On the computational complexity of algebra on lattices

We study the computational complexity of equivalence and minimization problems for expressions on many different lattices including each finite lattice and each distributive lattice. A general efficient expressibility condition $C$ on a lattice is presented such that 1. The equivalence problem is co$NP$ hard for constant-free expressions on any lattice with at least two elements that satisfies condition $C$. Each finite or distributive lattice is shown to satisfy condition $C$. Moreover, if a lattice $\mathcal{L}$ satisfies condition $C$ and $ \equiv $ is a congruence relation on $\mathcal{L}$, then ${\mathcal{L} / \equiv }$ also satisfies condition $C$. Several additional results are also presented. These results include the following: 2. In contrast to 1, the equivalence and operator minimization problems are solvable deterministically in polynomial time for disjunctive normal form and conjunctive normal form expressions on any lattice and for constant-free expressions on any free lattice with at least three generators: 3. Let $\mathcal{L}$ be a lattice. Then, the operator minimization problem and various approximate operator minimization problems for expressions on $\mathcal{L}$ are as hard as the problem of determining, for expressions $F$ and $G$ on $\mathcal{L}$, if $F \leqq G$.