*. ,

L. Hence, }. To-conclude,-a-k-?-a-p-?-{a-iq, and . Therefore, using the convention that i 0 = 0 and i j+1 = k, A p ? A k = ? implies that {a p+1 ,. .. , a k } A k or, if we take q ? [, the following conditions are verified: ? for all ? ?

S. Aaronson, Why philosophers should care about computational complexity. CoRR, abs/1108.1791, 2011.

M. Ajtai, ? 1 1-formulae on finite structures. Annals of pure and applied logic, vol.24
DOI : 10.1016/0168-0072(83)90038-6
URL : https://doi.org/10.1016/0168-0072(83)90038-6

E. Allender, A status report on the P versus NP question, Advances in Computers, vol.77, pp.117-147, 2009.

J. Almeida, A syntactical proof of locality of DA, IJAC, vol.6, issue.2, pp.165-178, 1996.

J. Almeida and P. Weil, Relatively free profinite monoids: an introduction and examples. Semigroups, Formal Languages and Groups, JB Fountain, vol.466, pp.73-117, 1995.
DOI : 10.1007/978-94-011-0149-3_4

N. Alon and U. Zwick, On Ne?iporuk's theorem for branching programs
DOI : 10.1016/0304-3975(89)90054-6
URL : https://doi.org/10.1016/0304-3975(89)90054-6

, Comput. Sci, vol.64, issue.3, pp.90054-90060, 1989.

S. Arora and B. Barak, Computational Complexity-A Modern Approach, 2009.
DOI : 10.1017/cbo9780511804090

A. M. David and . Barrington, Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1, J. Comput. Syst. Sci, vol.38, issue.1, pp.90037-90045, 1989.

A. M. David, D. Barrington, and . Thérien, Finite monoids and the fine structure of NC 1, J. ACM, vol.35, issue.4, pp.941-952, 1988.

A. M. David, H. Barrington, D. Straubing, and . Thérien, Non-uniform automata over groups, Inf. Comput, vol.89, issue.2, pp.90007-90012, 1990.

A. M. David, K. J. Barrington, H. Compton, D. Straubing, and . Thérien, Regular languages in NC 1, J. Comput. Syst. Sci, vol.44, issue.3, pp.478-499, 1992.

P. Beame, N. Grosshans, P. Mckenzie, and L. Segoufin, Nondeterminism and an abstract formulation of Ne?iporuk's lower bound method, TOCT, vol.9, issue.1, pp.1-5, 2016.

B. Ravi, M. Boppana, and . Sipser, The complexity of finite functions, Algorithms and Complexity (A), vol.A, pp.757-804, 1990.

A. Borodin, D. Dolev, F. E. Fich, and W. J. Paul, Bounds for width two branching programs, SIAM J. Comput, vol.15, issue.2, pp.549-560, 1986.

L. Chaubard, J. Pin, and H. Straubing, Actions, wreath products of C-varieties and concatenation product, Theor. Comput. Sci, vol.356, issue.1-2, pp.73-89, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00112843

L. Dartois, Méthodes algébriques pour la théorie des automates, 2014.

L. Dartois and C. Paperman, Adding modular predicates. CoRR, abs/1401, vol.6576, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00934622

S. Eilenberg, Automata, Languages, and Machines, volume A, 1974.

S. Eilenberg, Automata, Languages, and Machines, volume B, 1976.

L. Fortnow and S. Homer, A short history of computational complexity, Bulletin of the EATCS, vol.80, pp.95-133, 2003.

M. L. Furst, J. B. Saxe, and M. Sipser, Parity, circuits, and the polynomial-time hierarchy, Mathematical Systems Theory, vol.17, issue.1, pp.13-27, 1984.

R. Gavaldà and D. Thérien, Algebraic characterizations of small classes of Boolean functions, p.20, 2003.

, Annual Symposium on Theoretical Aspects of Computer Science, vol.2607, pp.331-342, 2003.

J. Goldsmith, M. A. Levy, and M. Mundhenk, Limited nondeterminism, SIGACT News, vol.27, issue.2, pp.20-29, 1996.

N. Grosshans, P. Mckenzie, and L. Segoufin, The power of programs over monoids in DA, 42nd International Symposium on Mathematical Foundations of Computer Science, vol.83, pp.1-2, 2017.

J. Håstad, The shrinkage exponent of de Morgan formulas is 2, SIAM J. Comput, vol.27, issue.1, pp.48-64, 1998.

J. Hromkovic and G. Schnitger, Nondeterministic communication with a limited number of advice bits, SIAM J. Comput, vol.33, issue.1, pp.43-68, 2003.

R. Impagliazzo, Computational complexity since 1980, FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science, 25th International Conference, vol.3821, pp.19-47, 2005.

S. Jukna, Boolean Function Complexity-Advances and Frontiers, Algorithms and combinatorics, vol.27, 2012.

M. Karchmer and A. Wigderson, On span programs, Proceedings of the Eigth Annual Structure in Complexity Theory Conference, pp.102-111, 1993.

H. Klauck, One-way communication complexity and the Ne?iporuk lower bound on formula size, SIAM J. Comput, vol.37, issue.2, pp.552-583, 2007.

O. Klíma and L. Polák, Hierarchies of piecewise testable languages, Int. J. Found. Comput. Sci, vol.21, issue.4, pp.517-533, 2010.

R. Knast, A semigroup characterization of dot-depth one languages, ITA, vol.17, issue.4, pp.321-330, 1983.

A. Krebs, K. Lange, and S. Reifferscheid, Characterizing TC 0 in terms of infinite groups, Theory Comput. Syst, vol.40, issue.4, pp.303-325, 2007.

K. Krohn and J. L. Rhodes, Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Transactions of the, vol.116, pp.450-464, 1965.

M. Kufleitner and P. Weil, On logical hierarchies within FO 2-definable languages, Logical Methods in Computer Science, vol.8, issue.3, p.2012
URL : https://hal.archives-ouvertes.fr/hal-00567171

R. E. Ladner and N. A. Lynch, Relativization of questions about log space computability, Mathematical Systems Theory, vol.10, pp.19-32, 1976.

C. Lautemann, P. Tesson, and D. Thérien, An algebraic point of view on the Crane Beach property, 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL, vol.4207, pp.426-440, 2006.

Y. Li, A. A. Razborov, and B. Rossman, On the AC 0 complexity of subgraph isomorphism, 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pp.344-353, 2014.

O. B. Lupanov, A method of circuit synthesis, Z. Radiofizika, vol.1, pp.120-140, 1958.

O. B. Lupanov, On the complexity of the realization of the functions of an algebra of logic by formulas, Problemy Kibernet. Vyp, vol.3, pp.61-80, 1960.

A. Maciel, P. Péladeau, and D. Thérien, Programs over semigroups of dot-depth one, Theor. Comput. Sci, vol.245, issue.1, pp.278-287, 2000.

M. Mahajan, Polynomial size log depth circuits: Between NC 1 and AC 1, Bulletin of the EATCS, vol.91, pp.42-56, 2007.

W. J. Masek, A fast algorithm for the string editing problem and decision graph complexity, 1976.

P. Mckenzie and D. Thérien, Automata theory meets circuit complexity, Automata, Languages and Programming, 16th International Colloquium, vol.89, pp.589-602, 1989.

P. Mckenzie, P. Péladeau, and D. Thérien, NC 1 : The automata-theoretic viewpoint, Computational Complexity, vol.1, pp.330-359, 1991.

E. I. Ne?iporuk, On the complexity of schemes in some bases containing nontrivial elements with zero weights, Problemy Kibernetiki, vol.8, pp.123-160, 1962.

E. I. Ne?iporuk, On a Boolean function, Doklady of the Academy of the USSR, vol.169, issue.4, pp.999-1000, 1966.

C. Paperman, Circuits booléens, prédicats modulaires et langages réguliers, 2014.

W. J. Paul, A 2.5 n-lower bound on the combinational complexity of Boolean functions, SIAM J. Comput, vol.6, issue.3, pp.427-443, 1977.
URL : https://hal.archives-ouvertes.fr/hal-01518903

P. Péladeau, Classes de circuits booléens et variétés de monoïdes, 1990.

P. Péladeau, H. Straubing, and D. Thérien, Finite semigroup varieties defined by programs, Theor. Comput. Sci, vol.180, issue.1-2, pp.297-303, 1997.

S. Perifel, Complexité algorithmique. Ellipses, 2014.

J. Pin, The dot-depth hierarchy, 45 years later, The Role of Theory in Computer Science-Essays Dedicated to Janusz Brzozowski, pp.177-202, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01614357

J. Pin, Varieties Of Formal Languages, vol.0306422948, 1986.

J. Pin, Syntactic semigroups, Handbook of formal languages, pp.679-746, 1997.
URL : https://hal.archives-ouvertes.fr/hal-00143946

J. Pin, Mathematical Foundations of Automata Theory, 2016.

J. Pin and H. Straubing, Some results on C-varieties, vol.ITA, pp.239-262, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00112835

P. Pudlák, The hierarchy of Boolean circuits, Computers and artificial intelligence, vol.6, issue.5, pp.449-468, 1987.

A. A. Razborov, Lower bounds on the size of bounded depth circuits over a complete basis with logical addition, Mathematical Notes, vol.41, issue.4, pp.333-338, 1987.

A. A. Razborov, Lower bounds for deterministic and nondeterministic branching programs, Fundamentals of Computation Theory, 8th International Symposium, FCT'91, vol.529, pp.47-60, 1991.

J. Reiterman, The Birkhoff theorem for finite algebras. Algebra universalis, vol.14, pp.1-10, 1982.

J. E. Savage, Computational work and time on finite machines, J. ACM, vol.19, issue.4, pp.660-674, 1972.

J. E. Savage, The Complexity of Computing, 1976.

M. Schützenberger, On finite monoids having only trivial subgroups, Information and Control, vol.8, issue.2, pp.90108-90115, 1965.

M. Schützenberger, Sur le produit de concaténation non ambigu, Semigroup Forum, vol.13, pp.47-75, 1976.

I. Simon, Piecewise testable events, Automata Theory and Formal Languages, 2nd GI Conference, vol.33, pp.214-222, 1975.

R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp.77-82, 1987.

,

H. Straubing, Families of recognizable sets corresponding to certain varieties of finite monoids, Journal of Pure and Applied Algebra, vol.15, issue.3, pp.305-318, 1979.

H. Straubing, Finite semigroup varieties of the form V * D, Journal of Pure and Applied Algebra, vol.36, pp.53-94, 1985.

H. Straubing, When can one finite monoid simulate another?, Algorithmic Problems in Groups and Semigroups, pp.267-288, 2000.

H. Straubing, Languages defined with modular counting quantifiers, Inf. Comput, vol.166, issue.2, pp.112-132, 2001.
DOI : 10.1007/bfb0028572

H. Straubing, On logical descriptions of regular languages, Theoretical Informatics, 5th Latin American Symposium, vol.2286, pp.528-538, 2002.
DOI : 10.1007/3-540-45995-2_46

P. Tesson, Computational Complexity Questions Related to Finite Monoids and Semigroups, 2003.

P. Tesson and D. Thérien, The computing power of programs over finite monoids, J. Autom. Lang. Comb, vol.7, issue.2, pp.247-258, 2001.

P. Tesson and D. Thérien, Diamonds are forever: the variety DA. Semigroups, algorithms, automata and languages, vol.1, pp.475-500, 2002.

D. Thérien, Programs over aperiodic monoids, Theor. Comput. Sci, vol.64, issue.3, pp.271-280, 1989.

B. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, Journal of Pure and Applied Algebra, vol.48, issue.1-2, pp.83-198, 1987.

H. Vollmer, Introduction to Circuit Complexity-A Uniform Approach, Texts in Theoretical Computer Science. An EATCS Series, 1999.

I. Wegener, The complexity of Boolean functions, 1987.

I. Wegener, Branching Programs and Binary Decision Diagrams. SIAM, 2000.

A. Wigderson, NP and mathematics-a computational complexity perspective, Proc. of the 2006 International Congress of Mathematicians, 2006.
DOI : 10.4171/022-1/25

R. Williams, Nonuniform ACC circuit lower bounds, J. ACM, vol.61, issue.1, 2014.
DOI : 10.1145/2559903
URL : http://www.cs.cmu.edu/%7Eryanw/acc-lbs.pdf