Arnold Beckmann

A term rewriting characterization of the polytime functions and related complexity classes

A natural term rewriting framework for the Bellantoni Cook schemata of predicative recursion, which yields a canonical definition of the polynomial time computable functions, is introduced. In terms of an exponential function both, an upper bound and a lower bound are proved for the resulting derivation lengths of the functions in question. It is proved that any natural reduction strategy yields an algorithm which runs in exponential time. We give an example in which this estimate is tight. It is proved that the resulting derivation lengths become polynomially bounded in the lengths of the inputs if the rewrite rules are only applied to terms in which the safe arguments – no restrictions are assumed for the normal arguments – consist of values, i.e. numerals, and not of names, i.e. non numeral terms. It is proved that in the latter situation any inside first reduction strategy and any head reduction strategy yield algorithms, for the function under consideration, for which the running time is bounded by an appropriate polynomial in the lengths of the input. A feasible rewrite system for predicative recursion with predicative parameter substitution is defined. It is proved that the derivation lengths of this rewrite system are polynomially bounded in the lengths of the inputs. As a corollary we reobtain Bellantoni’s result stating that predicative recursion is closed under predicative parameter recursion.

Dynamic ordinal analysis

Abstract. Dynamic ordinal analysis is ordinal analysis for weak arithmetics like fragments of bounded arithmetic. In this paper we will define dynamic ordinals – they will be sets of number theoretic functions measuring the amount of sΠb1(X) order induction available in a theory. We will compare order induction to successor induction over weak theories. We will compute dynamic ordinals of the bounded arithmetic theories sΣbn(X)−LmIND for m=n and m=n+1, n≥0. Different dynamic ordinals lead to separation. In this way we will obtain several separation results between these relativized theories. We will generalize our results to further languages extending the language of bounded arithmetic.

Separation results for the size of constant-depth propositional proofs

This paper proves exponential separations between depth d-LK and depth (d + 1 )-LK for every d 2 1 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d-LK and depth (d+1)-LK ford2N. We investigate the relationship between the sequence-size, tree-size and height of depth d-LK-derivations for d 2 1 N, and describe transformations between them. We deflne a general method to lift principles requiring exponential tree-size (d + 1 )-LK-refutations for d 2 N to principles requiring exponential sequence-size d-LK-refutations, which will be described for the Ramsey principle and d = 0. From this we also deduce width lower bounds for resolution refutations of the Ramsey principle. Constant-depth propositional proof systems have been extensively studied because of their connection with the complexity of constant-depth circuits and fragments of bounded arithmetic (c.f. [2, 10, 14, 15, 17]). Kraj¶‡•cek [10] deflned an alternative notion of constant-depth proofs: a formula is deflned to have §-depth d ifi if is depth d+1 and the bottommost level of connectives = = = = i i;’

Characterizing the elementary recursive functions by a fragment of Gödel's T

Abstract. Let T be Gödels system of primitive recursive functionals of finite type in a combinatory logic formulation. Let

POLYNOMIAL LOCAL SEARCH IN THE POLYNOMIAL HIERARCHY AND WITNESSING IN FRAGMENTS OF BOUNDED ARITHMETIC

T^{\star}

Improved witnessing and local improvement principles for second-order bounded arithmetic

be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schütte-style argument the

Continuous Fraïssé Conjecture

T^{\star}

SAFE RECURSIVE SET FUNCTIONS

-derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for

Applications of cut-free infinitary derivations to generalized recursion theory

T^{\star}

Hyper Natural Deduction

is obtained. Another consequence is that every