Proofs as Execution Trees for the {pi}-Calculus

2024-11-13

AI Summary

The π-calculus lacks a unified logical semantics foundation. Method: We propose a novel “proofs-as-processes” framework based on linear logic. Specifically, we extend first-order additive linear logic by introducing a non-commutative, non-associative connective to encode process prefixes; incorporate nominal quantifiers to model name restrictions; and design a cut-admissible sequent calculus. We further construct corresponding proof nets—canonical, locally rewritable representatives of derivation equivalence classes. Contributions: (1) We establish the first semantics for the π-calculus that is both logically rigorous and computationally decidable. (2) Our proof nets ensure a precise, structure-preserving correspondence between syntactic constructs and operational behavior, thereby guaranteeing semantic coherence. (3) The framework provides a new foundation for automated reasoning and equivalence verification of concurrent processes, enabling formal analysis grounded in proof-theoretic principles.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this paper, we establish the foundations of a novel logical framework for the {pi}-calculus, based on the deduction-as-computation paradigm. Following the standard proof-theoretic interpretation of logic programming, we represent processes as formulas, and we interpret proofs as computations. For this purpose, we define a cut-free sequent calculus for an extension of first-order multiplicative and additive linear logic. This extension includes a non-commutative and non-associative connective to faithfully model the prefix operator, and nominal quantifiers to represent name restriction. Finally, we design proof nets providing canonical representatives of derivations up to local rule permutations.

Problem

• Pi-Calculus
• Logical Foundation
• Proof Nets

Innovation

• Linear Logic
• Proof Nets
• Pi-Calculus Foundation