Automating Boundary Filling in Cubical Agda

2024-02-19

International Conference on Formal Structures for Computation and Deduction

AI Summary

This work addresses the challenge of automating proofs involving higher-dimensional homotopical structures in cubical type theory, focusing on boundary filling—the core task underlying both contortion solving (single-cube filling) and Kan solving (multi-cube gluing). Methodologically, it models contortion as a poset-mapping problem and introduces, for the first time, a constraint satisfaction programming (CSP) framework to solve Kan filling. To mitigate combinatorial explosion in high dimensions and undecidability inherent in the theory, it designs lightweight heuristic algorithms. The prototype solver, implemented in Haskell, integrates poset-based modeling, CSP solving, and an interface to Cubical Agda. It successfully automates the verification of the Eckmann–Hilton theorem and demonstrates efficiency across multiple benchmark suites. This work establishes a novel paradigm and delivers key technical foundations for practical proof automation in cubical type theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

Abstract

When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy Type Theory allows the user to reason about higher structures, such as topological spaces, using higher inductive types (HITs) and univalence. Cubical Agda is an extension of Agda with computational support for HITs and univalence. A difficulty when working in Cubical Agda is dealing with the complex combinatorics of higher structures, an infinite-dimensional generalisation of equational reasoning. To solve these higher-dimensional equations consists in constructing cubes with specified boundaries. We develop a simplified cubical language in which we isolate and study two automation problems: contortion solving, where we attempt to"contort"a cube to fit a given boundary, and the more general Kan solving, where we search for solutions that involve pasting multiple cubes together. Both problems are difficult in the general case - Kan solving is even undecidable - so we focus on heuristics that perform well on practical examples. We provide a solver for the contortion problem using a reformulation of contortions in terms of poset maps, while we solve Kan problems using constraint satisfaction programming. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve goals presented by Cubical Agda. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks - providing the ground for further study of proof automation in cubical type theories.

Problem

• Automating boundary filling in cubical type theories
• Solving contortion and Kan problems for higher structures
• Developing heuristics for practical cubical type theory applications

Innovation

• Simplified cubical language for automation problems
• Solver for Dedekind and De Morgan contortions
• Kan solving via constraint satisfaction programming