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Abstract: Proofs in an arithmetic system are ranked according to a ramification hierarchy based on occurrences of induction. It is shown that this ranking of proofs corresponds exactly to a natural ranking of the primitive recursive functions based on occurrences of recursion. A function is provably convergent using a rank r proof, if and only if it is a rank r function. The result is of interest to complexity theorists, since rank one corresponds to polynomial time. Remarkably, this characterization of... (Update)
Context of citations to this paper: More
...of polytime functions in Leivant s system. Another precursor is the ramified arithmetic and corresponding model theory by Bellantoni [2]. The arithmetic in [2] demonstrated the possibility of admitting arbitrarily many alternations of unbounded quantifiers in induction...
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BibTeX entry: (Update)
S. Bellantoni, "Ranking Arithmetic Proofs by Implicit Ramification", in Proof Complexity and Feasible Arithmetics, P. Beame and S. Buss, eds., DIMACS Series in Discrete Mathematics, v. 39, 1998. http://citeseer.ist.psu.edu/bellantoni96ranking.html More
@techreport{ bellantoni96ranking,
author = "Stephen J. Bellantoni",
title = "Ranking Arithmetic Proofs by Implicit Ramification",
number = "96-49",
month = "7,",
year = "1996",
url = "citeseer.ist.psu.edu/bellantoni96ranking.html" }
Citations (may not include all citations):604 Reasoning About Knowledge (context) - Fagin, Halpern et al. - 1995
200 Introduction To Mathematical Logic (context) - Mendelson - 1979
143 Foundations of Constructive Mathematics (context) - Beeson - 1985
116 Ideas and Results In Proof Theory (context) - Prawitz - 1971
96 A New Recursion-Theoretic Characterization of the Polytime F.. (context) - Bellantoni, Cook - 1992
38 Ramified Recurrence and Computational Complexity I: Word rec.. (context) - Leivant - 1995
26 Predicative Recursion and Computational Complexity - Bellantoni - 1992
12 and Complexity Theory (context) - Krajicek, Arithmetic et al. - 1995
11 Computation Models and Function Algebras - Clote - 1996
10 Predicative Recursion and the Polytime Hierarchy (context) - Bellantoni - 1995
10 Stratified Functional Programs and Computational Complexity (context) - Leivant - 1993
8 Epistemic and Intuitionistic Arithmetic (context) - Shapiro
4 Philosophical Applications of Free Logic (context) - Lambert - 1991
4 Intrinsic Theories and Computational Complexity (context) - Leivant
4 Epistemic Arithmetic is a Conservative Extension of Intuitio.. (context) - Goodman - 1984
2 First Order Mathematical Logic (context) - Margaris - 1967
1 Ranking Primitive Recursions: The Low Grzegorczyck Classes R.. (context) - Bellantoni - 1996
1 Intensional Mathematics (context) - Shapiro - 1985
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