Type Theory with Single Substitutions

2025-10-14

AI Summary

This paper addresses the high complexity and formal redundancy of parallel substitution and B-systems in type theory. We introduce the Single Substitution Calculus (SSC), a minimalist syntactic framework built solely upon single substitution and single weakening operations, governed by eight equational rules. SSC fully supports dependent function types and universe hierarchies, providing a lightweight foundation for algebraic type theory. Leveraging inductive–inductive quotient types, categorical semantics, and categories with families (CwFs), we establish—for the first time—a rigorous isomorphism between SSC syntax and CwF structure. Moreover, we give a constructive procedure to derive a CwF from any SSC model. Our key innovation lies in recasting core type-theoretic mechanisms using combinatorial simplicity, thereby elucidating the essential nature of substitution. This yields a novel paradigm for the design, verification, and implementation of type systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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.

Problem

• Defining a single substitution calculus for type theory
• Proving isomorphism between SSC and CwF syntaxes
• Providing minimalistic alternative to parallel substitution calculi

Innovation

• Single substitution calculus for type theory
• Minimalistic alternative to parallel substitution
• SSC syntax isomorphic to CwF syntax