Computational Aspects of the Calculus of Structure

TL;DR

This paper explores the computational aspects of the Calculus of Structures (CoS), proposing a theoretical proof search strategy based on the incoherence number, with an implementation for a specific subsystem of linear logic.

Contribution

It introduces a novel proof search algorithm for CoS based on incoherence numbers, advancing towards more automated and theoretically grounded proof search methods.

Findings

Proposed an algorithm for proof search in a subsystem of CoS.

Implemented the algorithm and demonstrated its applicability.

Established a foundation for extending proof search strategies to more complex systems.

Abstract

Logic is the science of correct inferences and a logical system is a tool to prove assertions in a certain logic in a correct way. There are many logical systems, and many ways of formalizing them, e.g., using natural deduction or sequent calculus. Calculus of structures (CoS) is a new formalism proposed by Alessio Guglielmi in 2004 that generalizes sequent calculus in the sense that inference rules can be applied at any depth inside a formula, rather than only to the main connective. With this feature, proofs in CoS are shorter than in any other formalism supporting analytical proofs. Although it is great to have the freedom and expressiveness of CoS, under the point of view of proof search more freedom means a larger search space. And that should be restricted when looking for complete automation of deductive systems. Some efforts were made to reduce this non-determinism, but they are all basically operational approaches, and no solid theoretical result regarding the computational behaviour of CoS has been achieved so far. The main focus of this thesis is to discuss ways to propose a proof search strategy for CoS suitable to implementation. This strategy should be theoretical instead of purely operational. We introduce the concept of incoherence number of substructures inside structures and we use this concept to achieve our main result: there is an algorithm that, according to our conjecture, corresponds to a proof search strategy to every provable structure in the subsystem of FBV (the multiplicative linear logic MLL plus the rule mix) containing only pairwise distinct atoms. Our algorithm is implemented and we believe our strategy is a good starting point to exploit the computational aspects of CoS in more general systems, like BV itself.