A very slow growing hierarchy for

Weiermann, Andreas; Eijck, Jan van; Oostrom, Vincent van; Visser, Albert

doi:10.1017/9781316755921.014

Cited by 6 publications

(5 citation statements)

References 16 publications

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“…But, as our first main theorem shows, the induced slow growing hierarchy (along the Howard-Bachmann ordinal) consists of elementary functions only. This generalises results from Weiermann (2004) where the resulting slow growing hierarchy along Γ 0 was shown to consist of elementary recursive functions only.…”

Section: Introductionsupporting

confidence: 84%

“…On the side of hierarchies of recursive functions this reduces to classifing the phase transition for the growth rates of the functions involved. In this article we are concerned with phase transitions for the slow growing hierarchy and we continue the investigations from [12][13][14][15]. From the pure logical side this article is motivated by the classical classification problem for the recursive functions and the resulting problem of comparing the slow and fast growing hierarchies.…”

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confidence: 99%

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An extremely sharp phase transition threshold for the slow growing hierarchy

Weiermann

2006

Math. Struct. in Comp. Science

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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 that the resulting slow growing hierarchy is eventually dominated by a fixed elementary recursive function if ε is equal to zero. We show further that the resulting slow growing hierarchy exhausts the provably recursive functions of ID1 if ε is strictly greater than zero. Finally we show that the resulting fast growing hierarchies exhaust the provably recursive functions of ID1 for all non negative values of ε. Our result is somewhat surprising since usually the slow growing hierarchy along the Howard Bachmann ordinal exhausts precisely the provably recursive functions of P A. Note that the elementary functions are a very small subclass of the provably recursive functions of PA and the provably recursive functions of PA are a very small subclass of the provably recursive functions of ID1. Thus the jump from ε equal to zero to ε greater than zero is one of the biggest jumps in growth rates for subrecursive hierarchies one might think of.

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Section: Introductionsupporting

confidence: 84%

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confidence: 99%

An extremely sharp phase transition threshold for the slow growing hierarchy

Weiermann

2006

Math. Struct. in Comp. Science

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“…These results can be described intuitively as phase transitions from provability to unprovability of an assertion by varying a threshold parameter [13,16,17,21]. Another face of this phenomenon is the transition from slow-growing to fastgrowing computable functions [15,18].…”

Section: Introductionmentioning

confidence: 99%

Classifying the phase transition threshold for Ackermannian functions

Omri

Weiermann

2009

Annals of Pure and Applied Logic

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It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy (of Grzegorczyk type) which is built on a start function like the successor function. In this paper we study for a given start function g iteration hierarchies with a sub-linear modulus h of iteration. In terms of g and h we classify the phase transition for the resulting diagonal function from being primitive recursive to being Ackermannian.

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“…These results can be described intuitively as phase transitions from provability to unprovability of an assertion by varying a threshold parameter [20,21,22]. Another face of this phenomenon is the transition from slow-growing to fast-growing computable functions [19,23].…”

Section: Introductionmentioning

confidence: 99%