Abstract
The main results of this paper are recursion-theoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fan-in circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, anda priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the “tiered recursion” techniques of Leivant and Bellantoni & Cook.
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References
William Allen, Arithmetizing uniformNC.Annals of Pure and Applied Logic 53(1) (1991), 1–50. See alsoDivide and Conquer as a Foundation of Arithmetic, Ph. D. thesis, University of Hawaii at Manoa, 1988.
Toshiyasu Arai, Frege systems, ALOGTIME and bounded arithmetic. Manuscript, 1992.
D. A. Barrington, Bounded-width polynomial-size branch programs recognize exactly those languages inNC 1.J. Comput. Sys. Sci. 38(1) (1989), 150–164. See alsoProc. Eighteenth Ann. ACM Symp. Theor. Comput., ACM Press 1986, 1–5.
David A. Mix Barrington, Quasipolynomial size circuit classes. InProceedings of the Seventh Annual Structure in Complexity Theory Conference. IEEE Computer Society Press, 1992, 86–93.
Stephen Bellantoni,Predicative Recursion and Computational Complexity. PhD thesis, University of Toronto, 1992.
Stephen Bellantoni andStephen Cook A new recursion-theoretic characterization of the polytime functions.computational complexity 2 (1992), 97–110.
Stephen Bloch, Alternating function classes withinP. Technical Report 92-16, University of Manitoba Computer Science Dept, 1992.
Samuel R. Buss,Bounded Arithmetic. Number 3 in Studies in Proof Theory. Bibliopolis (Naples), 1986. See also “The polynomial hierarchy and bounded arithmetic” inProc. Seventeenth Ann. ACM Symp. Theor. Comput., ACM Press 1985, 285–290.
Samuel R. Buss, The boolean formula value problem is in ALOGTIME. InProc. Nineteenth Ann. ACM Symp. Theor. Comput., 1987, 123–131.
A. Chandra, D. Kozen, andL. Stockmeyer, Alternation.J. Assoc. Comput. Mach. 28(1) (1981), 114–133.
Peter Clote, Sequential, machine-independent characterizations of the parallel complexity classesALOGTIME, AC k,NC k andNC. InProc. Workshop on Feasible Math., ed.Samuel R. Buss and P. Scott. Birkhäuser, 1989, 49–69.
Peter Clote, Polynomial size Frege proofs of certain combinatorial principles. InArithmetic, Proof Theory and Computational Complexity, ed.Peter Clote andJan Krajíček, 162–184. Oxford University Press, 1993.
Peter Clote andGaisi Takeuti, Bounded arithmetic forNC, ALogTIME, L andNL.Annals of Pure and Applied Logic 56 (1992), 73–117.
A. Cobham, The intrinsic computational difficulty of functions. InLogic, Methodology andPhilosophy of Science II, ed.Y. Bar-Hillel. North-Holland, 1965, 24–30.
Kevin Compton andClaude LaFlamme, An algebra and a logic forNC 1.Information and Computation 87 (1990), 241–263.
W. G. Handley, Bellantoni and Cook's characterization of polynomial time functions. Typescript, 1992.
Daniel Leivant, Subrecursion and lambda representation over free algebras. InFeasible Mathematics, ed.Samuel Buss andPhilip Scott,Perspectives in Computer Science, 281–291. Birkhäuser, 1990.
Daniel Leivant, A foundational delineation of computational feasibility. InProc. Sixth IEEE Conf. Logic in Computer Science. IEEE Computer Society Press, 1991.
J. Lind, Computing in logarithmic space. Technical Report 52, Project MAC, Massachusetts Inst. of Technology, 1974.
James Otto, A category-theoretic characterization of polytime. Contributed talk at MSI Feasible Mathematics Conference, 1992.
W. Ruzzo, On uniform circuit complexity.J. Comput. Sys. Sci. 22 (1981), 365–383.
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Bloch, S. Function-algebraic characterizations of log and polylog parallel time. Comput Complexity 4, 175–205 (1994). https://doi.org/10.1007/BF01202288
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DOI: https://doi.org/10.1007/BF01202288