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Pearce's Characterisation in an Epistemic Domain

Ezgi Iraz Su

TL;DR

This work addresses fragmentation in epistemic extensions of answer-set programming by introducing Epistemic Answer Set Programming (EASP), a unifying framework built on Pearce's equilibrium-model characterisation. It formalises a twofold world-view computation via $t$-minimality (truth) and $k$-minimality (knowledge), and generalises Ferraris' lemma to relate epistemic equilibrium logics (EEL) to EASP. The paper surveys major ES formalisms (ES_94, ES_20a, ES_20b, ES_21) and three EEL variants (AEEL, FAEEL, RAEEL), showing they can be embedded and compared within EASP's language and reducts. This correspondence clarifies how epistemic modalities interact with nonmonotonic reasoning, enabling robust encoding of incomplete information and self-belief in logic programs, with potential practical impact on knowledge representation and reasoning in AI systems.

Abstract

Answer-set programming (ASP) is a successful problem-solving approach in logic-based AI. In ASP, problems are represented as declarative logic programs, and solutions are identified through their answer sets. Equilibrium logic (EL) is a general-purpose nonmonotonic reasoning formalism, based on a monotonic logic called here-and-there logic. EL was basically proposed by Pearce as a foundational framework of ASP. Epistemic specifications (ES) are extensions of ASP-programs with subjective literals. These new modal constructs in the ASP-language make it possible to check whether a regular literal of ASP is true in every (or some) answer-set of a program. ES-programs are interpreted by world-views, which are essentially collections of answer-sets. (Reflexive) autoepistemic logic is a nonmonotonic formalism, modeling self-belief (knowledge) of ideally rational agents. A relatively new semantics for ES is based on a combination of EL and (reflexive) autoepistemic logic. In this paper, we first propose an overarching framework in the epistemic ASP domain. We then establish a correspondence between existing (reflexive) (auto)epistemic equilibrium logics and our easily-adaptable comprehensive framework, building on Pearce's characterisation of answer-sets as equilibrium models. We achieve this by extending Ferraris' work on answer sets for propositional theories to the epistemic case and reveal the relationship between some ES-semantic proposals.

Pearce's Characterisation in an Epistemic Domain

TL;DR

This work addresses fragmentation in epistemic extensions of answer-set programming by introducing Epistemic Answer Set Programming (EASP), a unifying framework built on Pearce's equilibrium-model characterisation. It formalises a twofold world-view computation via -minimality (truth) and -minimality (knowledge), and generalises Ferraris' lemma to relate epistemic equilibrium logics (EEL) to EASP. The paper surveys major ES formalisms (ES_94, ES_20a, ES_20b, ES_21) and three EEL variants (AEEL, FAEEL, RAEEL), showing they can be embedded and compared within EASP's language and reducts. This correspondence clarifies how epistemic modalities interact with nonmonotonic reasoning, enabling robust encoding of incomplete information and self-belief in logic programs, with potential practical impact on knowledge representation and reasoning in AI systems.

Abstract

Answer-set programming (ASP) is a successful problem-solving approach in logic-based AI. In ASP, problems are represented as declarative logic programs, and solutions are identified through their answer sets. Equilibrium logic (EL) is a general-purpose nonmonotonic reasoning formalism, based on a monotonic logic called here-and-there logic. EL was basically proposed by Pearce as a foundational framework of ASP. Epistemic specifications (ES) are extensions of ASP-programs with subjective literals. These new modal constructs in the ASP-language make it possible to check whether a regular literal of ASP is true in every (or some) answer-set of a program. ES-programs are interpreted by world-views, which are essentially collections of answer-sets. (Reflexive) autoepistemic logic is a nonmonotonic formalism, modeling self-belief (knowledge) of ideally rational agents. A relatively new semantics for ES is based on a combination of EL and (reflexive) autoepistemic logic. In this paper, we first propose an overarching framework in the epistemic ASP domain. We then establish a correspondence between existing (reflexive) (auto)epistemic equilibrium logics and our easily-adaptable comprehensive framework, building on Pearce's characterisation of answer-sets as equilibrium models. We achieve this by extending Ferraris' work on answer sets for propositional theories to the epistemic case and reveal the relationship between some ES-semantic proposals.

Paper Structure

This paper contains 16 sections, 2 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background: ASP and epistemic specifications (ES) in a nutshell
  3. Gelfond's epistemic specifications: ESdoksandort
  4. Truth conditions:
  5. Semantics:
  6. Motivation
  7. Epistemic Answer Set Programming (EASP)
  8. The Language of EASP (${\cal L}_{{ \mathsf{EASP} }}$)
  9. Semantics of EASP in terms of stable sfive-models
  10. Epistemic Extensions of Equilibrium Logic
  11. Su et al.'s approach (ESyirmia): autoepistemic equilibrium logic (AEEL)
  12. Epistemic here-and-there logic (EHT) and its equilibrium sfive-models w.r.t. truth
  13. Cabalar et al.'s approach (ESyirmib): founded autoepistemic equilibrium logic (FAEEL)
  14. Su's approach (ESyirmibir): reflexive autoepistemic equilibrium logic (RAEEL)
  15. Correspondence between EASP and EEL
  16. ...and 1 more sections

Key Result

Lemma 1

Given $I= \{ 1, \ldots , n \}$, let $\mathcal{T} = \{ T_i \} _{i \in I} = \{ T_i \ : \ i \in I \}$ be an $\mathsf{S5}$-model, and let $\texttt{s} \ : \ \mathcal{T} \rightarrow 2^{ \mathbb{P} }$ be a subset function s.t. $\texttt{s} (T_i)=H_i \subseteq T_i$ for every $i \in I$. For an $\mathsf{E

Theorems & Definitions (13)

  • Example 1
  • Example 2
  • Definition 1: weakening of a point in an $\mathsf{S5}$-model in terms of truth: functional approach
  • Definition 2: weakening of a point in an $\mathsf{S5}$-model in terms of truth: relational approach
  • Definition 3: generalisation of the truth-minimality ($\texttt{t}$-minimality) criterion of $\mathsf{ASP}$ to $\mathsf{EASP}$
  • Example 3
  • Example 4
  • Remark 1
  • Definition 4: generalisation of the conventional reduct definition of $\mathsf{ASP}$ to $\mathsf{EASP}$
  • Example 5
  • ...and 3 more