Stable-Unstable Semantics: Beyond NP with Normal Logic Programs

TL;DR

This paper introduces a novel logic programming paradigm that extends answer set programming to handle problems beyond NP by integrating oracles as logic programs, enabling reasoning on higher levels of the polynomial hierarchy.

Contribution

It proposes combined logic programs that embed oracles as logic programs, allowing ASP to model problems at arbitrary levels of the polynomial hierarchy.

Findings

Developed a new formalism for combined logic programs.

Implemented a proof-of-concept system demonstrating the approach.

Enables ASP to reason beyond NP using recursive oracle encodings.

Abstract

Standard answer set programming (ASP) targets at solving search problems from the first level of the polynomial time hierarchy (PH). Tackling search problems beyond NP using ASP is less straightforward. The class of disjunctive logic programs offers the most prominent way of reaching the second level of the PH, but encoding respective hard problems as disjunctive programs typically requires sophisticated techniques such as saturation or meta-interpretation. The application of such techniques easily leads to encodings that are inaccessible to non-experts. Furthermore, while disjunctive ASP solvers often rely on calls to a (co-)NP oracle, it may be difficult to detect from the input program where the oracle is being accessed. In other formalisms, such as Quantified Boolean Formulas (QBFs), the interface to the underlying oracle is more transparent as it is explicitly recorded in the quantifier prefix of a formula. On the other hand, ASP has advantages over QBFs from the modeling perspective. The rich high-level languages such as ASP-Core-2 offer a wide variety of primitives that enable concise and natural encodings of search problems. In this paper, we present a novel logic programming--based modeling paradigm that combines the best features of ASP and QBFs. We develop so-called combined logic programs in which oracles are directly cast as (normal) logic programs themselves. Recursive incarnations of this construction enable logic programming on arbitrarily high levels of the PH. We develop a proof-of-concept implementation for our new paradigm. This paper is under consideration for acceptance in TPLP.