🤖 AI Summary
This work addresses the challenge of automatically generating geometric rules and cut-free proofs from classical modal formulas in modal logic, focusing on analytic inductive formulas and their first-order correspondents. We introduce the MASSA algorithm—the first method to jointly synthesize analytic geometric rules and canonical cut-free derivations in the G3K labelled sequent calculus directly from modal formulas. We rigorously prove MASSA’s termination and correctness for all analytic inductive formulas and establish the logical equivalence between the generated geometric rules and their first-order frame correspondents. Furthermore, we demonstrate the completeness of the SCAN algorithm for second-order quantifier elimination on this class of formulas, and extend our approach to general inductive formulas and modal logics with first-order quantifiers. The results significantly advance automated theorem proving and semantic correspondence analysis in modal logic.
📝 Abstract
We introduce the algorithm MASSA which takes classical modal formulas in input, and, when successful, effectively generates: (a) (analytic) geometric rules of the labelled calculus G3K, and (b) cut-free derivations (of a certain `canonical' shape) of each given input formula in the geometric labelled calculus obtained by adding the rule in output to G3K. We show that MASSA successfully terminates whenever its input formula is a (definite) analytic inductive formula, in which case, the geometric axiom corresponding to the output rule is, modulo logical equivalence, the first-order correspondent of the input formula. In proving the correctness of MASSA, we also show that the algorithm for the elimination of second-order quantifiers SCAN is complete with respect to the class of inductive analytic formulas. Finally, we show how our algorithm can be extended to the class of inductive formulas and to modal logic with quantifiers.