Computing Stable Models by Using the ATMS

An algorithm is described which computes stable models of propositional logic programs with negation as failure using the Assumption Based Truth Maintenance mechanism. Since stable models of logic programs are closely connected to stable expansions of a class of autoepistemic theories, this algorithm points to a link between stable expansions of a class of autoepistemic theories and ATMS structures Introduction In this paper an algorithm is described which computes stable models of propositional logic programs with negation as failure [l], using the Assumption Based Truth Maintenance mechanism [2]. Since stable models of logic programs are closely connected to stable expansions of a class of auto-epistemic theories, this algorithm points to a link between stable expansions of a class of auto-epistemic theories and ATMS structures. Stable Models of Logic Programs Stable models of logic programs with negation as failure were introduced in [l] as a means of specifying the semantics of logic programs. They are defined as follows: A logic program is a set of clauses of the type P + Pl,PZ,**. q1,q2.. where N indicates negation as failure. In this paper we only consider propositional logic programs, where Pl,P2,*41,Q2** are propositions. We do not place any other restriction on the structure of the clasues. Definition 1 The answer set of a propositional Horn Clause program is the set of all propositions provable from it. We use Answer(n) to denote the answer set of the program II Definition 2 Let P be a propositional logic program with negation as failure, and I a set of propositions. The negation-free program PI is derived from P by a Deleting all the clauses in P which conditions such as N q where q E I have a negative m Deleting all the negative conditions in all the remaining clauses of P. I is a stable model of P ifl I = Answer(PI) Example: let P be the program Then Pjb} = {b +), and therefore {b} is a stable model of P. In general, a logic program may have any number of stable models. The algorithm described here computes all of them. Stable models of logic programs are closely connected to stable expansions of auto-epistemic theories. In fact, as observed in [l], if we replace every negative condition N p in the program P with the condition 4?(p), where B is the belief operator of auto-epistemic logic, the stable expansion of the resulting auto-epistemic theory is the same as the stable model of P. As such, the algorithm described in this paper can be considered to be a theorem prover for a restricted class of auto-epistemic theories.