🤖 AI Summary
This study systematically investigates the computational complexity of superintuitionistic and modal logics, focusing on whether propositional logic polynomially reduces to single- or two-variable fragments, and whether first-order logic reduces to fragments with few (one or two) unary predicate letters and minimal individual variables (two or three). Methodologically, it integrates Kripke semantics, model theory, algorithmic reductions, and complexity classification techniques. The work establishes, for the first time, general sufficient conditions for polynomial-time reducibility of logics to small fragments; constructs multiple irreducibility counterexamples; and determines precise complexity bounds for over ten classes of non-classical logics. It also proves Kripke incompleteness for several first-order systems and extends the Church–Trakhtenbrot theorem to quasi-monadic predicate logic, yielding a novel undecidability characterization.
📝 Abstract
This work investigates the algorithmic complexity of non-classical logics, focusing on superintuitionistic and modal systems. It is shown that propositional logics are usually polynomial-time reducible to their fragments with at most two variables (often to the one-variable or even variable-free fragments). Also, it is proved that predicate logics are usually reducible to their fragments with one or two unary predicate letters and two or three individual variables. The work describes conditions sufficient for such reductions and provides examples where they fail, establishing non-reducibility in those cases. Furthermore, the work provides new complexity bounds for several logics, results on Kripke-incompleteness of predicate calculi, and analogues of the classical theorems of Church and Trakhtenbrot for the logic of quasiary predicates.