A logic program Π 1 is said to be equivalent to a logic program Π 2 in the sense of the answer set semantics if Π 1 and Π 2 have the same answer sets. We are interested in the following stronger condition: for every logic program, Π, Π 1 , ∪ Π has the same answer sets as Π 2 ∪ Π. The study of strong equivalence is important, because we learn from it how one can simplify a part of a logic program without looking at the rest of it. The main theorem shows that the verification of strong equivalence can be accomplished by cheching the equivalence of formulas in a monotonic logic, called the logic of here-and-there, which is intermediate between classical logic and intuitionistic logic.

No abstract

No abstract

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in artificial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's first-order logic as a basis for defining nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for first-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QN c 5 , as a basis for defining a first-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QN c 5 , including Skolem forms and Herbrand theorems and Interpolation, and show that the first-oder version of equilibrium logic can be used as a foundation for answer set inference.

No abstract