We formalize the algorithms computing primitive recursive (PR) functions as the abstract state machines (ASMs) whose running length is computable by a PR function. Then we show that there exists a programming language (implementing only PR functions) by which it is possible to implement any one of the previously defined algorithms for the PR functions in such a way that their complexity is preserved.

We extend Meyer and Ritchie's LOOP language with higher-order procedures and procedural variables and we show that the resulting programming language (called LOOP ω ) is a natural imperative counterpart of Gödel System T. The argument is two-fold:(1) we define a translation of the LOOP ω language into System T and we prove that this translation actually provides a lock-step simulation,(2) using a converse translation, we show that LOOP ω is expressive enough to encode any term of System T.Moreover, we define the "iteration rank" of a LOOP ω program, which corresponds to the classical notion of "recursion rank" in System T, and we show that both translations preserve ranks. Two applications of these results in the area of implicit complexity are described.

No abstract

We investigate the structure of "worst-case" quasi reduced ordered decision diagrams and Boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of "hard" Boolean functions as functions whose QROBDD are "worstcase" ones. So we exhibit the relation between hard functions and the Storage Access function (also known as Multiplexer).

No abstract