Barrington, David A. Mix; Immerman, Neil; Straubing, Howard On uniformity within \(NC^ 1\). (English) Zbl 0719.68023 J. Comput. Syst. Sci. 41, No. 3, 274-306 (1990). The class \(NC^ 1\) of languages recognized by log-depth polynomial-size circuits is considered. The main goal is to study circuit complexity classes within \(NC^ 1\) in a uniform setting. It is shown that families of circuits defined by first-order formulas and a uniformity corresponding to deterministic log-time reductions are equivalent. This leads to a natural notion of uniformity for low-level circuit complexity classes. In this connection the expressive power of the first-order logic with and without the BIT pedicate is explored. New quantifiers (modular, counting, majority and group quantifiers) are also introduced to express languages in larger complexity classes. Recent results on the characterization of \(NC^ 1\) in terms of constant-width branching programs are shown to still hold true in uniform setting. Reviewer: S.P.Yukna (Vilnius) Cited in 3 ReviewsCited in 148 Documents MSC: 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010) 03D20 Recursive functions and relations, subrecursive hierarchies Keywords:uniform complexity; regular languages; low-level circuit complexity classes × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Ajtai, M., \(Σ_1^i\) formulae on finite structures, Ann. Pure Appl. Logic, 24, 1-48 (1983) · Zbl 0519.03021 [2] Allender, E., \(P\)-uniform circuit complexity, J. Assoc. Comput. Mach., 36, No. 4, 912-928 (1989) · Zbl 0697.68031 [3] Barrington, D. A., Bounded-Width Branching Programs, (Ph.D. thesis, M.I.T. Dept. of Mathematics, Technical Report TR-361 (May 1986), M.I.T. Laboratory for Computer Science) · Zbl 0837.68056 [4] Barrington, D. A., Bounded-width polynomial-size branching recognize exactly those languages in \(NC^1\), J. 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