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Quantifying over Optimum Answer Sets

Giuseppe Mazzotta, Francesco Ricca, Mirek Truszczynski

TL;DR

The paper addresses the challenge of encoding optimization problems within Answer Set Programming extended by Quantifiers (ASP(Q)) by introducing ASP^ω(Q), which integrates weak constraints both locally in quantified subprograms and globally for cross-subprogram optimization. It develops a formal semantics for coherence and optimal quantified answer sets, demonstrates modeling capabilities with Minmax Clique and Logic-Based Abduction, and provides a systematic rewriting from ASP^ω(Q) to plain ASP(Q) to enable complexity analysis and potential implementation. The authors establish upper and lower complexity bounds across PH levels (e.g., Σ_{n+1}^P, Π_{n+1}^P, Δ_{n+1}^P, Θ_{n+1}^P) and present several completeness results, revealing nuanced behavior when local constraints interact with quantifier structure. They also outline a rewriting strategy that preserves semantics while reducing quantifier depth, advancing both theoretical understanding and practical applicability for optimization in higher PH classes.

Abstract

Answer Set Programming with Quantifiers (ASP(Q)) has been introduced to provide a natural extension of ASP modeling to problems in the polynomial hierarchy (PH). However, ASP(Q) lacks a method for encoding in an elegant and compact way problems requiring a polynomial number of calls to an oracle in $Σ_n^p$ (that is, problems in $Δ_{n+1}^p$). Such problems include, in particular, optimization problems. In this paper we propose an extension of ASP(Q), in which component programs may contain weak constraints. Weak constraints can be used both for expressing local optimization within quantified component programs and for modeling global optimization criteria. We showcase the modeling capabilities of the new formalism through various application scenarios. Further, we study its computational properties obtaining complexity results and unveiling non-obvious characteristics of ASP(Q) programs with weak constraints.

Quantifying over Optimum Answer Sets

TL;DR

The paper addresses the challenge of encoding optimization problems within Answer Set Programming extended by Quantifiers (ASP(Q)) by introducing ASP^ω(Q), which integrates weak constraints both locally in quantified subprograms and globally for cross-subprogram optimization. It develops a formal semantics for coherence and optimal quantified answer sets, demonstrates modeling capabilities with Minmax Clique and Logic-Based Abduction, and provides a systematic rewriting from ASP^ω(Q) to plain ASP(Q) to enable complexity analysis and potential implementation. The authors establish upper and lower complexity bounds across PH levels (e.g., Σ_{n+1}^P, Π_{n+1}^P, Δ_{n+1}^P, Θ_{n+1}^P) and present several completeness results, revealing nuanced behavior when local constraints interact with quantifier structure. They also outline a rewriting strategy that preserves semantics while reducing quantifier depth, advancing both theoretical understanding and practical applicability for optimization in higher PH classes.

Abstract

Answer Set Programming with Quantifiers (ASP(Q)) has been introduced to provide a natural extension of ASP modeling to problems in the polynomial hierarchy (PH). However, ASP(Q) lacks a method for encoding in an elegant and compact way problems requiring a polynomial number of calls to an oracle in (that is, problems in ). Such problems include, in particular, optimization problems. In this paper we propose an extension of ASP(Q), in which component programs may contain weak constraints. Weak constraints can be used both for expressing local optimization within quantified component programs and for modeling global optimization criteria. We showcase the modeling capabilities of the new formalism through various application scenarios. Further, we study its computational properties obtaining complexity results and unveiling non-obvious characteristics of ASP(Q) programs with weak constraints.
Paper Structure (17 sections, 37 theorems, 49 equations, 2 algorithms)

This paper contains 17 sections, 37 theorems, 49 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Answer Set Programming
  3. Quantified Answer Set Programming with Weak Constraints
  4. Modeling examples
  5. Rewriting into plain ASP(Q)
  6. Complexity issues
  7. Related Work
  8. Conclusion
  9. Preliminaries on complexity classes
  10. Modeling $\Delta_2^P$ in ASP$^{\omega}$(Q)
  11. Stratified definition assumption
  12. Rewriting into plain ASP(Q)
  13. Rewriting uniform plain subprograms.
  14. Rewriting uniform notplain-plain subprograms.
  15. Rewrite subprograms with weak constraints.
  16. ...and 2 more sections

Key Result

Lemma 1

Let program $\Pi$ be such that $n \geq 2$ and the first two subprograms are plain and uniform, i.e., $\Box_1 = \Box_2$, and $\mathcal{W}(P_1) = \mathcal{W}(P_{2})=\emptyset$, then $\Pi$ is coherent if and only if $col_1(\Pi) = \Box_1 P_1 \cup P_{2}\Box_{3} P_{3}\ldots\Box_n P_n:C$ is coherent.

Theorems & Definitions (77)

  • Example 3.1: Impact of local weak constraints
  • Example 3.2: Optimal quantified answer sets
  • Lemma 1: Correctness $col_1(\cdot)$ transformation
  • Definition 1: Collapse notplain-plain existential subprograms
  • Lemma 2: Correctness $col_2(\cdot)$ transformation
  • Definition 2: Collapse notplain-plain universal subprograms
  • Lemma 3: Correctness $col_3(\cdot)$ transformation
  • Definition 3: Transform weak constraints
  • Definition 4: Transform existential not-plain subprogram
  • Lemma 4: Correctness $col_4(\cdot)$ transformation
  • ...and 67 more