A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of<br> Automata

TL;DR

This paper introduces a finite semantics for simply-typed lambda-trees based on automata, enabling decidability of automaton runs on regular infinite lambda-trees, including recursion schemes of any level.

Contribution

It develops a finite semantics for simply-typed lambda-trees using automata-based classification, and proves a soundness and completeness calculus that ensures decidability for arbitrary recursion schemes.

Findings

Decidability of automaton runs on regular infinite lambda-trees.

Finite semantics for simply-typed lambda-trees based on automata.

Applicability to recursion schemes of any level.

Abstract

Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees. A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.