Apt, K.R. (Krzysztof)

Blair, H.A.

English

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Fundamenta lnformaticae XIV (1991) 339-343 !OS Press ARITHMETIC CLASSIFICATION OF PERFECT MODELS OF STRATIFIED PROGRAMS Krzysztof R. APT 339 Centre for Mathematics and Computer Sciences, P.O. Box 4079, 1009 AB Amsterdam, Netherlands and Department of Computer Sciences, University of Texas at Austin, Austin, Texas 78712-1188, USA and Howard A. BLAIR School of Computer and Information Science, 4-116 CST, Syracuse University, Syracuse, NY 13244-4100, USA RECURSION-FREE PROGRAMS The following section completes the analysis of arithmetic complexity of perfect models and has been inadvertently omitted in the previous version of the paper. We say that a general program P is recursionjree if in its dependency graph Dp there is no cycle. Clearly recursion-free programs form a subclass of stratified programs. Recursion-free programs form a very simple generalization of the class of hierarchical programs introduced in [C78]. Hierarchical programs satisfy an additional condition on variable occurrences in clauses that prevents floundering, i.e. a forced selection of a non-ground negative literal in an SLDNF- derivation. In this section we study the complexity of perfect models of recursion-free programs. 1. Hierarchical stratifications We call a stratification P = P1 U · · · U Pn of a general program hierarchical if for i = 1, ... ,n and for every relation symbol which occurs in a body of a general clause from P;, its definition is contained within some P1 forj<i. . . . . The following lemma shows that general programs adrruttmg hierarchical stratifications and recursion-free programs coincide. It is in fact a special case of the well known fact that a finite relation can be topologically sorted iff it is acyclic. Therefore we omit the proof. LEMMA 1: A general program P is recursionjree iff there is a hierarchical stratification ef ~ 0 *)to the version of this paper that appeared in Fundamenta lnformaticae 13, pp.1-18, 1990. 340 K.R. Apt and HA. Blair I Perfect Models of Stratified Programs 2. Completions of recursionjree programs In the sequel we shall study comp(P), CLARK'S [C78] completion of a general pro-gram P. Its definition can be found in [Ll87]. comp(P) is a theory whose language is the first order language L(P) of the general program P augmented by the equality relation symbol"=". Given a general program P we denote by L(=) the language obtained by deleting from L (P) all relation symbols and by adding "=" to it. From now on in presence of a general program P we denote L (P) by L. comp(P) is a set of formulas which consists of free equality axioms, which we denote by Eq, together with certain other formulas, about which we only need to know the following two properties. PROPERTY 1: For every m-ary relation symbol q of L with the empty definition in P, the formula Vx 1 ••. 'v'xm--, q(x 1' ... , Xm) is in comp (P). PROPERTY 2: For every m-ary relation symbol q of L with the non-empty definition in P, there is in comp (P) a formula of the form q(x J, · · . , Xm)-fq such that every relation symbol occurring in fq, other than "= ", occurs in the definition of q in P. The equality axioms in Eq are the usual axioms of first-order logic with equality that say that = is a congruence, together with axioms that say that the function sym-bols of L, including the 0-ary ones, denote one-one functions with disjoint ranges, and axioms which say that all functions definable by composition from the given function symbols have no fixed points. As pointed out by KuNEN [K87], these are the axioms required for justifying the soundness of both success and failure of unification. In the proof of the next lemma we shall need the following result from mathemati-cal logic (see [Sh67] p. 34). THEOREM 2: (Equivalence Theorem). Let T be a theory and q, a formula. Suppose that q,' is obtained from q, by replacing some, possibly all occurrences of subformulas f i. ... , fn by o/' J, ... , f' n respectively. Then if for i = 1, ... , n, Trf;-o/';, then Trq,-q,'. D Here replacing involves an appropriate renaming of variables performed in order to avoid variable clashes. LEMMA 3: Let P be a recursionjree program. For every atom A of L there exists a for-mula q, A of L ( =) all of whose free variables occur in A such that comp(P)rA -'i>A· (1) PROOF. Let P = P 1 U · · · U Pn be a hierarchical stratification of P whose existence is guaranteed by lemma 1. We define a mapping height from relation symbols of L K.R. Apt and H.A. Blair I Perfect Models of Stratified Programs 341 into {O, l, ... ,n} as follows. ~et r be a relation. symbol of L_ whose definition within P is empty. Then we put he1ght(r)=O. Otherwise we put he1ght(r)=i iff the definition of r is contained in P·. Suppose that A-~(ti..:., tm) for a_relation symbol rand terms tJ, ... ,tm. 'we prove the lemma by mduction on the height of r. If height(r)=O, then by property 1 comp(P) r A ~false, sow~ can take false for <f>A· We can regard false as an abbreviation of -,'Vx(x =x). If height (r)= 1, then by property 2 lji, is a formula from L(=) with free variables X1> ... , Xm· Let lji', stand for lji,{x1ltJ, ... , Xmltm}· Then fr is a formula from L ( =) all of whose free variables occur in A such that comp(P)rA ~lji'n (2) so we can take 1// r for </>A. Assume n~w that the claim holds for all relation symbols with height <k and sup-pose that he1ght(r)=k. By property 2, (2) holds, but where, instead of lji', being a for-mula of L(=), every relation symbol q occurring in lji', and different from occurs in the definition of r in P. But the stratification P=P1U···UPn of P is hierarchical, so every such relation symbol q is of height <k. Thus by the induction hypothesis, for every atom B occurring in o/' r and whose relation symbol differs from "= ", there exists a formula <f>B if L ( =) all of whose free variables occur in B such that comp(P)rB ~<f>B· (3) Now, replace each occurrence of such an atom B in lji', by <f>B and call the resulting formula <f>A· Note that <f>A is a formula of L(=) and that all its free variables appear in A. Now by theorem 2 we get (1) by virtue of (2) and (3). D COROLLARY 4: Let P be a recursionfree program. For every formula</> of L there exists a formula lji of L ( =) all of whose free variables occur as free variables in </> and such that PROOF. By lemma 3 for every atom A occurring in </> there exists a formula <f>A of L(=) all of whose free variables occur in A and such that (1) holds. Now, replace each occurrence of an atom A in</> by <f>A and call the resulting formula \jiq,. Then o/.p is a formula of L ( =) all of whose free variables occur as free variables in cJ>. By theorem 2 we now get the desired conclusion by virtue of (1). D 3. Domain closure axiom In the sequel we shall refer to a number of basic concepts from mathematical logic which we now briefly recall. By a closed formula we mean a formula without free variables. A theory T is called complete if for all closed formulas </> either Tr</> ?r Tr-,cp. A theor~ T is calle~ con-sistent if for no formula cp both TrcJ> and Tr-,<J>. Fmally, a theory T is called decidable if (after the standard encoding) the set { <J>:Tr</>} is recursive. Let L be a first order language with finitely many function symbols and constants. 342 K.R. Apt and H.A. Blair I Perfect Models of Stratified Programs By DCA (the domain closure axiom) we mean the following first order formula of L: '<Ix j3Y1 ... 3yn(x=f(Y1, · · · ,yn)), where n is the number of arguments off, and is 0 if f is a constant. Thanks to the restriction on L, DCA is indeed a first order formula. For example, if L contains one constant a, one unary function symbol f and one binary function symbol g, then DCA can be taken as '<lx(x =a V 3y(x =/(JI) V 3y 13y2(x = g(Y 1,yz))). We now need the following result due to MAHER [M88]. THEOREM 5: Let L be a first order language with = and with finitely many function sym-bols and constants but at least with one constant. Then Eq U {DCA} is a complete and decidable theory. D Note that the Herbrand base corresponding to the language L augmented by "=" is a model of Eq U {DCA}, so Eq U {DCA} is also a consistent theory. Recall that we originally assumed that each general program P contains at least one constant and one function symbol. So we can apply the above theorem here. We can now prove the main result of this section. THEOREM 6: Let P be a recursionfree program. Then comp(P) U {DCA} is a complete and decidable theory. PROOF. Let cp be a closed formula of the language L(P) augmented by =. Then the formula ifi<P from corollary 4 is closed, as well. By corollary 4 comp(P)U{DCA}r</> iff comp(P)U {DCA}rl/iq,. Moreover by theorem 5 comp(P) U {DCA} r t/iq, !ff Eq U {DCA }ri/Jq, , since comp(P) U {DCA} is consistent. Combining these two equivalences we get comp(P) U {DCA } r </> !ff Eq U {DCA} r'l/iq, . (4) But by the form of i/Jq, we have that i/J..,q, is identical to -.i/Jq,, so since --,cf> is a closed formula, as well, comp(P) U {DCA }h</> !ff Eq U {DCA }hi/Jq,. (5) Now by virtue of theorem 5, (4) implies that comp(P) U {DCA} is decidable and (4) and (5) imply that comp(P) U {DCA} is complete. COROLLARY 7: Let P be a recursionfree program. Then for every ground atom A A EMp !ff comp(P) U {DCA }rA . D PROOF. Mp is a model of comp(P)U{DCA}. Thus A EMp implies that comp(P) U {DCA} r -,A does not hold which by theorem 6 implies comp(P)U{DCA}rA. Also, At/.Mp implies that comp (P)U{DCA}rA does not hold. D K.R. Apt and H.A. Blair I Perfect Models of Stratified Programs 343 We can obtain the desired conclusion. COROLLARY 8: Let P be a recursionjree program. Then Mp is recursive. PROOF. By corollary 7 and theorem 6. D 4. Discussion It is straightforward to define the completion of a program, hence Eq, and in the case of stratified programs, the standard model Mp, with respect to a language L with possibly infinitely many function and relation symbols, where L is any particular extension of the smallest language L(P) in which the clauses of P can be expressed. It is easy to see that Mp is in fact dependent on the underlying language L. The point of view where all programs are taken as sets of clauses over the same denumer-able (effectively presented) language Lis extensively discussed by Maher [M88a]. If comp(P) and Mp are defined in this way with respect to L, the fundamental results that Mp is independent of the stratification of P, and that Mp is a model of comp(P) continue to hold. If the set of function symbols of L is infinite (and the sets of function and relation symbols of L are suitably effectively presented), then the theorem and corollaries of section 3 continue to hold, provided DCA is deleted from their statements and proofs. This holds because for such a language L, the equality theory Eq, (without a domain closure axiom, which would require an infinite disjunction to express) is com-plete and decidable, (cf. [K87]). REFERENCES [C78] [Ll87] [K87] [M88] [M88a] K.L. CLARK, Negation as Failure, in: Logic and Databases, (H. Gallaire and J. Minker, eds.), Plenum Press, New York, 1978, pp. 293-322. J.W. LLOYD, Foundations of Logic Programming, 2nd edition, Springer-Verlag, 1987. K. KUNEN, Negation in Logic Programming. J. of Logic Programming, Vol. 4, 1987, pp. 289-308. M.J. MAHER, Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees, in: Proc. of the 3rd Logics in Computer Sci-ence Conference, 1988, pp. 348-357. M.J. MAHER, Equivalences of Logic Programs, in: Foundations of Deductive Databases and Logic Programming, (Jack Minker, ed.) Morgan-Kaufmann, Los Altos, CA. 1988, pp. 627-658. 

Arithmetic classification of perfect models of stratified programs

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