This paper gives an implicit characterization of the class of functions computable in polynomial space by deterministic Turing machines – PSPACE. It gives an inductive characterization of PSPACE with no ad-hoc initial functions and with only one recursion scheme. The main novelty of this characterization is the use of pointers to reach PSPACE. The presence of the pointers in the recursion on notation scheme is the main difference between this characterization of PSPACE and the well-known Bellantoni-Cook characterization of the polytime (...) functions – PTIME. (shrink)

In 2000, a draft note of David Hilbert was found in his Nachlass concerning a 24th problem he had consider to include in the his famous problem list of the talk at the International Congress of Mathematicians in 1900 in Paris. This problem concerns simplicity of proofs. In this paper we review the traces of this problem which one can find in the work of Hilbert and his school, as well as modern research started on it after its publication. We (...) stress, in particular, the mathematical nature of the problem.1. (shrink)

A two-sorted term system characterizing NC implicitly is described. The term system is defined over the tree algebra [MATHEMATICAL DOUBLE-STRUCK CAPITAL T], the free algebra generated by 0, 1 and ∗, and the recursion scheme uses pointers over tier 0. This differs from previous characterizations of NC, where tier 1 pointers were used or full parameter substitution over tier 0 was allowed.

In this paper we introduce applicative theories which characterize the polynomial hierarchy of time and its levels. These theories are based on a characterization of the functions in the polynomial hierarchy using monotonicity constraints, introduced by Ben-Amram, Loff, and Oitavem.

We give recursion-theoretic characterizations of the counting class \(\textsf {\#P} \), the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels \(\{\textsf {\#P} _k\}_{k\in {\mathbb {N}}}\) of the counting hierarchy of functions \(\textsf {FCH} \), which result from allowing queries to functions of the previous level, and \(\textsf {FCH} \) itself as a whole. This is done in the style of (...) Bellantoni and Cook’s safe recursion, and it places \(\textsf {\#P} \) in the context of implicit computational complexity. Namely, it relates \(\textsf {\#P} \) with the implicit characterizations of \(\textsf {FPTIME} \) (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and \(\textsf {FPSPACE} \) (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of \(\textsf {FPSPACE} \). (shrink)

We discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-rule.

An implicit characterization of the class NP is given, without using any minimization scheme. This is the first purely recursion-theoretic formulation of NP.

We give a term rewriting characterization of the polyspace functions. Our work follows investigations on term rewriting characterizations of some classes of (sub-) recursive functions as initiated by Cichon and Weiermann [4] and continued by Beckmann and Weiermann [1].The main novelty of this paper is a technique for reformulating recursion schemes. The aim of this technique is to provide rewriting rules which give rise to rewriting chains whose terms are suitably bounded. This bounding is crucial when dealing with computational classes (...) with space constraints. (shrink)