Good-for-games $\omega$-Pushdown Automata

Abstract

We introduce good-for-games $\omega$-pushdown automata ($\omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $\omega$-GFG-PDA are more expressive than deterministic $\omega$- pushdown automata and that solving infinite games with winning conditions specified by $\omega$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $\omega$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $\omega$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $\omega$-GFG- PDA and decidability of good-for-gameness of $\omega$-pushdown automata and languages. Finally, we compare $\omega$-GFG-PDA to $\omega$-visibly PDA, study the resources necessary to resolve the nondeterminism in $\omega$-GFG-PDA, and prove that the parity index hierarchy for $\omega$-GFG-PDA is infinite.

History-Deterministic Timed Automata Are Not Determinizable

“…We show that as is the case in the regular [HP06], pushdown [LZ22], cost function [Col09], and quantitative [BL21] settings, synthesis games with winning conditions given by historydeterministic TA are no harder to solve than those with for winning condition given by deterministic TA. Theorem 7.2.…”

“…The proof below follows a similar reduction to one in [LZ22], in which the nondeterminism of the automaton is moved into Player 2's output alphabet, forcing her to simultaneously build a word in the winning condition and an accepting run witnessing this. Since accepting runs are recognised by deterministic automata, this reduces the problem to the synthesis problem for deterministic timed automata.…”

“…For transition-based history-deterministic automata, minimisation is PTime [ARK19], while for state-based ones, it is NP-complete [Sch20]. Recently, the notion has been extended to richer automata models, such as pushdown automata [LZ22,GJLZ21] and quantitative automata [BL21,BL22], where deterministic and nondeterministic models have different expressivity, and therefore, allowing a little bit of nondeterminism can, in addition to succinctness, also provide more expressivity.…”

Using the past for resolving the future

“…HD alternating automata for ω-regular languages are further studied in Boker and Lehtinen (2019). History-deterministic ω-pushdown automata were studied in Guha et al ( 2021) and Lehtinen and Zimmermann (2022). Their definition is similar to the definition of HD ω-regular automata, except that now, the nondeterminism that the strategy resolves corresponds to the different choices in the transition function of pushdown automata; thus, the strategy maps the history to both the next state of the automaton and the operations on the stack.…”

An Objective Improvement Approach to Solving Discounted Payoff Games

“…Most research has focused on parity games: as the most special class of games, algorithms have the option to use the special structure of their problems, and they are most directly linked to the synthesis and verification problems mentioned earlier. Parity games have thus enjoyed a special status among graph games and the quest for efficient algorithms [13,11,26,38,7,37,27,4,3] for solving them has been an active field of research during the last decades, which has received further boost with the arrival of quasi-polynomial techniques [8,18,15,23,24,10].…”