Complexity of Finitely Presented Algebras

Abstract

An algebra $\cal A$ is finitely presented if there is a finite set G of generator symbols, a finite set O of operator symbols, and a finite set $\Gamma$ of defining relations $x \equiv y$ where $x$ and $y$ are well-formed terms over G and O, such that $\cal A$ is isomorphic to the free algebra on G and O modulo the congruence induced by $\Gamma$. The uniform word problem, the finiteness problem, the triviality problem (whether $\cal A$ is the one element algebra), and the subalgebra membership problem (whether a given element of $\cal A$ is contained in a finitely generated subalgebra of $\cal A$) for finitely presented algebras are shown to be $\leq^{m}_{\log}$-complete for P. The schema satisfiability problem and schema validity problem are shown to be $\leq^{m}_{\log}$-complete for NP and co-NP, respectively, Finally, the problem of isomorphism of finitely presented algebras is shown to be polynomial time many-one equivalent to the problem of graph isomorphism.