First-order logic as a constraint satisfaction problem

In this paper, we discourse an analysis of classical first-order predicate logic as a constraint satisfaction problem, CSP. First, we will offer our general framework for CSPs, and then apply it to first-order logic. We claim it would function as a new semantics, constraint semantics, for logic. Then, we prove the soundness and completeness theorems with respect to the constraint semantics. The latter theorem, which will be proven by a proof-search method, implies the cut-elimination theorem. Furthermore, using the constraint semantics, we make a new proof of the Craig interpolation theorem. Also, we will provide feasible algorithms to generate interpolants for some decidable fragments of first-order logic: the propositional logic and the monadic fragments. The algorithms, reflecting a ‘projection’ of an indexed relation, will show how to transform given formulas syntactically to obtain interpolants.