Non-deterministic approximation operators: ultimate operators, semi-equilibrium semantics and aggregates (full version)

TL;DR

This paper extends approximation fixpoint theory to non-deterministic operators, introducing ultimate approximations, algebraic semi-equilibrium semantics, and generalizations to disjunctive logic programs with aggregates.

Contribution

It introduces ultimate approximations for non-deterministic operators, formalizes semi-equilibrium semantics algebraically, and extends disjunctive logic program characterizations to include aggregates.

Findings

Defined and analyzed ultimate approximations.

Provided algebraic formulation of semi-equilibrium semantics.

Generalized disjunctive logic program characterizations to include aggregates.

Abstract

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, i.e.\ operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of non-deterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola, et al., and (3) we generalize the characterisations of disjunctive logic programs to disjunctive logic programs with aggregates.