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.