[Paper Review] On Winning Conditions of High Borel Complexity in Pushdown Games

Olivier Finkel|ArXiv.org|Mar 12, 2008

semigroups and automata theory21 references3 citations

TL;DR

This paper investigates winning conditions of high Borel complexity in pushdown games, showing that the sets of winning positions for players can be non-deterministic, non-ambiguous, inherently ambiguous, or even non-context-free ω-languages. It resolves open questions by proving that such winning sets are not restricted to deterministic context-free languages, establishing their precise expressive power within the hierarchy of ω-languages.

ABSTRACT

Some decidable winning conditions of arbitrarily high finite Borel complexity for games on finite graphs or on pushdown graphs have been recently presented by O. Serre in [ Games with Winning Conditions of High Borel Complexity, in the Proceedings of the International Conference ICALP 2004, LNCS, Volume 3142, p. 1150-1162 ]. We answer in this paper several questions which were raised by Serre in the above cited paper. We first show that, for every positive integer n, the class C_n(A), which arises in the definition of decidable winning conditions, is included in the class of non-ambiguous context free omega languages, and that it is neither closed under union nor under intersection. We prove also that there exists pushdown games, equipped with such decidable winning conditions, where the winning sets are not deterministic context free languages, giving examples of winning sets which are non-deterministic non-ambiguous context free languages, inherently ambiguous context free languages, or even non context free languages.

Motivation & Objective

• To resolve open questions from Serre (2005) regarding the expressive power of winning conditions in pushdown games.
• To analyze the classes $\mathbb{C}_n(A)$ and $\mathbb{C}_n^\lambda(A)$, showing they consist of non-ambiguous context-free ω-languages and their complements.
• To determine whether winning sets in pushdown games can exceed the class of deterministic context-free languages.
• To investigate closure properties of $\mathbb{C}_n(A)$ under union and intersection.
• To construct explicit examples of pushdown games where the winning set for Eve is not a deterministic context-free language.

Proposed method

• Define the classes $\mathbb{C}_n(A)$ as the set of languages $L(\mathcal{A}_1 \rhd \cdots \rhd \mathcal{A}_n \rhd \mathcal{A}_{n+1})$ for deterministic pushdown automata with stack unboundedness and acceptance conditions.
• Prove that languages in $\mathbb{C}_n(A)$ are non-ambiguous context-free ω-languages using the structure of stack limit languages.
• Introduce the larger class $\mathbb{C}_n^\lambda(A)$ to analyze complements, showing they are also non-ambiguous context-free.
• Construct explicit pushdown games with winning conditions $\Omega_{\mathcal{A}_1 \rhd \mathcal{A}_2}$ to demonstrate that winning sets can be inherently ambiguous or non-context-free.
• Use the pumping lemma for context-free languages to prove that $\{a^n b^n c^n \mid n \geq 1\}$ is not context-free, applying it to show non-context-freeness of winning sets.
• Analyze configurations in pushdown games where Eve’s winning positions depend on multiple infinite plays, using stack unboundedness and limit acceptance conditions.

Experimental results

Research questions

• RQ1 Are the sets of winning positions in pushdown games with high Borel complexity restricted to deterministic context-free languages?
• RQ2 Can the classes $\mathbb{C}_n(A)$ and their complements be characterized as non-ambiguous context-free ω-languages?
• RQ3 Are the classes $\mathbb{C}_n(A)$ closed under union and intersection?
• RQ4 Can winning sets in pushdown games be inherently ambiguous or even non-context-free?
• RQ5 What is the expressive power of winning conditions $\Omega_{\mathcal{A}_1 \rhd \cdots \rhd \mathcal{A}_{n+1}}$ in terms of ω-language classes?

Key findings

• The classes $\mathbb{C}_n(A)$ are contained within the class of non-ambiguous context-free ω-languages.
• The complements of languages in $\mathbb{C}_n(A)$ are also non-ambiguous context-free ω-languages, as shown via the larger class $\mathbb{C}_n^\lambda(A)$.
• The classes $\mathbb{C}_n(A)$ are neither closed under union nor under intersection.
• There exist pushdown games with winning conditions $\Omega_{\mathcal{A}_1 \rhd \mathcal{A}_2}$ where the winning set for Eve is an inherently ambiguous context-free language, specifically $\{a^n b^m c^p \mid n,m,p \geq 1 \text{ and } (n=m \text{ or } m=p)\}$.
• There exist pushdown games where the winning set for Eve is a non-context-free language, such as $\{a^n b^n c^n \mid n \geq 1\}$, proven via closure properties and the pumping lemma.
• The paper confirms that winning sets in pushdown games can be non-deterministic, non-ambiguous, inherently ambiguous, or non-context-free, thus exceeding the class of deterministic context-free languages.