on the relationship between cwa, minimal model, and minimal herbrand model semantics

On the Relationship between CWA, Minimal Model, and Minimal Herbrand Model Semantics* Michael Gelfoindt, Halina Przymusinskat, and Teodor PrzymusinskiS ?Computer Science Department and $Department of Mathematics, The University of Texas at El Paso, El Paso, TX 79968

The purpose of ithis article is to compare three types of nonmonotonic semantics: (a) proof-theoretic semantics based on the closed world assumption, (b) model-theoretic semantics based on the notion of a minimal model, and (c) model-theoretic semantics based on the notion of a minimal Herbrand model. All of these semantics capture the nonmonotonicity of commonsense reasoning, that is, the ability to withdraw conclusions after some new information is added to the original theories, and proved to be powerful enough to handle most examples of such reasoning presented in the literature. However, since ithese formalizations are based on different intuitions and often produce different results, the problem of understanding the relationship between them is especially important. In the first part of the article we concentrate on the class of positive logic programs, also called dejnite theories. Although the three semantics usually differ for universal sentences, our main result shows that they always coincide for existential queries. This result is particularly significant in view of the fact that in many applications existential queries are of main interest. It also plays an important role in the problem of finding a suitable declarative semantics for logic programs. In the second part we investigate arbitrary universal theories and we show that subtle differences exist between the threee approaches and therefore no straightforward generalization of the results from the first part can be obtained.

I. INTRODUCTION

A nonmonotonic declarative semantics of a given theory T-whether it is a logic program or a deductive database-can be defined in several different ways, among which the following two are most common. One that can be called proof-theoretic, associates with T a first-order theory COMP(T) called a completion of T and declares that a given sentence F is true iff it is logically implied

* The extended abstract of this article appeared in the Proceedings of the Third International Symposium on Methodologies for Intelligent Systems, Torino, Italy, Oc- tober 1988.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 5, 549-564 (1990) 8 1990 John Wiley & Sons, Inc. CCC 0884-8 173/90/050549-16$04.00

550 GELFOND, PRZYMUSINSKA, AND PRZYMUSINSKI

by COMP(T), that is, if COMP(T) F. An important example of such a definition is Reiter’s Closed World Assumption CWA(T),’ obtained by adding to T negations of all ground atoms not provable from T. Like its more broadly applicable generalizations-such as GCWA(T) and ECWA(T)2-6- it is based on the idea of adding to the theory a suitably selected set of ground formulae, which are not derivable from the theory itself.

Another approach, that can be called model-theoretic, associates with T a set MOD(T) of one or more models of T and declares that a given sentence F is true iff it is satisfied in all models from MOD(T). Important examples of such an approach are the minimal model semantics based on the set MIN(T) of all minimal models of the t h e ~ r y ~ . ~ - ~ and the feast model semantics based on the least Herbrand model M T of the theory.iO,ll The semantics of circumscription CIRC( T) and domain circumscription DCIRC( Z‘)7,12,13 allow powerful generalizations of these two semantics, by using more specialized classes of minimal (or minimal Herbrand) models.

The purpose of our article is to compare three of the above discussed types of semantics:

I The proof-theoretic semantics based on the closed world assumption; I1 The model-theoretic semantics based on the notion of a minimal

model; I11 The model-theoretic semantics based on the notion of a minimal

Herbrand model.

In general, these approaches lead to essentially different semantics. Since semantics of type I are obtained by adding negations of only ground formulae and semantics of type I11 use only Herbrand models, they tend to handle formulae with variables differently f roh the way they are handled by semantics of type 11. Semantics of type I and I11 do not coincide either, because Herbrand model semantics imply the Domain Closure Axiom, while semantics based on the closed world assumption do not.

All of these semantics capture the nonmonotonicity of commonsense reasoning, that is, the ability to withdraw conclusions after some new information is added to the original theories and proved to be powerful enough to handle most examples of such reasoning presented in the literature. However, since these formalizations are based on different intuitions and often produce different results, the problem of understanding the relationship between them is especially important.

Throughout the article we assume a suitable form of the Unique Names Assumption, namely the so-called Clark’s Equality Theory CET. We first concentrate on the important class of positive logic programs P , also called definite theories. We compare Reiter’.s Closed World Assumption CWA(P), the minimal model semantics MINCP), and the least model semantics M p . Although these semantics usually differ for universal sentences, our main result shows that they always coincide for existential sentences. This shows that for an important class of queries-namely existential-the results produced by any

COMPARING SEMANTICS 55 1

one of these semantics are exactly identical. This result is not only important in the context of norimonotonic reasoning, but it also plays an important role in the problem of finding a proper declarative semantics for logic program^.^^^^^^^ It is also significant un view of the fact that in many applications existential queries are of main interest.

In the second part of the article we discuss the relationship between semantics of types 1-111 mentioned above for arbitrary universal theories T. The most general formalization of the semantics of type I is called the Extended Closed World Assumption ECWA( T ) and was introduced and investigated in Refs. 5 and 6 . It generalizes Minker’s GCWA(T). On the other hand, general forms of semantics of type I1 and I11 were captured by the notions of Mc- Carthy’s Parallel Circumscription CIRC( T ) and Domain Circumscription DCIRC(T).7,12.13 We give a semantic characterization of ECWA and use it to compare the deductive powers of ECWA, Parallel Circumscription, and Do- main Circumscription. We show that subtle differences exist between these semantics and therefore no straightforward generalization of the results from the first part can be obtained. We prove, however, several results relating the three semantics for ground queries.

The work presented in this article can be viewed as a continuation of the work started in Refs. 4,5, and 6 (see also Ref. 16), where it was shown that, for a theory T without function symbols, with finite number of constants, and the Domain Closure ,4ssumption (DCA),” the Extended Closed World Assump- tion ECWA(T), Parallel Circumscription CIRC( T ) , and Domain Circumscrip- tion DCIRC(T) are all equivalent. However, the assumption that a theory T has no function symbols, only a finite number of constants, and satisfies the DCA is very strong and effectively restricts us to propositional theories, which are clearly inadequate for many applications. Throughout this article we allow functional symbols and infinitely many constants and we do not make any domain closure assumptions.

11. NOTATION AND DEFINITIONS

By a positive logic program we mean a finite set of universally quantified clauses of the form

A t A 1 , . . . , A ,

where m 2 0 and A and A , are atoms. The alphabet of a program P consists of all the constant, predicate, and

function symbols that appear in P , a countably infinite set of variable symbols, and the usual punctuation symbols, connectives (A, v, i j , and quantifiers (3, Vj. We assume that equality predicate = does not occur in P and if there are no constants in P , we add one to the alphabet. The language of P consists of all the well-formed formulae of the so obtained first-order theory.

Names of variables are capitalized and functions, constants, and predicates are written in lowercase. Constants are identified with functions of arity


zero. A formula is called positive if it does not contain the negation symbol 1. For a formula F, by 3 F and V F we denote its existential or universal closure, respectively. Unless stated otherwise, all formulae are universally quantified. By an existential (resp. universal) formula we mean a formula F of the form F = 3G(resp. F = VG) , where G is a quantifier-free formula.

By Clark's Equational Theory of P (CET(P)),18 we mean the theory P augmented with the equality predicate = and the following set of axioms, called CET axioms:

CETl. X = X; CET2.X= Y j Y = X , CET3. X = Y A Y = Z + X = Z ; CET4. X I = Y1 A . . . AX, = Y, j f(Xl, . . . , X,) = f( Yl , . . . , Y,),

CET5. X I = Y1 A . . . A X, = Y, =$ (p(X1, . . . , X,) j p(Y1, . . . , for any functionfi

Y,), for any predicate p ;

symbols f and g;

for any function f;

CET6. f (XI, . . . , X,) # g( Y1 , . . . , Y J , for any two different function

CEV.f(X1,. . . ,X,)=f(Yi, . . . , Y , ) J X I = YIA. . . A X m = Y, ,

C E n . t[Xl # X, for any term t [ X ] different from X , but containing X .

The first five axioms describe the usual equality axioms and the remaining three axioms are called unique names axioms or freeness axioms. The signifi- cance of these axioms to logic programming is widely recognized. Conse- quently, instead of talking about the theory P we will talk about the theory CET(P) = P + CET.

The equality axioms (CETI) - (CET5) ensure that we can always assume that the equality predicate = is interpreted as identity in all models of CET(P). Consequently, models of CET(P) can be identified with precisely those models of P in which the equality predicate-when interpreted as identity-satisfies the unique names axioms (CET6) - (CET8). This means that models of CET(P) can simply be viewed as a subclass of the class of all models of P.

We do not assume any domain closure axioms and we consider all-not necessarily Herbrand-models of the theory CET(P). Since the unique names axioms are automatically satisfied in Herbrand models, Hebrand models of CET(P) can be identified with Herbrand models of P , in which equality is interpreted as identity. Herbrand models of P are as usual considered to be subsets of the Herbrand base of P, that is, the set of all ground atoms of the theory P.

If M and N are two models of CET(P) with the same universe and the same interpretation of functions (and constants) then we say that M 5 N, if the extension of every predicate in M is contained in its extension in N. A model N of CET(P) is called minimal if there is no model M of CET(P) such that M N and M # N. It is well-known that for every model M of CET(P) there is a minimal model N such that N M.8 It is easy to see that a model M is a minimal model of CET(P) iff it is a model of CET(P) and a minimal model of P.

COMPARING SEMANTICS 553

It is also well-known that every positive logic program P has exactly one minimal Herbrand model Mp, which is called the feast Herbrand model of P.lo,l1 This model-with equality = interpreted as identity-can be thought of as the least Herbrand model of CET(P).

By MIN(P) we denote the set of all minimal-not necessarily Herbrand- models of CET(P). By a minimal model semantics of P we mean the semantics induced by the set MIN(P). Under this semantics a formula F of the language of P is considered to be true iff F is satisfied in all minimal models from MIN(P). In this case we write MIN(P)

By the least model semantics of a positive program P we mean the semantics induced by the model Mp. Under this semantics a formula F of the language of P is coinsidered to be true iff F is satisfied in Mp, that is, if M P k F.

By the CWA-semantics of a positive program P' we mean the semantics induced by the completion CWA(P) of P defined by CWA(P) = P U CET U { l A : A is a ground atom and P A}. Under this semantics a formula F of the language of P is considered to be true iff CWA(P)

Note that all formulae are supposed to belong to the language of P and therefore they do not contain the equality predicate.

Suppose that M is any model of CET(P) with universe U and equality = interpreted as identity. For every tuple (u , , . . . , u,) of elements of U and an n-ary function f we denote by f the interpretation f: U" H U off in M and thereforef(u1, . . . , u,) E U denotes the image of ( u l , . . . , u,) under$. Similarly, by A we denote the interpretation a : U" H {T, I} of a predicate symbol A. By Terms(X) we denote the set of all terms of the language whose variables belong to the set X .

F .

F .

111. TECHNICAL LEMMAS

In the next section the following two technical lemmas will play a crucial role. Their proofs are fairly complex and therefore they were included in a separate section.

Lemma 3.1. Suppose that M is a model of CET(P) with universe U and X and Yare finite sets of variables. Suppose also that, for i 5 n , Oi : X I + Terms( Y) is a substitution and /3 : Y H U is a U-instantiation such that for every i, j 5 n:

eip = ejp.

Then, there exists a unification 7: Y H Terrns(Y) such that for every i , j 5 n:

Proof. We will prove the lemma for n = 2. The general case can then be obtained by easy induction. Let us suppose therefore that X = { x I , . . . , x,} and let Oi(x,) = to. We have


We will show that the standard unification algorithm-applied consecutively to pairs of terms { t l i , t2i)-always succeeds and therefore produces the desired unification. For a more detailed description of the unification algorithm, see Ref. 23.

Let us start with an empty substitution 8, take the first pair of terms T1 = { t l l , t2]}, and try to find the first disagreement set of T , that is, a pair {sl, s2) of subterms of terms t l l and t 2 ] , respectively, located by finding the first symbol at which terms t l l and t21 disagree and then extracting the terms beginning with those symbols.

If such a disagreement set cannot be found, then the terms t11 and ~ Z I are already identical and we can move to the next pair T2 = {t12, t 2 2 ) and continue in the same fashion. Otherwise, observe that from formula (1) and axiom (CET7) it follows that

Consequently-in view of axiom (CET6)-these two terms cannot differ by having different principal function symbols and therefore at least one of them, say sI , must be a variable, say, y. It follows from axiom (CET8) and formula (2) that the variable y does not occur in the term s2 . Therefore, we add the substitution { y ) s z } of the term s2 for the variable y to our substitution 6 and replace variable y by s2 in all the terms t i J . Now, we try to find the first disagreement set in the newly obtained, substituted terms t i l and ti, and continue the algorithm.

It is we1l:known that the unification algorithm always terminates and since it never fails, from the Unification Theorem (see Ref. 23) it follows that it produces the desired unification. The equality 6p = p follows immediately formula (2).

Lemma 3.2. Suppose that M is a minimal model of CET(P) with universe U. Suppose that A(tI , . . . , t,) is an atom, whose variables belong to the finite set X and a : X H U is a U-instantiation such that & I , . . . , t,)a = T. Then there exists a set Y of variables, a substitution 8:X H Terms(Y) and a U- instantiation p : Y H U such that:

P (V)A(t I , . . . , t,)O and 6p = a.

Proof. We will call the set

93 = {A(ul , . . . , u,) : A is a predicate symbol, u1 , . . . , u, E U }

of formal terms A ( u l , . . , u,) the base of M. I1 is clear that-assuming fixed universe U and fixed interpretation of functions-we can identify the model M with the following subset of 93:

U = { A ( u ~ , . . . , u,) E % : A ( u I , . . . , u,) = T}.


We will now define an increasing infinite sequence %, i = 1, 2 , . . . of subsets of B such that

Jl/l = u 93i.

Let CJio = 0 and having defined PArn let us define as follows:

= 93m U { A ( t l , . . . , t,)a: there is a clause in P A ( f i , . . . , t,) +Ai( t i i , . . . , f n d r . . . , A d t i k , . . . , t n d , k 2 0, and a U-instantiation a such that Vi A i ( t ~ i , . . . , t,,)a E am}.

Now we show that

U A = X, where X = i<E ai,

It follows easily from the fact that M is a model of P that

Jtd 3 SIT.

Since M is a minimal model of CET(P), in order to show that Jl/l C A" it suffices to show !hat the unique interpretation N of CET(P) determined by the set JY" is a model of CET(Pj.

Suppose that

is a clause in P. lm order to show that the above clause is satisfied in N we have to show that for every U-instantiation a either Actl , . . . , t,ja = T in N or there is an i such that ai(tIi, . . . , tnija = I in N . Suppose therefore that for every i we have .&(rli, . . , , t,Ja = T in N or-in other words-suppose that for every i we ha.ve:

There exists therefore an m such that for all i we have:

From the definition of am+ I it follows now that A ( [ ! , . . . , t,)a E %,,I C X and therefore A(rl, . . . , t,)a = T in N , which shows that At = X.

Now we can complete the proof of Lemma 3.2 . Suppose that A ( t l , . . . , t , ) is an atom, whose variables belong to the finite set X and a : X H U is an U- instantiation such that A ( t l , . . . , t,)a = T in M . This means that E = A ( t l , . . . , t,)a E A = U 93, and therefore E E am, for some m. We have to show that there exists a substitution 8 : X H Terms( Y) and a U-instantiation /3 : Y H U such that:


P ,‘= (V)A(?l, . , . , ?,)O and Op = a. (3)

The proof is by induction on m. If m = 0, then-since 30 = 0-there is nothing to prove. Suppose now that the above fact has been proven for E E a,,,. We will prove it for E E . By definition, there exists in P a clause

whose variables belong to a finite set X’ and a Uinstantiation a’ : X’ H U such that

ViAi(sli , . . . , s,;)a’ E and A(tl, . . . , ?,)a = A(s1, , . . , s,)a‘. (4)

Consequently,

Vi t;a = s;a’. ( 5 )

First of all, we will show that there exists a substitution w ’ :X’ w Terms( Y’) and a U-instantiation p‘ : Y‘ H’ U such that:

From formula (4) and the inductive assumption it follows that for every i there exists a substitution 0,’ : X’ H Terms( Y,’) and a U-instantiation pf : Y; H

U such that:

P ,‘= (V)Ai(sli, . . . , s,;)Of and Oipf = a’. (7)

We can clearly assume that the sets Yi are mutually disjoint and disjoint from X (otherwise, we can use renaming substitutions) and let Y’ = U Yf. Define p’ : Y ‘ H U as a combination (union) of instantiations p i . Then for every i, j 5 k we have:

By Lemma 3.1 there exists a substitution 0’ : Y’ H Terms( Y‘) such that for every i , j 5 n:

and

By formula (7) P (V)Aj(sli, . . . , sni)Oi and therefore also P (V)Ai (V)Ai(sli, . , . , s,;)w’ and ( S l i , . . . , s,i)OfO’ and thus Vi we have that P

therefore


which proves c!laim (6).

fromulae ( 5 ) anld (6) it follows that for all i Now, to complete the proof we define ui = siw’ and observe that from

Let W = {wl , . . . , w,} be any set of n variables disjoint from Y, where Y = X U Y’ (recall, that X and Y’ are disjoint sets). Define two substitutions : W H Terms(X) and O2 : W t-+ Terms( Y’) as follows:

el(wi) = t i ; e2(wi) = ui = siof

and let /3 : Y H U be a combination (union) of U-instantiations a and p’ . Then by formula (8) ‘we have:

and therefore by Lemma 3.1 there exists a substitution 8’ : Y H Terms( Y) such that:

elef = e2er =

and

e’p = p.

Since by formula (6) P 1 (V)A(sl, . . . , s,)w’ we therefore also have that P k: (V)A(sl, . . . , s,)w‘f?‘ and therefore since siw’@’ = ti@’ we have:

P (V)A( t , , . . . , t,)e’.

Define 8 : X H Terms( Y) to be the restriction of 8’ to the set X. Then from the previous formula and the fact that by definition the restriction of p to X is equal to a we get:

P ( V ) A ( t l , . . . , t , )O and Op = p = a,

which shows that formula (3) holds and thus completes the proof of the lemma.

IV. MINIMAL MODEL SEMANTICS, LEAST MODEL SEMANTICS, AND CWA FOR POSITIVE LOGIC PROGRAMS

Throughout this section we assume that P is a positive logic program. Our first theorem states that the minimal model semantics MIN(P) for P is categori- cal for positive existential formulae F, in the sense that either P k F and then obviously also MIN(P) F, or otherwise MIN(P) i F .


Theorem 4.1. Suppose that F is a positive existential formula. Then

P F e MIN(P) k 1 F .

Proof. Suppose that F is a positive existential formula and suppose that MIN(P) p 1 F . We have to show that P k F. Without loss of generality, we may assume that F = (3)G, where G = G I A . . . A G , is represented as a conjunction of positive clauses Gi and let X be the set of all variables occurring in G . There must exist a minimal model M of P with universe U such that M k F. Consequently, there is a &instantiation a : XH U such that G a is true in M. For every i < m we can therefore find an atom Aj(tli, . . . , t,i) belonging to Gi and such that

By Lemma 3.2, for every i there exists a substitution O i : X H Terms(Ui) and a U-instantiation p i : K I+ U such that:

P (V)Ai( t I j , . . . , tn i )d i and &pi = a.

We can clearly assume that the sets Yi are disjoint (otherwise, we can use renaming substitutions) and let Y = U Yi . Let /3 : Y w U be a combination of U- instantiations pi .

By Lemma 3.1 there exists a substitution 8 : Y H Terms( Y) such that for every i, j I n:

d i e = oje = w.

Since P k (V)Aj(tli, . . . , t,,i)di therefore also P k (V)Ai(tli, . . . , tni)8i8 and thus for all i we have that P k (V)Ai(tl i , . . . , t,Jw and therefore P k (V)Gw, which implies that P (3)G and therefore P k F , which completes the proof.

The following example illustrates the assumptions used. Example 4.1:

1. The assumption that F is positive is essential. Indeed, if P is given by a single clause p ( a ) and F = 3 x i p ( x ) , then P p F and yet MIN(P) p i F .

2 . The assumption that F is existential is also essential. Indeed, if P is as above and F = Vxp(x), then P p F and yet MIN(P)

3. The assumption that P is a positive logic program is essential. Indeed, if P consists of clauses p(a) and q(a) c i p ( x ) and F is a positive ground formula q(a), then P /d F and yet MIN(P) p 1 F .

4 . The assumption that only models of CET(P) are considered is also essential. Indeed, if P consists of clauses p ( a ) and q(a) t p ( f ( x ) ) and F is a positive ground formula q(a), then P p F and yet there exist minimal models of P in which F holds, because in some minimal models of P we may have, for example, a = f(a).

1 F .


The following corollary shows that for positive existential formulae F the minimal model semantics MIN(P) of P is equivalent to the least model semantics M P and that both are fully determined by the provability or nonprovability of F from P itself.

Corollary 4.2. Suppose that F is a positive existential formula. Then

P k F e MIN(P) k F e M p k F

and

Proof. This is an easy consequence of Theorem 4.1. Example 4.1 shows that all assumptions in the above Corollary are essen-

tial. Our main result shows that for all existential-positive or negative- formulae F all thr'ee semantics-the minimal model semantics MIN(P j, the least Herbrand model semantics M P , and the CWA-semantics CWA(P)-are equivalent. As shown by Example 4.1, though, they may no longer be determined simply by the provability or nonprovability of F from P.

Theorem 4.3 (Main). Suppose that F is an existential formula. Then

MIN(P) k F e M P k F @ CWA(P) k F .

-+ Proof. Let F = 3xG and let G be represented as a conjunction of clauses

G , , i 5 n . Clearly, if MIN(P) k F then M p k F . Suppose now that M P k F. Since M p represents a unique model, there must exist a ground substitution 8 such that M P k GO. For every i 5 n there exist literals I , in G, such that M P k 1,8. If 1, is positive, then-by Corollary 4.2-MIN(P) 1,8. If 1, is negative, then I , = l A , , where A , is an atom and M P l= i A , 8 . Again-by Corollary 4.2- it follows that MIN(Pj k i A , B and therefore MIN(P) k 1,8, for all i's. Conse- quently, MIN(P) I= G8 and therefore MIN(P) k F .

If CWA(P) k F , then clearly M P 1 F, because Mp is a model of CWA(P). On the other hand, if M P 1 F then there is a ground substitution 8 such that M P k FO and thereforle CWA(P) k F.

Notice, that tlhe above proof shows that for an existential formula F we have that MIN(P) /= F iff there is a ground substitution 8 such that MIN(P) F8.

Corollary 4.4. Suppose that F is a ground formula. Then

lMIN(P) k F G Mp k F CWA(P) k F

and

The above two results are important, because they confirm that, us far as ground or existential information is concerned, using CWA(Pj, all minimal


models or just the unique least Herbrand model produces exactly the same results, On the other hand, it is easy to see that for universal formulae the three approaches are no longer equivalent (cf. Example 4.1).

V. CIRCUMSCRIPTION, DOMAIN CIRCUMSCRIPTION, AND ECWA FOR UNIVERSAL THEORIES

Throughout this section we will consider an arbitrary universal theory T augmented by the axioms CET1-CET8 from Section 11. Such theories will be called CET theories.

Suppose that T is a CET theory and P = {pr , . . . , p m } and 2 = {zl , . . . , z,} are disjoint lists of predicate symbols from the language of T. The predicate symbols from 2 are called variables. Literals with predicate symbols not in 2 U P and positive literals with predicate symbols from P will be called marked literals.

Definition 5.1. Let K be an arbitrary ground formula not containing predicate symbols from 2. K is called free for negation in T if there exists no disjunction B = B , v . . . v B,, consisting of marked literals, such that

(9 TI= K V B ; (ii) T B.

Definition 5.2. The Extended Closed World Assumption of T w.r.t. P and 2 is the theory ECWA( T; P; 2) defined as follows: ECWA(T; P; 2) = T U CET U { i K : K is free for negation in T} .

Whenever it does not lead to a confusion we will use ECWA instead of ECWA(T; P; 2). To clarify the above definitions let us consider the following example.

Example 5.1. Let T be a CET theory consisting of the following state- ments:

Hot(x, s) --j Hot(x, result(move, x , s))

iHot(x, s) --j Moved(x, result(move, x, s))

These axioms describe a system of blocks A I , A2 , A3. The second axiom says that in the initial situation So, at least one of the blocks Al and A2 is believed to be hot, the third axiom states that movement of blocks has no impact on their temperature, while the fourth axiom guarantees that if a block x is not hot in a situation s then the operation of moving x will be successful (here result(move, x , s) is a situation term denoting the situation in which the system will find itself after the operation move is performed on x). We would like to assume that a block x is not hot unless we have some evidence to the contrary. This informal assumption will allow us to conclude that statement lHot(A3, s) is true while preventing us from making any conclusions about the temperature of the re-

COMPARING SEMANTICS 56 1

maining blocks. It is easy to see that if P = {Hot} and Z = {Moved} then for any situation term S, Hot(A 3 , S) is free for negation in T while Hot(A 1 , S), Hot(A2, S) are not. Consequently, unlike lHot(A I , S) and lHot(Az, S), lHot(A3, S) belongs to ECWA(T; P; Z). Consider now a statement Hot(A1) A Hot(A2). It is easy to see that it is also free for negation in T, hence its negation is in ECWA. This shows that ECWA turns an inclusive disjunction in the second axiom of T into an exclusive one.

To review the notion of Circumscription let us recall the following definitions. By p M we denote the extension of the predicate p in the model M.

Definition 5.3.u9u924 For any two models M and N of T we write M 5 N modulo (P, Z) if models M and N differ only in how they interpret predicates from P and Z and if for every predicate p from P, p M C P N .

Definition 5.4. A model M of T is (P, 2)-minimal if there is no model N such that N < M (i.e., such that N I M but not M I N).

Definition SS., A second order theory CIRC(T; P; Z) is called a Circum- scription (resp. Domain Circumscription) of T with P minimized and Z vaned if a structure (resp. Herbrande structure) M is a model of CIRC(T; P; 2) (resp. DCIRC(T; P; Z)) iff M is a (P, Z)-minimal model (resp. Herbrand model) of T.

For more detailed discussion see Ref. 13. The following theorem relates a syntactic definition of ECWA to the notion of a ( P I Z)-minimal model of T.

Theorem 5.1. For any ground formula F not containing predicates from 2, F is free for negation in T iff 1 F is true in all (P; 2)-minimal Herbrand models of T.

Proof. (+) Let us assume that F is not free for negation and therefore there exists a disjunction B = B 1 V . . . v B , consisting of marked literals, such that T k F v B and T p B. We will show that this implies the existence of a (P, Z)-minimal Herbrand model for the theory Tin which F is true. Since T B and the theory T is universal, in virtue of Lemma 3.3 in Ref. 4 there exists a Herbrand model N for Tin which l l 3 is satisfied. Let M by any (P, Z)-minimal Herbrand model of T such that M 5 N modulo (P, Z). The existence of such an M is guaranteed by Theorem 4.2 in Ref. 25. Since 1 B contains only marked literals and M is ( P , 2) minimal the sentence 1 B also has to be satisfied in M. Our assumption that T F v B now implies that M

(+) We assume now that there is a ( P , Z)-minimal Herbrand model M o of T such that Mo F and we will show that this assumption implies that F is not free for negation. First we show that for an arbitrary Herbrand model N for T such that N BN and Mo p B N .

If for every predicate symbol not in Z its extensions in N and in Mo are identical, then N F because no predicates from Z are in F. Since this is impossible by the definition of N , there is a predicate symbol A not in Z, whose extensions in N and Mo are not equal.

Consider two cases. If we can find such an A which does not belong to P then we can find al literal B N , whose predicate symbol does not belong to P U Z and such that N B N and Mo p B N . Otherwise, we can find such an A in P and in this case for each predicate symbol not in P U Z its extensions in N and Mo are identical. Since Mo is (P, 2)-minimal, there must exist a positive literal BN with predicate symbol in P, such that N k BN and M o p BN . If this were not the

F.

F w e can find a marked literal B N such that N


case then we would have N I Mo and since Mo is a (P, Z)-minimal model, this would imply that N and M have the same extensions for all predicate symbols not belonging to 2.

F} and let L be a possibly infinite disjunction of all literals in B . Clearly, for every Herbrand model N of T, F v L holds in N and therefore by Lemma 3.4 in Ref. 4 there is a finite subdisjunction E of L such that T k F v E. If E is empty then F is true in all Herbrand models and therefore in all models of T and cannot be free for negation. On the other hand, if E is not empty then T E, because Ma /= 3 E . This shows that F is not free for negation and completes the proof.

As the following example shows the assumption that there are no predicates from Z in F is essential.

Example 5.2. Consider a theory T = {p(x) vpcf (x ) ) , p(x) v q(a)}, where p belongs to P, q belongs to 2, and F is i q ( a ) . It is easy to see that for any ground term t , there is a disjunction B = p( t ) v p ( f ( t ) ) which satisfies the conditions from Definition 5.1 and therefore p(t) is not free for negation in T and hence ECWA q(a). On the other hand, any (P, Z)-minimal model M of T contains q(a), since otherwise for all ground terms t, p(t) would belong to M which contradicts (P, 2)-minimality of M. Therefore 1 F is true in all minimal (P, Z) models of T.

Now we will discuss the relationship between ECWA, CIRC, and DCIRC for CET theories. We will start with CIRC and DCIRC. By definition, for every formula F if CIRC k F then DCIRC k F. The following example shows that even for ground formulas DCIRC is essentially stronger than CIRC.

Example 5.3. Let T = { p ( a ) , p ( x ) v r(a)}, where both p and r belong to P. It is easy to see that DCIRC /= ~ ( a ) , while CIRC 1 ~ r ( a ) . To see why the latter is true let us consider the universe U = {a, w} and a structure M such that p~ = {a} and r M = {a}. This structure is a minmal model of T and therefore CIRC ~ r ( a ) .

Next, we discuss the relationship between ECWA and DCIRC. The following theorem shows that for ground formulae DCIRC is always stronger than ECWA.

For any ground formula F, if ECW A /= F then DCIRC

Let F be an arbitrary ground formula such that ECWA k F. It follows from the Definition 4.2 that F is true in any Herbrand model of T in which all free for negation formulas are false. In virtue of Theorem 4.1, all free for negation formulas are false in all (P, Z)-minimal Herbrand models of T and therefore DCIRC k F.

Example 5.2 shows that in the above Theorem the implication in the oppo- site direction does not always hold.

However, the following result shows that ECWA and DCIRC coincide for ground formulae not containing predicates from 2.

Theorem 5.3. ECWA and DCIRC coincide for ground formulae not containing predicates from 2.

Proof. Implication in one direction follows from Theorem 4.2. We have to show that for an arbitrary ground formula F not containing predicates from Z

Let B = { B N : N is a model of T such that N

Theorem 5.2.

Proof. k F-


ZDCIRC F. However, by Theorem 4. I , if DCIRC /= F then i F is free for negation and therefore ECW A k F.

Finally, we compare ECWA to CIRC. Corollary 4.3 shows that for definite theories with empty Q and 2, ECWA and CIRC coincide on ground formulae. Examples given in this section and in Section 111 show that, in general, this is not the case, that is, there are ground formulae which are implied by ECWA, but not by CIRC and vice versa.

F then ECW A

VI. CONCLUSION

The determination of the existing relationships between different formalizations of nonmonotonic reasoning not only clarifies relative power of different approaches and makes their semantics clearer, but it also may provide us with insights necessary to discover more general, unifying principles of nonmonotonic reasoning.

In this article we compared the deductive power of three nonmonotonic formalisms: circumscription (CIRC), domain circumscription (DCIRC), and the extended dosed world assumption (ECWA). In particular, we discussed their special cases, namely, the minimal model semantics (MIN), the least model semantics ( M T ) , and the closed world assumption (CWA). Table I sum- marizes the positiw results proved in the article, while the examples presented above indicate that these results cannot be significantly strengthened.

Observe, that for any dejinite theory T (i.e., for any positive logic program T ) and for any sentence F we have:

Also, note that any relationship that holds for all existential sentences automatically applies to ground sentences.

The article demonstrates subtle, and sometimes unexpected, differences between the corresponding formalisms. It also indicates the extent to which inference engines based on one of them can be applied to answering queries in systems based on the others. We believe that future work will discover areas of applicability for each of these formalisms.

Table I. Relationships between the three semantics. Theory T Sentence F Relationship Definite Positive Existential P k F = CIRC F - DCIRC F

Definite Existential ECWA F = CIRC + F - DCIRC F Definite Positive Existential P F = CIRC 1 F = DCIRC k i F

Universal Arbitrary CIRC F .$ DCIRC F Universal Ground ECWA k F 3 DCIRC F Universal Ground (no Z s ) ECWA k F = DCIRC k F


References

1 . R. Reiter, “On closed-world data bases,” In Logic andData Bases, H. Gallaire and J. Minker (Eds.), Plenum Press, New York, 1978, pp. 55-76.

2. J. Minker, “On indefinite data bases and the closed world assumption,” Proc. 6th Cogerence on Automated Deduction, Springer-Verlag, 1982, pp. 292-308.

3. A. Yahya and L. Henschen, “Deduction in non-Horn databases,” Journal of Auto- mated Reasoning, l(2) 141-160 (1985).

4. M. Gelfond and H. Przymusinska, “Negation as failure: Careful closure proce- dure,” Artificial Intelligence, 30, 273-287 (1986).

5. M. Gelfond, H. Przymusinska, and T. Przymusinski, “The extended closed world assumption and its relationship to parallel circumscription,” Proceedings ACM SZGACT-SIGMOD Symposium on Principles of Database Systems, Cambridge,

6. M. Gelfond, H. Przymusinska, and T. Przymusinski, “On the relationship between circumscription and negation as failure,” Artificial Intelligence, 38, 75-94 (1989).

7. J. McCarthy, “Circumscription-A form of nonmonotonic reasoning,” Artgcial Intelligence, 13, 27-39 (1980).

8. G. Bossu and P. Siegel, “Saturation, nonmonotonic reasoning and the closed world assumption,” Artificial Intelligence, 25, 13-63 (1985).

9. T. Przymusinski, “Perfect model semantics,” Proceedings of the Fourth Interna- tional Cogerence on Logic Programming, Seattle, WA, August 1988, pp. 1081- 1096.

10. M.H. Van Emden and R.A. Kowalski, “The semantics of predicate logic as a programming language,” Journ. ACM, 3(4), 733-742 (1976).

11. K. Apt and M. Van Emden, “Contributions to the theory of logic programming,”

12. J. McCarthy, “Applications of circumscription to formalizing common sense

13. V. Lifschitz, “Computing circumscription,” Proceedings IJCAZ-85, Los Angeles,

14. T. Przymusinski, “On the Declarative Semantics of Stratified Deductive Databases and Logic Programs,” In Foundations of Deductive Databases and Logic Program- ming, J. Minker (Ed), Morgan Kaufmann, Inc., Los Altos, CA, 1988, pp. 193-216.

15. T. Przymusinski, “On the relationship between logic programming and nonmonotonic reasoning,” Proceedings of the Seventh National Conference on Artificial Intelligence AAAZ’88, St. Paul, MN, August 1988, pp. 444-448.

16. V. Lifschitz, “Closed world data bases and circumscription,” Artificial Intelli- gence, 27, 229-235 (1985).

17. R. Reiter, “Towards a logical reconstruction of relational database theory,” In On Conceptual Modeling, M . Brodie et al. (Eds.), Springer-Verlag, 1984, pp. 191-233.

18. K. Kunen, “Negation in logic programming,” J. of Logic Programming, 4(4), 289- 308 (1987).

19. J.W. Lloyd, Foundations of Logic Programming, Springer-Verlag, Berlin, 1984. 20. J. Shepherdson, “Negation as failure,” J. Logic Programming, 2, 185-202 (1985). 21. J. Shepherdson, “Negation in logic programming,” In Foundations of Deductive

Databases and Logic Programming, J. Minker (Ed.), Morgan Kaufmann, Inc., LOS

22. J.W. Lloyd, and R.W. Topor, “Making Prolog more expressive,” Journ. of Logic Programming, 1, 225-240 (1984).

23. C. Chang and R.C. Lee, Symbolic Logic and Mechanical Theorem Proving, Aca- demic Press, New York, 1973.

24. D. Etherington, “Reasoning with Incomplete Information. Investigations of Non- Monotonic Reasoning,” Ph.D. Thesis, Dept. of Computer Science, University of British Columbia, 1986.

25. V. Lifschitz, “On the satisfiability of circumscription,” Artificial Intelligence, 28,

MA, March 1986, pp. 133-139.

JACM, 29, 841-862 (1982).

knowledge,” J. Artificial Intelligence, 28, 89-1 16 (1986).

1985, pp. 121-127.

Altos, CA, 1988, pp. 19-88.

17-27 (1986).

on the relationship between cwa, minimal model, and minimal herbrand model semantics

Documents