Real or natural number interpretation and their effect on complexity

Interpretation methods have been introduced in the 70s by Lankford [1] in rewriting theory to prove termination. Actually, as shown by Bonfante et al. [2], an interpretation of a program induces a bound on its complexity. However, Lankford's original analysis depends deeply on the Archimedean p...

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Published in:Theoretical computer science Vol. 585; pp. 25 - 40
Main Authors: Bonfante, Guillaume, Deloup, Florian, Henrot, Antoine
Format: Journal Article
Language:English
Published: Elsevier B.V 20.06.2015
Subjects:
Algebra
Algebraic geometry
Complexity
Implicit computational complexity
Integers
Number theory
Polynomial interpretation
Polynomials
Real numbers
Term rewriting
Term rewriting
Polynomial interpretation
Algebraic geometry
Implicit computational complexity
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary:Interpretation methods have been introduced in the 70s by Lankford [1] in rewriting theory to prove termination. Actually, as shown by Bonfante et al. [2], an interpretation of a program induces a bound on its complexity. However, Lankford's original analysis depends deeply on the Archimedean property of natural numbers. This goes against the fact that finding a real interpretation can be solved by Tarski's decision procedure over the reals (as described by Dershowitz in [3]), and consequently interpretations are usually chosen over the reals rather than over the integers. Doing so, one cannot use anymore the (good) properties of the natural (well-)ordering of N used to bound the complexity of programs. We prove that one may take benefit from the best of both worlds: the complexity analysis still holds even with real numbers. The reason lies in a deep algebraic property of polynomials over the reals. We illustrate this by two characterizations, one of polynomial time and one of polynomial space.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2015.03.004