Adaptive Synchronisation of Pushdown Automata

TL;DR

This paper studies adaptive synchronization in pushdown automata, establishing complexity bounds for non-deterministic and deterministic cases, and introduces new variants of subset-synchronization to analyze the problem.

Contribution

It defines adaptive synchronization for pushdown automata, proves its complexity bounds, and relates it to subset-synchronization problems with new polynomial-time equivalences.

Findings

Non-deterministic case is 2-EXPTIME-complete.

Deterministic case is EXPTIME-complete.

Polynomial-time algorithms for certain subset-synchronization variants.

Abstract

We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an \emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is 2-EXPTIME-complete and for deterministic pushdown automata, we show EXPTIME-completeness. To prove the lower bounds, we first introduce (different variants of) subset-synchronisation and show that these problems are polynomial-time equivalent with the adaptive synchronisation problem. We then prove hardness results for the subset-synchronisation problems. For proving the upper bounds, we consider the problem of deciding if a given alternating pushdown system has an accepting run with at most $k$ leaves and we provide an $n^{O(k^2)}$ time algorithm for this problem.