Back and Forth between Rules and SE-Models

Abstract

Rules in logic programming encode information about mutual interdependencies between literals that is not captured by any of the commonly used semantics. This information becomes essential as soon as a program needs to be modified or further manipulated. We argue that, in these cases, a program should not be viewed solely as the set of its models. Instead, it should be viewed and manipulated as the set of sets of models of each rule inside it. With this in mind, we investigate and highlight relations between the SE-model semantics and individual rules. We identify a set of representatives of rule equivalence classes induced by SE-models, and so pinpoint the exact expressivity of this semantics with respect to a single rule. We also characterise the class of sets of SE-interpretations representable by a single rule. Finally, we discuss the introduction of two notions of equivalence, both stronger than strong equivalence [1] and weaker than strong update equivalence [2], which seem more suitable whenever the dependency information found in rules is of interest. ⋆ This is an extended version of the paper accepted for publication at LPNMR 2011.Changes on March 1, 2011: minor substitutions to be in line with the LPNMR version. SE-Models.Given a rule r and an interpretation J, we define the reduct of r relative to J, denoted by r J , as follows: If some atom from H(r) − is false under J or some atom from B(r) − is true under J, then r J is ε; otherwise r J is H(r) + ← B(r) + . Intuitively, the reduct r J is the positive part of a rule r that "remains" after all its negative literals are interpreted under interpretation J. The two conditions in the definition check whether the rule is satisfied based on the negative atoms in its head and body, interpreted under J. If this is the case, the reduct is by definition the canonical tautology. If none of these conditions is satisfied, the positive parts of r are kept in the reduct, discarding the negative ones.An SE-interpretation is a pair of interpretations I, J such that I is a subset of J. The set of all SE-interpretations is denoted by I SE . We say that an SE-interpretationThe set of all SE-models of a rule r is denoted by mod SE (r). The SE-models of a program P are the SE-models of all rules in P. A set of SE-interpretations S is called rule-representable if there exists a rule r such that S = mod SE (r).We say that a rule r is SE-tautological if mod SE (r) = I SE . Note that the canonical tautology ε (c.f. Definition 1) is SE-tautological. We say that two rules r, r ′ are strongly equivalent, or SE-equivalent, if they have the same set of SE-models. Corollary 25. Let r be a canonical rule different from ε, put I = B(r) + , J = H(r) − ∪ B(r) + and J ′ = L \ B(r) − , and let p be an atom. Then the following holds: (1) The SE-interpretation I, J is not an SE-model of r. (2) The SE-interpretation I, J ∪ { p } is an SE-model of r if and only if p belongs to B(r) − . (3) The SE-interpretation I ∪ { p } , J ∪ { p } is an SE-model of r if and only if p belongs to H(r) + ∪ B(r...

Forgetting in Answer Set Programming with Anonymous Cycles

“…To simplify the presentation and the cases that are considered in the construction, and also to reduce the size of the input, we reduce programs to a normal form, similar to [16] and previous related work [14,15,4,20]. There are two essential differences to the normal form considered in [16].…”

Exception-Based Knowledge Updates

“…This led us to the study of stronger notions of program equivalence. In [65] we proposed to view a program as the set of sets of models of its rules in order to acknowledge rules as the atomic pieces of knowledge and, at the same time, abstract away from unimportant differences between their syntactic forms, focusing on their semantic content. In this paper we develop these ideas further and arrive at a unifying perspective on both classical and rule updates.…”

“…This observation resulted in the definition of strong rule equivalence (or SR-equivalence) and strong minimal rule equivalence (or SMR-equivalence) in [65] that, in terms of strength, fall between SE-equivalence and SU-equivalence. It is based on the idea of viewing a program P as the set of sets of SE-models of its rules…”

“…In [65] we studied the expressivity of SE-models with respect to a single rule. On the one hand, SE-models turned out to be a useful means of stripping away irrelevant syntactic details.…”

“…Further properties of RE-models, analogous to those established in [65] for SE-models, can be found in Appendix A starting on page 31.…”