A Curry-Howard Approach to Church's Synthesis

TL;DR

This paper introduces SMSO, an intuitionistic variant of MSO, and demonstrates its soundness and completeness for Church's synthesis, connecting logic, automata, and game theory.

Contribution

It presents SMSO, a new logical framework for Church's synthesis, bridging Curry-Howard correspondence with automata-based realizability.

Findings

SMSO is sound and complete for synthesis.

Automata-based realizability model supports SMSO.

Provides a new logical perspective on Church's synthesis.

Abstract

Church's synthesis problem asks whether there exists a finite-state stream transducer satisfying a given input-output specification. For specifications written in Monadic Second-Order Logic (MSO) over infinite words, Church's synthesis can theoretically be solved algorithmically using automata and games. We revisit Church's synthesis via the Curry-Howard correspondence by introducing SMSO, an intuitionistic variant of MSO over infinite words, which is shown to be sound and complete w.r.t. synthesis thanks to an automata-based realizability model.