Shuffles of Context-Free Languages along Regular Trajectories

TL;DR

This paper investigates how context-free languages behave under constrained shuffles dictated by regular automata, revealing conditions under which CFLs remain or become non-CFLs, especially for deterministic CFLs.

Contribution

It introduces new tools to determine when shuffles of CFLs under regular trajectories result in non-CFLs, and characterizes the behavior for deterministic CFLs with a novel trichotomy.

Findings

Certain trajectories always produce non-CFLs when shuffling nonregular CFLs.

A trichotomy classifies how deterministic CFLs are affected by different trajectories.

New lemmata describe how pushdown automata invoke stacks when accepting CFLs.

Abstract

In single-core processors, concurrency requires that multiple processes be interleaved into a single thread of execution by a scheduler. The language-theoretic operation that corresponds to this is the shuffle of two languages: the set of words obtained by interleaving a word from each language in an arbitrary, letter-wise fashion. It is well known that regular languages are closed under shuffles, while context-free languages (CFLs) are not. Following an established line of research, this paper considers shuffles according to regular ``trajectories,'' that is, subject to scheduling constraints expressed by an automaton. Unsurprisingly, some trajectories allow for CFLs to be shuffled into CFLs (e.g., simple concatenation of the two words), while others do not. This paper provides a robust toolset to show that a given trajectory would always shuffle two nonregular CFLs into a nonCFL. In the case of deterministic CFLs (DCFLs), a salient trichotomy of trajectories depending on how they shuffle DCFLs is provided. Notably, these results are based on lemmata of independent interest regarding how pushdown automata (PDA) must invoke the stack when accepting a nonregular CFL or DCFL. The latter case relies on a recent result of Jan\v{c}ar and \v{S}\'ima (MFCS'2021); answering an open question therein, it is demonstrated that said result cannot be generalized to arbitrary CFLs, leading to dedicated machinery for both cases.