[Paper Review] Type theory and rewriting

Frédéric Blanqui|arXiv (Cornell University)|Oct 11, 2006

Logic, programming, and type systems77 references3 citations

TL;DR

This PhD thesis presents a formal framework integrating type theory with rewriting systems, establishing conditions under which type systems extended with rewriting rules preserve strong normalization and logical consistency. It introduces Reduction Type Systems (RTS) and Algebraic Type Systems (ATS), proving that confluence and specific ordering constraints on rewrite rules ensure correctness, enabling safe program extraction from proofs and supporting advanced features like quotient types and inductive definitions.

ABSTRACT

We study the properties, in particular termination, of dependent types systems for lambda calculus and rewriting.

Motivation & Objective

• To formalize the interaction between type theory and rewriting in a way that preserves logical consistency and strong normalization.
• To extend the Calculus of Constructions with rewriting while ensuring that proofs remain valid and computationally meaningful.
• To provide a foundation for practical proof assistants like Coq by enabling safe use of rewrite rules in type systems.
• To address challenges in defining inductive types, quotient types, and recursive predicates under rewriting.
• To explore the impact of rewriting on provability, conservativity, and type safety in higher-order type systems.

Proposed method

• Introduces Type Systems Modulo (TSMs) as a general framework for defining type systems with equational theories.
• Defines Reduction Type Systems (RTS) as a formalism combining β-reduction and user-defined rewrite rules, with conditions for confluence and strong normalization.
• Proposes Algebraic Type Systems (ATS) as a subclass of RTS that supports inductive types and constructors with rewrite rules.
• Employs reductibility candidates and interpretation schemes to prove strong normalization of terms under the defined systems.
• Applies the General Schema of Jouannaud and Okada to handle higher-order recursive definitions, extended with ordering constraints on arguments.
• Establishes correctness via a semantic interpretation model that maps terms to reducibility candidates, ensuring subject reduction and consistency.

Experimental results

Research questions

• RQ1 Under what conditions does the addition of rewrite rules to a type system preserve strong normalization?
• RQ2 How can one ensure that the combination of β-reduction and rewriting remains confluent and logically consistent?
• RQ3 What syntactic and structural constraints on rewrite rules allow for safe and meaningful inductive and recursive definitions?
• RQ4 Can the framework support advanced features like quotient types and local definitions while maintaining type safety?
• RQ5 How does rewriting affect the conservativity of type theories—i.e., whether proofs using rewrite rules can be reconstructed without them?

Key findings

• The paper establishes sufficient conditions—confluence of the combined β and rewrite reduction, and specific ordering constraints on recursive calls—under which type systems with rewriting preserve strong normalization.
• It proves that the General Schema, extended with higher-order recursive path ordering, can capture a wide class of recursive definitions while ensuring termination.
• The framework supports the formalization of quotient types via rewrite rules on constructors, provided normal forms are well-understood and reduction is confluent.
• The interpretation model based on reductibility candidates successfully validates subject reduction and ensures that all typable terms are strongly normalizing.
• The results are applicable to practical systems like the Calculus of Inductive Constructions (CIC) and Natural Deduction Modulo, enabling richer specification and proof capabilities.
• The work identifies key open problems such as extending results to rewriting modulo equational theories, handling η-reduction, and relaxing simplicity constraints on rewrite rules.