On the complexity of reasoning with negation as failure

2 Preliminaries
In this section we introduce some notation, and recall some preliminary de nitions and properties of the logic MBNF . We use L to denote a xed propositional language built in the usual way from an alphabet A of propositional symbols (atoms) and the propositional connectives _; ^; :; . We use the symbol ` to denote entailment in propositional logic. We denote with LM the modal extension of L with the modalities B and not . Moreover, we denote with L1 the set of at MBNF formulas, that is the set of formulas from LM M in which each propositional symbol appears in the scope of exactly one modality, and with LS the set of subjective MBNF formulas, i.e. the subset of formulas from LM in which each M propositional symbol appears in the scope of at least one modality. We call a modal formula ' from LM positive (resp. negative) if the modality not (resp. B) does not occur in '. The symbol LB denotes the set of positive formulas from LM . We denote with 'B the positive formula obtained from ' by substituting each occurrence of not with :B . Let LM . We denote with : the set f:' : ' 2 g and with not the set fnot ' : ' 2 g. We now recall the notion of MBNF model 16]. An interpretation is a set of propositional symbols. Satis ability of a formula in a structure (I; Mb; Mn ), where I is an interpretation and Mb ; Mn are sets of interpretations, is de ned inductively as follows: 1. if ' is an atom, ' is true in (I; Mb; Mn ) i ' 2 I ; 2. :' is true in (I; Mb; Mn ) i ' is not true in (I; Mb; Mn ); 3. '1 ^ '2 is true in (I; Mb; Mn ) i '1 is true in (I; Mb; Mn ) and '2 is true in (I; Mb; Mn ); 4. '1 _ '2 is true in (I; Mb; Mn ) i '1 is true in (I; Mb; Mn ) or '2 is true in (I; Mb; Mn);
Riccardo Rosati
In this work we study the computational properties of the propositional fragment of MBNF , the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms. In particular, the logic MBNF has been used in order to give a declarative semantics to very general classes of logic programs with negation. We characterize the complexity and present algorithms for entailment in various subclasses of propositional MBNF . In particular, we show that reasoning in propositional MBNF is p -complete, whereas it is 3 p 2 -complete in the case of MBNF theories without nested occurrences of the modalities.
Although several aspects of the logic MBNF have been thoroughly investigated 4, 3], the existent studies concerning the computational properties of MBNF are limited to subclasses of propositional MBNF theories 13] or to a very restricted subset of the rst-order case 1]. In this work we give a computational characterization and provide \optimal" algorithms for deduction in three subclasses of MBNF . In particular, we show that skeptical reasoning in the propositional fragment of MBNF is a p -complete problem. Moreover, we study the 3 subclass of subjective propositional MBNF theories, proving that in this case reasoning is still a p -complete problem, and of at MBNF theories, showing that in this case entailment 3 is p -complete. 2 This last case is the most interesting from the deductive database viewpoint. Indeed, it implies that, under the stable model semantics, increasing the syntax of the database, by allowing propositional formulas as goals in the rules, does not a ect the worst-case complexity of deduction for disjunctive databases with negation as failure. In the next section, we brie y recall the logic MBNF . Then, we study the problem of reasoning in propositional MBNF theories. We rst tackle the case of at MBNF theories, then we deal with subjective MBNF theories and nally we study the computational properties of general propositional MBNF theories.
Abstract
1 Introduction
The stable model (or answer set) semantics for logic programs 8, 9, 10] is one of the best known semantics for logic programs and deductive databases. Such a semantics, which gives a \negation by default" interpretation to the operator not , has deep connections with the realm of nonmonotonic logics, in particular with default logic 2, 20] and autoepistemic logic 17, 4, 13]. Recently, Lifschitz 19, 16] has de ned a modal formalism which can be viewed as a \generalization" of the framework of logic programming under the stable model semantics, the logic of minimal belief and negation as failure (MBNF ). Roughly speaking, such a logic is built by adding to rst-order logic two distinct modalities, a \minimal belief" modality B and a \negation as failure" modality not . The logic thus obtained can be characterized in terms of a nice model-theoretic semantics. Therefore, MBNF has been used in order to give a declarative semantics to very general classes of logic programs 18, 17, 16]. In particular, MBNF is able to formalize programs which allow two kinds of negation (explicit negation and negation as failure), disjunction of goals in the head of the rules and even positive occurrences of negation as failure 13]. The logic MBNF can also be viewed as an extension of the theory of epistemic queries to databases due to Levesque and Reiter 15, 24], which deals with the problem of querying a rst-order database about its own knowledge. More generally, the logic MBNF can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, circumscription, epistemic queries and logic programming.
On the complexity of reasoning with negation as failure
Dipartimento di Informatபைடு நூலகம்ca e Sistemistica Universita di Roma \La Sapienza" Via Salaria 113, 00198 Roma, Italy email: rosati@dis.uniroma1.it