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CATEGORICAL COMPLEXITY

Part of: Theory of computing General theory of categories and functors Computational aspects in algebraic geometry

Published online by Cambridge University Press:  30 June 2020

SAUGATA BASU Open the ORCID record for SAUGATA BASU [Opens in a new window]  and
UMUT ISIK
SAUGATA BASU
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA; sbasu@math.purdue.edu
UMUT ISIK
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA; isik@math.uci.edu

Abstract

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We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

MSC classification

Primary: 18A10: Graphs, diagram schemes, precategories 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.)
Secondary: 14Q20: Effectivity, complexity
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e34
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s) 2020

References

Andradas, C. and Ruiz, J. M., ‘Ubiquity of Lojasiewicz’s example of a nonbasic semialgebraic set’, Michigan Math. J. 41(3) (1994), 465–472. MR 1297702.Google Scholar
Awodey, S., Category Theory (Oxford University Press, Oxford, UK, 2010).Google Scholar
Basu, S., ‘A complexity theory of constructible functions and sheaves’, Found. Comput. Math. 15(1) (2015), 199–279.CrossRefGoogle Scholar
Basu, S. and Patel, D., ‘Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda’s theorem’, Preprint, 2018, arXiv:1812.07483.Google Scholar
Basu, S., Pollack, R. and Roy, M.-F., ‘Betti number bounds, applications and algorithms’, inCurrent Trends in Combinatorial and Computational Geometry: Papers from the Special Program at MSRI (Cambridge University Press, Cambridge, UK, 2005), 87–97. MSRI Publications, 52.Google Scholar
Basu, S., Pollack, R. and Roy, M.-F., Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 10 (Springer, Berlin, 2006), MR 1998147 (2004g:14064).CrossRefGoogle Scholar
Basu, S. and Zell, T., ‘Polynomial hierarchy, Betti numbers, and a real analogue of Toda’s theorem’, Found. Comput. Math. 10(4) (2010), 429–454. MR 2657948.CrossRefGoogle Scholar
Bierstone, E., Grigoriev, D., Milman, P. and Włodarczyk, J., ‘Effective Hironaka resolution and its complexity’, Asian J. Math. 15(2) (2011), 193–228. MR 2838220.CrossRefGoogle Scholar
Bürgisser, P., Clausen, M. and Amin Shokrollahi, M., Algebraic Complexity Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 315 (Springer, Berlin, 1997), With the collaboration of Thomas Lickteig. MR 1440179.CrossRefGoogle Scholar
Blum, L., Cucker, F., Shub, M. and Smale, S., Complexity and Real Computation (Springer, New York, 1998), With a foreword by Richard M. Karp. MR 1479636 (99a:68070).CrossRefGoogle Scholar
Bürgisser, P., Completeness and Reduction in Algebraic Complexity Theory, Algorithms and Computation in Mathematics, 7 (Springer, Berlin, 2000).CrossRefGoogle Scholar
Gale, D., Neighborly and Cyclic Polytopes, Proceedings of Symposia in Pure Mathematics, VII (American Mathematical Society, Providence, RI, 1963), 225–232. MR 0152944.Google Scholar
Immerman, N., ‘Descriptive complexity: a logician’s approach to computation’, Not. Amer. Math. Soc. 42(10) (1995), 1127–1133. MR 1350010.Google Scholar
Umut Isik, M., ‘Complexity classes and completeness in algebraic geometry’, Found. Comput. Math. 19(2) (2019), 245–258. MR 3937954.CrossRefGoogle Scholar
Johnstone, P. T., Sketches of an Elephant: a Topos Theory Compendium, Oxford Logic Guides, 43 1 (The Clarendon Press, Oxford University Press, New York, 2002), MR 1953060.Google Scholar
Kaltofen, E., ‘Greatest common divisors of polynomials given by straight-line programs’, J. ACM (JACM) 35(1) (1988), 231–264.CrossRefGoogle Scholar
Kreuzer, M. and Robbiano, L., Computational Commutative Algebra. 1 (Springer, Berlin, 2000), MR 1790326.CrossRefGoogle Scholar
Lambek, J. and Scott, P. J., Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics, 7 (Cambridge University Press, Cambridge, 1988), Reprint of the 1986 original. MR 939612.Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, 3rd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34 (Springer, Berlin, 1994), MR 1304906.CrossRefGoogle Scholar
Lane, S. M., Categories for the Working Mathematician, 2nd edn, Graduate Texts in Mathematics, 5 (Springer, New York, 1998), MR 1712872.Google Scholar
Lane, S. M. and Moerdijk, I., Sheaves in Geometry and Logic, Universitext (Springer, New York, 1994), A first introduction to topos theory, Corrected reprint of the 1992 edition. MR 1300636 MR 1300636.CrossRefGoogle Scholar
Mumford, D. and Oda, T., Algebraic Geometry. II, Texts and Readings in Mathematics, 73 (Hindustan Book Agency, New Delhi, 2015), MR 3443857.Google Scholar
Poizat, B., Les petits cailloux, Nur al-Mantiq wal-Ma’rifah [Light of Logic and Knowledge], 3, Aléas, Lyon, 1995, Une approche modèle-théorique de l’algorithmie. [A model-theoretic approach to algorithms]. MR 1333892.Google Scholar
Simmons, H., An Introduction to Category Theory (Cambridge University Press, Cambridge, 2011), MR 2858226.CrossRefGoogle Scholar
Strassen, V., ‘Vermeidung von divisionen’, J. Reine Angew. Math. 264 (1973), 184–202.Google Scholar
Valiant, L. G., ‘Completeness classes in algebra’, inProceedings of the Eleventh Annual ACM Symposium on Theory of Computing (ACM, New York, NY, USA, 1979), 249–261.Google Scholar
Valiant, L. G., ‘The complexity of computing the permanent’, Theoret. Comput. Sci. 8(2) (1979), 189–201.CrossRefGoogle Scholar
von zur Gathen, J., ‘Feasible arithmetic computations: Valiant’s hypothesis’, J. Symbolic. Comput. 4(2) (1987), 137–172.CrossRefGoogle Scholar
Yanofsky, N. S., ‘Computability and complexity of categorical structures’, Preprint, 2015, CoRR arXiv:1507.05305.Google Scholar