Abstract
In our ISMVL 2012 paper, we introduced the notion of max-co-clone as a set of relations closed under a new type of quantification, max-quantification. This new concept was motivated by its connections to approximation complexity of counting constraint satisfaction problems. In this paper, we go beyond scattered examples of max-co-clones and describe all max-co-clones on a 2-elements set (Boolean max-co-clones). It turns out that there are infinitely many Boolean max-co-clones and that all of them are regular co-clones, although it is not true for larger sets. Also, there are many usual co-clones that are not closed under max-quantification, and therefore are not max-co-clones.
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Ahlswede R., Daykin D.E.: An inequality for the weights of two families of sets, their unions and intersections. Z. Wahrsch. Verw. Gebiete 43, 183–185 (1978)
Bodnarchuk, V.G., Kaluzhnin, L.A., Kotov, V.N., Romov, B.A.: Galois theory for Post algebras. I. Kibernetika, 3, 1–10 (1969) (Russian)
Börner, F.: Basics of Galois connections. In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.) Complexity of Constraints, pp. 38–67, LNCS, vol. 5250. Springer (2008)
Börner F., Bulatov A., Chen H., Jeavons P., Krokhin A.: The complexity of constraint satisfaction games and QCSP. Inf. Comput. 207, 923–944 (2009)
Börner F., Pöschel R., Sushchansky V.: Boolean systems of relations and Galois connections. Acta Sci. Math. (Szeged) 68, 535–560 (2002)
Bulatov, A.: The complexity of the counting constraint satisfaction problem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7–11, 2008, Proceedings, Part I: Track A: Algorithms, Automata, Complexity, and Games, pp. 646–661, LNCS, vol. 5125. Springer (2008)
Bulatov, A.: Counting problems and clones of functions. In: ISMVL 2009, 39th International Symposium on Multiple-Valued Logic, 21–23 May 2009, Naha, Okinawa, Japan, pp. 1–6, IEEE Computer Society (2009)
Bulatov, A.: The complexity of the counting constraint satisfaction problem. J. of the ACM, 60 (2013)
Bulatov, A., Dyer, M.E., Goldberg, L.A., Jerrum, M.: Log-supermodular functions, functional clones and counting CSPs. In: Dürr, C., Wilke, T. (eds.) The 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th–March 3rd, 2012, Paris, France, pp. 302–313, LIPIcs, vol. 14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)
Bulatov, A., Dyer, M.E., Goldberg, L.A., Jerrum, M., Mcquillan, C.: The expressibility of functions on the boolean domain, with applications to counting CSPs J. of the ACM, 60 (2013)
Bulatov, A., Hedayaty, A.: Counting quantifiers, subset surjective functions, and counting CSPs. In: Miller, D.M, Gaudet, V.C. (eds.) The 42nd IEEE International Symposium on Multiple-Valued Logic, ISMVL 2012, Victoria, BC, Canada, May 14–16, 2012, pp. 331–336, IEEE (2012)
Bulatov A., Hedayaty A.: Counting problems and clones of functions. Multiple-Valued Logic and Soft Computing 18, 117–138 (2012)
Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of Boolean constraint satisfaction problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM (2001)
Creignou N., Kolaitis P.G., Zanuttini B.: Structure identification of Boolean relations and plain bases for co-clones. J. Comput. Syst. Sci. 74, 1103–1115 (2008)
Dyer M.E., Goldberg L.A., Greenhill C., Jerrum M.: On the relative complexity of approximate counting problems. Algorithmica 38, 471–500 (2003)
Dyer M.E., Goldberg L.A., Jerrum M.: An approximation trichotomy for Boolean #CSP. J. Comput. Syst. Sci. 76, 267–277 (2010)
Geiger D.: Closed systems of function and predicates. Pacific J. of Math. 27, 95–100 (1968)
Jeavons P.G., Cohen D.A., Gyssens G.: Closure properties of constraints. J. of the ACM 44, 527–548 (1997)
Madelaine, F.R., Martin, B., Stacho, J.: Constraint satisfaction with counting quantifiers. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) Computer Science—Theory and Applications—7th International Computer Science Symposium in Russia, CSR 2012, Nizhny Novgorod, Russia, July 3–7, 2012, pp. 253–265, LNCS, vol. 7353, Springer (2012)
Pöschel, R.: Galois connection for operations and relations. Technical Report MATH-AL-8-2001, Technische Universität Dresden, Germany (2001)
Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren. DVW, Berlin (1979)
Post, E.L.: The two-valued iterative systems of mathematical logic. Annals Mathematical Studies, vol. 5, Princeton University Press (1941)
Schaefer, T.: The complexity of satisfiability problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1–3, 1978, San Diego, California, USA, pp. 216–226, ACM (1978)
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Presented by M. Jackson.
Dedicated to Brian Davey on the occasion of his 65th birthday
This research was supported by NSERC Discovery Grant.
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Bulatov, A.A. Boolean max-co-clones. Algebra Univers. 74, 139–162 (2015). https://doi.org/10.1007/s00012-015-0336-1
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DOI: https://doi.org/10.1007/s00012-015-0336-1