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Domains for Higher-Order Games

Authors Matthew Hague, Roland Meyer, Sebastian Muskalla

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Matthew Hague
Roland Meyer
Sebastian Muskalla

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Matthew Hague, Roland Meyer, and Sebastian Muskalla. Domains for Higher-Order Games. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.MFCS.2017.59

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.

Subject Classification

Keywords
  • higher-order recursion schemes
  • games
  • semantics
  • abstract interpretation
  • fixed points

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