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 the algebraic structure of the target language or evaluation semigroup. Our main contributions are therefore to show (deterministic) space complexity characterizations, which we do for out-of-order evaluation of monoids and semigroups. For monoids, we establish a trichotomy: the space complexity is either Θ(1), Θ(log n), or Θ(n), where n = |w|. More specifically, the problem admits a constant-space solution for commutative monoids, while all non-commutative monoids require Ω(log n) space. We further identify a class of monoids admitting an O(log n)-space algorithm, and show that all remaining monoids require Ω(n) space. For general semigroups, the situation is more intricate. We characterize a class of semigroups admitting constant-space algorithms for out-of-order evaluation, and show that semigroups outside this class require at least Ω(log n) space.

翻译：本文引入形式语言L的无序成员判定任务，其中单词w的字母以对抗顺序逐个揭示。单词长度|w|已知，但w的内容以数据流形式传输为(i, w[i])对，每个位置i恰好接收一次，且顺序任意。我们研究当L为正则语言时该任务的高效算法，旨在针对固定目标语言建立关于|w|的紧致复杂度界限。我们的多数结果适用于代数定义的变体——无序求值问题：该问题针对固定有限幺半群或半群S定义，目标是计算流式传输的w元素的有序乘积。我们证明，对于任意固定的正则语言或有限半群，两个问题均可在每个流式符号的常数时间内解决，且所需空间为线性。然而，精确的空间复杂度高度依赖于目标语言或求值半群的代数结构。因此，我们的主要贡献在于给出（确定性）空间复杂度刻画，针对幺半群和半群的无序求值问题展开分析。对于幺半群，我们建立三分定理：空间复杂度为Θ(1)、Θ(log n)或Θ(n)，其中n = |w|。具体而言，交换幺半群存在常数空间解法，而所有非交换幺半群需要Ω(log n)空间。我们进一步识别出允许O(log n)空间算法的幺半群类，并证明其余幺半群需要Ω(n)空间。对于一般半群，情况更为复杂。我们刻画了允许常数空间算法进行无序求值的半群类，并证明不属于该类的半群至少需要Ω(log n)空间。