Technical Foundations of a DPLL-Based SAT Solver for Propositional Gödel Logic
We provide the foundations of automated deduction in the propositional Gödel logic. The propositional Gödel logic is one of the simplest infinitely valued fuzzy logics, which generalizes classical propositional logic. We propose an extension of the Davis–Putnam–Logemann–Loveland (<italic>DPLL</italic>) procedure to this logic and prove its refutational soundness and finite completeness. Using the <italic>DPLL</italic> procedure, we solve the deduction problem <inline-formula><tex-math notation="LaTeX">${T\models \phi }$</tex-math> </inline-formula> (<inline-formula><tex-math notation="LaTeX">$\boldsymbol{T}$</tex-math></inline-formula> is a finite theory and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\phi }$</tex-math></inline-formula> a formula), which covers the finite <italic>SAT</italic> problem for a theory and the <italic>VAL</italic> problem for a formula, obviously. This paper serves, on the one side, as a technical basis for the design of a <italic>SAT</italic> solver; on the other side, gives some preliminary theoretical results concerning the logical and computational foundations of fuzzy inference, which is our main aim.