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37
An extremely sharp phase transition threshold for the slow growing hierarchy
 Mathematical Structures in Computer Science
"... Abstract. We investigate natural systems of fundamental sequences for ordinals below the Howard Bachmann ordinal and study growth rates of the resulting slow growing hierarchies. We consider a specific assignment of fundamental sequences which depends on a non negative real number ε. We show that t ..."
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Cited by 4 (3 self)
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Abstract. We investigate natural systems of fundamental sequences for ordinals below the Howard Bachmann ordinal and study growth rates of the resulting slow growing hierarchies. We consider a specific assignment of fundamental sequences which depends on a non negative real number ε. We show
Classifying the phase transition of Hydra games and Goodstein sequences, 2006. Manuscript, available at http://www.math.uu.nl/people/weierman/goodstein.pdf
"... It is well known from the work of Kirby and Paris that the hydra game and the Goodstein process yield combinatorial statements which are true but unprovable in first order Peano arithmetic PA. In this note we characterize the phase transition from provability to unprovability for these assertions. A ..."
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Cited by 2 (0 self)
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. As a byproduct we classify those non pointwise descent recursions (along the standard system of fundamental sequences for the ordinals below ε0) which are still in the scope of the so called slow growing hierarchy. 1 Introduction and statement of the results This article is part of a general
ProofTheoretic Analysis of Termination Proofs
 APAL
, 1994
"... Introduction In [Cichon 1990] the question has been discussed (and investigated) whether the order type of a termination ordering places a bound on the lengths of reduction sequences in rewrite systems reducing under . It was claimed that at least in the cases of the recursive path ordering rpo a ..."
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Cited by 11 (0 self)
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and the lexicographic path ordering lpo the following theorem holds. (0) If is the order type of a termination ordering for a nite rewrite system R then the function G from the SlowGrowing Hierarchy bounds the lengths of reduction sequences in R. From (0) together with Girard's Hierarchy Comparison Theorem
A Relationship among Gentzen’s ProofReduction
 KirbyParis’ Hydra Game, and Buchholz’s Hydra Game, Math. Logic Quarterly
, 1997
"... KirbyParis [9] found a certain combinatorial game called Hydra Game whose termination is true but cannot be proved in $PA $. Cichon [4] gave a new proof based on Wainer’s finite characterization of the $\mathrm{P}\mathrm{A}$provably recursive functions by the use of Hardy functions. Both KirbyP ..."
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Cited by 3 (0 self)
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Paris and Cichon’s proofs on the unprovability result were obtained by a certain investigation on the ordinals less than $\epsilon_{0} $ , the critical ordinal of $\mathrm{P}\mathrm{A} $ , with the help of the fast and slow growing hierarchies, respectively. On the other hand, Jervell [7] proposed a combinatorial
Weak Length Induction and Slow Growing Depth Boolean Circuits. submitted
, 1999
"... Abstract. We define a hierarchy of circuit complexity classes LD i, whose depth are the inverse of a function in Ackermann hierarchy. Then we introduce extremely weak versions of length induction called L (m) IND and construct a bounded arithmetic theory L i 2 whose provably total functions exactly ..."
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Cited by 2 (1 self)
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Abstract. We define a hierarchy of circuit complexity classes LD i, whose depth are the inverse of a function in Ackermann hierarchy. Then we introduce extremely weak versions of length induction called L (m) IND and construct a bounded arithmetic theory L i 2 whose provably total functions exactly
Semantics and Syntax for Ordinal Notations and Hierarchies
"... We give a complete description of a class of free infinitary algebras in the category of posets. They are represented as functors to the ordinal\Omega\Gamma One of these algebras is used to define an ordinal notation system based on the simply typed lambda calculus, and prove its soundness. As an ap ..."
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. As an application the Slow Growing and Hardy hierarchies are described in that calculus. The literature on ordinal notations is already quite extensive. There is a particular class of ordinal notations which is based on the notion of fundamental sequence: a countable ordinal ff is defined in terms of a sequence
What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 56 (2 self)
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, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization
PERSPECTIVES
"... A more likely explanation is the involvement of one or more distinct semistable epigenetic states that cannot be directly mapped to a differentiation hierarchy. In bacterial cell populations, a fraction of cells randomly assumes a distinct phenotype that is characterized by resistance to stress, ..."
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, including antibiotic treatment, at the expense of reduced proliferation rates ( 6). The reemergence of previously minor clones after oxaliplatin treatment and their ability to initiate new tumors (although at smaller size) suggest the presence of such slowgrowing dormant clones. Similarly, a
On the number of permutations avoiding a given pattern
 J. Comb. Theory, Ser. A
, 1999
"... Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 <... < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = {τ ∈ Sn  τ avoids σ}. Stanley and Wilf conjectured that for any σ ∈ Sk ..."
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there exists a constant c = c(σ) such that F (n, σ) ≤ c n for all n. Here we prove the following weaker statement: For every fixed σ ∈ Sk, F (n, σ) ≤ c nγ ∗ (n) , where c = c(σ) and γ ∗ (n) is an extremely slow growing function, related to the Ackermann hierarchy. 1
acQuiremacros: An Algorithm for Automatically Learning Macroactions
 In NIPS'98 Workshop on Abstraction and Hierarchy in Reinforcement Learning
, 1998
"... ion and Hierarchy in Reinforcement Learning 1 acQuiremacros: An Algorithm for Automatically Learning Macroactions Amy McGovern amy@cs.umass.edu Computer Science Department University of Massachusetts, Amherst Amherst, MA 01003 November 23, 1998 Abstract We present part of a new algorithm for a ..."
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Cited by 15 (1 self)
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Introduction Much of the current research in reinforcement learning focuses on temporal abstraction, modularity, and hierarchy in learning. Although learning at the level of the most primitive actions allows the agent to discover the optimal policy, learning can be very slow. Temporal abstraction can enable
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