Andreas Weiermann

Proof-theoretic investigations on Kruskal's theorem

Abstract In this paper we calibrate the exact proof-theoretic strength of Kruskals theorem, thereby giving, in some sense, the most elementary proof of Kruskals theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.

A Uniform Approach to Fundamental Sequences and Hierarchies

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.

An application of graphical enumeration to PA

For α less than e 0 let N α be the number of occurrences of ω in the Cantor normal form of α . Further let ∣ n ∣ denote the binary length of a natural number n , let ∣ n ∣ h denote the h -times iterated binary length of n and let inv( n ) be the least h such that ∣ n ∣ h ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈 α 0 , …, α n 〉 of ordinals less than e 0 which satisfy the condition that the Norm Nα i of the i -th term α i is bounded by K + ∣ i ∣ · ∣ i ∣ i . As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈 α 0 , …, α n 〉 of ordinals less than e 0 which satisfies the condition that the Norm N αi of the i -th term α i is bounded by K + ∣ i ∣ · inv( i ). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations. Using results from Otter and from Matousek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otters tree constant 2.9557652856.…

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.

Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results☆

Abstract This paper is intended to give for a general mathematical audience (including non-logicians) a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below e 0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws.

Phase transition thresholds for some Friedman-style independence results

We classify the phase transition thresholds from provability to unprovability for certain Friedman-style miniaturizations of Kruskals Theorem and Higmans Lemma. In addition we prove a new and unexpected phase transition result for e0. Motivated by renormalization and universality issues from statistical physics we finally state a universality hypothesis. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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

The strength of infinitary Ramseyan principles can be accessed by their densities

T^{\star}

Sharp thresholds for the phase transition between primitive recursive and Ackermannian Ramsey numbers

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

Streamlined subrecursive degree theory

T^{\star}