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.

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 the stable models of a safe sentence can be characterized by a formula of a simple syntactic form.

Answer set programming is a prominent declarative programming paradigm used in formulating combinatorial search problems and implementing distinct knowledge representation formalisms. It is common that several related and yet substantially different answer set programs exist for a given problem. Sometimes these encodings may display significantly different performance. Uncovering {\em precise formal} links between these programs is often important and yet far from trivial. This paper claims the correctness of a number of interesting program rewritings.

Generalized relational theories with null values in the sense of Reiter are first-order theories that provide a semantics for relational databases with incomplete information. In this paper we show that any such theory can be turned into an equivalent logic program, so that models of the theory can be generated using computational methods of answer set programming. As a step towards this goal, we develop a general method for calculating stable models under the domain closure assumption but without the unique name assumption.

Circumscription and logic programs under the stable model semantics are two well-known nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis of a family of action languages, such as language A and several of its descendants. Based on the discovery that circumscription and the stable model semantics coincide on a class of canonical formulas, we reformulate the situation calculus and the event calculus in the general theory of stable models. We also present a translation that turns the reformulations further into answer set programs, so that efficient answer set solvers can be applied to compute the situation calculus and the event calculus.

Lin and Zhao's theorem on loop formulas states that in the propositional case the stable model semantics of a logic program can be completely characterized by propositional loop formulas, but this result does not fully carry over to the first-order case. We investigate the precise relationship between the first-order stable model semantics and first-order loop formulas, and study conditions under which the former can be represented by the latter. In order to facilitate the comparison, we extend the definition of a first-order loop formula which was limited to a nondisjunctive program, to a disjunctive program and to an arbitrary first-order theory. Based on the studied relationship we extend the syntax of a logic program with explicit quantifiers, which allows us to do reasoning involving non-Herbrand stable models using first-order reasoners. Such programs can be viewed as a special class of first-order theories under the stable model semantics, which yields more succinct loop formulas than the general language due to their restricted syntax.

This paper focuses on computing first-order theories under either stable model semantics or circumscription. A reduction from first-order theories to logic programs under stable model semantics over finite structures is proposed, and an embedding of circumscription into stable model semantics is also given. Having such reduction and embedding, reasoning problems represented by first-order theories under these two semantics can then be handled by using existing answer set solvers. The effectiveness of this approach in computing hard problems beyond NP is demonstrated by some experiments.

We provide reformulations and generalizations of both the semantics of logic programs by Faber, Leone and Pfeifer and its extension to arbitrary propositional formulas by Truszczynski. Unlike the previous definitions, our generalizations refer neither to grounding nor to fixpoints, and apply to first-order formulas containing aggregate expressions. In the same spirit as the first-order stable model semantics proposed by Ferraris, Lee and Lifschitz, the semantics proposed here are based on syntactic transformations that are similar to circumscription. The reformulations provide useful insights into the FLP semantics and its relationship to circumscription and the first-order stable model semantics.

We present an approach to integrating rules and ontologies on the basis of the first-order stable model semantics proposed by Ferraris, Lee and Lifschitz. We show that some existing integration proposals can be uniformly reformulated in terms of the first-order stable model semantics. The reformulations are simpler than the original proposals in the sense that they do not refer to grounding.

We provide reformulations and generalizations of both the semantics of logic programs by Faber, Leone and Pfeifer and its extension to arbitrary propositional formulas by Truszczynski. Unlike the previous definitions, our generalizations refer neither to grounding nor to fixpoints, and apply to first-order formulas containing aggregate expressions. Similar to the first-order stable model semantics by Ferraris, Lee and Lifschitz, the reformulations presented here are based on syntactic transformations that are similar to circumscription. The reformulations provide useful insights into the FLP semantics and its relationship to circumscription and the first-order stable model semantics.