. A result claimed without proof by R. Constable in a STOC73 paper is here corrected: a strictly increasing function f is presented for which Constable's class K(f) is properly contained in FP (f ), the collection of functions polynomial time computable in f .

This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1

. The goal of this paper is to study tractable iteration mechanisms for bags. The presence of duplicates in bags prevents iteration mechanisms developed in the context of sets to be directly applied to bags without losing tractability. We study two constructs for controlling tractability of iteration over bags. The deflationary fixpoint construct keeps removing elements from a bag until a fixpoint is reached. The bounded fixpoint construct is an inflationary iteration mechanism that never exceeds some predefined bounding bag. We study these constructs in the context of a standard (nested) bag algebra. We show that the deflationary and bounded inflationary fixpoint constructs are equally expressive and strictly more expressive than their set-based counterparts. We also show that, unlike in the set case, the bag algebra with bounded fixpoint fails to capture all PTIME queries over databases with ordered domains. We then show that adding just one construct, which can be used...

. We give function algebraic characterizations of the polynomial time computable functions using unbounded recursion schemes without introducing extra notions such as safe and normal in variables and tiering in word algebras. Key words. polytime functions, function algebras, recursion schemes. Subject classications. 68Q15. 1. Introduction Cobham (1965) characterized the class Fptime of polynomial time computable functions as the smallest class of functions containing certain initial functions, and closed under composition and the following variant of primitive recursion: the function f is dened by bounded recursion on notation (brn) from g; h 0 ; h 1 ; k if f(0; ~y) = g(~y); f(s 0 (x); ~y) = h 0 (x; ~y; f(x; ~y)); (if x 6= 0) f(s 1 (x); ~y) = h 1 (x; ~y; f(x; ~y)) provided that f(x; ~y) k(x; ~y) for all x; ~y, where s 0 and s 1 are the binary successor functions with s 0 (x) = 2 x and s 1 (x) = 2 x + 1. Although Cobham's characterization has been fruitful and yielded a ...

A fragment of second-order lambda calculus (System F) is defined that characterizes the elementary recursive functions. Type quantification is restricted to be non-interleaved and stratified, i.e., the types are assigned levels, and a quantified variable can only be instantiated by a type of smaller level, with a slightly liberalized treatment of the level zero.

Abstract We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization. The levels of the digital polynomial hierarchy correspond to safe recursion with digital predicative minimization. Also, we show that polynomial alternating time corresponds to safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions. 1

We introduce two hierarchies of unknown ordinal height. The hierarchies are induced by natural fragments of a calculus based on finite types and Gödel’s T, and all the classes in the hierarchies are uniformly defined without referring to explicit bounds. Deterministic complexity classes like logspace, p, pspace, linspace and exp are captured by the hierarchies. Typical subrecursive classes are also captured, e.g. the small relational Grzegorczyk classes E 0 ∗, E 1 ∗ and E 2 ∗.

References

In this paper we continue Feferman’s unfolding program initiated in [11] which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried through for a schematic system of non-finitist arithmetic NFA in Feferman and Strahm [13] and for a system FA (with and without Bar rule) in Feferman and Strahm [14]. The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apart from the operational unfolding U0(FEA) of FEA, we study two full unfolding notions, namely the predicate unfolding U(FEA) and a more general truth unfolding UT(FEA) of FEA, the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth TPT over combinatory logic, which has recently been introduced in Eberhard and Strahm [7] and Eberhard [6] and whose involved proof-theoretic analysis is due to Eberhard [6]. The results of this paper were first announced in [8].

We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new two-sorted algebra which is very similar to Cook and Bellantoni’s famous two-sorted algebra B for polynomial time [4]. The second theory describes logarithmic space by justifying concatenation- and sharply bounded recursion. All theories contain the predicates W representing words, and V representing temporary inaccessible words. They are inspired by Cantini’s theories [6] formalising B. 1.