Ratajczyk, Z., Subsystems of true arithmetic and hierarchies of functions, Annals of Pure and Applied Logic 64 95–152. The combinatorial method coming from the study of combinatorial sentences independent of PA is developed. Basing on this method we present the detailed analysis of provably recursive functions associated with higher levels of transfinite induction, I, and analyze combinatorial sentences independent of I. Our treatment of combinatorial sentences differs from the one given by McAloon [18] and gives more natural sentences. The same (...) method give also a combinatorial technique with no use of the cut-elimination theorem which is appropriate to study proof-theoretic strength of subsystems of first order arithmetic and some of their expansions. It was used to analyze iterated reflection principle and system of transfinite induction with a satisfaction class. (shrink)

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms I Δ₀(π) + def(π), where π(x) is the number of primes not exceeding x, IΔ₀(π) denotes the theory of Δ₀ induction for the language of arithmetic including the new function symbol π, and de f(π) is an axiom expressing the usual recursive definition of π. We prove a modified version in which π is replaced by a more general (...) function ξ that counts some of the primes below x (which primes depends on the values of parameters in ξ), and has the property that π is provably Δ₀(ξ) definable. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

In his 1953's paper, Grzegorczyk proved that a certain kind of relation classes of Grzegorczyk's hierarchy could be characterized inductively. We give a simpler version of this characterization.

We consider the complexity of the integration operator on real functions with respect to the subrecursive class M^2 . We prove that the definite integral of a uniformly M^2-computable analytic real function with M^2-computable limits is itself M^2-computable real number. We generalise this result to integrals with parameters and with varying limits. As an application, we show that the Euler-Mascheroni constant is M^2-computable.

This paper deals with a philosophical question that arises within the theory of computational complexity: how to understand the notion of INTRINSIC complexity or difficulty, as opposed to notions of difficulty that depend on the particular computational model used. The paper uses ideas from Blum's abstract approach to complexity theory to develop an extensional approach to this question. Among other things, it shows how such an approach gives detailed confirmation of the view that subrecursive hierarchies tend to rank functions in (...) terms of their intrinsic, and not just their model-dependent, difficulty, and it shows how the approach allows us to model the idea that intrinsic difficulty is a fuzzy concept. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give a new proof, (...) which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares. (shrink)

Two simply typed term systems $\sf {PR}_1$ and $\sf {PR}_2$ are considered, both for representing algorithms computing primitive recursive functions. $\sf {PR}_1$ is based on primitive recursion, $\sf {PR}_2$ on recursion on notation. A purely syntactical method of determining the computational complexity of algorithms in $\sf {PR}_i$ , called $\mu$ -measure, is employed to uniformly integrate traditional results in subrecursion theory with resource-free characterisations of sub-elementary complexity classes. Extending the Schwichtenberg and Müller characterisation of the Grzegorczyk classes ${\mathcal{E}}_n$ for $n\ge (...) 3$ , it is shown $\mathcal{E}_{n+1} = \mathcal{R}^n_1 $ for $ n\ge 1 $, where $ \mathcal{R}^n_i$ denotes the \emph{ $n$ th modified Heinermann class} based on $\mu$ . The proof does not refer to any machine-based computation model, unlike the Schwichtenberg and Müller proofs. This is due to the notion of modified recursion lying on top of each other provided by $\mu$ . By Ritchie's result, $\mathcal{R}^1_1$ characterises the linear-space computable functions. Using the same method, a short and straightforward proof is presented, showing that $\mathcal{R}^1_2$ characterises the polynomial time computable functions. Furthermore, the classes $\mathcal{R}^n_2$ and $\mathcal{R}^n_1$ coincide at and above level 2. (shrink)

We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the (...) new interpretation to theories of classical arithmetic and analysis. (shrink)

In this article, we study some new characterizations of primitive recursive functions based on restricted forms of primitive recursion, improving the pioneering work of R. M. Robinson and M. D. Gladstone. We reduce certain recursion schemes (mixed/pure iteration without parameters) and we characterize one-argument primitive recursive functions as the closure under substitution and iteration of certain optimal sets.

The termination of rewrite systems for parameter recursion, simple nested recursion and unnested multiple recursion is shown by using monotone interpretations both on the ordinals below the first primitive recursively closed ordinal and on the natural numbers. We show that the resulting derivation lengths are primitive recursive. As a corollary we obtain transparent and illuminating proofs of the facts that the schemata of parameter recursion, simple nested recursion and unnested multiple recursion lead from primitive recursive functions to primitive recursive functions.

We survey a selection of results about majorization hierarchies. The main focus is on classical and recent results about the comparison between the slow and fast growing hierarchies.

A family of formal intuitionistic theories is proposed with realizability of proved formulas in several subrecursive classes, e.g. Grzegorczyk classes, polynomial-time computable functions class, etc. xA) Algorithm extraction forxyA is shown for various classes of bounded complexity. The results on polynomial computability are closely connected to work on the Bounded Arithmetic by S. Buss.

The paper studies a domain theoretical notion of primitive recursion over partial sequences in the context of Scott domains. Based on a non-monotone coding of partial sequences, this notion supports a rich concept of parallelism in the sense of Plotkin. The complexity of these functions is analysed by a hierarchy of classes ${\cal E}^{\bot}_n$ similar to the Grzegorczyk classes. The functions considered are characterised by a function algebra ${\cal R}^{\bot}$ generated by continuity preserving operations starting from computable initial functions. Its (...) layers ${\cal R}^{\bot}_n$ are related to those above by showing $\forall n \ge 2.{\cal E}^{\bot}_{n+1} ={\cal R}^{\bot}_n$ , thus generalising results of Schwichtenberg/Müller and Niggl. (shrink)

Subrecursive hierarchy classifications are used to compare the complexities of recursive functions according to their derivations in a version of Kleene's equation calculus, and their computations by term-rewriting. In each case ordinal bounds are assigned, and it turns out that the respective complexity measures are given by a version of the Fast Growing Hierarchy, and the Slow Growing Hierarchy. Known comparisons between the two hierarchies then provide ordinal trade-offs between derivation and computation. Characteristics of some well-known subrecursive classes are also (...) read off. (shrink)

A key problem in implicit complexity is to analyse the impact on program run times of nesting control structures, such as recursion in all finite types in functional languages or for-do statements in imperative languages.Three types of programs are studied. One type of program can only use ground type recursion. Another is concerned with imperative programs: ordinary loop programs and stack programs. Programs of the third type can use higher type recursion on notation as in functional programming languages.The present approach (...) to analysing run time yields various characterisations of fptime and all Grzegorczyk classes at and above the flinspace level. Central to this approach is the distinction between “top recursion” and “side recursion”. Top recursions are the only form of recursion which can cause an increase in complexity. Counting the depth of nested top recursions leads to several versions of “the μ-measure”. In this way, various recent solutions of key problems in implicit complexity are integrated into a uniform approach. (shrink)