Effective Synthesis of Strongly Equivalent Logic Programs via Craig Interpolation in First-Order Logic

Core Concepts

The existence of a logic program R that is strongly equivalent to the combination of a given program P and another given program Q, under a restricted vocabulary, can be characterized as a first-order entailment. Moreover, such a program R can be effectively constructed from a Craig interpolant of the entailment.

Abstract

The paper presents a technique for synthesizing logic programs that are strongly equivalent to the combination of two given programs, under a restricted vocabulary. The key ideas are:

Characterizing first-order formulas that encode logic programs, and extracting a logic program from such a formula.
Defining a variation of Craig interpolation, called LP-interpolation, that produces an interpolant that also encodes a logic program.
Showing that the existence of a strongly equivalent program R under a restricted vocabulary can be expressed as a first-order entailment, and that R can be effectively constructed from an LP-interpolant of this entailment.

The paper also presents a more fine-grained version of this result, where the allowed positions of predicates within rules are also constrained.
The technical approach relies on a known translation of logic programs to first-order logic for verifying strong equivalence, which is used in reverse to extract programs from formulas. The authors implement a prototype of the synthesis technique using first-order theorem provers for computing Craig interpolants.

Stats

None.

Quotes

None.

Key Insights Distilled From

Synthesizing Strongly Equivalent Logic Programs: Beth Definability for Answer Set Programs via Craig Interpolation in First-Order Logic

by Jan Heuer,Ch... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2402.07696.pdf

Deeper Inquiries

What are some potential applications of the proposed technique for synthesizing strongly equivalent logic programs beyond the stable model semantics, e.g., in other non-monotonic reasoning frameworks

The proposed technique for synthesizing strongly equivalent logic programs using interpolation could have various applications beyond the stable model semantics. One potential application is in the field of non-monotonic reasoning frameworks, such as default logic and autoepistemic logic. In these frameworks, the ability to synthesize logic programs that are strongly equivalent under specific semantics is crucial for tasks like belief revision, reasoning about incomplete information, and handling conflicting information. By extending the technique to these frameworks, it could facilitate more efficient and accurate reasoning processes by ensuring the equivalence of different program representations.

How could the proposed technique be extended to handle more complex constraints on the structure of the synthesized logic program, beyond just restricting the vocabulary and predicate positions

To handle more complex constraints on the structure of the synthesized logic program, the proposed technique could be extended in several ways. One approach could involve incorporating additional constraints on the rules of the logic programs, such as specifying the ordering of predicates within rules or imposing restrictions on the combination of predicates in different rules. This could be achieved by introducing more sophisticated encoding schemes in the first-order formulas used for interpolation, allowing for a finer control over the structure of the synthesized programs. Additionally, the technique could be enhanced to handle constraints on the arity of predicates, the presence of specific literals, or the use of certain logical connectives within rules.

What are the key challenges in scaling the proposed interpolation-based synthesis approach to larger and more complex logic programs, and how could these challenges be addressed

Scaling the proposed interpolation-based synthesis approach to larger and more complex logic programs poses several challenges. One key challenge is the computational complexity associated with handling a large number of rules, predicates, and constraints in the interpolation process. This can lead to increased memory and time requirements, making the synthesis process inefficient for large-scale programs. To address this challenge, optimization techniques such as parallel processing, distributed computing, and algorithmic improvements could be employed to enhance the scalability of the approach. Additionally, developing more efficient interpolation algorithms tailored for logic programs and leveraging advancements in automated reasoning systems could help overcome the challenges of scaling the technique to handle larger and more complex programs.