Skip to main content

Many-one reductions and the category of multivalued functions

  • ARNO PAULY (a1)

Multivalued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).

A more general understanding is possible, if the category-theoretic properties of multivalued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch).

Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multivalued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multivalued functions.

Hide All
Ambos-Spies, K. (1987). Minimal pairs for polynomial time reducibilities. In: Computation Theory and Logic. Springer Lecture Notes in Computer Science 270 113.
Bauer, A. (1998). Topology and computability. Thesis proposal (for Bauer (2000)), Carniege Mellon University.
Bauer, A. (2000). The Realizability Approach to Computable Analysis and Topology. PhD thesis, Carnegie Mellon University.
Bauer, A. (2004). Realizability as the connection between computable and constructive mathematics. Tutorial at CCA 2004 (notes).
Beame, P., Cook, S., Edmonds, J., Impagliazzo, R. and Pitassi, T. (1998). The relative complexity of NP search problems. Journal of Computer and System Science 57 319.
Bellare, M. and Goldwasser, S. (1994) The complexity of decision versus search. SIAM Journal on Computing 23 97119.
Blass, A. (1995). Questions and answers – a category arising in linear logic, complexity theory, and set theory. In: Girard, J. Y., Lafont, Y., and Regnier, L. (eds.) Advances in Linear Logic, London Mathematical Society Lecture Note Series, Cambridge University Press, volume 222, 6181. arXiv:math/9309208.
Borodin, A., Constable, R. L. and Hopcroft, J. E. (1969). Dense and non-dense families of complexity classes. In: Proceedings of the 10th Annual Symposium on Switching and Automata Theory 7–19.
Brattka, V., de Brecht, M. and Pauly, A. (2012a). Closed choice and a uniform low basis theorem. Annals of Pure and Applied Logic 163 (8) 9681008.
Brattka, V. and Gherardi, G. (2011a). Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 1 73117 arXiv:0905.4685.
Brattka, V. and Gherardi, G. (2011b). Weihrauch degrees, omniscience principles and weak computability. Journal of Symbolic Logic 76 143176 arXiv:0905.4679.
Brattka, V., Gherardi, G. and Hölzl, R. (2015). Probabilistic computability and choice. Information and Computation 242, 249286.
Brattka, V., Gherardi, G. and Marcone, A. (2012b). The Bolzano-Weierstrass Theorem is the jump of Weak König's Lemma. Annals of Pure and Applied Logic 163 (6) 623625 also arXiv:1101.0792.
Brattka, V. and Hertling, P. (1994). Continuity and computability of relations. Informatik Berichte 164, FernUniversität Hagen.
Brattka, V., LeRoux, S. and Pauly, A. (2012c). On the computational content of the Brouwer fixed point theorem. In: Barry Cooper, S., Dawar, A. and Löwe, B. (eds.) How the World Computes, Lecture Notes in Computer Science volume 7318, 5667.
Brattka, V. and Pauly, A. On the algebraic structure of Weihrauch degrees. forthcoming.
Brattka, V. and Pauly, A. (2010). Computation with advice. Electronic Proceedings in Theoretical Computer Science 24 CCA 2010.
Chen, X. and Deng, X. (2005). Settling the complexity of 2-player Nash-equilibrium. Technical Report 134, Electronic Colloquium on Computational Complexity.
Cockett, J. R. B. and Hofstra, P. J. W. (2008). Introduction to Turing categories. Annals of Pure and Applied Logic 156 (2–3) 183209.
Cockett, J. R. B. and Lack, S. (2002). Restriction categories I: Categories of partial maps. Theoretical Computer Science 270 (1–2) 223259.
Cockett, J. R. B. and Lack, S. (2003). Restriction categories II: Partial map classification. Theoretical Computer Science 294 (1–2) 61102. Category Theory and Computer Science.
Cockett, R. and Garner, R. (2014). Restriction Categories As Enriched Categories. Theoretical Computer Science, 523, 3755.
Cockett, R. and Lack, S. (2007). Restriction categories III: Colimits, partial limits and extensivity. Mathematical Structures in Computer Science 17 775817.
dePaiva, V. (1989). A Dialectica-like model of linear logic. In: Pitt, D. et al. (ed.) Categories in Computer Science and Logic. Springer Lecture Notes in Computer Science 389.
Daskalakis, C. and Papadimitriou, C. (2011). Continuous local search. In: Proceedings of SODA 790–804.
DiPaola, R. and Heller, A. (1987). Dominical categories: Recursion theory without elements. Journal of Symbolic Logic 52 594635.
Dorais, F. G., Dzhafarov, D. D., Hirst, J. L., Mileti, J. R. and Shafer, P. On uniform relationships between combinatorial problems. Transactions of the AMS, to appear. arXiv 1212.0157.
Downey, R. and Fellows, M. (1999). Parameterized Complexity, Springer.
Etessami, K. and Yannakakis, M. (2007). On the complexity of Nash equilibria and other fixed points (extended abstract). In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science 113–123.
Fabrikant, A., Papadimitriou, C. and Talwar, K. (2004). The complexity of pure Nash equilibria. In: STOC '04: Proceedings of the 36th Annual ACM Symposium on Theory of Computing 604–612.
Flum, J. and Grohe, M. (2006). Parameterized Complexity Theory, Springer.
Gherardi, G. and Marcone, A. (2009). How incomputable is the separable Hahn-Banach theorem. Notre Dame Journal of Formal Logic 50 (4) 393425.
Gottlob, G. (2005). Computing cores for data exchange: New algorithms and practical solutions. In: PODS 148–159.
Higuchi, K. and Pauly, A. (2013). The degree-structure of Weihrauch-reducibility. Logical Methods in Computer Science 9 (2).
Hoyrup, M., Rojas, C. and Weihrauch, K. (2012). Computability of the Radon-Nikodym derivative. Computability 1 (1) 313.
Johnson, D. S., Papadimtriou, C. H. and Yannakakis, M. (1988). How easy is local search. Journal of Computer and System Sciences 37 (1) 79100.
Jurdzinski, M., Paterson, M. and Zwick, U. (2008). A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38 (4) 15191532.
Karp, R. M. (1972). Reducibility among combinatorial problems. In: Miller, R. E. and Thatcher, J. W. (eds.) Complexity of Computer Computations, Plenum 85103.
Kawamura, A. and Cook, S. (2012). Complexity theory for operators in analysis. ACM Transactions on Computation Theory 4 (2).
Kawamura, A. and Pauly, A. (2014a). Function spaces for second-order polynomial time. In: Beckmann, A., Csuhaj-Varjú, E. and Meer, K. (eds.) Language, Life, Limits, Lecture Notes in Computer Science volume 8493, 245254.
Kawamura, A. and Pauly, A. (2014b). Function spaces for second-order polynomial time. arXiv 1401.2861.
Kozen, D. (1990). On Kleene algebras and closed semirings. In. Proceedings of the Mathematical Foundations of Computer Science. Springer Lecture Notes in Computer Science 452.
Ladner, R. E. (1975). On the structure of polynomial time reducibility. Journal of the ACM 22 (1) 155171.
Longo, G. and Moggi, E. (1984). Cartesian closed categories of enumerations and effective type structures. In: Khan, P. and MacQueen, , (eds.) Symposium on Semantics of Data Types. Springer Lecture Notes in Computer Science 173.
Medvedev, Y. T. (1955). Degrees of difficulty of mass problems. Doklady Akademii Nauk SSSR 104 501504.
Papadimitriou, C. H. (1994). On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and Systems Science 48 (3) 498532.
Pauly, A. (2010a). How incomputable is finding Nash equilibria. Journal of Universal Computer Science 16 (18) 26862710.
Pauly, A. (2010b). On the (semi)lattices induced by continuous reducibilities. Mathematical Logic Quarterly 56 (5) 488502.
Pauly, A. (2012a). Multi-valued functions in computability theory. In: Cooper, S., Dawar, A. and Löwe, B. (eds.) How the World Computes, Lecture Notes in Computer Science volume 7318, 571580.
Pauly, A. (2012b). On the topological aspects of the theory of represented spaces.
Pauly, A. and Ziegler, M. (2013). Relative computability and uniform continuity of relations. Journal of Logic and Analysis 5 139.
Robinson, E. and Rosolini, G. (1988). Categories of partial maps. Information and Computation 79 (2) 95130.
Weihrauch, K. (July 1992a). The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen.
Weihrauch, K. (September 1992b). The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen.
Weihrauch, K. (2000). Computable Analysis, Springer-Verlag.
Yates, C. E. M. (1966). A minimal pair of recursively enumerable degrees. The Journal of Symbolic Logic 31 (2) 159168.
Yoshimura, K. (2014). From Weihrauch lattice to logic: Part I. unpublished notes.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


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