Finite Monoids: From Word to Circuit Evaluation

• Authors: Martin Beaudry;Pierre McKenzie;Pierre Péladeau;Denis Thérien
• Venue: SIAM Journal on Computing
• Year: 1997

The problem of evaluating a circuit whose wires carry values from a finite monoid $M$ and whose gates perform the monoid operation provides a meaningful generalization to the well-studied problem of evaluating a word over $M$. Evaluating words over monoids is closely tied to the fine structure of the complexity class $NC^1$, and in this paper analogous ties between evaluating circuits over monoids and the structure of the complexity class $P$ are exhibited. It is shown that circuit evaluation in the case of any nonsolvable monoid is $P$ complete, while circuits over solvable monoids can be evaluated in $DET \subseteq NC^2$. Then the case of aperiodic monoids is completely elucidated: their circuit evaluation problems are either in $AC^0$ or $L$- or $NL$-complete, depending on the precise algebraic properties of the monoids. Finally, it is shown that the evaluation of circuits over the cyclic group ${\Bbb Z}_q$ for fixed $q \geq 2$ is complete for the logspace counting class $co$-$MOD_qL$, that the problem for $p$-groups ($p$ a prime) is complete for $MOD_pL$, and that the more general case of nilpotent groups of exponent $q$ belongs to the Boolean closure of $MOD_qL$.