[Paper Review] Trakhtenbrot's Theorem in Coq, A Constructive Approach to Finite Model Theory

Dominik Kirst, Dominique Larchey-Wendling|arXiv (Cornell University)|Apr 15, 2020

Logic, programming, and type systems20 references6 citations

TL;DR

This paper presents a constructive, machine-checked proof of Trakhtenbrot's theorem in Coq, establishing that finite first-order satisfiability (FSAT) is undecidable for signatures containing at least one binary relation or a binary function with a unary relation. The authors use a many-one reduction from the bounded Post correspondence problem (BPCP) to FSAT over a custom signature, while also proving decidability for monadic logic and enumerability for enumerable signatures, all within a synthetic framework of constructive type theory.

ABSTRACT

We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.

Motivation & Objective

• To provide a fully constructive, machine-verified classification of finite first-order satisfiability (FSAT) in dependent type theory.
• To overcome the limitations of classical proofs by using a reduction from the bounded Post correspondence problem (BPCP) instead of Turing machines, enabling better mechanization in Coq.
• To formalize signature transformations and treat equality constructively, addressing subtleties often neglected in classical treatments.
• To establish decidability of FSAT for monadic first-order logic and enumerability for arbitrary enumerable signatures within a synthetic undecidability framework.

Proposed method

• Employing synthetic undecidability in constructive type theory, where all definable functions are computable by construction, ensuring the computability of reductions.
• Encoding each instance of the bounded Post correspondence problem (BPCP) as a first-order formula such that a solution exists iff the formula has a finite model.
• Using a chain of signature transformations to reduce arbitrary signatures to minimal forms, eliminating or compressing function and relation symbols step by step.
• Formalizing finite models and satisfiability using a synthetic approach that avoids reliance on classical set-theoretic models.
• Abstracting over logical connectives via $ullet\square$ and $ullet\nabla$ to streamline structural definitions and proofs in Coq.
• Avoiding external dependencies like Autosubst 2 and functional extensionality to maintain axiom-free correctness in the Coq development.

Experimental results

Research questions

• RQ1 Can Trakhtenbrot’s theorem be proven constructively in a proof assistant like Coq using a reduction from a simpler undecidable problem than the halting problem?
• RQ2 What is the precise classification of finite satisfiability depending on the signature’s function and relation symbols in a constructive setting?
• RQ3 How can signature transformations be formally justified and mechanized to reduce complex signatures to minimal forms while preserving equisatisfiability?
• RQ4 Can decidability of monadic first-order logic and enumerability of FSAT for enumerable signatures be formally established in a synthetic, constructive framework?
• RQ5 What are the practical and foundational benefits of using a synthetic approach to undecidability proofs in dependent type theory?

Key findings

• FSAT is undecidable for any first-order signature containing at least one binary relation symbol or one unary relation symbol together with a function symbol of arity at least two.
• The undecidability of FSAT is established via a many-one reduction from the bounded Post correspondence problem (BPCP), which is more amenable to mechanization than reductions from the halting problem.
• FSAT is decidable for monadic signatures—those with only unary functions and relations or only nullary relations—providing a sharp boundary for decidability.
• FSAT is enumerable for any enumerable signature, meaning there exists a computable enumeration of all finitely satisfiable formulas.
• The Coq development, consisting of 10,000 lines of code with 3,000 lines of reusable utilities, is fully axiom-free and integrates into a growing library of synthetic undecidability proofs.
• The formalization includes novel treatment of interpreted equality and signature transformations, which are often omitted or handwaved in classical presentations.