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)

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)

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to (...) the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. (shrink)

We study a hierarchy of Kalmàr elementary functions on integers based on a classification of LOOP programs of limited complexity, namely those in which the depth of nestings of LOOP commands does not exceed two. It is proved that -place functions in can be enumerated by a single function in , and that the resulting hierarchy of elementary predicates (i.e., functions with 0,1-values) is proper in that there are predicates that are not in . Along the way the rudimentary predicates (...) of Smullyan are classified as. (shrink)

A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π₁⁰-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π₁⁰ formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π₁⁰-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount (...) of computational resources required can be bounded in advance. (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)

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)

Allen, B., Arithmetizing Uniform NC, Annals of Pure and Applied Logic 53 1–50. We give a characterization of the complexity class Uniform NC as an algebra of functions on the natural numbers which is the closure of several basic functions under composition and a schema of recursion. We then define a fragment of bounded arithmetic, and, using our characterization of Uniform NC, show that this fragment is capable of proving the totality of all of the functions in Uniform NC. Lastly, (...) in the spirit of Buss, we show that any function which is definable by a ∑b1-formula in our theory is a function which is in Uniform NC. (shrink)