# Equivalence in Answer Set Programming

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3 use of such transformations for theoretical purposes. Some popular transformations for normal programs (RED +, RED, SUC, Failure, Contra) had been generalized to disjunctive programs and proved to preserve equivalence. We present here a generalization of the Loop transformation, defined in [9], and prove that it also preserves equivalence. We also continue and generalize some of these transformations to free programs (which allow default negation in the head of clauses) and prove that they preserve strong equivalence. We also show a theoretical application of such transformations. They are used, in particular, to prove and construct a linear time computable transformation of free programs (with negation in the head) to disjunctive ones. This simple transformation allows us to compute answer sets for programs up to the augmented type (with nested expressions) using current software implementations, which can deal with disjunctive clauses and constraints only. Of course they also propose a way to implement new software capable to compute answer sets in more general circumstances. We think our results will contribute to a better understanding of the notion of ASP in general propositional theories. It is a standard procedure in answer set programming to work with finite propositional theories. Variables are removed from programs with predicates by grounding. As Lifschitz noted in [14] We treat programs as propositional objects; rules with variables are viewed as schemata that represent their ground instances. Function symbols are not allowed so that the ground instance of a program is always finite. This is why our attention is mainly restricted to finite propositional theories. However, it is also discussed how some of our results can be applied to programs with predicates before making these programs ground. Our paper is structured as follows: in Section 2 we present the syntax of disjunctive, general, free and augmented programs, we also include the official definition of answer sets for augmented logic programs. In Section 3 we present the definition of some si-logics and our results for relations between Jankov and Intuitionistic logics. In Section 4 we present an application of our results to ASP. We prove some results on strong equivalence. Some transformations for disjunctive and free programs are discussed. Also reductions of programs to simplify their structure are proposed. In Section 6 we present our conclusions. Finally in Section 7 we give the proofs of our results. 2 Background A signature L is a finite set of elements that we call atoms, or propositional symbols. The language of propositional logic has an alphabet consisting of proposition symbols: p 0, p 1,... connectives:,,, auxiliary symbols: (, ). Where,, are 2-place connectives and is a 0-place connective. Formulas are built up as usual in logic. If F is a formula we will refer to its signature L F as the set of atoms that occur in F. The formula F is introduced as an abbreviation for F (default negation), and F G as an abbreviation for

4 (F G) (G F ), also abbreviates. Observe that can be equivalently defined as a a where a is any atom. A literal is either an atom a, or the negation of an atom a. The complement of a literal is defined as (a) c = a and ( a) c = a, analogously given M a set of literals M c = {(a) c a M}. Given a set of formulas F, we define the set F = { F F F}. Also, for a finite set of formulas F = {F 1,..., F n }, we define F = F 1 F n and F = F1 F n. If F = then we define F = and F =. When a formula is constructed as a conjunction (or disjunction) of literals, F = l (or F = l) with l a set of literals, we denote by Lit(F ) such set of literals l. Elementary formulas are atoms and the connectives and [22], and an formula is a formula constructed by using (only) the logical connectives {,, } arbitrarily nested. For instance (a b) ( c (p q)) is an formula. A clause is a formula of the form H B where H and B, arbitrary formulas in principle, are known as the head and body of the clause respectively. If H = the clause is called a constraint and may be written as B. When B = the clause is known as a fact and can be noted just by H. An augmented clause is a clause where H and B are any formulas. A basic formula is either or an formula, which does not contain the negation as failure operator ( ). Note that a broader form of this type of clauses is considered in [22] where they also include two kinds of negation as well as implications in the body. We only include one kind of negation that corresponds to their default negation, however they use the symbol not for our negation. A free clause is a clause of the form (H + H ) (B + B ) where H +, H, B +, B are, possibly empty, sets of atoms. Sometimes such clause might be written as H + H B +, B following typical conventions for logic programs. When H =, there is no negation in the head, the clause is called a general clause. If, moreover, H + (i.e. it is not a constraint) the clause is called a disjunctive clause. When the set H + contains exactly one element the clause is called normal. Finally, a program is a finite set of clauses. If all the clauses in a program are of a certain type we say the program is also of this type. For instance a set of augmented clauses is an augmented program, a set of free clauses is a free program and so on. For general programs, and proper subclasses, we will use HEAD(P ) to denote the set of all atoms occurring in the head of clauses in P. 2.1 Answer sets We now define the background for answer sets, also known as stable models. This material is taken from [22] with minor modifications since we do not consider their classical negation. Definition 2.1. [22] We define when a set X of atoms satisfies a basic formula F, denoted by X = F, recursively as follows: for elementary F, X = F if F X or F =.

5 X = F G if X = F and X = G. X = F G if X = F or X = G. Definition 2.2. [22] Let P be a basic program. A set of atoms X is closed under P if, for every clause H B in P, X = H whenever X = B. Definition 2.3. [22] Let X be a set of atoms and P a basic program. X is called an answer set for P if X is minimal among the sets of atoms closed under P. Definition 2.4. [22] The reduct of an augmented formula or program, relative to a set of atoms X, is defined recursively as follows: for elementary F, F X = F. (F G) X = F X G X. (F G) X = F X G X. ( F ) X = if X = F X and ( F ) X = otherwise. (H B) X = H X B X. P X = {(H B) X H B P } Definition 2.5 (Answer set). [22] Let P be an augmented program and X a set of atoms. X is called an answer set for P if it is an answer set for the reduct P X. As can be seen, answer sets are defined for propositional logic programs only. However this definition can be extended to predicate programs, which allow the use of predicate symbols in the language, but without function symbols to ensure the ground instance of the program to be finite. So a term can only be either a variable or a constant symbol. A substitution is a mapping from a set of variables to a set of terms. The symbol θ is used to denote a substitution. The application of a substitution θ over an atom p will be denoted as pθ. For more formal definitions and discussions related to substitutions we refer to [16]. Suppose for example we have a substitution θ where θ := [X/a, Y/b, Z/a]. Then p(x, Y, Z, c)θ is the ground instance p(a, b, a, c). The ground instance of a predicate program, Ground(P ), is defined in [14] as the program containing all ground instances of clauses in P. Then M is defined as an answer set of a predicate program P if it is an answer set for Ground(P ). We want to stress the fact that the general approach for calculating answer sets of logical programs is to work with their ground instances. However we will see that some of our proposed transformations can be applied to programs before grounding, so that less computational effort might be required. 3 Results on logic A si-logic is any logic stronger (or equal) than intuitionistic (I) and weaker than classical logic (C). For an axiomatic theory, like I, we also use its name to denote the set of axioms that define it. Jankov Logic (Jn) is the si-logic

6 Jn = I { p p}, where the axiom schema p p characterizing it is also called weak law of excluded middle. The HT logic (or G 3 ) can be defined as the multivalued logic of three values as defined by Gödel, or axiomatically as I {( q p) (((p q) p) p)}. We write X to denote the provability relation in a logic X. Two formulas F and G are said to be equivalent under a given logic X if X F G, also denoted as F X G. We present now several results on intuitionistic and Jankov logics, which will be used later to prove and construct other properties and transformations related to Answer Set Programming. However notice that the following results and definitions in this section are not restricted to the syntax of clauses or logical programs. Formulas are as general as they can be built typically in logic. Lemma 3.1. Let T be any theory, and let F, G be a pair of equivalent formulas under any si-logic X. Any theory obtained from T by replacing some occurrences of F by G is equivalent to T (under X). Definition 3.1. The set P of positive formulas is the smallest set containing all formulas that do not contain the connective. The set N of two-negated formulas is the smallest set X with the properties: 1. If a is an atom then ( a) X. 2. If A X then ( A) X. 3. If A, B X then (A B) X. 4. If A X and B is any formula then (A B), (B A), (A B) X. For a given set of formulas Γ, the positive subset of Γ, denoted as Pos(Γ ), is the set Γ P. Proposition 3.1. Let Γ be a subset of P N, and let A P be a positive formula. If Γ I A iff Pos(Γ ) I A. Proposition 3.2. Let a 1,..., a n be all the atoms occurring in a formula A. Then Jn A iff ( a 1 a 1 ),..., ( a n a n ) I A. Lemma 3.2. Let A be a positive formula and Γ be a subset of P N. Γ Jn A iff P os(γ ) I A. 4 Applications to ASP In this section we return to the context of Answer Set Programming. So far we can define several types of equivalence between programs. For example, two logic programs P 1 and P 2 are said to be equivalent under stable, denoted P 1 stable P 2, if they have the same answer sets. Similarly P 1 and P 2 are equivalent under logic X if the formula X P1 P 2 is provable. Another useful definition of equivalence, known as strong equivalence, can be defined in terms of any equivalence relation for logic programs.

7 Definition 4.1 (Strong equivalence). Two programs P 1 and P 2 are said to be strongly equivalent (under X) if for every program P, P 1 P and P 2 P are equivalent (under X). This definition is given in [15] in the context of stable semantics where it is found to be particularly useful. The main purpose of our paper is now to stand some relations between these different types of equivalence. In general, it is clear that strong equivalence implies equivalence. But the converse is not always true. Example 4.1. Consider the programs P 1 = {a b} and P 2 = {a}. They are equivalent in stable because {a} is the unique stable model for both programs. However P 1 {b a} has no stable models, while P 2 {b a} has {a, b} as a stable model. One first important connection of Answer Set Programming with Logic is given in the following theorem. Theorem 4.1 ([15]). Let P 1 and P 2 be two augmented programs. Then P 1 and P 2 are strongly equivalent under stable iff P 1 and P 2 are equivalent under G 3 logic. In [13] a different version of this theorem is proved using Jankov Logic instead of the G 3 logic. Theorem 4.2 ([13]). Let P 1 and P 2 be two augmented programs. Then P 1 and P 2 are strongly equivalent under stable iff P 1 and P 2 are equivalent in Jankov logic. Using the machinery of logic and results from previous section we can propose alternative ways of testing whether two programs are strong equivalent, and some simple transformations that preserve strong equivalence. Lemma 4.1. Let P 1 and P 2 be two augmented programs, and let {a 1,..., a n } be the atoms in L P1 P 2. Then the programs P 1 and P 2 are strongly equivalent iff ( a 1 a 1 ),..., ( a n a n ) I P1 P 2. Consider the following program P 1 that shows that intuitionistic logic cannot be used in theorem 4.2 (the program, as well as the observation, was given in [15]). P 1 : q p. p q. r p q. s p. s q. Let P 2 be P 1 {s r}. We can show, by Theorem 4.2, that they are strongly equivalent by showing that they are equivalent in Jankov logic. Note that by using our lemma 4.1 we can use intuitionistic logic to get the desired result.

8 Furthermore, it is possible to use an automatic theorem prover, like porgi implemented in standard ML and available at allen/porgi.html, to help us to solve our task. We can also use lemma 3.2 to reduce a proof of strong equivalence to a proof in intuitionistic logic without having to add the extra assumptions ( a i a i ). Of course, we need the hypothesis of the lemma to hold. An interesting case is the following. Lemma 4.2. Let P be a disjunctive program and a an atom. P is strongly e- quivalent to P {a} iff P os(p ) I a. Lemma 4.3. Given an augmented program P and a negated atom a, then P is strongly equivalent to P { a} iff P C a The above two lemmas allow us to add known or provable facts to programs in order to simplify them and make easier the computation of answer sets. Lemma 4.4 (Negative Cumulativity). Stable satisfies negative cumulativity, namely: for every atom a, if a is true in every stable model of P, then P and (P { a}) have identical stable models. 5 Program Transformations In this section we will study some transformations that can be applied to logic programs. We verify some properties and applications, and stress the use of such transformations for theoretical reasons. We show how they can be used to define semantics and impose desirable properties to them. 5.1 Disjunctive Programs We will first discuss some transformations defined for disjunctive programs. Given a clause C = A B +, B, we write dis-nor(c) to denote the set of normal clauses {a B +, (B (A \ {a})) a A}. This definition is extended to programs as usual, applying the transformation to each clause. For a normal program P, we write Definite(P ) to denote the definite program obtained from P removing every negative literal in P. For a disjunctive program, Definite(P ) = Definite(dis-nor(P )). And given a definite program P, by MM (P ) we mean the unique minimal model of P (which always exists for definite programs, see [16]). The following transformations are defined in [4] for disjunctive programs. Definition 5.1 (Basic Transformation Rules). A transformation rule is a binary relation on the set of all programs defined within the signature L. The following transformation rules are called basic. Let a disjunctive program P be given. RED + : Replace a rule A B +, B by A B +, (B HEAD(P )).

9 RED : Delete a clause A B +, B if there is a clause A in P such that A B. SUB: Delete a clause A B +, B if there is another clause A B +, B such that A A, B + B + and B B. Example 5.1. Let P be the program: P : a b c c d. a c b. c d e. b c d e. then HEAD(P ) = {a, b, c, d} and we can apply RED + on the 4th clause to get the program P 1 : a b c c d. a c b. c d. b c d e. If we apply RED + again we will obtain P 2 : a b c c d. a c b. c d. b c d. And now we can apply SUB to get the program P 3 : a c b. c d. b c d. Finally we can remove the third clause using RED P 4 : a c b. c d. We observe that the transformations just mentioned above are among the minimal requirements a well-behaved semantics should have (see [8]). The following are two other transformations as defined in [4]. GPPE a : (Generalized Principle of Partial Evaluation) If P contains a clause A B +, B, B +, and B + contain a distinguished atom a, where a / (A B + ), replace such clause by the following n clauses (i = 1,..., n): A (A i \ {a}) (B + \ {a}) B i +, (B B i ), where A i B i +, B i are all clauses with a A i. If no such clauses exist, we simply delete the former clause. If the transformation cannot be applied then GPPE a behaves as the identity function over A B +, B.

10 TAUT a : (Tautology) If there is a clause A B +, B such that A B + and A B + contain a distinguished atom a then remove such clause. If the transformation cannot be applied then TAUT a behaves as the identity function over A B +, B. The rewriting system that contains, besides the basic transformation rules, the rules GPPE and TAUT is introduced in [4] and denoted as CS 1. This system is also used in [5] to define a semantic known as the disjunctive wellfounded semantics or D-WFS. However, the computational properties of the CS 1 system are not very good. In fact, computing the normal form of a program is exponential, whereas it is known that the WFS can be computed in quadratic time. Some other transformations need to be defined in order to solve this computational problem. Definition 5.2 (Dsuc). If P contains the clause a and there is also a clause A B +, B such that a B +, then replace it by A (B + \ {a}), B. It is not hard to verify that the transformations RED, DSuc, TAUT and SUB preserve strong equivalence, while RED + and GPPE preserve equivalence only. The transformation Dloop is defined in [9] with the purpose of generalizing the useful Loop transformation [6]. Recall that the well-founded model computation of SMODELS can be considered as an efficient implementation of a specific strategy where the use of Dloop is essential [6]. The authors in [17] claimed that Dloop preserves equivalence under stable, but they only provided a vague hint for the proof. We formally prove that indeed DLoop preserves equivalence. Our proof follows the spirit of the proof of this proposition in the case of normal programs [6], but more technical difficulties are now involved. Definition 5.3 (Dloop). Let unf (P ) = L P \MM (Definite(P )). Then we define Dloop(P ) = {A B +, B P B + unf (P ) = }. Proposition 5.1. Let P be a disjunctive program. If a unf (P ) is an atom and P 1 is obtained from P by an application of TAUT a, then unf (P ) = unf (P 1 ) and Dloop(P ) = Dloop(P 1 ). Let TAUT + a denote the transitive closure of TAUT a. Thus, TAUT + a (P ) is the program obtained by the application as long as possible of the transformation TAUT a to P. The next corollary follows immediately. Corollary 5.1. Let P be a program. If a unf (P ) and P 1 = TAUT + a (P ) then unf (P ) = unf (P 1 ) and Dloop(P ) = Dloop(P 1 ). Similarly we will now prove that, if the program does not contain any more tautologies with respect to a, then unf and Dloop does not change after a single application of GPPE a. This result will also extend immediately for GPPE + a, the transitive closure of GPPE a.

11 Proposition 5.2. Let P be a disjunctive program such that P = TAUT + a (P ). If a unf (P ) and P 1 is obtained from P by an application of GPPE a, then unf (P ) = unf (P 1 ) and Dloop(P ) = Dloop(P 1 ). Corollary 5.2. Let P be a disjunctive program such that P = TAUT + a (P ). If a unf (P ) let P 1 = GPPE + a (P ), then unf (P ) = unf (P 1 ) and Dloop(P ) = Dloop(P 1 ). Theorem 5.1 (Dloop preserve stable). Let P 1 be a disjunctive program. If P 2 = Dloop(P 1 ) then P 1 and P 2 are equivalent under the stable semantics. If we consider CS 2 as the rewriting system based on the transformations SUB, RED +, RED, Dloop, Dsuc as defined in [1], it is possible to define a semantic called D1-WFS [18]. Consider again example 5.1. As we noticed before program P reduces to P 4. Observe unf (P 4 ) = {a, b}, and by an application of Dloop we can obtain P 5, which contains the single clause: c d. For normal programs both systems D-WFS and D1-WFS are equivalent since they define WFS. However the residual programs w.r.t CS 1 and CS 2 are not necessarily the same, and for disjunctive programs they may give different answers. An advantage of CS 2 over CS 1 is that residual programs are polynomial-time computable, as it can be easy seen by the definition of the tranformations. We will show also how to generalize the Dloop transformation to work with predicate programs. The definitions of MM (P ) and Definite(P ) are generalized for predicate programs applying corresponding definitions to the ground instance of P. Recall that θ denotes a substitution of variables over terms. Definition 5.4 (Predicate Dloop). Let P be a disjunctive predicate program, and let L be the language of Ground(P ). We define: unf (P ) = L \ MM (Definite(P )). P 1 = {A B +, B P θ, p unf (P ), b B + we have bθ p}. P 2 = {A B +, B Ground(P ) \ Ground(P 1 ) B + unf (P ) = }. The transformation Dloop reduces a program P to Dloop(P ) := P 1 P 2. Theorem 5.2 (Dloop preserves stable). Let P and P be two disjunctive predicate programs. If P is obtained of P by an application of Dloop then P and P are equivalent under stable. Example 5.2. We illustrate a Dloop reduction for a disjunctive predicate program P : m(b). h(a). p(x) q(x) f(x). f(x) q(x). r(x) k(x) p(x). k(x) m(x). k(x) h(x). w(x) m(x). = m(b). h(a). r(x) k(x) p(x). k(b) m(b). k(a) h(a). w(b) m(b).

12 Also observe that unf (P ) = {m(a), h(b), p(a), p(b), q(a), q(b), f(a), f(b), w(a)}. 5.2 Free Programs Now we propose a set of Basic Transformation Rules for free programs, which defines a rewriting system that we will call CS++. These transformations are generalizations of notions already given for disjunctive programs. Definition 5.5. A transformation rule is a binary relation on free programs over L. The following transformation rules are called basic. Let P be a program. (a) Delete a clause H B if there is another clause H in P such that (Lit(H )) c Lit(B). (b) Delete a clause H B if there is another clause H B in P such that Lit(H ) Lit(H) and Lit(B ) Lit(B). (c) If there is a clause of the form l where l is a literal, then in all clauses of the form H B such that l Lit(B) delete l from B. (d) If there is a clause of the form l where l is a literal, then in all clauses of the form H B such that l c Lit(H) delete l c from H. (e) Delete a clause H B if it is a simple theorem in Jn. By definition, H B is a simple Jn theorem only if: Lit(H) Lit(B) or Lit(B) is inconsistent (it contains both b and b for some atom b). Lemma 5.1 (CS++ is closed under Jn Logic). Let P 1 and P 2 be two programs related by any transformation in CS++. Then P 1 Jn P 2. It has been found that CS++ is very useful in ASP. The transformations proposed for this rewriting system can be applied without much computational effort. They are also generalizations of transformations given for normal programs. In the case of normal programs CS++, without reducing inconsistency in the body as a theorem and plus two more transformations (Loop and Failure), is strong enough to compute the WFS semantics efficiently, see [6]. In many useful cases the stable semantics has only one model that corresponds to the WFS semantics. Moreover, sometimes these transformations can transform a program to a tight one [17]. For tight programs the stable semantics corresponds to the supported semantics. Recent research [2] has shown that, in this case, a satisfiability solver (e.g. SATO [24]) can be used to obtain stable models. Interestingly, some examples are presented in [2] where the running time of SATO is approximately ten times faster than SMODELS, one of the leading stable model finding systems [21]. Example 5.3. Consider the following example that illustrates some reductions obtained applying CS++ transformations to a program: a b c d a c t k p p q t = a c t

13 5.3 Transformations Finally we would like to present some other transformations that can be used to simplify the structure of programs. The concept of an answer set for a program has been generalized to work with augmented programs, however current popular software implementations do not support more than disjunctive programs. These transformations will allow us to compute answer sets for programs with more complicated logical constructions. Lifschitz, Tang and Turner offered [22] a generalization of stable semantics for augmented programs. They also showed that it is possible to transform an augmented program into a free one without altering the resulting stable models. Theorem 5.3 ([22]). Any augmented program is strongly equivalent under stable to a free program. Inoue and Sakama showed [20] that every free program P can be transformed into disjunctive one P such that P is a conservative extension of P. We say that P is a conservative extension of P to denote the fact that M is a stable model of P iff M is a stable model of P such that M = M L P. However their transformation is very expensive and involves a cubic increase in the size of programs. We show this same goal can be achieved with a linear time computable transformation whose increase of program s size is also linear. Lemma 5.2. Let P be a free program. For a given set S L P let ϕ : S Σ be a bijective function, where Σ is a set of atoms such that Σ L P =. Let S = a S {ϕ(a) a., a ϕ(a).}. Then P S is a conservative extension of P. If we take S as the set of atoms appearing negated in the head of clauses in P, lemma 5.2 can allow us to eliminate such negations building a general program. Formally: Theorem 5.4. Let P be a free program and let S be the set containing all atoms a such that a appears in the head of some clause in P. Let ϕ and S be defined as in lemma 5.2. Let P be the general program obtained from P by replacing each occurrence of a with ϕ(a) for all a S. Then P S is a general program that is a conservative transformation of P. In particular M is stable model of P iff M S is a stable model of P S, where M S = M ϕ(s \ M). It is well known that a general program P is equivalent to a disjunctive program under a simple translation. Namely, replace every constraint clause A by p A p, where p is an atom not in L P. Notice that Theorem 5.3 together with Theorem 5.4 stand an equivalence between programs that shows augmented programs are not more expressive than disjunctive ones with respect to the stable semantics. This transformation can be applied also to logical programs with predicates before grounding.

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