Pavel E. Alaev

We show that natural structures related to the so called homomorphism preorder (or h -preorder) on the iterated labeled forests have isomorphic copies computable in polynomial time. Moreover, the polynomials in the upper bounds are of low degree which makes the computational content of the whole theory feasible. We discuss possible applications of these results to relevant questions of automata and computability theory.

We give some simple description for the finitely generated structures with P-computable isomorphic presentation; i.e., presentation computable in polynomial time. The description is close to the formulation of a Remmel and Friedman theorem. We prove that each finitely generated substructure of a P-computable structure also has a P-computable presentation. The description is applied to the classes of finitely generated semigroups, groups, unital commutative rings and fields, as well as ordered unital commutative rings and fields. We prove that every finitely generated commutative ring or a field has a P-computable presentation.