Abstract
Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets 
satisfying a certain closure property, a one-to-one correspondence between \(FO[<,\ensuremath{\mathfrak{P}}]\)-uniform linear circuits, two-variable formulae with \(\ensuremath{\mathfrak{P}}\) predicates, and weak block products of monoids. In particular, we consider the case of linear TC0, majority quantifiers, and finitely typed monoids. This correspondence will hold for any numerical predicate set which is FO[ < ]-closed and whose predicates do not depend on the input length.
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References
Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1. J. Comp. System Sci. 38, 150–164 (1989)
Barrington, D.A., Immerman, N., Straubing, H.: On uniformity within NC 1. J. Comp. System Sci. 41, 274–306 (1990)
Barrington, D., Immerman, N., Lautemann, C., Schweickardt, N., Thérien, D.: The Crane Beach Conjecture. In: Proc. of the 16th IEEE Symposium On Logic in Computer Science, pp. 187–196. IEEE Computer Society Press, Los Alamitos (2001)
Barrington, D., Thérien, D.: Finite Monoids and the Fine Structure of NC 1. Journal of ACM 35(4), 941–952 (1988)
Behle, C., Krebs, A., Reifferscheid, S.: A 5 not in FO+MOD+MAJ2[reg], http://www-fs.informatik.uni-tuebingen.de/publi/a5notinltc0.pdf (to appear)
Behle, C., Lange, K.-J.: FO[ < ]-Uniformity. In: IEEE Conference on Compuatational Complexity (2006)
Furst, M., Saxe, J.B., Sipser, M.: Parity circuits and the polynomial-time hierarchy. In: Proc. 22th IEEE Symposium on Foundations of Computer Science, pp. 260–270 (1981)
Krebs, A., Lange, K.-J., Reifferscheid, St.: In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, Springer, Heidelberg (2005)
Koucký, M., Lautemann, C., Poloczek, S., Thérien, D.: Circuit lower bounds via Ehrenfeucht-Frai̇ssé games. In: Proc. 21st Conf. on Compuatational Complexity (CCC’06) (2006)
Lange, K.-J.: Some results on majority quantifiers over words. In: Proc. of the 19th IEEE Conference on Computational Complexity, pp. 123–129. IEEE Computer Society Press, Los Alamitos (2004)
Lautemann, C., McKenzie, P., Schwentick, T., Vollmer, H.: The descriptive complexity approach to LOGCFL. J. Comp. System Sci. 62, 629–652 (2001)
Lawson, M.: Finite Automata. Chapman & Hall/CRC (2004)
Rhodes, J., Tilson, B.: The Kernel of Monoid Morphisms. J. Pure Applied Alg. 62, 227–268 (1989)
Roy, A., Straubing, H.: Definability of Languages by Generalized First-Order Formulas over (N,+). In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, Springer, Heidelberg (to appear 2006)
Ruhl, M.: Counting and addition cannot express deterministic transitive closure. In: Proc. of 14th IEEE Symposium On Logic in Computer Science, pp. 326–334. IEEE Computer Society Press, Los Alamitos (1999)
Schweikardt, N.: On the Expressive Power of First-Order Logic with Built-In Predicates. In: Dissertation, Universität Mainz (2001)
Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser (1994)
Straubing, H., Thérien, D., Thomas, W.: Regular languages defined by generalize quantifiers. Information and Computation 118, 289–301 (1995)
Straubing, H., Thérien, D.: Regular Languages Defined by Generalized First-Order Formulas with a Bounded Number of Bound Variables. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 551–562. Springer, Heidelberg (2001)
Straubing, H., Thérien, D.: Weakly Iterated Block Products of Finite Monoids. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 91–104. Springer, Heidelberg (2002)
Thérien, D., Wilke, T.: Over Words, Two Variables are as Powerful as One Quantifier Alternation. In: Proc. 30th ACM Symposium on the Theory of Computing, pp. 256–263 (1998)
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Behle, C., Krebs, A., Mercer, M. (2007). Linear Circuits, Two-Variable Logic and Weakly Blocked Monoids. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_15
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DOI: https://doi.org/10.1007/978-3-540-74456-6_15
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