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The Formalization of two Computational Models in Dafny

Ştefan Ciobâc\b{a}, Diana-Elena Gratie, Dragoş-Irinel Rotariu

arXiv:2609.34883v1cs.LOcs.FLcs.PL

TL;DR

Formalizing computational models can make their mathematical foundations more understandable and trustworthy, while existing formalizations may be less accessible to programmers seeking practical explanations. This paper formalizes Turing Machines and Lambda Calculus in Dafny and applies the formalizations to termination and Church encodings. The authors conclude that the formalizations deepen understanding and support useful applications.

  • Problem

    Formalizing computational models can improve understanding and trust, while ordinary theorem provers may be harder for programmers seeking practical explanations of the models.

    Source: §1 · p. 1, §2 · p. 2

  • Method

    The paper formalizes Turing Machines and Lambda Calculus in Dafny, using finite transition paths to model termination and mechanically verifying Church encodings.

    Source: §3 · pp. 3–4, §4 · p. 7

  • Results

    The authors report that the formalizations deepen understanding and support useful applications, including proving that a transformed machine halts for every input and verifying Church encodings.

    Source: §3 · p. 4, §4 · p. 7, §6 · p. 9

  • Takeaways & Limitations

    The authors conclude that Dafny can serve programmers as an intuitive and clear formal verifier for proofs involving data structures, mechanics, and complicated theorems.

    Source: §6 · p. 9

    Using alpha-equivalence as the fundamental relation between Lambda Calculus terms proved difficult in Dafny and required many helper lemmas.

    Source: §4 · p. 8

Abstract

from arXiv · show

We describe the formalization in Dafny of two computational models, Turing Machines and the Lambda Calculus. We present several application of the formalizations: machine proofs of termination for Turing machines, Dafny proofs for Church encodings, and a mechanized proof of the Church-Rosser theorem.

1 Introduction

The paper formalizes Turing Machines and Lambda Calculus in Dafny, a verification-aware programming language. It motivates formalization as a way to build understanding and trust in computational models and introduces Dafny’s specification and verification features.

1 Introduction

  • Formalizing mathematical definitions can increase understanding and trust in computational models and their properties.Computability can also provide mathematical approaches to proving problem complexity and explaining programming-language evaluation differences.Source: §1 · p. 1
  • The paper formalizes Turing Machines and Lambda Calculus in Dafny, which supports specifications and static verification.Its verification approach uses preconditions, postconditions, assertions, and lemmas.Source: §1 · p. 1
  • The authors present Dafny as a practical middle ground between conventional programming languages and interactive theorem provers.They cite its combination of mathematical proofs with practical code structure as the reason for selecting it.Source: §1 · p. 1
  • The paper proceeds from contributions and related work to Turing Machine and Lambda Calculus formalizations, then future work and conclusion.The Lambda Calculus section also covers Church encoding and beta reduction.Source: §1 · pp. 1–2

2 Contributions

The paper positions its contribution as a practical, method-oriented formalization of computational models in Dafny. It covers both deterministic and nondeterministic Turing Machines and uses named variables for Lambda Calculus, while also reporting AI-assisted formalization as a case study.

2 Contributions

  • The formalization covers both deterministic and nondeterministic Turing Machines, clarifying their differences and similarities.Earlier formalizations cited by the authors use deterministic machines [8] and include work using Dafny.Source: §2 · p. 2
  • The authors use named variables for Lambda Calculus rather than de Bruijn indices, which they say may be less intuitive for novices.They note that named variables also appear in some mathematical proofs.Source: §2 · p. 2
  • The project formalizes standard theoretical foundations without omitting elements that may be unnecessary for proving theorems but matter to the foundations.The authors frame the work as raising understanding of Lambda Calculus and Turing Machines.Source: §2 · p. 2
  • The contribution is primarily methodological and theoretical, using a formal verifier closer to programming languages than standard theorem provers.The authors connect this approach to illuminating concepts in computational models and computational theory.Source: §2 · p. 2
  • The paper analyzes conversational and autonomous agents’ capabilities in formal verification through AI use in parts of the formalization.It presents this experience as a case study in using AI to formally prove parts of a theorem.Source: §2 · p. 2

3 Turing Machines and their formalization in Dafny

The paper formalizes Turing Machines in Dafny using data structures that represent their configurations and transitions. It links nondeterministic and deterministic representations and models termination through finite paths, then proves that a simple machine halts for every input.

3 Turing Machines and their formalization in Dafny

  • The tape is represented as a function from positions to symbols, allowing an unbounded tape while retaining only non-empty cells.Unrepresented cells return Blank.Source: §3 · p. 3
  • A configuration stores the machine state, head position, and tape; a transition uses the state and pointed-to symbol to select an action that updates the machine.The action changes the pointed-to cell, changes state, and moves the head left or right; a missing transition returns None.Source: §3 · p. 3
  • A predicate checks whether each transition key has only one available action, while conversion functions translate between nondeterministic and deterministic transition representations.Corresponding functions and predicates are linked by lemmas proving similar functionality.Source: §3 · p. 3
  • Using a DTM representation simplifies simulation, while simulating any NTM with a modified DTM and certificate is noted as possible but outside the paper’s scope.The NTM simulation requires a position and is described as more demanding and complex.Source: §3 · p. 3
  • Termination is modeled by a finite path of an exact number of transitions, with the natural-number step count decreasing to zero.An existence predicate supplies the number of steps; Dafny uses it as a decreasing measure.Source: §3 · p. 4
  • The example machine receives strings of 1s, accepts inputs of even length, rejects odd-length inputs, and is proved to halt.The halting proof applies to every input for the machine defined by the figure.Source: §3 · p. 4
  • The main method proves the transition-relation translation correct and shows the resulting deterministic machine halts for every input.The proof links intermediate configurations with the initial and final steps.Source: §3 · p. 4

4 Lambda Calculus and its formalization in Dafny

The section formalizes lambda terms in Dafny and applies the formalization to Church encodings and a mechanized Church–Rosser proof. It details the challenges of substitution, alpha-equivalence, and formally bridging reduction relations.

Term representation and substitution

  • Lambda terms use algebraic data types with natural-number variable identifiers, and the formalization includes ordinary substitution and capture-avoiding substitution.The latter renames a binder when it occurs free in the substituted term, preventing variable capture.Source: §4 · p. 6

Substitution

  • Capture-avoiding substitution is difficult to verify because renaming complicates the recursion and requires a separate lemma showing the renamed term preserves height.Fresh variables are generated by assigning one more than the highest variable number in the identifier list; the list acts as an annotated set with helper lemmas supporting recursion.Source: §4 · p. 6

Alpha-equivalence

  • Alpha-equivalence is checked structurally by tracking bound variables in traversal order and comparing their binding positions, while requiring free variables to match.Repeated alpha-conversions were too expensive for a frequently used predicate, motivating an alternative to that approach.Source: §4 · p. 7

Beta-reduction

  • Beta-reduction selects a redex by its index in left-outermost-to-right-innermost order, and normal order repeatedly reduces the leftmost outermost redex until none remain.The full normal-order reduction strategy is also used in the Church-encoding proofs.Source: §4 · p. 7

Church encodings

  • Mechanical proofs verify that Church-encoded logical operators behave as expected, including ((AND TRUE) TRUE) reducing to TRUE.The proofs link normal-order reduction steps; termination uses the same technique as the Turing-machine halting proof.Source: §4 · p. 7

Church–Rosser theorem

  • The Church–Rosser formalization uses parallel reduction to reduce selected beta-redexes together, proves its reflexive, transitive closure has the diamond property, then connects it to beta-reduction closure.The proof follows the Waterloo lecture notes and requires seven main lemmas.Source: §4 · pp. 8–9
  • The Dafny proof establishes the Church–Rosser theorem by proving the diamond property for parallel reduction and bridging it to the reflexive, transitive closure of beta-reduction.The formal proof requires many lemmas and explicit bridges between reduction relations, details that typical mathematical proofs may omit.Source: §4 · pp. 8–9

5 Future work

The paper identifies extensions for both formalizations: Turing Machine work could address NP-completeness, while Lambda Calculus work could examine encodings, equivalence, and evaluation strategies.

5 Future work

  • Turing Machine formalization could be extended to NP-completeness and used to prove Dafny predicates NP-complete.Source: §5 · p. 9
  • Lambda Calculus formalization could explore Church encodings and prove equivalence between Lambda Calculus and Turing Machines.Source: §5 · p. 9
  • Different beta-reduction strategies could be studied to compare their advantages and disadvantages.Source: §5 · p. 9

6 Conclusion

The authors conclude that formalizing Turing Machines and Lambda Calculus deepens understanding and supports useful applications. They also describe Dafny as accessible for formal verification and report both strengths and weaknesses from using AI agents.

6 Conclusion

  • The formalizations deepen understanding, support useful applications, and leave several avenues for future work.Source: §6 · p. 9
  • Dafny can support formal proofs of data structures, mechanics, and complicated theorems while remaining intuitive for programmers.Source: §6 · p. 9
  • AI agents handled concise helper-lemma tasks effectively but often missed code context and struggled with abstract topics.The authors report that this could lead to unnecessary functions or non-optimal solutions.Source: §6 · p. 9
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