Abstract
We introduce the space function s(n) of a finitely presented semigroup \({S =\langle A \mid R \rangle}\). To define s(n) we consider pairs of words w,w′ over A of length at most n equal in S and use relations from R for the derivations \({w = w_0 \rightsquigarrow \dots \rightsquigarrow w_t = w'; s(n)}\) bounds from above the lengths of the words w i at intermediate steps, i.e., the space sufficient to implement all such transitions \({w \rightsquigarrow \dots \rightsquigarrow w'}\). One of the results obtained is the following criterion: A finitely generated semigroup S has decidable word problem of polynomial space complexity if and only if S is a subsemigroup of a finitely presented semigroup H with polynomial space function.
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Olshanskii, A.Y. Space Functions and Space Complexity of the Word Problem in Semigroups. comput. complex. 22, 771–830 (2013). https://doi.org/10.1007/s00037-012-0058-0
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DOI: https://doi.org/10.1007/s00037-012-0058-0