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Explorability in Pushdown Automata

Ayaan Bedi, Karoliina Lehtinen

TL;DR

This work introduces explorability as a graded measure of nondeterminism for pushdown automata, extending history-determinism to a family of k-explorable PDAs. It establishes an infinite expressiveness hierarchy Expl_k-CFL ⊊ Expl_{k+1}-CFL ⊊ CFL and, with a parameterized notion f(n)-explorability, shows that Expl_{exp}-CFL equals CFL. The authors also prove strong succinctness gaps: explorable PDAs can be doubly exponentially more succinct than HD-PDAs, and the deterministic versus 2-explorable gap is not recursively enumerable. Overall, explorability provides a robust, operationally meaningful spectrum between determinism and nondeterminism for pushdown systems, with rich implications for language classes and representation efficiency.

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.

Explorability in Pushdown Automata

TL;DR

This work introduces explorability as a graded measure of nondeterminism for pushdown automata, extending history-determinism to a family of k-explorable PDAs. It establishes an infinite expressiveness hierarchy Expl_k-CFL ⊊ Expl_{k+1}-CFL ⊊ CFL and, with a parameterized notion f(n)-explorability, shows that Expl_{exp}-CFL equals CFL. The authors also prove strong succinctness gaps: explorable PDAs can be doubly exponentially more succinct than HD-PDAs, and the deterministic versus 2-explorable gap is not recursively enumerable. Overall, explorability provides a robust, operationally meaningful spectrum between determinism and nondeterminism for pushdown systems, with rich implications for language classes and representation efficiency.

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.

Paper Structure

This paper contains 14 sections, 20 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Explorability of Pushdown Automata
  4. Expressiveness
  5. (k+1)-explorable PDAs are strictly more expressive than k-explorable ones
  6. Expl-CFL is a proper subset of CFL
  7. Parametrized Explorability and Its Expressiveness
  8. Succinctness
  9. Explorable vs. HD-PDA
  10. k-Explorable PDA vs. Explorable PDA
  11. Deterministic vs. Explorable PDA
  12. Conclusion
  13. Turing Machine
  14. Pumping Proofs

Key Result

Theorem 7

$\text{DCFL} \subsetneq \text{HD-CFL} \subsetneq \text{CFL}$

Figures (1)

  • Figure 1: The figure illustrates two $k$-runs of a $k$-explorable PDA $P$. On the left, we depict the $k$-run $\overline\rho_1$, where at a position $p$ an accepting state is reached along the run. On the right, we show $\overline\rho_2$, in which an identical modes appears at the same index modulo $n$, allowing a suffix of $\rho_1$ (beginning from the first run's step following the matching mode) to be replayed on $\rho_2$.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2: k-Explorability Game for PDAs
  • Definition 2: k-Explorability Game for PDAs
  • Definition 3
  • Definition 4
  • Definition 5: k-Run in a k-Explorable PDA
  • Definition 6: $f(n)$-Parameterized Explorability Game
  • Theorem 7: Theorem 4.1 Guha2021-sv
  • Theorem 8
  • Lemma 9
  • ...and 18 more