Out-of-Order Membership in Regular Languages
Antoine Amarilli, Sebastien Labbe, Charles Paperman

TL;DR
This paper studies the complexity of recognizing regular languages and evaluating finite algebraic structures when input symbols are revealed in an adversarial, out-of-order manner, providing tight space bounds.
Contribution
It introduces the out-of-order membership problem and characterizes its space complexity for regular languages and algebraic structures like monoids and semigroups.
Findings
Constant space algorithms exist for certain classes of monoids and semigroups.
Non-commutative monoids require logarithmic space, while commutative ones require constant space.
The paper provides a detailed space complexity classification for out-of-order evaluation tasks.
Abstract
We introduce the task of out-of-order membership to a formal language L, where the letters of a word w are revealed one by one in an adversarial order. The length |w| is known in advance, but the content of w is streamed as pairs (i, w[i]), received exactly once for each position i, in arbitrary order. We study efficient algorithms for this task when L is regular, seeking tight complexity bounds as a function of |w| for a fixed target language. Most of our results apply to an algebraically defined variant dubbed out-of-order evaluation: this problem is defined for a fixed finite monoid or semigroup S, and our goal is to compute the ordered product of the streamed elements of w. We show that, for any fixed regular language or finite semigroup, both problems can be solved in constant time per streamed symbol and in linear space. However, the precise space complexity strongly depends on…
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