An Effective Fixpoint Semantics for Linear Logic Programs

TL;DR

This paper develops a new bottom-up fixpoint semantics for linear logic programs, using constraints for symbolic representation, and explores its relation to disjunctive logic programming and Petri Nets.

Contribution

It introduces the first effective fixpoint semantics for linear logic programs using constraints and analyzes its relation to disjunctive logic programming.

Findings

Guarantees convergence for propositional LO

Establishes correctness and completeness of the abstraction for Petri Nets

Provides a formal foundation for linear logic program semantics

Abstract

In this paper we investigate the theoretical foundation of a new bottom-up semantics for linear logic programs, and more precisely for the fragment of LinLog that consists of the language LO enriched with the constant 1. We use constraints to symbolically and finitely represent possibly infinite collections of provable goals. We define a fixpoint semantics based on a new operator in the style of Tp working over constraints. An application of the fixpoint operator can be computed algorithmically. As sufficient conditions for termination, we show that the fixpoint computation is guaranteed to converge for propositional LO. To our knowledge, this is the first attempt to define an effective fixpoint semantics for linear logic programs. As an application of our framework, we also present a formal investigation of the relations between LO and Disjunctive Logic Programming. Using an approach based on abstract interpretation, we show that DLP fixpoint semantics can be viewed as an abstraction of our semantics for LO. We prove that the resulting abstraction is correct and complete for an interesting class of LO programs encoding Petri Nets.