Explorability in Pushdown Automata

TL;DR

This paper introduces and analyzes the concept of explorability in pushdown automata, revealing a hierarchy of expressiveness, its relation to context-free languages, and implications for automata succinctness.

Contribution

It defines explorability as a new measure of nondeterminism, establishes a hierarchy of automata classes, and connects explorability to language expressiveness and automata succinctness.

Findings

Explorable PDAs form a strict hierarchy between history-deterministic and nondeterministic PDAs.

Exponential explorability characterizes context-free languages.

Explorable PDAs can be exponentially more succinct than history-deterministic ones.

Abstract

We study explorability, a measure of nondeterminism in pushdown automata, which generalises history-determinism. An automaton is k-explorable if, while reading the input, it suffices to follow k concurrent runs, built step-by-step based only on the input seen so far, to construct an accepting one, if it exists. We show that the class of explorable PDAs lies strictly between history-deterministic and fully nondeterministic PDAs in terms of both expressiveness and succinctness. In fact increasing explorability induces an infinite hierarchy: each level k defines a strictly more expressive class than level k-1, yet the entire class remains less expressive than general nondeterministic PDAs. We then introduce a parameterized notion of explorability, where the number of runs may depend on input length, and show that exponential explorability precisely captures the context-free languages. Finally, we prove that explorable PDAs can be doubly exponentially more succinct than history-deterministic ones, and that the succinctness gap between deterministic and 2-explorable PDAs is not recursively enumerable. These results position explorability as a robust and operationally meaningful measure of nondeterminism for pushdown systems.