An explicit algorithm for normal forms in small overlap monoids
| العنوان: | An explicit algorithm for normal forms in small overlap monoids |
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| المؤلفون: | James D. Mitchell, Maria Tsalakou |
| المساهمون: | University of St Andrews. Pure Mathematics |
| المصدر: | Journal of Algebra. 630:394-433 |
| بيانات النشر: | Elsevier BV, 2023. |
| سنة النشر: | 2023 |
| مصطلحات موضوعية: | FOS: Computer and information sciences, Algebra and Number Theory, Formal Languages and Automata Theory (cs.FL), Semigroup, Computer Science - Formal Languages and Automata Theory, Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing), DAS, Mathematics - Rings and Algebras, Monoid, Normal forms, Rings and Algebras (math.RA), Finite presentation, MCP, Computer Science - Data Structures and Algorithms, FOS: Mathematics, Small cancellation, Data Structures and Algorithms (cs.DS), QA Mathematics, Small overlap, QA, 20M05, 08A50, 20F10, Computer Science::Formal Languages and Automata Theory |
| الوصف: | We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the $C(4)$ condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in $C(4)$ monoids: the word problem is solvable in $O(\min\{|u|, |v|\})$ time in the size of the input words $u$ and $v$; the uniform word problem for $\langle A|R\rangle$ is solvable in $O(N ^ 2 \min\{|u|, |v|\})$ where $N$ is the sum of the lengths of the words in $R$; and a normal form for any given word $u$ can be found in $O(|u|)$ time. Although Kambites' algorithm for solving the word problem in $C(4)$ monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in $C(4)$ monoids that has time complexity $O(|u| ^ 2)$ for input word $u$, but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in $O(N \min\{|u|, |v|\})$ time. Comment: 31 pages, 6 figures (some further minor corrections from the referee are resolved, to appear in Journal of Algebra) |
| وصف الملف: | application/pdf |
| تدمد: | 0021-8693 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::7a29b6477ff831abc6ad7b27334a5bc3 https://doi.org/10.1016/j.jalgebra.2023.04.019 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi.dedup.....7a29b6477ff831abc6ad7b27334a5bc3 |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 00218693 |
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