Problem
Research questions and friction points this paper is trying to address.
interaction trees
domain theory
coinduction
program equivalence
Rocq
Domain Theory Meets Interaction Trees in Rocq
🤖 AI Summary
This study addresses the limitations of coinductive formalizations of interaction trees in the Rocq theorem prover, specifically their high complexity and the difficulty of proving equivalences for non-terminating programs. To overcome these challenges, this work proposes a domain-theoretic formalization framework that eschews built-in coinduction mechanisms. Instead, it defines program equivalence through inductive reasoning over terminating approximations, thereby circumventing coinductive pitfalls and simplifying the treatment of monad laws. As a key demonstration, the framework successfully verifies the equivalence of programs encoding the Syracuse sequence—a property whose termination is equivalent to an open mathematical conjecture. This result substantiates the effectiveness and practicality of the proposed approach in rigorously handling the semantics of non-terminating computations within interactive theorem proving environments.
📝 Abstract
We present a domain-theoretical formalization of interaction trees in the Rocq prover. Unlike existing formalizations, ours does not rely on Rocq's built-in coinduction. Hence, we avoid complications occurring in earlier works, such as artificially including silent steps to comply with Rocq's productivity checker, treating monad laws as weak bisimulations, and coinductive bisimulation reasoning. We define an inductive program equivalence relation as the congruence closure of a base relation on primitive effects with respect to action sequencing and least upper bounds. This enables reasoning about possibly non-terminating programs by reducing their equivalences to equivalences of their terminating approximations, which are then proved by induction. This relation is proved correct: provided the base relation is correct, equivalent computations have equal denotations in any monad that faithfully implements the effects. We illustrate the framework by showing the equivalence of two programs encoding the Syracuse sequence, whose termination is an open mathematical conjecture.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Interaction Trees
Domain Theory
Inductive Equivalence
Coinduction-free Formalization
Rocq Prover
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