Abstract The input language of the answer set solver clingo is based on the definition of a stable model proposed by Paolo Ferraris. The semantics of the ASP-Core language, developed by the ASP Standardization Working Group, uses the approach to stable models due to Wolfgang Faber, Nicola Leone, and Gerald Pfeifer. The two languages are based on different versions of the stable model semantics, and the ASP-Core document requires, “for the sake of an uncontroversial semantics,” that programs avoid the use of recursion through aggregates. In this paper we prove that the absence of recursion through aggregates does indeed guarantee the equivalence between the two versions of the stable model semantics, and show how that requirement can be relaxed without violating the equivalence property.

This paper introduces first-order modular logic programs, which provide a way of viewing answer set programs as consisting of many independent, meaningful modules. We also present conservative extensions of such programs. This concept helps to identify strong relationships between modular programs as well as between traditional programs. For example, we illustrate how the notion of a conservative extension can be used to justify the common projection rewriting. This is a short version of a paper was presented at the 32nd International Conference on Logic Programming (Harrison and Lierler, 2016).

Abstract We argue that turning a logic program into a set of completed definitions can be sometimes thought of as the “reverse engineering” process of generating a set of conditions that could serve as a specification for it. Accordingly, it may be useful to define completion for a large class of Answer Set Programming (ASP) programs and to automate the process of generating and simplifying completion formulas. Examining the output produced by this kind of software may help programmers to see more clearly what their program does, and to what degree its behavior conforms with their expectations. As a step toward this goal, we propose here a definition of program completion for a large class of programs in the input language of the ASP grounder gringo, and study its properties.

Abstract Modular logic programs provide a way of viewing logic programs as consisting of many independent, meaningful modules. This paper introduces first-order modular logic programs, which can capture the meaning of many answer set programs. We also introduce conservative extensions of such programs. This concept helps to identify strong relationships between modular programs as well as between traditional programs. We show how the notion of a conservative extension can be used to justify the common projection rewriting.

Abstract The definition of stable models for propositional formulas with infinite conjunctions and disjunctions can be used to describe the semantics of answer set programming languages. In this note, we enhance that definition by introducing a distinction between intensional and extensional atoms. The symmetric splitting theorem for first-order formulas is then extended to infinitary formulas and used to reason about infinitary definitions.

Abstract The infinitary propositional logic of here-and-there is important for the theory of answer set programming in view of its relation to strongly equivalent transformations of logic programs. We know a formal system axiomatizing this logic exists, but a proof in that system may include infinitely many formulas. In this note we describe a relationship between the validity of infinitary formulas in the logic of here-and-there and the provability of formulas in some finite deductive systems. This relationship allows us to use finite proofs to justify the validity of infinitary formulas.

Abstract This paper defines the syntax and semantics of the input language of the ASP grounder gringo . The definition covers several constructs that were not discussed in earlier work on the semantics of that language, including intervals, pools, division of integers, aggregates with non-numeric values, and lparse-style aggregate expressions. The definition is abstract in the sense that it disregards some details related to representing programs by strings of ASCII characters. It serves as a specification for gringo from Version 4.5 on.

Abstract Propositional formulas that are equivalent in intuitionistic logic, or in its extension known as the logic of here-and-there, have the same stable models. We extend this theorem to propositional formulas with infinitely long conjunctions and disjunctions and show how to apply this generalization to proving properties of aggregates in answer set programming.