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SAFE RECURSIVE SET FUNCTIONS

Published online by Cambridge University Press:  22 July 2015

ARNOLD BECKMANN ,
SAMUEL R. BUSS  and
SY-DAVID FRIEDMAN
ARNOLD BECKMANN
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE COLLEGE OF SCIENCE SWANSEA UNIVERSITYSWANSEA SA2 8PP, UKE-mail:a.beckmann@swansea.ac.uk
SAMUEL R. BUSS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA SAN DIEGO, LA JOLLA CA 92093-0112, USAE-mail:sbuss@ucsd.edu
SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA A-1090 VIENNA, AUSTRIAE-mail:sdf@logic.univie.ac.at
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Abstract

We introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.

We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.

We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

Keywords

Safe recursiveset functionsalternating Turing machinesinfinite time Turing machinespolynomial timerudimentary functionsJensen hierarchy

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 3 , September 2015 , pp. 730 - 762
Copyright
Copyright © The Association for Symbolic Logic 2015 

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