2 results
Termination orderings and complexity characterisations
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- By E.A. Cichon, Bedford New College
- Edited by Peter Aczel, University of Manchester, Harold Simmons, University of Manchester, Stanley S. Wainer, University of Leeds
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- Book:
- Proof Theory
- Published online:
- 05 November 2011
- Print publication:
- 11 February 1993, pp 171-194
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Summary
INTRODUCTION
This paper discusses proof theoretic characterisations of termination orderings for rewrite systems and compares them with the proof theoretic characterisations of fragments of first order arithmetic.
Rewrite systems arise naturally from systems of equations by orienting the equations into rules of replacement. In particular, when a number theoretic function is introduced by a set of defining equations, as is the case in first order systems of arithmetic, this set of equations can be viewed as a rewrite system which computes the function.
A termination ordering is a well-founded ordering on terms and is used to prove termination of a term rewriting system by showing that the rewrite relation is a subset of the ordering and hence is also well founded thus guaranteeing the termination of any sequence of rewrites.
The successful use of a specific termination ordering in proving termination of a given rewrite system, R, is necessarily a restriction on the form of the rules in R, and, as we show here in specific cases, translates into a restriction on the proof theoretical complexity of the function computed by R. We shall mainly discuss two termination orderings. The first, the so-called recursive path ordering (recently re-christened as the multiset path ordering) of [Der79], is widely known and has been implemented in various theorem provers. The second ordering is a derivative of another well known ordering, the lexico-graphic path ordering of [KL80]. This derivative we call the ramified lexicographic path ordering. We shall show that the recursive path ordering and the ramified lexicographic path ordering prove termination of different algorithms yet characterise the same class of number-theoretic functions, namely the primitive recursive functions.
The slow-growing and the Graegorczyk hierarchies
- E.A. Cichon, S.S. Wainer
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- Journal:
- The Journal of Symbolic Logic / Volume 48 / Issue 2 / June 1983
- Published online by Cambridge University Press:
- 12 March 2014, pp. 399-408
- Print publication:
- June 1983
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We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α < β ∈ I then the βth function dominates the αth one at all but finitely-many positive integers x.
Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)
The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 < λ1 < λ2 < … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.
Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.