Skip to main content Accessibility help




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.



Hide All
[1]Arai, Toshiyasu, Predicatively computable functions on sets. Archive for Mathematical Logic, vol. 54 (2015), pp. 471485.
[2]Bellantoni, Stephen and Cook, Stephen, A new recursion-theoretic characterization of the polytime functions. Computational Complexity, vol. 2 (1992), no. 2, pp. 97110.
[3]Berman, Leonard, The complexity of logical theories, Theoretical Computer Science, vol. 11 (1980), pp. 7177.
[4]Blum, Lenore, Shub, Mike, and Smale, Steve, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society (N.S.), vol. 21 (1989), no. 1, pp. 146.
[5]Bruss, Anna R. and Meyer, Albert R., On the time-space classes and their relation to the theory of real addition. Theoretical Computer Science, vol. 11 (1980), pp. 5969.
[6]Chandra, Ashok K., Kozen, Dexter C., and Stockmeyer, Larry J., Alternation, Journal of the Association for Computing Machinery, vol. 28 (1981), pp. 114133.
[7]Devlin, Keith J., Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984.
[8]Ferrante, Jeanne and Rackoff, Charles W., A decision procedure for the first order theory of real addition with order. SIAM Journal on Computing, vol. 4 (1975), no. 1, pp. 6976.
[9]Friedman, Sy-David and Welch, Philip D., Two observations concerning infinite time Turing machines, BIWOC 2007 Report (Dimitriou, I., editor), Hausdorff Centre for Mathematics, Bonn, January 2007, pp. 4447.
[10]David Hamkins, Joel and Lewis, Andy, Infinite time Turing machines, this Journal, vol. 65 (2000), no. 2, pp. 567604.
[11]Jech, Thomas, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[12]Björn Jensen, Ronald, The fine structure of the constructible hierarchy. Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.
[13]Björn Jensen, Ronald and Karp, Carol, Primitive recursive set functions, Axiomatic Set Thoory (Proceedings of Symposia in Pure Mathematics, vol. XIII, Part 1, University of California, Los Angeles, California, 1967), American Mathematical Society, Providence, R.I., 1971, pp. 143176.
[14]Leivant, Daniel, Subrecursion and lambda representation over free algebras (preliminary summary), Feasible mathematics (Buss, S. and Scott, P., editors), Birkhäuser, 1990, pp. 281292.
[15]Leivant, Daniel, A foundational delineation of computational feasibility, Proceedings of 6th Annual Symposium on Logic in Computer Science (LICS'91), IEEE Computer Society, 1991, pp. 211.
[16]Sazonov, Vladimir Yu., On bounded set theory, Logic and Scientific Methods (Florence, 1995), Synthese Library, vol. 259, Kluwer Academic Publishers, Dordrecht, 1997, pp. 85103.
[17]Schindler, Ralf, P ≠ NP for infinite time Turing machines. Monatshefte für Mathematik, vol. 139 (2003), no. 4, pp. 335340.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed