# Monotone boolean formulas, distributive lattices, and the complexities of logics, algebraic structures, and computation structures (preliminary report)

## Abstract

We refine and significantly extend the known results on the complexities of the Satisfiability, Tautology, Equivalence, and ≤-Problems for classes of Boolean formulas. Results are presented on the relationships between the number of repetitions of variables occurring in formulas and the complexities of decision problems for the formulas. Results are also presented on the complexities of very simple monotone Boolean formulas and of implication.

A variety of applications of these results are presented to a number of logics studied in the literature, to computational probability and counting, to algebraic structures, and to program schemes and Binary Decision Diagrams.

Assuming P ↮ NP, a number of the results are “best” possible or are close to “best” possible.

## Keywords

Polynomial Time Distributive Lattice Algebraic Structure Propositional Calculus Equivalence Problem## Preview

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