### The membership problem in aperiodic transformation monoidsThe membership problem in aperiodic transformation monoids

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 Author Beaudry, M. ♦ McKenzie, P. ♦ Thrien, D. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1992 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Aperiodic ♦ Membership ♦ Monoids ♦ Varieties Abstract The problem of testing membership in aperiodic or “group-free” transformation monoids is the natural counterpart to the well-studied membership problem in permutation groups. The class $\textbf{A}$ of all finite aperiodic monoids and the class $\textbf{G}$ of all finite groups are two examples of $\textit{varieties},$ the fundamental complexity units in terms of which finite monoids are classified. The collection of all varieties $\textbf{V}$ forms an infinite lattice under the inclusion ordering, with the subfamily of varieties that are contained in $\textbf{A}$ forming an infinite sublattice. For each $\textbf{V}$ ⊆ $\textbf{A},$ the associated problem $MEMB(\textbf{V})$ of testing membership in transformation monoids that belong to $\textbf{V},$ is considered. Remarkably, the computational complexity of each such problem turns out to look familiar. Moreover, only five possibilities occur as $\textbf{V}$ ranges over the whole aperiodic sublattice: With one family of NP-hard exceptions whose exact status is still unresolved, any such $MEMB(\textbf{V})$ is either PSPACE-complete, NP-complete, P-complete or in AC0. These results thus uncover yet another surprisingly tight link between the theory of monoids and computational complexity theory. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1992-07-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 39 Issue Number 3 Page Count 18 Starting Page 599 Ending Page 616

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Source: ACM Digital Library