This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the first-order applicative theories introduced in Strahm [19, 20].

This paper is a direct successor to 12. Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in 12 by the so-called join axiom of explicit mathematics.

In Spescha and Strahm [15], a system of explicit mathematics in the style of Feferman [6] and [7] is introduced, and in Spescha and Strahm [16] the addition of the join principle to is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of are the polytime functions on binary words. However, although strongly conjectured, it remained open whether the same holds true for the corresponding theory with classical logic. This note supplements a (...) proof of this conjecture. (shrink)