Computing Modular Paracoherent Answer Sets: Preliminary Result

Abstract

Answer Set Programming (ASP) is a well-established logic programming formalism. Problem solving in ASP requires to write an ASP program whose answers sets correspond to solutions. Despite the non-existence of answer sets for some ASP programs can be a modeling feature, it turns out to be a weakness in many other cases, and especially for query answering. Paracoherent answer set semantics extend the classical semantics of ASP to draw meaningful conclusions also from incoherent logic programs, with the result of increasing the range of applications of ASP. State of the art implementations of paracoherent ASP adopt the semi-stable and the semi-equilibrium models semantics, but cannot compute split semi-equilibrium semantics, also known as modular paracoherent semantics, that discards undesirable semi-equilibrium models. In this paper, we introduce and discuss a rst approach for the computation of modular paracoherent answer sets.

Keywords

• Answer Set Programming
• Paracoherent Reasoning
• Semi- equilibrium models

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Answer Set Programming (ASP) [ 22,23,34 ] is a well-established logic programming language, based on the stable model (or answer set) semantics [ 40 ], with a robust solving technology [ 43,3,24,35,36,37,39,12,16,17,18,47,46,38,42,41 ]. As a matter of fact, ASP has been applied to solve complex problems in several areas of knowledge such as AI [ 21,31,32,11,1,7 ]; Bioinformatics [ 25 ]; Databases [ 19,45 ]; Game Theory [ 15,6 ]; and industrial applications [ 27,28 ].

The non-existence of answer sets for some ASP programs can be a modeling feature, but, as argued in [ 13 ], it turns out to be a weakness in many other applications, such as: debugging, model building, inconsistency-tolerant query answering, diagnosis, planning and reasoning about actions. To remedy to the non-existence of answer sets, paracoherent semantics extend the classical answer set semantics to draw meaningful conclusions also from incoherent programs. This ASP variant has been termed paracoherent reasoning [ 13 ]. In particular, [ 13 ] improved the paracoherent semantics of semi-stable models [51] avoiding some anomalies with respect to basic modal logic properties by resorting to equilibrium logic [48]. Thus, this paracoherent semantics is called semi-equilibrium model (SEQ) semantics [ 13 ].

More recently, [ 13 ] noticed that, although the SEQ semantics has nice properties, it may select models that do not respect modular structure of the program. SEQ semantics use 3-valued interpretations where a third truth value besides true and false expresses that an atom is believed true. For instance, the incoherent logic program P = fb not a; c not a; not cg admits two SEQ models, say M1 and M2. In M1, b is true, c is believed true, and a is false; whereas in M2 a is believed true and both b and c are false. Now, M1 appears preferable to M2, as, according with a layering (strati cation) principle, which is widely agreed in logic programming, one should prefer b rather than a, as there is no way to derive a (note that a does not appear in the head of any rule of the program). Therefore, [ 14 ] re ne SEQ-models using splitting sequences [ 44 ], the major tool for modularity in modeling and evaluating answer set programs. In particular, the re ned semantics, called Split SEQ model semantics, also known as modular paracoherent semantics, is able to discard model M2.

The rst e cient implementations of paracoherent semantics were proposed recently [ 8,10 ], but they only support semi-stable and semi-equilibrium semantics. Although the Split SEQ semantics discards some undesirable SEQ models, the existing methods for computing SEQ models are not able to compute the re ned semantics. In this paper, we ll this lack presenting and discussing a rst strategy for computing a modular paracoherent answer set. 2

Preliminaries

We start with recalling answer set semantics, and then present the paracoherent semantics of semi-equilibrium models, and its re ned version based on splitting sequences. 2.1

Answer Set Programming

We concentrate on logic programs over a propositional signature . A disjunctive rule r is of the form a1 _ _ al b1; :::; bm; not c1; :::; not cn; (1) where all ai, bj , and ck are atoms (from ); l > 0, m; n 0; not represents negation-as-failure. The set H(r) = fa1; :::; alg is the head of r, while B+(r) = fb1; :::; bmg and B (r) = fc1; : : : ; cng are the positive body and the negative body of r, respectively; the body of r is B(r) = B+(r) [ B (r). We denote by At(r) = H(r) [ B(r) the set of all atoms occurring in r. A rule r is a fact, if B(r) = ; (we then omit ); normal, if jH(r)j 1; and positive, if B (r) = ;. A (disjunctive logic) program P is a nite set of disjunctive rules. P is called normal [resp. positive] if each r 2 P is normal [resp. positive]. We let At(P ) = Sr2P At(r), that is the set of all atoms occurring in the program P .

The dependency graph of a program P is the directed graph DG(P ) = hVP ; EP i whose nodes VP are the atoms in P and EP contains an edge (a; b) if a occurs in H(r) and either b occurs in B(r) or in H(r) n f g a . The strongly connected components (SCCs) of P , denoted SCC(P ), are the SCCs of DG(P ), which are the maximal sets of nodes C such that every pair of nodes is connected by some path in DG(P ) with nodes only from C.

Any set I is an interpretation; it is a model of a program P (denoted I j= P ) if and only if for each rule r 2 P , I \ H(r) 6= ; if B+(r) I and B (r) \ I = ; (denoted I j= r). A model M of P is minimal, if and only if no model M 0 M of P exists. We denote by MM (P ) the set of all minimal models of P and by AS(P ) the set of all answer sets (or stable models) of P , i.e., the set of all interpretations I such that I 2 MM (P I ), where P I is the well-known Gelfond-Lifschitz reduct [ 40 ] of P w.r.t. I, i.e., the set of rules a1 _ ::: _ al b1; :::; bm, obtained from rules r 2 P of form (1), such that B (r) \ I = ;. We say that a program P is coherent, if it admits some answer set (i.e., AS(P ) 6= ;), otherwise, it is incoherent. 2.2

Semi-Equilibrium Models

Here, we introduce the paracoherent semantics of the semi-equilibrium (SEQ) models introduced in [ 13 ]. Consider an extended signature = [ fKa j a 2 g. Intuitively, Ka can be read as a is believed to hold. The SEQ models of a program P are obtained from its epistemic HT -transformation P HT , de ned as follows.

De nition 1. Let P be a program over . Then its epistemic HT -transformation P HT is obtained from P by replacing each rule r of the form (1) in P , such that B (r) 6= ;, with:

r;1 _ : : : _ r;l _ Kc1 _ : : : _ Kcn for 1 i; k l and 1 j n, where the r;i, r;k are fresh atoms; and by adding the following set of rules:

Ka1 _ ::: _ Kal _ Kc1 _ ::: _ Kcn

Kb1; :::; Kbm; for a 2

, respectively for every rule r 2 P of the form (1).

Note that for any program P , its epistemic HT -transformation P HT is positive. For every interpretation I over 0 , let G(I ) = fKa 2 I j a 62 I g denote the atoms believed true but not assigned true, also referred to as the gap of I . Given a set F of interpretations over 0, an interpretation I 2 F ai r;i b1; : : : ; bm; r;i; r;i; cj ; ai; r;k; Ka a; (2) (3) (4) (5) (6) (7) is maximal canonical in F , if no J 2 F exists such that G(I ) G(J ). By mc(F ) we denote the set of maximal canonical interpretations in F . SEQ models are then de ned as maximal canonical interpretations among the answer sets of P HT .

De nition 2. Let P be a program over , and let I be an interpretation over . Then, I 2 SEQ (P ) if, and only if, I 2 fM \ j M 2 mc(AS(P HT ))g, where SEQ (P ) is the set of semi-equilibrium models of P . 2.3

Split Semi-equilibrium Models

A set S At(P ) is a splitting set of P , if for every rule r in P such that H(r)\S 6= ; we have that At(r) S. We denote by bS (P ) = fr 2 P j At(r) Sg the bottom part of P , and by tS (P ) = P n bS (P ) the top part of P relative to S. A splitting sequence S = (S1; : : : ; Sn) of P is a sequence of splitting sets Si of P such that Si Sj for each i < j. Let SCC(P ) be the set of all strongly connected components of P , and let (C1; : : : ; Cn) be a topological ordering of SCC(P ). It is known that = ( 1; : : : ; n), where j = C1 [ : : : [ Cj for j = 1; : : : ; n, is a splitting sequence of P . So that, we obtain a strati cation for P in subprograms (P1; : : : ; Pn) such that P1 = b 1 (P ), and Pj = b j (P ) n Pj 1, for j = 2; : : : ; n. Given an interpretation Mi over Ci, we denote by info(Mi) the set of rules fa j a 2 Mig [ f not a j Ka 2 Mig [ f a j a 2 Ci n Mig. De nition 3. Given a topological ordering (C1; : : : ; Cn) of SCC(P ), an interpretation M over At(P ) is a semi-equilibrium model of P relative to if there is a sequence of interpretations M1; : : : ; Mn over 1; : : : n, respectively, such that (1) M = Mn; (2) M1 2 SEQ (P1); (3) Mj 2 SEQ (Pj [ info(Mj 1)), for j = 2; : : : ; n; and (4) M is maximal canonical among the interpretations over At(P ) satisfying conditions (1), (2) and (3). The set of all semi-equilibrium model of P relative to is denoted by SEQ (P ).

Since SEQ (P ) is independent by the given topological ordering of SCC(P ) (see, Theorem 5 in [ 13 ]), the SCC-models of P have been de ned as the set M SCC (P ) = SEQ (P ) for an arbitrary topological ordering of SCC(P ). We will refer to them as split semi-equlibrium models. Finally, note that M SCC (P ) SEQ(P ). Example 1. Consider the program

P = fb not a; d b; not c; c dg: Then, (fag; fbg; fc; dg) is a topological ordering of SCC(P ), so that = (fag; fa; bg; fa; b; c; dg) is a splitting sequence for P . Hence, SEQ (P ) = ffb; Kb; Kcgg. Indeed P1 = b 1 (P ) = ; and thus SEQ (P1) = f;g. Then, P2 [ info(;) = fb not a; ag and thus SEQ (P2 [ info(;)) = ffbgg. Finally, P3 [ info(fbg) = fd b; not c; c d; b; ag and thus SEQ (P3 [ info(fbg)) = ffb; Kb; Kcgg.

In the following, we will refer to semi-equilibrium models as paracoherent answer sets, and to split semi-equilibrium models as modular paracoherent answer sets.

On the Computation of Split Semi-Equilibrium

Models

In this section, we describe a strategy to compute a split semi-equilibrium model by exploiting the computation of a semi-equilibrium model.

First, we consider a possible path that can be generated through the splitting sequence. Note that, each path leads to obtain a paracoherent answer set of the last program (i.e., Pn [ info(Mn 1)), as stated in the following theorem. Theorem 1. Let P be a program and let (P1; :::; Pn) be a strati cation for P . Then, for each i = 1; : : : ; n 1, M 2 SEQ(Pi) implies SEQ(Pi+1[info(M )) 6= ;. Proof. It is known that whenever P has a classical model, then SEQ(P ) 6= ;. Hence, it is enough to show that Pi+1 [ info(M ) admits a classical model. Recall that info(M ) = fa: j a 2 M g [ f not a j Ka 2 M g [ f a j a 2 Ci n M g. Moreover, by construction of Pi+1, no atom appearing in a constraint of info(M ), appears in the head of a rule in Pi+1. Then, by setting as true each atom in the head of the rule in Pi+1, we obtain a classical model of Pi+1 [ info(M ).

Therefore, we could choose a paracoherent answer set M1 of the rst subprogram P1 (that always exists as P1 is constraint-free), then we move to nd a model M2 of the subprogram P2 [ info(M1) (that always exists by Theorem 1), and so on. At the end of the procedure, we obtain a paracoherent answer set Mn of the program Pn [ info(Mn 1). However, it could not be a paracoherent answer set of the original program.

Example 2. Consider the program P = fa not b; b not a; c a; not cg. In the rst layer of P , we have the subprogram P1 = fa not b; b not ag whose (paracoherent) answer sets are fa; Kag and fb; Kbg. So that, considering info(fa; Kag) [ fc a; not cg, we obtain the paracoherent answer set fa; Ka; Kcg, while considering info(fb; Kbg) [ fc a; not cg, we obtain the (paracoherent) answer set fb; Kbg. Since G(fa; Ka; Kcg) G(fb; Kbg), then fa; Ka; Kcg cannot be a paracoherent answer set of P .

Intuitively, since each subprogram could have more than one answer set, we need to explore all possible paths by enumerating all possible paracoherent models obtainable from each path to make feasible a nal phase of gap minimization.

More formally, let (P1; : : : ; Pn) be a strati cation for a program P . We denote by PAS 1 the set of all semi-equilibrium models of P1, i.e., PAS 1 = SEQ(P1); and, for each i = 2; : : : ; n, we denote by PAS i the set of all semi-equilibrium models of Pi[info(Mi 1), where Mi 1 varies among the semi-equilibrium models of PAS i 1, i.e., PAS i = fM 2 SEQ(Pi [info(Mi 1)) j Mi 1 2 PAS i 1g. Hence, the computation of a split semi-equilibrium model is given as follows. (1) For each i = 1; : : : ; n, we compute PAS i. (2) Then, we look for a model M in PAS n that is gap-minimal, with respect to subset inclusion, among all models in PAS n.

Theorem 2. Let P be a logic program. If M 2 PAS n is gap-minimal with respect to subset inclusion, among all models in PAS n, then M is a split semiequilibrium model of P .

Proof. Let M 2 PAS n. Hence, by de nition of PAS n, M 2 SEQ(Pn[info(Mn 1)), for some model Mn 1 2 PAS n 1. Moreover, by de nition of PAS n 1, Mn 1 2 SEQ(Pn 1 [ info(Mn 2)), for some model Mn 2 2 PAS n 2. By repeatedly applying the de nition of PAS i, for i = 1; : : : ; n 1, and considering models named Mi, at the end, we obtain that, by de nition of PAS 2, M2 2 SEQ(P2[info(M1)), for some model M1 2 PAS 1 = SEQ(P1). Hence, condition (2) and condition (3) in De nition 3 are satis ed. Finally, as M is gap-minimal with respect to subset inclusion, among all models in PAS n, then M is maximal canonical among the interpretations over At(P ) satisfying the three conditions in De nition 3. Therefore, M is a split semi-equilibrium model of P .

Intuitively, this approach could provide an improvement in the practical computation of a (modular) paracoherent aswer set, as the search of a semiequilibrium model is relative to small parts of the whole program. If n is su ciently large, the size of each subprogram Pi, in the strati cation of P , decreases enormously. In particular, the average of the size will be kPik = kP k=n, where k k can be the number of atoms or the number of rules of a logic program. However, this observation must be counterbalanced by the fact, repeatedly observed, that it is now necessary to calculate all possible paths leading to a split semiequilibrium model candidate, and these paths are in an exponential number. In fact, if for example we assume that the original program is strati ed into three subprograms, say P1, P2 and P3, such that each computation produces two semiequilibrium models, we will have that: at step (1), SEQ(P1) = fM11; M21g; at step (2), SEQ(P2 [ info(M11)) = fM12; M22g and SEQ(P2 [ info(M21)) = fM32; M42g; and at step (3), SEQ(P3 [ info(M12)) = fM13; M23g, SEQ(P3 [ info(M22)) = fM33; M43g, SEQ(P3[info(M32)) = fM53; M63g, SEQ(P3[info(M42)) = fM73; M83g. Now, assume for instance that the computation time of a semi-equilibrium model of a program P is directly proportional to the size of P . We denote by time(P ) the time to compute a semi-equilibrium model of P . Hence, since we could compute an exponential number of semi-equilibrium models, say 2n (like in the example), intuitively, the computation time will be about time(P1) 21 +: : :+time(Pn) 2n that will be proportional to time(P ) 21=n + : : : + time(P ) 2n=n, that is time(P ) 2n+n1 1 . Therefore, it is desirable for the future to identify more sophisticated techniques, to try to overcome such a computational explosion. 4

Related Work

Semantics for non-monotonic logic programs [ 50,53,54,51,30,52,20,49,2,33,13,26 ] that relax the de nition of answer set to overcome the absence of answer sets can be considered in broader terms paracoherent semantics. Nonetheless, the term paracoherent answer set was used for the rst time by Inoue and Sakama in [51], where they introduced the semi-stable semantics as a remedy to the absence of answer sets due to cyclic negation. Later, in [ 14,5 ] some anomalies of semi-stable semantics with respect to some epistemic properties were evidenced, and the semi-equilibrium semantics was proposed as a remedy. In [ 13 ] it was demonstrated that semi-equilibrium semantics features a number of highly desirable theoretical properties for a knowledge representation language (for instance, minimal unde nedness [ 4 ]), and at the same time, it was observed that semi-equilibrium models do not enjoy the same nice modular composition properties of stable models (e.g., the splitting set [ 44 ] modularity tool cannot be used straightforwardly). Notably, modular composition is used in ASP for simplifying the modeling of problems (actually, the guess and check programming methodology [ 29 ] is based on this property) and is a principle underlying the architectures of ASP systems [ 43 ]. The split semi-equilibrium semantics [ 13 ] solves this problem by using splitting sequences to decompose the program into hierarchically organized subprograms. Split semi-equilibrium models are semi equilibrium models that enjoy a modularity property.

Concerning the implementation of semi-stable and semi-equilibrium semantics, we observe that they have been implemented e ciently only recently. In particular, in [ 8 ] a number of algorithms has been proposed, that compute paracoherent answer sets in two steps: (i) an epistemic transformation of programs is applied, and (ii) a strategy for computing answer sets of minimum gap is implemented by calling (possibly multiple times) an ASP solver. The same strategy has been improved in [ 10 ] by replacing the classic epistemic transformations by more parsimonious ones (that we also adopt). The new transformations are based on the characterization of paracoherent answer sets in terms of externally supported models. Neither [ 8 ] nor [ 10 ] support split semi-equilibrium semantics that is the focus of this paper. 5

Conclusion

Paracoherent answer set semantics can draw meaningful conclusions also from incoherent programs, and in this way increase the applicability of ASP for solving AI problems [ 13 ]. Practical applications are possible once e cient implementations are available, and the complex task of computing e ciently a paracoherent answer set has been approached only recently [ 8,10,9 ]. State of the art solutions supported the semi-equilibrium semantics but cannot compute the split semiequilibrium semantics. In this paper we presented a rst approach to compute a split semi-equilibrium model.

As future work, we plan to identify more e cient evaluation strategies by exploiting the computational complexity properties of the modular paracoherent semantics, that could allow for computing a split semi-equilibrium model using a plain ASP solver. Finally, we will implement our approaches and compare them against existing implementations for semi-equilibrium models. 46. Maratea, M., Pulina, L., Ricca, F.: A multi-engine approach to answer-set programming. TPLP 14(6), 841{868 (2014) 47. Maratea, M., Ricca, F., Faber, W., Leone, N.: Look-back techniques and heuristics in DLV: implementation, evaluation, and comparison to QBF solvers. J. Algorithms 63(1-3), 70{89 (2008) 48. Pearce, D.: Equilibrium logic. Ann. Math. Artif. Intell. 47(1-2), 3{41 (2006) 49. Pereira, L.M., Pinto, A.M.: Approved models for normal logic programs. In: LPAR.

pp. 454{468 (2007) 50. Przymusinski, T.C.: Stable semantics for disjunctive programs. New Generation

Comput. 9(3/4), 401{424 (1991) 51. Sakama, C., Inoue, K.: Paraconsistent stable semantics for extended disjunctive programs. J. Log. Comput. 5(3), 265{285 (1995) 52. Seipel, D.: Partial evidential stable models for disjunctive deductive databases. In:

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