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Ordinal Bounds for Programs
"... this paper we use methods of proof theory to assign ordinals to classes of (terminating) programs, the idea being that the ordinal assignment provides a uniform way of measuring computational complexity "in the large". We are not concerned with placing prior (e.g. polynomial) bounds on computation ..."
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this paper we use methods of proof theory to assign ordinals to classes of (terminating) programs, the idea being that the ordinal assignment provides a uniform way of measuring computational complexity "in the large". We are not concerned with placing prior (e.g. polynomial) bounds on computationlength, but rather with general methods of assessing the complexity of natural classes of programs according to the ways in which they are constructed. We begin with an overview of the method in section 2, the crucial idea being supplied by Buchholz's ! +
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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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 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 one derives (I) If the rules of a nite rewrite system R are reducing under rpo then the lengths of reduction sequences in R are bounded by some primitive recursive function. (II) If the rules of a nite rewrite system R are reducing under lpo then the lengths of reduction sequences in R are bounded by some function F from the fastgrowing hierarchy below ! . Unfort
Accessible Recursive Functions
 Bulletin of Symbolic Logic
, 1999
"... . The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (prooftheoretically significant) subrecursive classes do. This paper attempts to measure the limit of predicative generation in this context, by ..."
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. The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (prooftheoretically significant) subrecursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over wellorderings which have already been "coded" at previous levels. The question is: how can a recursion code a wellordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite di#erent and simplified framework specific to our purpose. The "accessible" recursive functions thus generated turn out to be those provably recursive in (# 1 1  CA) 0 . Introduction. Before one accepts a computable function as being recursive, a proof of totality is required. This will generally ...
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.