[Paper Review] Surface proofs for linear logic.

Lawrence H. Dunn, Jamie Vicary|arXiv (Cornell University)|Jan 20, 2016

Logic, programming, and type systems13 references3 citations

TL;DR

This paper proposes a geometric representation of proofs in multiplicative linear logic using decorated surfaces, where logical equivalence of proofs corresponds exactly to geometric equivalence of their surfaces. The approach relies on a coherence theorem for Frobenius pseudomonoids to establish this correspondence, providing a topological foundation for proof equivalence in linear logic.

ABSTRACT

We show that a proof in multiplicative linear logic can be represented as a decorated surface, such that two proofs are logically equivalent just when their surfaces are geometrically equivalent. The technical basis is a coherence theorem for Frobenius pseudomonoids.

Motivation & Objective

• To establish a geometric semantics for proofs in multiplicative linear logic.
• To address the problem of determining when two proofs are logically equivalent.
• To develop a topological framework—using decorated surfaces—that captures proof equivalence through geometric means.
• To prove that logical equivalence of proofs coincides precisely with geometric equivalence of their surface representations.

Proposed method

• Represent proofs in multiplicative linear logic as decorated surfaces, where surface structure encodes logical structure.
• Use the algebraic framework of Frobenius pseudomonoids to formalize the algebraic properties of proof composition.
• Apply a coherence theorem for Frobenius pseudomonoids to ensure that all valid surface representations correspond to well-formed proofs.
• Establish a bijection between proof equivalence and surface geometric equivalence via the coherence result.
• Use topological invariance to show that deformations of surfaces preserve logical meaning.
• Demonstrate that surface isomorphism captures exactly the same equivalence as proof equivalence in the logic.

Experimental results

Research questions

• RQ1 Can proofs in multiplicative linear logic be represented as geometric objects such that logical equivalence corresponds to geometric equivalence?
• RQ2 What algebraic structure underlies the correspondence between proof composition and surface topology?
• RQ3 How can a coherence theorem for Frobenius pseudomonoids be used to justify the geometric representation of proofs?
• RQ4 Is every valid proof representation captured by a unique geometric surface up to topological equivalence?
• RQ5 Does the geometric representation preserve the full logical content of proofs while abstracting away syntactic redundancy?

Key findings

• Logical equivalence of proofs in multiplicative linear logic is precisely captured by geometric equivalence of their corresponding decorated surfaces.
• The coherence theorem for Frobenius pseudomonoids ensures that all surface representations respect the algebraic structure of proof composition.
• Proofs that are logically equivalent can be transformed into one another via a sequence of geometric moves on the surface.
• The surface representation provides a canonical form for proofs under logical equivalence.
• The method establishes a complete and sound correspondence between proof-theoretic and topological notions of equivalence.
• The framework enables a topological characterization of proof equivalence without relying on syntactic manipulation.