[Paper Review] A Topological Application of Labelled Natural Deduction

Tiago M. L. de Veras, Arthur Ramos+2 more|arXiv (Cornell University)|Jun 19, 2019

Data Management and Algorithms27 references6 citations

TL;DR

This paper introduces a labelled natural deduction system based on computational paths—sequences of rewrites—to formalize equality in type theory and apply it to compute fundamental groups in algebraic topology. By treating paths as explicit syntactic objects, the authors derive the fundamental groups of the circle, torus, and real projective plane using only rewrite rules and path composition, showing that this approach provides a direct, computationally grounded alternative to traditional or homotopy type theory methods.

ABSTRACT

We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We then proceed to show the main result here: using this system to obtain the calculation of the fundamental group of the circle, of the torus and the real projective plane.

Motivation & Objective

• To address the lack of a syntactic counterpart for homotopical paths in type theory, particularly the absence of explicit path constructors beyond reflexivity.
• To formalize equality in type theory using computational paths as explicit rewrite sequences between terms, enabling syntactic reasoning about equality and equality of equalities.
• To provide a computationally grounded, alternative method to traditional algebraic topology or the encoding-decoding technique in homotopy type theory for computing fundamental groups.
• To demonstrate that fundamental groups of standard topological spaces can be derived using only labeled natural deduction and path rewriting, avoiding complex topological machinery.

Proposed method

• Employ a labelled natural deduction (LND) system where labels represent computational paths—sequences of rewrites—between terms of the same type.
• Use path composition and path equality rules (e.g., τ(α,β) for composition, ρ for reflexivity) to model path operations and derive group structures.
• Define path equivalence via rewrite rules (rw) and use normal forms to compare paths, establishing when two paths are equal.
• Apply the system to model loops on topological spaces: the circle via a single generator, the torus via two commuting generators, and the real projective plane via a single non-trivial loop with order 2.
• Use isomorphism theorems to map path groups to known algebraic groups (e.g., ℤ for the circle, ℤ×ℤ for the torus, ℤ₂ for the real projective plane).
• Prove group axioms (identity, associativity, inverse) syntactically using path rewriting and label manipulation, showing that the resulting structure forms a group under composition.

Experimental results

Research questions

• RQ1 Can computational paths be formalized as explicit syntactic entities in a natural deduction system to represent equality between terms?
• RQ2 How can path composition and equality of paths be axiomatized in a way that supports reasoning about higher-order equalities?
• RQ3 Can this formalism be used to derive the fundamental group of topological spaces without relying on advanced algebraic topology or homotopy type theory techniques?
• RQ4 What is the algebraic structure of the fundamental group when computed via path rewriting and labeled deduction?
• RQ5 Is it possible to establish isomorphisms between the syntactically derived path groups and standard algebraic groups (e.g., ℤ, ℤ₂) using this method?

Key findings

• The fundamental group of the circle is isomorphic to ℤ, derived via a single generator loop and path composition.
• The fundamental group of the torus is isomorphic to ℤ×ℤ, obtained by modeling two independent, commuting loops.
• The fundamental group of the real projective plane is isomorphic to ℤ₂, with a single non-trivial loop of order 2.
• The group axioms (identity, associativity, inverse) are verified syntactically using path rewriting and label transformations, confirming the group structure.
• The path group of the real projective plane is generated by ρ (reflexivity) and α (a non-trivial loop), with α∘α ≡ ρ, confirming ℤ₂ structure.
• An isomorphism is explicitly constructed between ℤ₂ and Π₁(ℝP²), showing that every path is rw-equivalent to either ρ or α, and the map toPathℤ₂ is a group isomorphism.