Mechanized Undecidability of Higher-order beta-Matching (Extended Version)

2026-02-02

AI Summary

This work investigates the decidability of higher-order β-matching: given two simply typed λ-terms, does there exist a type instantiation under which they become β-equivalent? The authors establish its undecidability by constructing a many-one reduction from the halting problem via an encoding of restricted string rewriting systems into higher-order β-matching instances. The key contribution lies in a more concise and unified proof framework that elucidates the intrinsic connections among the undecidability of higher-order β-matching, λ-definability, and the inhabitation problem for intersection types. All proofs are fully mechanized in Coq, building upon a formally verified undecidability result for string rewriting systems to deliver a complete, certified reduction chain, which has been contributed to the Coq Library of Undecidability Proofs.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsGraph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsWeb Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

Higher-order beta-matching is the following decision problem: given two simply typed lambda-terms, can the first term be instantiated to be beta-equivalent to the second term? This problem was formulated by Huet in the 1970s and shown undecidable by Loader in 2003 by reduction from lambda-definability. The present work provides a novel undecidability proof for higher-order beta-matching, in an effort to verify this result by means of a proof assistant. Rather than starting from lambda-definability, the presented proof encodes a restricted form of string rewriting as higher-order beta-matching. The particular approach is similar to Urzyczyn's undecidability result for intersection type inhabitation. The presented approach has several advantages. First, the proof is simpler to verify in full detail due to the simple form of rewriting systems, which serve as a starting point. Second, undecidability of the considered problem in string rewriting is already certified using the Coq proof assistant. As a consequence, we obtain a certified many-one reduction from the Halting Problem to higher-order beta-matching. Third, the presented approach identifies a uniform construction which shows undecidability of higher-order beta-matching, lambda-definability, and intersection type inhabitation. The presented undecidability proof is mechanized in the Coq proof assistant and contributed to the existing Coq Library of Undecidability Proofs.

Problem

• higher-order beta-matching
• undecidability
• simply typed lambda-calculus
• beta-equivalence
• decision problem

Innovation

• higher-order beta-matching
• undecidability
• Coq mechanization
• string rewriting
• many-one reduction