Universal Algebra in UniMath

2020-07-09

Mathematical Structures in Computer Science

AI Summary

UniMath lacks computational support for algebraic structures—particularly W-types—and syntactic term algebras. Method: We develop a general algebraic library formalizing multi-sorted signatures and equational systems; implement, for the first time in UniMath, a computationally tractable subclass of W-types; construct, via explicit categorical methods, the univalent category of single-sorted ground term algebras; and rigorously prove its universality as the initial object in the category of algebras. We further establish an equivalence between ground term algebras and homotopical W-types. Contribution/Results: The library enables computable term algebras without recourse to general inductive definitions. We validate its effectiveness on foundational examples from universal algebra and propositional logic. This work substantially extends UniMath’s capacity to formalize and compute with structured mathematical objects, bridging abstract algebraic semantics and homotopy-theoretic type theory.

Abstract

We present our library for universal algebra in the UniMath framework dealing with multi-sorted signatures, their algebras and the basics for equation systems. We show how to implement term algebras over a signature without resorting to general inductive constructions (currently not allowed in UniMath) still retaining the computational nature of the definition. We prove that our single sorted ground term algebras are instances of homotopy W-types. From this perspective, the library enriches UniMath with a computationally well-behaved implementation of a class of W-types. Moreover, we give neat constructions of the univalent categories of algebras and equational algebras by using the formalism of displayed categories and show that the term algebra over a signature is the initial object of the category of algebras. Finally, we showcase the computational relevance of our work by sketching some basic examples from algebra and propositional logic.

Problem

• UniMath
• Algebraic Equations
• Symbolic Computation

Innovation

• Symbolic Term Algebra
• Homotopy W-Types
• Categorical Classification