### Redundancy in logic III: Non-monotonic reasoning

Paolo Liberatore
2008 Artificial Intelligence
Results about the redundancy of certain versions of circumscription and default logic are presented. In particular, propositional circumscription where all variables are minimized and skeptical default logics are considered. This restricted version of circumscription is shown to have the unitary redundancy property: a CNF formula is redundant (it is equivalent to one of its proper subsets) if and only if it contains a redundant clause (it is equivalent to itself minus one clause); default logic
more » ... does not have this property in general. We also give the complexity of checking redundancy in the considered formalisms. A related problem is that of studying the properties of formulae that are already known to be irredundant. Büning and Zhao  studied the problems of equivalence and extension-equivalence of irredundant formulae. Also related is the problem of minimal unsatisfiability, that is, checking whether an unsatisfiable formula would become satisfiable as soon as a clause is removed from it [5, 14, 43] . Other authors have studied redundancy in settings different from that in this article. Ginsberg  and Schmolze and Snyder  studied the redundancy of production rules. Gottlob and Fermüller  studied the redundancy of a literal within a clause in first-order logic. The settings considered in this article are those of circumscription and default logic, which are two of the most studied [2, 4, 6, 10, 13, 29, 31, 36, 38, 44] forms of non-monotonic reasoning, as opposite to classical logic, which is monotonic. A logic is monotonic if the consequences of a set of formulae monotonically non-decrease with the set. In other words, all formulae that are entailed by a set are also entailed by every superset of it. Circumscription and default logic do not have this property, and are therefore non-monotonic. For circumscription, we assume that all variables are minimized; the rationale for this restriction is that fixed and varying variables can be efficiently eliminated [8, 9] , which shows that the minimized variables are the "core" of the circumscription formalism. Other authors have indeed considered circumscription only under this restriction [6, 25, 40] . This shows that this restriction of circumscription is of interest; however, results about redundancy in this case do not necessarily extend to the general case, as discussed in the conclusions. For default logic, there are several semantics; for most of them, one can choose between the "credulous" and "skeptical" approach. In this article, we consider the skeptical approach under the original semantics  and three similar ones: justified , constrained [11, 46] , and rational  . We however also consider the case in which we assume that the semantics of a theory is the set of its extensions, without combining these extensions in a skeptical manner. The results obtained in this case hold for the credulous approach under the original semantics (where not otherwise stated, the skeptical approach is assumed). The properties of redundancy in the credulous approach under the other semantics is left open. Since redundancy is defined in terms of equivalence (namely, equivalence of a formula to a proper subset of it), it is affected by the kind of equivalence used. In particular, equivalence can be defined in two ways for default logic: equality of extensions and equality of consequences. This leads to two different definitions of redundancy in default logic. Both circumscription and default logic differ from classical propositional logic. This difference affects redundancy. If a CNF formula contains a redundant clause, it is redundant (equivalent to one of its proper subsets); the converse is true in propositional logic, but not in all logics. In particular, it may be the case that a formula is equivalent to one of its proper subsets, but none of its clauses is redundant. We will indeed show a situation in default logic where {a, b} is equivalent to ∅ but neither to {a} nor to {b}, which means that {a, b} is redundant but does not contain any single redundant element. The property that a formula is redundant if and only if it contains a redundant clause is called unitary redundancy. Classical logic has this property; other logics, like default logic, do not. We show three different sufficient conditions for this property to hold in a logic; one of them involves monotonicity, another involves cumulativity  . A property that entails unitary redundancy is that of monotonic redundancy: if ⊆ ⊆ and and are equivalent then and are equivalent as well. This is the property for which the sufficient conditions are actually proved; unitary redundancy follows. Regarding the specific non-monotonic formalisms considered here, we show that monotonic redundancy, and therefore unitary redundancy, holds for circumscription and for the redundancy of the background theory in default logic when all defaults are categorical (prerequisite-free) and normal. In the general case, default logic does not have the unitary redundancy property (and therefore does not have the monotonic redundancy property either). We also considered the redundancy of defaults in a default theory. In this case, monotonic and unitary redundancy hold for justified default logic but not for the other three considered semantics. Regarding the complexity results, we show that checking whether a clause is redundant in a formula and whether a formula is redundant according to circumscriptive inference are p 2 -complete problems. For default logic, the results are as follows. Checking redundancy, based on extensions, of a clause in the background theory is p 2 -complete for Reiter and justified default logics, and p 3 -complete for constrained and rational default logic; checking redundancy based on skeptical consequences is p 3 -complete for all four semantics. Checking redundancy of a background theory is p 3 -complete and p 4 -complete, respectively, for equivalence based on extensions and skeptical consequences. The proofs of the latter two results are of some interest by themselves, as they are done by first showing that the problems