Year: 2025
Type: article
Abstract: Type theory can be described as a generalised algebraic theory.This automatically gives a notion of model and the existence of the syntax as the initial model, which is a quotient inductive-inductive type.Algebraic definitions of type theory include Ehrhard's definition of model, categories with families (CwFs), contextual categories, Awodey's natural models, C-systems, B-systems.With the exception of B-systems, these notions are based on a parallel substitution calculus where substitutions form a category.In this paper we define a single substitution calculus (SSC) for type theory and show that the SSC syntax and the CwF syntax are isomorphic for a theory with dependent function space and a hierarchy of universes.SSC only includes single substitutions and single weakenings, and eight equations relating these: four equations describe how to substitute variables, and there are four equations on types which are needed to typecheck the other equations.SSC provides a simple, minimalistic alternative to parallel substitution calculi or B-systems for defining type theory.SSC relates to CwF as extensional combinatory calculus relates to lambda calculus: there are more models of the former, but the syntaxes are equivalent.If we have some additional type formers, we show that an SSC model gives rise to a CwF.* This research was supported by the European Union (ERC, HOTT, 101170308).Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council.
DOI: 10.4204/eptcs.431.4
Authors: Ambrus Kaposi, Szumi Xie
Institution: Eötvös Loránd University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Funder: European Commission
Award: G2759981103
OpenAlex ID: W4415102167
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