# Easy solutions are hard to find

## Abstract

There are two main results in this paper.

A new NP-complete problem is found: given a system (S) of recursion equations, determine whether (S) has a solution in a non-trivial "contraction algebra" in which one of the components is a projection. This problem, which arose in [3] where all solutions of a system of recursion equations in a contraction algebra A were found, is related to the equivalence problem for deterministic pushdown automata.

Secondly, for signatures Σ with a finite number of function symbols of positive rank, the free complete contraction Σ-algebras are shown to be isomorphic to algebras of "Σ-trees". When Σ has an infinite number of function symbols of positive rank, it is shown that there are no free complete contraction Σ-algebras.

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