[Paper Review] Abduction, ASP and Open Logic Programs

Piero A. Bonatti|ArXiv.org|Jul 7, 2002

Logic, Reasoning, and Knowledge12 references3 citations

TL;DR

This paper establishes open logic programs as a unifying framework that generalizes both abduction and answer set programming (ASP), demonstrating that open inference subsumes and extends both paradigms. It proves that open programs can simulate abduction and ASP under generalized stable model semantics, enabling new forms of abductive reasoning—such as nonground inference and domain extension with new individuals—while also characterizing the computational complexity of credulous and skeptical open inference for finite, function-free programs.

ABSTRACT

Open logic programs and open entailment have been recently proposed as an abstract framework for the verification of incomplete specifications based upon normal logic programs and the stable model semantics. There are obvious analogies between open predicates and abducible predicates. However, despite superficial similarities, there are features of open programs that have no immediate counterpart in the framework of abduction and viceversa. Similarly, open programs cannot be immediately simulated with answer set programming (ASP). In this paper we start a thorough investigation of the relationships between open inference, abduction and ASP. We shall prove that open programs generalize the other two frameworks. The generalized framework suggests interesting extensions of abduction under the generalized stable model semantics. In some cases, we will be able to reduce open inference to abduction and ASP, thereby estimating its computational complexity. At the same time, the aforementioned reduction opens the way to new applications of abduction and ASP.

Motivation & Objective

• To formalize the relationships between open logic programs, abduction, and answer set programming (ASP) under generalized stable model semantics.
• To extend the expressive power of abduction by enabling the abductive inference of new individuals and domain cardinality constraints, which are not supported in standard frameworks.
• To investigate whether open inference can be reduced to ASP and abduction, thereby enabling the use of existing ASP engines for abductive reasoning under bounded domains.
• To explore new computational strategies, such as lemma generation and nonground inference, through the generalized framework of open programs.
• To characterize the computational complexity of credulous and skeptical open inference for finite, function-free open programs.

Proposed method

• Proposes open logic programs as a generalization of both abduction and ASP, where open predicates are not fully specified and can be completed in multiple ways.
• Introduces a formal embedding of abduction and ASP into open inference by encoding abduction frameworks as open programs with empty open predicate sets and normal logic programs as degenerate open programs.
• Applies the generalized stable model semantics to open programs, allowing for the abductive inference of new individuals and upper bounds on domain size.
• Reduces credulous and skeptical open inference to standard ASP inference via a translation that preserves stable models under finite, function-free constraints.
• Uses the Smodels ASP engine to compute abductive explanations in open programs, demonstrating practical applicability of the framework.
• Introduces a new form of abduction—generalized skeptical consequences—inspired by diagnosis-as-entailment, analogous to mixed inference in open programs.

Experimental results

Research questions

• RQ1 Can open logic programs serve as a unifying framework that generalizes both abduction and ASP under the generalized stable model semantics?
• RQ2 What are the key differences and similarities between open inference, abduction, and ASP, particularly regarding domain handling and predicate specification?
• RQ3 Can open inference be reduced to standard ASP inference, and if so, under what conditions (e.g., finite, function-free programs)?
• RQ4 Can the open framework support nonground abductive reasoning and the abductive inference of new individuals, which are not supported in standard abduction?
• RQ5 What is the computational complexity of credulous and skeptical open inference in finite, function-free open programs?

Key findings

• Open logic programs generalize both abduction and ASP, with any abduction framework ⟨T, A⟩ being representable as an open program ⟨T, ∅, A⟩ and any normal logic program P as ⟨P, ∅, ∅⟩.
• The framework enables abductive inference of new individuals and domain cardinality bounds—features absent in standard abduction but supported in 3-valued completion and stratified negation.
• Credulous and skeptical open inference for finite, function-free open programs can be reduced to standard ASP inference, enabling the use of ASP engines like Smodels for abductive reasoning.
• The complexity of skeptical and credulous open inference is characterized as co-NP-complete for finite, function-free open programs, based on the embedding into ASP.
• The open framework supports nonground inference and suggests new computation strategies, such as lemma generation, via its nonground skeptical open resolution calculus.
• A new form of abduction—generalized skeptical consequences—emerges from the framework, analogous to diagnosis-as-entailment, and may be used to pre-compute lemmata for efficiency.