t E I ( e i , ei) ) v (3i f E V[R](ei)),

and thus f E Vs[E](d). Therefore E ~ E'. (4) The cases for E of the form -n(atleast n R) and for E of the form -7(fillers R

C~ . . . C,) are similar to the previous two cases.

Therefore

::]e 1 . . .8 m


Suppose R = (restrict S C) and R' = (restrict S' C') are canonical form roles, and that t -R =), R'. Then $'C_ S and t -C :~ C', so, f rom the induction hypothesis, c=> c'.

If t@ Vs[R](d, e), then tE Vs[P](d, e) for each P in S and t E Vs[C](e ). Thus t E Vs[P'](d, e) for each P' in S' and, from the definition of subsumption, t E Vs[C'](e ), so t E Vs[R'](d, e). Also, f @ Vs[R'](d, e) implies f ~ Vs[P'](d, e) for some P' in S' o r f E Vs[C'](e ). Thus f E Vs[P](d, e) for some P in S or, f rom the definition of subsumption, f ~ Vs[C](e ), so f ~ Vs[R](d, e). Therefore R ::~, R'. [ ]

The completeness part of the proof is more complex. First, a completeness proof for sets of canonical form primaries is shown, and then this is extended to canonical form concepts and roles.

Lemma 1. Let E and E' be sets of canonical form primaries. Then, ~ /~ E V E' implies /~ E F)~ V E'.

Proof. The proof proceeds by using structural induction to construct semantic structures S' and S~E, such that EE'

t E Vs~,L,[EI(0 ) but t~Vs, E,[E'I(O )

and

f ~ " V ~ E [E](0) but f E V f [E'](0) SEE,

for all E E E and for all E ' E E ' , where 0 is a special domain element , thus showing that /~ E ~)¢ V E' .

The first step of the construction is to construct a group of special semantic structures. For each E E E of the form (atleast n R), the semantic structures S' E and S~ will be constructed. For each E' E E ' of the form -7(atleast m R'), the semantic structures S~, and s f l will be constructed. For each E E E of the form (fillers S' CI . • • C,), the semantic structures S'E4 and S~4 will be constructed, for each 1 ~< j ~< n. For each E' C E of the form ~(fillers S' C'~ . . . C ' ) , the semantic structures S~',i and S~I i, will be constructed, for each 1 ~< i ~< m.

For E of the form (atleast n (restrict S C)), the semantic structure StE has the property that t E Vs~[C](0'E), where 0' E is a special e lement of the domain of S ' E. However , for E' E E ' of the form (atleast m (restrict S' C')), with m ~< n and

t , t E' E ' ' . . . ' S' C_ S, ,~Vs,[C ](0E). Also, for E of the form (fillers S' C~ C,,), with m ~< n and S' C_ S, t~Vs,[C~](OtE), for some C' i. This semantic structure is used to make t @ Vs~E. [(atleast n (restrict S C))](0) without making t E Vs, E ,[E'](O) for any E' E E' .

t For E of the form (fillers S C ~ . . . Cn), the semantic structure Sed has the


property that t C Vs, j[Cj](O'E,j). However, for E ' E E ' of the form (fillers S' C ~ . . . C',), with m ~< n and S' C_ S, t ~ V s , ~ [ e l i ( 0 ' E j), for those C'~ such that C~

,1

C's Also, for E' ~ E ' of the form (atleast m (restrict S' C')), with m ~< n and S' C S, I{J : t ~ Vs, ~ [C'](0~ ~)}l < m. This semantic structure is used to make t ~ Vs~,[(fillers S C~ ~ . . C,,)]'(0) without making t ~ Vs,e~,[E'](O ) for any E' ~ E'.

The other semantic structures are used similarly. Throughout the construction it is convenient to be able to refer to the set of

"top-level" primaries in a canonical form concept, written T(C). This set is defined as follows:

T((and C, . . . C,,)) = {C 1 . . . . . On} r ( -~c) = r ( c ) T((atleast n (restrict S C))) = T(C) T((fillers S C , . . . C,,)) = U o f ( C j ) -

The construction of S' E and S~, for E = (atleast n (restrict S C)) ~ E, proceeds as follows: Let T be the union of T(E') , for E' E E ' of the form (atleast m (restrict $' C')) or (fillers S' C ~ . . . C ' , ) with 9 ' C $ . Let C ' = {D'E T : ~ C ~ O ' } and let C = T(C). Finally, let S' E = S'cc, and S~ = S~.c,, and let 0' E and 0~ be the special domain elements of S'cc, and SIcc ,, respectively.

Now t E Vs~[D](0'E) for all O ~ C and t~'V%[D'](0'E) for all D' E C', from the f , f inductive hypothesis• Similarly, f % V ~[D](0E" ) for all D E C, a n d f E V j[D ](0E)

S E S E

for all D ' E C'. Therefore t C v, ICl(0'E) and fe'V [Cl(0 ). Consider (atleast m (restrict S' C')) E E ' with m ~< n and S' C $. There exists

some primary, O', in C', such that ~ C ~ D' (or else ~- A E ~ V E ' ) . Therefore D' e C', so t.e'V~dC'](0'E) and f E V {[C'](0~).

Also, consider (fillers S' C'~ . . . C,',) E E ' with m ~< n and S' C S. There exists some primary, D', in some C;, such that ~ C ~ D ' (or else ~- /k E ~ V E ' ) . Therefore D ' E C', so t~'Vs.~[C;l(0tE) and f e V ~-[C;](0~), for some C;.

• S / f , E The construction of SE.j and S E j , for E = (fillers S C~ . . . C,,) ~ E, proceeds as

follows: Let T be the union of T(E') , for E' E E ' of the form (atleast m (restrict S' C')) or (fillers S' C'j . . . C ' ) with S' C S. Let C' = {D' E T : ~ C j ~ D'} and let

• t t - - f f C = T(C~). Finally, let SE, j = Scc , and SE, j = Scc , , and let 0' E j and 0~ j be the special domain elements of S'cc, and S~:c,, respectively. ' '

Now t E V s, ,[D](0' E j) for all D ¢ C and ty~Vs, ~ j[D'](0' E y) for all D' C C', from

the induct ivehypothesis . Similarly, f ~ V s , r [I3"](0£ j) for all D E C and f E t f t t " E . j t Vj [D] (OE) for all D C C . Therefore t E V s, [Cz](OE. ) and f~Z" SE, j "1 E.I "1

vL[cjl(0~,j) . Consider E' ~ E ' of the form (atleast m (restrict S' C')), with m ~< n and S' C_ S.

If ~ - C i ~ C ' , then = ID 'ET(E ' ) such that ~ / C i ~ D ' . Therefore D ' E C ' and thus t fZVsLj[D'](O'E,/) and f ~ V£,fiD'](0~4). Since ~ A E ~ V E', thus


• ¢ ~ t t I { / : [ - C j = > C } [ m, and thus ] { j : t E V s , [C](OE,i )}]<m and ] { j : f ~ vL[c'](0{,j)}l < m•

Also, consider (fillers S' C '~. . . C ' ) E E ' with S' C_ S. Since ~ / ~ E ~ V E', for each matching from C'~ . . . C" to C~ . . . C, there is a C~ and corresponding Cj such that ~ / C j ~ C I. Thus, 3 D ' E T(C'i) such that ~ . C y ~ D ' , so D ' E C' and

t t t f t ~ V s , ~j[C ](OE•j) and f E V¢ [C ](OE4 ). Thus, for each matching from • SE, j

C'~...C,~, to C ~ . . . C , there is a C' i and corresponding Cj such that t ~ ¢ l ¢ f Vs, ~ j[Ci](OE,i) and f E V.I [Ci](OE,j).

• E , /

• " f ' E' The construcnon of S E, and S E,, for = 7(atleast rn (restrict S' C ' ) )E E', proceeds as follows: Let T be the union of T(E), for E E E of the form -7(atleast n (restrict S C)) or -7(fillers S C~ . . . C,) with S C S'. Let C = {D E T" [-C' ~ D} and let C ' = T(C'). Finally, let S r, = S'c, c and S{; = S~,, c, and let 0~' and 0{; be the special domain elements of Sc, c and S~:, c, respectively.

' " D' C' " Now t ~ Vsf[D ](OE, ) for all E and t ~V s , , [D](OE, ) for all D E C, from E ' ,

the inductive hypothesis. Similarly, f ~ V j , [ D ' ] ( O { , ) for all D ' ~ C' and f ~ V f:[D](O~:) for all D E C. Therefore t E V~'[C'](0~',) and f~v{:[c'](o{;).

Consider (atleast n (restrict S C)) @ E with n ~< m and S C_ S'. There exists some primary, D, in C, such that ~zC'@ D (or else ~-/~ E ~ V E') . Therefore D e C, so t~E~V%[C](O'E',) and f e V ~:[C](Of;).

Also, consider (fillers S C~ . . . C,) E E with n ~< m and S C S'. There exists some primary, D, in some Ci, such that ~ / C ' ~ D (or else ~ - / ~ E ~ V E ' ) . Therefore D E C, so t~Vsf,[Cj](O'~', ) and f E Vs~;[Cj](O{;), for some Cj.

The construction of S'E',•i and S{I,, for E'=-7(fillers S' C ' ~ . . . C ' , ) ~ E ' , proceeds as follows: Let T be the union of T(E), for E ~ E of the form 7(atleast n (restrict S C)) or -7(fillers S C~ . . . C,) with S C S'. Let C = {D E T: bzCi ~ D} and let C' = T(CI). Finally, let S~',j = S'c, c and S~;~ = S~., c, and let Or,,i and 0{;~ be the special domain elements of S'c, c and S~,', c, respectively•

Now t E Vsf,,[D'](O'E',~ ) for all D' ~ C' and t~Vsf,,[D](O'E',, ) for all D ~ C, from

the inductive hypothesis• Similarly, f ~ V / ; [D'](O{;~) for all D 'C C' and t t ' / ~ V ~ ; [ D ] ( O ~ ; i ) for all D ~ C . Therefore t~Vsf,,[C~](O~,~ ) and f ~

v , SE', i

Consider E ~ E of the form -7(atleast n (restrict S C)), with n ~< m and S C_ S'. If ~ z C I ~ C , then ZlD~ T(E) such that ~O'~=>D. Therefore D ~ C and thus t " ~Vsf,~[D](O~,~ ) and f~V~;.[D](O{;~). Since ~ A E © V E , thus I{i :

t ' } - C ; ~ O } ] < n , and thus I{i : t~Vs , ; .Jc](o~, ,~)} l<n and I{i " f ~ V ~,, [C](O{:i)} I < n.

Also, consider -7(fillers S C 1 . . . C,,)@ E with S C_ S'. Since [-//~ E O V E', for each matching from C~ . . . C, to C~ . . . C', there is a Cj and corresponding C I such that ~C~:::>Cj. Thus, ~ D ¢ T(Cj) such that ~ C I O D , so D ~ C and


t~Vsc,,[C](O'E',, ) a n d f E V ~; [C](O~,',). Thus, for each matching from C, . . . C,,

to C~ . . C', there is a Cj and corresponding Cj such that " ' . ' I ~Vs,,.,,[C /](OE,j) and f E V ~ [C;](O~i,/).

The niain step of the proof is the construct ion of S~. e, as follows: First, let 5#be the disjoint union of

. : S '

l { s b , , . . . . .

for each E = (atleast n (restrict S C)) ~ E for each E = (fillers S C l . . . C~) ~E E for each E ' = ~(at least m (restrict S' C'))~E E for each E' = ~(f i l lers S' C'~ . . . C,',,) ~ E

Then S~v ~, is the ex tended disjoint union of o w that meets the following constraints:

t ~ Vs~ [A](0 )

f ~E V%E [A](0 )

Vs,eE[P](O, O'E)= {t, f }

Vsk, [P](0, 0'~.,) = {t , f }

Vsk~,[P](0 , 0~,') = { }

f ' V,%E,[P](0, 0E',i) = (}

Vs%~ [Pl(O,d ) = { f } I(O,d) = {}

i t j t g 0E) = ( f } l(0'e.~,0'Eu ) = {f} l(d, e) = {t}

l(d, e ) = {}

for A ~ E, A an atomic concept ,

for -TA E E, A an atomic concept , for E = (atleast n (restrict S C)) ~ E

and P in S, for E = (fillers S C~ . . . C , ) ~E E and P in S,

for E' = q(at least m (restrict S' C')) ~ E' and P in S',

for E' = -7(fillers S' C I . . . C,',,) ~ E ' and P in S',

o therwise, for P an atomic role, for d # 0 , for i # j , for i # j , for d, e distinguished elements of

different S{I i or S~I i, for E ' ~ E' , for o ther distinguished elements of

const i tuents of 5 ~'.

Because S' is an ex tended disjoint union, l ( ~ . [ C ] ( d ) = Vs, ~,~,[C](d) for any E E ' =.

¢, any S ~ 5 e, and for any d in the domain of S. It remains to show that t E Vs~:~.[E](0 ) but t~Vsk~,[E'](O), for all E ~ E and

for all E' ~ E' . Define O(E) as follows: If E C E is of the form (atleast n (restrict S C)), then

O ( E ) = { ( n , 0'E) }. If E E E is of the form (fillers S C j . . , C , , ) , then O ( E ) = {0'E, ~ . . . . . 0'E,,, ). If E ~ E ' is of the form ~(atleast m (restrict S' C')), then O ( E ) = { ( m , 0~')}. If E E E ' is of the form (fillers S' C ' j . . . C , ' , ) , then O(E) = {0~i~ . . . . . 0~i,,,}. Otherwise , O ( E ) = {}.


For E E E: (1) Suppose E is an atomic role. Then t E Vs~E,[E](O ) directly. (2) Suppose E is of the form 7A, where A is an atomic role. Then

f vs E [A](0), so t e vs E [El(0). (3) Suppose E is of the form (atleast n (restrict S G)). Then {(n, StE)} E 5eand

• i t thus tEVs~E.[C]('OtE) and tEVs , E,[S](O, OE), for l<~i<~n, and f E I(i0tE,J0'E) for i ~ j. Therefore t ~ Vs~E,[E](0 ).

(4) Suppose E is of the form (fillers S G 1 . . . Cn). Then StE.i @ 9 9, for 1 ~<j ~< n, and thus t E Vs~E.[S](0, 0~Eu) and tE Vs~E.[Gj](O~E,j), for all j, and f E I t t (0E,i,0E,j) for i ~ j . Therefore t E Vs~E,[E](0 ).

(5) Suppose E is of the form -7(atleast m (restrict S' C')). If f C~Vs,EE,[(restdct S C)](0, e) then e E O(E') for some E ' E E'. Thus

maxcard{e : f~Vs,EE.[(restrict S C)](0, e)} = maxE,~E,maxcard{e E O(E' ) : f~Vs~E,[(restrict S C)](0, e)},

since otherwise there will be a pair of entities in the set which are known to be identical. Consider E' E E ' of the form (atleast m (restrict S' G')). I fn <~ m and S C_ S', then, from the construction of S~I, f ~ Vs~e,[C](e), for e E O(E') . Thus maxcard{e E O(E' ) : f , f~Vs,[(restrict S C)](0, e)} < n. Consider E' E E ' of the form -q(flllers S' C[ . . . C ' ) . If S C_ S' then from the construction of S~I, I { e E O ( E ' ) : Thus

maxcard{e E O(E' ) : f~Vs,ee,[(restrict S C)](0, e)} < n. Therefore t E Vs2E,[E](0 ).

(6) Suppose E is of the form -q(fillers S Cl . • • O,). Iff,f~VstEE.[S](O, e) then e E O(E' ) for some E ' E E'. Also, if {e~ . . . . . em} has e i ¢: ej and t,~I(e~, ej) for all i # j and f ,~Vs, ,[S](O, ei) , then {e 1 . . . . , em} C_ O(E') for some E' E E' . Consider E' E E ' of the form -q(atleast m (restrict S' C')). If S C_ S' and m>~n then, from the construction of S£;, fEVs,EE.[Cj](e ), for all e ~ O(E' ) and some Cj. Consider E' E E ' of the form -n(fillers S' C~ . . . C ' ) . If S C S' then from the construction of S£I, for any matching from C~ . . . C, to C' 1 . . . C ' , there is some Cj and a corresponding C I such that f E Vs,[Cj](O'E,i ). Thus, if for each j there exists a distinct i, then for some j and corresponding i, f E Vs2~ [C j](Ot~, ~). Thus, if

=le~ . . . e m (Vi " / = j e i :/= ej /x t~L~I(ei, ei) ) /x (Vi f ,~V[S'](e~)),

then =lj f E Vs, E,[Cjl(ej).


T h e r e f o r e t E Vsg,~,[E](0 ).

For E' ~ E ' : (1) Suppose E' is an a tomic role. Then t~Vs , [E'](O), since E' ~ E . (2) Suppose E' is of the fo rm ~A, where A is an a tomic role. Then

f~Vs~E. [A] (0 ), since -hA ~ E , so t f~Vs , [E ' l (O ). (3) Suppose E' is of the fo rm (atleast m (restrict S ' C')).

If t C Vsk~[(restrict S ' C')](0, e) then e E O(E) for some E E E. Thus

mincard{e • t C Vs~F.[(restrict S ' C')](0, e)} = maxE~L.mincard{e E O ( E ) " t E Vs%,[(restrict S ' C')](0, e)},

since o therwise there will be a pair of entit ies in the set which are not known to be distinct. Cons ider E ~ E of the form (atleast n (restrict S C)). If m ~< n and S ' G S, then, f rom the const ruct ion of S' E, tf~Vs, x..[C'](e ), for e E O(E) . There - fore mincard{e E O(E) • t E Vs~,~,[(restrict S ' C')](0, e)} < m. Cons ider E ~ E of the fo rm (fillers S Gx • • • C,,). If S ' C_ 8 then f rom the construct ion of S ' E, ]{e E O(E) " t E V%:t.[C'](e)} ] < m. T h e r e f o r e min-

card{e E O ( g ) • t E Vs1:~:.[(restrict S ' C')](0, e)} < rn.

T h e r e f o r e tfZVs, [E'](O ). (4) Suppose E' is of the fo rm (tillers S ' C I . . . G,',,).

If t E V%E[S ' ] (0 , e) then e E O(E) for some E@ E. Also, if {ej . . . . . e,,} has e~# ej and f ~ l(e~, ei) for all i ¢ j and t E V%~[S ' ] (0 , e~), then {e~ . . . . . e,,} C_ O(E) for some E ~ E. Consider E E E of the fo rm (atleast n (restrict S C)). If S ' _C S and n ~> rn then, f rom the const ruct ion of S~E, t~Vs,t, ,[Ci](e ), for all e E O(E) and some C' i . Cons ider E E E of the form (fillers S G 1 . . . C,,). I f S' C_ S then f rom the

• t construct ion of S E, for any match ing f rom C L . . . C,I , to C~ . . . C,,, there is some C' i and a cor responding Gj such that t~Vs,t~E.[C'i](O'E.j). Thus , if for each i there exists a distinct j, then for some i and match ing j,

t~Vs, lc',l(O'~.j). Thus, if

::le I . . . e , , ( V i ¢ j e , : ~ e j A f ~ l ( e i, e i ) ) A (Vi t E V[S](ei)),

then Eli t,_~'V%L.[C;](ei). The re fo re t ~ Vs~:~.[E'](0).

(5) Suppose E' is of the fo rm ~(at least m (restrict S ' C')). T h e n { ( m, S~I } } E ,9 ~,


and thus f~Vs,eE,[C'](io{;) and f~Vs~E.[S'](o,iofE:), for all 1<~ i ~ < m,

i t 0E, ), for i ~ j . Therefore tf~Vs,EE,[E'](O). and t ~ I( 0~,, J ' (6) Suppose E' is of the form -n(fillers S' C ' l . . . C ' ) . Then S~ii E ow, for

1 ~< i ~< m, and thusf~Vs, E,[S'](O, 0{;i) and t E Vske,[Ci](0{;i), for all i, and t ~/I(0~;,,0~;4) for i ~ j . Therefore t~Vs,eE.[E](O).

Therefore t E Vs,eE,[/~ E](0) but t~Vs,eE,[ V E'](0). The construction of SY'EE, is similar. []

Theorem 6. Let C and C' be canonical form concepts or roles. Then ~/C ~ C' implies G ~ C'.

Proof (Concepts). Let C and C' be canonical form concepts. If ~/C ~ C', then, for some D' in C', ~/C ~ V { D'}. Therefore G 5~ V { D'}, so

there exists some semantic structure, S, such that t E Vs[O](0) for each D in G, but ty~Vs[D'](O).

Then tEVs[C](O ) but t~Vs[C'](O ) so C 7~C'. []

Proof (Roles). Let R = (restrict S C) and R ' = (restrict S' C') be canonical form roles. If R ~ R', then either $' ~ ' $ or ~z C ~ G'. In the former case, construct a semantic structure, S, such that t E Vs[C](0 ).

(This can be done by using the construction that shows that C 7~ A, where A is an atomic concept not appearing in C.) Let S' be an extended disjoint union of the singleton set containing S, with

Vs,[P](O, Os) = {t} for P E S Vs,[P](O , Os) = {} otherwise.

Then t E Vs,[R](0, 0s) but t~Vs,[R'](O, Os), so R ~)~ R'. In the latter case, C ~ C' so there exists some semantic structure, S, such that

t E Vs[C](0 ) but t~Vs[C'](O ). Let S' be an extended disjoint union of the singleton set containing S, with

Vs,[P](O, Os)={t} for P E S Vs,[P](O , Os) = {} otherwise.

Then tEVs,[R](O, Os) but t~vs,[n'l(O, Os), so a # a ' . []

This completes the proof of Theorem 1.


ACKNOWLEDGMENT

Hector Levesque and Ron Brachman, through their investigation of the complexity of computing subsumption in standard semantics for frame-based description languages, provided the impetus for this research. Several anonymous reviewers made suggestions that served to improve both the content and the style of the paper.

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Received February 1988; revised version received June 1988

a four-valued semantics for terminological logics

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