. As-an-illustration, . Consider-schaefer-'s-dichotomy, and . Theorem, which, as described in detail in Section 2.2, asserts that, for every finite set of Boolean relations, either Sat(S) is in P or Sat(S) is NP-complete; moreover, the tractable cases of Sat(S) are the cases in which S is Horn, or S is dual Horn, or S is bijunctive, or every S is affine. Thus, the quadratic algorithm for Min co-Clone implies that, given a finite set S of Boolean relations, we can decide, quadratic time whether or not Sat(S) is in P. A similar result holds for the metaproblem associated with the InvSat problem studied in [KS98]. Earlier known algorithms for these meta-problems were cubic

. However, assume that whenever every relation in S ? can be expressed from the relations in S using only finite Cartesian products and identification of variables, then ?(S ? ) is polynomial- References, AHV95] S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases, 1995.

G. [. Bergman and . Slutzki, Complexity of Some Problems Concerning Varieties and Quasi-Varieties of Algebras, SIAM Journal on Computing, vol.30, issue.2, pp.359-382, 2000.
DOI : 10.1137/S0097539798345944

E. Böhler, N. Creignou, S. Reith, and H. Vollmer, Playing with Boolean Blocks, Part I: Post's Lattice with Applications to Complexity Theory, Complexity Theory Column 42, pp.38-52, 2003.

E. Böhler, N. Creignou, S. Reith, and H. Vollmer, Playing with Boolean Blocks, Part II: Constraint satisfaction problems, Complexity Theory Column 43, pp.22-35, 2004.

E. Böhler, S. Reith, H. Schnoor, and H. Vollmer, Bases for Boolean co-clones, Information Processing Letters, vol.96, issue.2, pp.59-66, 2005.
DOI : 10.1016/j.ipl.2005.06.003

N. Creignou, S. Khanna, and M. Sudan, Complexity classifications of Boolean constraint satisfaction problems, SIAM Monographs on Discrete Mathematics and Applications, 2001.
DOI : 10.1137/1.9780898718546

P. [. Creignou, B. Kolaitis, and . Zanuttini, Preferred representations of Boolean relations, technical report TR05-119, 2005.

]. V. Dal00 and . Dalmau, Computational complexity of problems over generalized formulas, 2000.

]. R. Dec03 and . Dechter, Constraint Processing, 2003.

J. [. Dechter and . Pearl, Structure identification in relational data, Artificial Intelligence, vol.58, issue.1-3, pp.237-270, 1992.
DOI : 10.1016/0004-3702(92)90009-M

]. D. Gei68 and . Geiger, Closed systems of functions and predicates, Pac. J. Math, vol.27, issue.2, pp.228-250, 1968.

J. [. Garcia-molina and J. Ullman, Widom Database Systems: The Complete Book, 2002.

M. [. Kavvadias and . Sideri, The Inverse Satisfiability Problem, SIAM Journal on Computing, vol.28, issue.1, pp.152-163, 1998.
DOI : 10.1137/S0097539795285114

]. R. La75 and . Ladner, On the structure of polynomial time reducibility, Journal of the ACM, vol.22, issue.1, pp.155-171, 1975.

]. N. Pip97 and . Pippenger, Theories of Computability, 1997.

]. E. Pos41 and . Post, The two-valued iterative systems of mathematical logic, Annals of Mathematical Studies, vol.5, pp.1-122, 1941.

]. A. Sze86 and . Szendrei, Clones in Universal Algebra, 1986.

J. [. Zanuttini and . Hébrard, A unified framework for structure identification, Information Processing Letters, vol.81, issue.6, pp.335-339, 2002.
DOI : 10.1016/S0020-0190(01)00247-2
URL : https://hal.archives-ouvertes.fr/hal-00995240