Logical Primes, Metavariables and Satisfiability

Abstract

For formulas F of propositional calculus I introduce a "metavariable" MF and show how it can be used to define an algorithm for testing satisfiability. MF is a formula which is true/false under all possible truth assignments iff F is satisfiable/unsatisfiable. In this sense MF is a metavariable with the "meaning" 'F is SAT'. For constructing MF a group of transformations of the basic variables ai is used which corresponds to 'flipping" literals to their negation. The whole procedure corresponds to branching algorithms where a formula is split with respect to the truth values of its variables, one by one. Each branching step corresponds to an approximation to the metatheorem which doubles the chance to find a satisfying truth assignment but also doubles the length of the formulas to be tested, in principle. Simplifications arise by additional length reductions. I also discuss the notion of "logical primes" and show that each formula can be written as a uniquely defined product of such prime factors. Satisfying truth assignments can be found by determining the "missing" primes in the factorization of a formula.