Autoepistemic answer set programming☆
Introduction
Epistemic reasoning [31], [18] constitutes a crucial feature for any agent to be considered intelligent. The capacity of representing and reasoning about knowledge and beliefs has proved to be a key property in different domains such as planning under incomplete information, speech acts in natural language understanding, software verification of security protocols, formalisation of multi-agent systems or foundations of game theory (see [13], [17], [3], [15], [50], [58]). An important field in Knowledge Representation (KR) where epistemic reasoning has played a relevant role since its inception is non-monotonic reasoning. There, default rules have been frequently addressed in terms of modal constructions expressing the agent's own knowledge and beliefs, as a kind of introspection. There exists a vast literature on non-monotonic modal logics (see for instance [43], [39], [42], [37]) among which Moore's Autoepistemic Logic (AEL) [45] is perhaps the most prominent and well-studied approach for non-monotonic epistemic introspection. Moreover, AEL has been commonly used in translations or encodings for other non-monotonic approaches, like default negation in logic programming [27], [38].
Despite of its clear significance in KR, the impact of epistemic reasoning in practical applications has been moderate so far. One possible reason is that dealing with the agent's knowledge normally implies an increase in computational complexity. However, a more important obstacle appears when we extend an existing KR formalism with epistemic constructs and there exist multiple options for their interpretation without a clear orientation or agreement about which one preserves the main features of the extended formalism in the best way. This is precisely the situation in the case of Answer Set Programming (ASP) [41], [47], one of the most popular paradigms for practical KR and problem solving based on the stable model [25] semantics for disjunctive logic programs.
The first steps towards an epistemic extension of answer set programming can be traced back to the language of epistemic specifications. This language was proposed by Gelfond in three consecutive papers [23], [26], [28] and extends ASP with epistemic operators K and M. Using these constructs, it is possible to check whether a regular literal l is true in every stable model (written ) or in some stable model (written ) of the program. For instance, the rule: means that a must hold if we cannot prove that all the stable models contain b. The definition of a “satisfactory” semantics for epistemic specifications has proved to be a non-trivial enterprise, as shown by the list of different attempts proposed so far [21], [23], [24], [33], [54], [56], [57], [60]. The main difficulty arises because subjective literals query the set of stable models but, at the same time, occur in rules that determine those stable models. As an example, the program consisting of: and (1) has now two rules defining atoms a and b in terms of the presence of those same atoms in all the stable models. To solve this kind of cyclic interdependence, the original semantics by Gelfond [23], [26], [28] (abbreviated1as G94) considered different alternative world views or sets of stable models. In the case of program (1)-(2), G94 yields two alternative world views,2 and , each one containing a single stable model, and this is also the behaviour obtained in the remaining approaches developed later on. The feature that made G94 unconvincing, though, was the generation of self-supported world views. A prototypical example for this effect is the epistemic program consisting of the single rule: whose world views under G94 are and . The latter is considered counter-intuitive by all authors3 because it relies on a self-supported derivation: a is derived from by rule (3), but the only way to obtain is rule (3) itself. Although the rejection of world views of this kind seems natural, the truth is that all approaches in the literature have concentrated on studying the effects on individual examples, rather than capturing the absence of self-supportedness as a formal property. To achieve such a goal, we would need to establish some kind of derivability condition in a very similar fashion as done with unfounded sets [59] for standard logic programs. To understand the similarity, think about the (tautological) rule . The classical models of this rule are ∅ and , but the latter cannot be a stable model because a is not derivable applying the rule. Intuitively, an unfounded set is a collection of atoms that is not derivable from a given program and a fixed set of assumptions, as happens to in the last example. As proved by Leone et al. [35], the stable models of any disjunctive logic program are precisely its classical models that are founded, that is, that do not admit any unfounded set. As we can see, the situation in (3) is pretty similar to but, this time, involves derivability through subjective literals. An immediate option is, therefore, extending the definition of unfounded sets for the case of epistemic programs – this constitutes, indeed, the first contribution of this paper.
Once the property of founded world views is explicitly stated, the paper proposes a new semantics for epistemic specifications, called Founded Autoepistemic Equilibrium Logic (FAEEL), that satisfies that property. In the spirit of [21], [56], [60], our proposal actually constitutes a full modal non-monotonic logic where K becomes the usual necessity operator applicable to arbitrary formulas. Formally, FAEEL is a combination of Pearce's Equilibrium Logic [48], [49], a well-known logical characterisation of stable models, with Moore's AEL, one of the most representative approaches among modal non-monotonic logics. The reason for choosing Equilibrium Logic is quite obvious, as it has proved its utility for characterising other extensions of ASP [1], [2], [4], [5], [7], [8], [9], [10], [12], [14], [20], [29], [51], including the already mentioned epistemic approaches [21], [56], [60]. As for the choice of AEL, it shares with epistemic specifications the common idea of agent's introspection where means that φ is one of the agent's beliefs. The only difference is that those beliefs are just classical models in the case of AEL whereas epistemic specifications deal with stable models instead. Interestingly, the problem of self-supported models has also been extensively studied in AEL [34], [40], [46], [52], where the formula , analogous to (3), also yields an unfounded world view4 . Our solution consists in combining the monotonic bases of AEL and Equilibrium Logic (the modal logic KD45 and the intermediate logic of Here-and-There (HT) [30], respectively), but defining a two-step models selection criterion that simultaneously keeps the agent's beliefs as stable models and avoids unfounded world views from the use of the modal operator K. As expected, we prove that FAEEL guarantees the property of founded world views, among other properties lifted from standard ASP. Our main result, however, goes further and asserts that the FAEEL world views of an epistemic program are precisely the set of founded G94 world views. We reach, in this way, an analogous situation to the case of standard logic programming, where stable models are the set of founded classical models of the program. These results suggest that FAEEL is a solid formal basis for the development of an autoepistemic extension of ASP.
The rest of the paper is organised as follows. Sections 2 and 3 respectively revisit the background knowledge about equilibrium logic and epistemic specifications necessary for the rest of the paper. Section 4 introduces the foundedness property for epistemic logic programs and then, Section 5 provides a pair of counterexamples that suffice to prove that it does not hold for any of the previously existing semantics. In Section 6 we introduce FAEEL and explain some of its properties, making special emphasis on its relation to G94, showing that the latter can be captured as a special subset of FAEEL models. Section 7, contains the proof of the main result, that is, FAEEL-world views are precisely the founded G94-world views. The proof relies on an alternative characterisation of FAEEL that starts from G94 semantics and imposes an additional semantic condition which can be considered as a semantic counterpart of foundedness. In Section 8 we make a comparison among the different semantics, using several examples from the literature and including a table where, apart from foundedness, we also consider other four formal properties recently proposed, showing that only FAEEL satisfies all of them so far. Finally, Section 9 concludes the paper.
Section snippets
Background
We begin recalling the basic definitions of equilibrium logic and its relation to stable models. We start from the syntax of propositional logic, with formulas built from combinations of atoms in a set At with operators and → in the usual way. We define the derived operators , , and .
A propositional interpretation T is a set of atoms . We write to represent that T classically satisfies formula φ. An HT-interpretation is a pair
G94 semantics for epistemic theories
In this section we provide a straightforward generalisation of G94 allowing its application to arbitrary modal theories. Formulas are extended with the necessity operator K according to the following grammar: An (epistemic) theory is a set of formulas. In our context, the epistemic reading of is that “ψ is one of the agent's beliefs.” Thus, a formula φ is said to be subjective if all its atom occurrences (having at least one) are in the scope of
Founded world views of epistemic specifications
As we explained in the introduction, world view of is considered to be “self-supported” in the literature but, unfortunately, there is no formal definition for such a concept, to the best of our knowledge. To cover this lack, we proceed to extend here the idea of unfounded sets from disjunctive logic programs to the epistemic case. For this purpose, we focus next on the original language of epistemic specifications [23] (a fragment of epistemic theories closer to logic programs) on
Foundedness in the previously existing semantics
As we discussed in Example 5, our introduced definition of unfounded world view allowed disregarding the self-supported solution for program obtained by G94. This immediately implies that G94 does not satisfy foundedness, as expected. In fact, this is not surprising, since all the remaining approaches previously existing in the literature were precisely proposed to disregard (among other cases) world view for that example. What is more striking, however, is that none of those
Founded autoepistemic equilibrium logic
We present now the semantics proposed in this paper, introducing Founded Autoepistemic Equilibrium Logic (FAEEL). As suggested by the similarity in their names, FAEEL follows the same spirit as F15, that is, it combines Equilibrium Logic with a modal approach, but replaces S5 by Moore's Autoepistemic Logic (AEL). Note that this implies combining two non-monotonic formalisms, since AEL is non-monotonic too.7
Characterisation as founded G94-world views
In this section, we review an alternative characterisation of FAEEL introduced by Fandinno [19] that will allow us to prove the main result in this paper, namely, that FAEEL precisely obtains those G94-world views that are founded. According to this alternative characterisation, FAEEL-word views are those G94-word views that are equilibrium models on a new logic we will call S5-Equilibrium Logic (or S5-EL ). This logic is similar to FAEEL, but without the “autoepistemic” minimisation of
Comparison to other approaches
To illustrate the effect of FAEEL when compared to other approaches in the literature, we begin providing a list of usual examples taken from Table 4 in [21]. The left table in Fig. 1 contains programs where all semantics agree. The right table contains examples where the different semantics are divided into two groups: one in which G94, G11 and FAEEL coincide, and another in which the other three, K15, F15 and S17, coincide. Note that, in these programs, all subjective literals are negative
Conclusions
In order to characterise self-supported world-views, already present in Gelfond's original semantics [23], [26], [28] (G94), we have extended the definition of unfounded sets from standard logic programs to epistemic specifications. As a result, we proposed the foundedness property for epistemic semantics, which is not satisfied by other approaches in the literature. Our main contribution has been the definition of a new semantics, based on the so-called Founded Autoepistemic Equilibrium Logic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We are grateful to David Pearce, Michael Gelfond and the anonymous reviewers from the journal and, previously, from the 15th International Conference on Logic Programming and Nonmonotonic Reasoning (where a preliminary version of this work was presented), for their comments and suggestions that have helped us to improve the paper. We are also thankful to the organisers of LPNMR'19 for their support in preparing this extended version. This research was partially supported by MINECO, Spain, grant
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This paper is an invited revision of a paper which first appeared at the 2019 International Conference on Logic Programming and Non-Monotonic Reasoning (LPNMR-19).