A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

TL;DR

This paper investigates the expressiveness and succinctness of history-deterministic pushdown automata (HD-PDA), showing they recognize more languages than deterministic PDA but less than all context-free languages, and analyzing their properties and complexity.

Contribution

It characterizes the language recognition power and succinctness of HD-PDA and HD-VPA, compares them with deterministic automata, and studies their complexity and resolvers.

Findings

HD-PDA recognize more languages than DPDA but not all CFL.

HD-PDA can be exponentially more succinct than DPDA.

Deciding HDness in VPA is ExpTime-complete.

Abstract

We study the expressiveness and succinctness of history-deterministic pushdown automata (HD-PDA) over finite words, that is, pushdown automata whose nondeterminism can be resolved based on the run constructed so far, but independently of the remainder of the input word. These are also known as good-for-games pushdown automata. We prove that HD-PDA recognise more languages than deterministic PDA (DPDA) but not all context-free languages (CFL). This class is orthogonal to unambiguous CFL. We further show that HD-PDA can be exponentially more succinct than DPDA, while PDA can be double-exponentially more succinct than HD-PDA. We also study HDness in visibly pushdown automata (VPA), which enjoy better closure properties than PDA, and for which we show that deciding HDness is ExpTime-complete. HD-VPA can be exponentially more succinct than deterministic VPA, while VPA can be exponentially more succinct than HD-VPA. Both of these lower bounds are tight. We then compare HD-PDA with PDA for which composition with games is well-behaved, i.e. good-for-games automata. We show that these two notions coincide, but only if we consider potentially infinitely branching games. Finally, we study the complexity of resolving nondeterminism in HD-PDA. Every HDPDA has a positional resolver, a function that resolves nondeterminism and that is only dependant on the current configuration. Pushdown transducers are sufficient to implement the resolvers of HD-VPA, but not those of HD-PDA. HD-PDA with finite-state resolvers are determinisable.