On the complexity of the disjunction property in intuitionistic and modal logics

In this paper we study the complexity of disjunction property for Intuitionistic Logic, the modal logics S3, S4.1, Grzegorczyk Logic, Godel-Lob Logic and the intuitionistic counterpart of the modal logic K. For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs. The techniques we use do not require to prove structural properties of the calculi in hand, such as the Cut-elimination Theorem or the Normalization Theorem. This is a key-point of our approach, since it allows us to treat logics for which only Hilbert-style characterizations are known

A proof-theoretical analysis of semiconstructive intermediate theories

In the 80's Pierangelo Miglioli, starting from motivations in the framework of Abstract Data Types and Program Synthesis, introduced semiconstructive theories, a family of "large subsystems" of classical theories that guarantee the computability of functions and predicates represented by suitable formulas. In general, the above computability results are guaranteed by algorithms based on a recursive enumeration of the theorems of the whole system. In this paper we present a family of semiconstructive systems, we call uniformly semiconstructive, that provide computational procedures only involving formulas with bounded complexity. We present several examples of uniformly semiconstructive systems containing Harrop theories, induction principles and some well-known predicate intermediate principles. Among these, we give an account of semiconstructive and uniformly semiconstructive systems which lie between Intuitionistic and Classical Arithmetic and we discuss their constructive incompatibility