First-Order Stable Model Semantics with Intensional Functions

In classical logic, nonBoolean fluents, such as the location of an object, can be naturally described by functions. However, this is not the case in answer set programs, where the values of functions are pre-defined, and nonmonotonicity of the semantics is related to minimizing the extents of predicates but has nothing to do with functions. We extend the first-order stable model semantics by Ferraris, Lee, and Lifschitz to allow intensional functions -- functions that are specified by a logic program just like predicates are specified. We show that many known properties of the stable model semantics are naturally extended to this formalism and compare it with other related approaches to incorporating intensional functions. Furthermore, we use this extension as a basis for defining Answer Set Programming Modulo Theories (ASPMT), analogous to the way that Satisfiability Modulo Theories (SMT) is defined, allowing for SMT-like effective first-order reasoning in the context of ASP. Using SMT solving techniques involving functions, ASPMT can be applied to domains containing real numbers and alleviates the grounding problem. We show that other approaches to integrating ASP and CSP/SMT can be related to special cases of ASPMT in which functions are limited to non-intensional ones.

A Functional View of Strong Negation in Answer Set Programming

The distinction between strong negation and default negation has been useful in answer set programming. We present an alternative account of strong negation, which lets us view strong negation in terms of the functional stable model semantics by Bartholomew and Lee. More specifically, we show that, under complete interpretations, minimizing both positive and negative literals in the traditional answer set semantics is essentially the same as ensuring the uniqueness of Boolean function values under the functional stable model semantics. The same account lets us view Lifschitz's two-valued logic programs as a special case of the functional stable model semantics. In addition, we show how non-Boolean intensional functions can be eliminated in favor of Boolean intensional functions, and furthermore can be represented using strong negation, which provides a way to compute the functional stable model semantics using existing ASP solvers. We also note that similar results hold with the functional stable model semantics by Cabalar.