Safe Formulas in the General Theory of Stable Models (Preliminary Report)

Abstract

Abstract. Safe first-order formulas generalize the concept of a safe rule, which plays an important role in the design of answer set solvers. We show that any safe sentence is equivalent, in a certain sense, to the result of its grounding-to the variable-free sentence obtained from it by replacing all quantifiers with multiple conjunctions and disjunctions. It follows that a safe sentence and the result of its grounding have the same stable models, and that stable models of a safe sentence can be characterized by a formula of a simple syntactic form.

A Revised Concept of Safety for General Answer Set Programs

“…By expanding the language to include new predicates, this logic can be embedded in classical first-order logic, [19], and this permits an alternative but equivalent formulation of the concept of stable model for first-order formulas, expressed in terms of classical, second-order logic [6]. The latter definition of answer set has been further studied in [10,11] where the basis of a first-oder programming language, RASPL-1, is described. An alternative approach to a first-order ASP language is developed in [8,7].…”

“…In this manner, NFN programs can be compiled into the standard DLV system. A second example is the definition of safety introduced by Lee, Lifschitz and Palla in [11] (we will call it here LLP-safety) defined essentially for arbitrary first-order formulas in such a way that the stable model of such a formula can be obtained by considering its ground version. It is also shown that the stable models of safe formulas form a (first-order) definable class.…”

“…Here the motivation is not so much to compile safe programs into existing ASP languages, but rather to find tractable, first-order extensions of the customary languages. 1 On the face of it, the approach followed in [11] is much more general than that of [2] and so we might expect that the safety of general formulas from [11] embraces also the safe NFN rules of [2]. But this is not the case: safe NFN rules need not be safe formulas in the sense of [11].…”

“…1 On the face of it, the approach followed in [11] is much more general than that of [2] and so we might expect that the safety of general formulas from [11] embraces also the safe NFN rules of [2]. But this is not the case: safe NFN rules need not be safe formulas in the sense of [11]. In fact, there are simple examples of formulas that would normally be regarded as safe that fail to be so under the [11] definition.…”

“…But this is not the case: safe NFN rules need not be safe formulas in the sense of [11]. In fact, there are simple examples of formulas that would normally be regarded as safe that fail to be so under the [11] definition. Consider the following example.…”