Verified Model Checking for Conjunctive Positive Logic

Abstract

We formalize, in the Dafny language and verifier, a proof system PS for deciding the model checking problem of the fragment of first-order logic, denoted FO(∀, ∃, ∧), known as conjunctive positive logic (CPL). We mechanize the proofs of soundness and completeness of PS ensuring its correctness. Our formalization is representative of how various popular verification systems can be used to verify the correctness of rule-based formal systems on the basis of the least fixpoint semantics. Further, exploiting Dafny's automatic code generation, from the completeness proof we achieve a mechanically verified prototype implementation of a proof search mechanism that is a model checker for CPL. The model checking problem of FO(∀, ∃, ∧) is equivalent to the quantified constraint satisfaction problem (QCSP), and it is PSPACE-complete. The formalized proof system decides the general QCSP and it can be applied to arbitrary formulae of CPL.