On the stable model semantics for intensional functions

Abstract

Several extensions of the stable model semantics are available to describe "intensional" functions-functions that can be described in terms of other functions and predicates by logic programs. Such functions are useful for expressing inertia and default behaviors of systems, and can be exploited for alleviating the grounding bottleneck involving functional fluents. However, the extensions were defined in different ways under different intuitions. In this paper we provide several reformulations of the extensions, and note that they are in fact closely related to each other and coincide on large syntactic classes of logic programs.

A Free Logic for Stable Models with Partial Intensional Functions

“…The term "free" logic refers to a family of formalisms where syntactic terms may denote objects that are outside the domain of quantification, something that can be used to capture partial functions. 10 We show that this feature can be naturally accommodated 9 In fact, as explained in [4], the difference total/partial between the two semantics is not essential. In Cabalar's semantics, any function can always be forced to be total by adding an axiom ¬¬∃y f (x) = y.…”

“…There currently exist two different ways of understanding intensional functions. On the one hand, Bartholomew and Lee introduced a variant [3] (we will call BL semantics) that repairs some counterintuitive features of Lifschitz's approach. Like the latter, BL semantics exclusively deals with total functions defining their "stability" in terms of value uniqueness among values stemming from possible models.…”

“…In this way, the intuition behind BL intensional functions is clearly different from predicate-based ASP and relies on an idea of selecting a function value when there is no other way to vary that value. This idea was captured in [4] and [12] defining in the latter a flexible extension of QEL together with a Gentzen calculus for the "flexible" version of its monotonic basis, the so-called logic of Quantified Here-and-There (QHT).…”

“…As a result, nested functions can be safely "unfolded" until all atoms involving functions eventually have the form f (t) = t where t and t are functionfree terms. This syntactic form is called c-plain in [4] and there it was shown that both BL and Cabalar's semantics coincide for this form of theory, under the assumption of total functions. 9 Unfortunately, the unfolding transformation (2) is not safe in BL semantics and the question whether any theory can be equivalently reduced to c-plain form under BL is still unanswered.…”

Logic Programming and Nonmonotonic Reasoning

“…We adopt a definition of stable models based on syntactic transformations [2] which is a generalization of the previous definitions from [14] [19] and [13]. For predicate symbols (constants or variables) u and c, expression u ≤ c is defined as shorthand for ∀x(u(x) → c(x)).…”