# Domains for Higher-Order Games

## TL;DR

This paper introduces a novel finite domain for analyzing two-player inclusion games over higher-order recursion schemes, enabling an optimal algorithm for program synthesis verification.

## Contribution

It develops a new domain based on Boolean formulas and automaton states, providing a direct fixed-point algorithm for two-player inclusion games with proven optimal complexity.

## Key findings

- Finite domain representation of winning regions.
- Algorithm with (k+1)EXP complexity for order-k schemes.
- Matching lower bound proving optimality.

## Abstract

We study two-player inclusion games played over word-generating higher-order recursion schemes. While inclusion checks are known to capture verification problems, two-player games generalize this relationship to program synthesis. In such games, non-terminals of the grammar are controlled by opposing players. The goal of the existential player is to avoid producing a word that lies outside of a regular language of safe words. We contribute a new domain that provides a representation of the winning region of such games. Our domain is based on (functions over) potentially infinite Boolean formulas with words as atomic propositions. We develop an abstract interpretation framework that we instantiate to abstract this domain into a domain where the propositions are replaced by states of a finite automaton. This second domain is therefore finite and we obtain, via standard fixed-point techniques, a direct algorithm for the analysis of two-player inclusion games. We show, via a second instantiation of the framework, that our finite domain can be optimized, leading to a (k+1)EXP algorithm for order-k recursion schemes. We give a matching lower bound, showing that our approach is optimal. Since our approach is based on standard Kleene iteration, existing techniques and tools for fixed-point computations can be applied.