On Uniformity Within NC David A Mix Barrington Neil Immerman HowardStraubing University of Massachusetts University of Massachusetts Boston Col lege Journal of Computer and System Science Abstract In order to study circuit complexity classes within NC in a uniform setting we need a uniformity condition which is more restrictive than those in common use Twosuch conditions stricter than NC uniformity RuCo have app eared in recent research Immermans families of circuits dened by rstorder formulas ImaImb and a unifor mity corresp onding to Buss deterministic logtime reductions Bu We show that these two notions are equivalent leading to a natural notion of uniformity for lowlevel circuit complexity classes Weshow that recent results on the structure of NC Ba still hold true in this very uniform setting Finallyweinvestigate a parallel notion of uniformity still more restrictive based on the regular languages Here we givecharacterizations of sub classes of the regular languages based on their logical expressibility extending recentwork of Straubing Therien and Thomas STT A preliminary version of this work app eared as BIS Intro duction Circuit Complexity Computer scientists have long tried to classify problems dened as Bo olean predicates or functions by the size or depth of Bo olean circuits needed to solve them This eort has Former name David A Barrington Supp orted by NSF grant CCR Mailing address Dept of Computer and Information Science U of Mass Amherst MA USA Supp orted by NSF grants DCR and CCR Mailing address Dept of Computer and Information Science U of Mass Amherst MA USA Supp orted by NSF grant CCR Mailing address Dept of Computer Science Boston College Chestnut Hill MA USA develop ed into the eld of circuit complexity theory where classes of problems are dened in terms of constraints up on circuits solving them This study has b ecome more imp ortant recently b ecause of the connections b etween size and depth of Bo olean circuits and number of pro cessors and running time on a parallel computer see Co ok Co for a general survey The complexityclass NC consists of those Bo olean functions functions from f g to f g which can b e computed by circuits of fanin two and depth O log n That is f is in NC if for each n there is a circuit C which computes f correctly on inputs of size n n and each C has depth at most c log n for some constant c This is nonuniform NC n we discuss uniformity b elow Problems in NC are usually considered particularly easy to solve in parallel and thus NC is considered a small complexity class for example it is the smallest of the ten surveyed by Co ok Co But it lies ab ove a certain frontier our current techniques for proving lower b ounds on circuit complexityhave not allowed us to proveany signicant problems to b e outside of it for example even any NP complete problems This motivates a study of sub classes of NC whichmight lie b elow this frontier in an eort to develop new techniques and new understanding There is a sub class of NC for which separation results are known AC is the class of problems whichhave circuits of p olynomial size and constant depth in a mo del with unb ounded fanin Furst Saxe and Sipser FSS and indep endently Ajtai Aj proved that the exclusive OR function is not in AC separating this class from NC Laterwork has attempted to extend the frontier upward from AC byproving lower b ounds for more powerful sub classes Razb orov Ra considered the extension of AC obtained by also allowing unb ounded fanin exclusive OR gates and showed that the ma jority function dened by f x x n i the ma jorityofthex are is not in this class Barrington Ba dened the class i AC C AC with counters which further extends Razb orovs class by allowing unb ounded fanin gates whichcount their inputs mo dulo some constant He conjectured that the ma jority problem was not in AC C and hence that AC C NC This remains op en though Smolensky Sm has proved some imp ortant partial results in this direction intro ducing what promises to b e a p owerful new pro of technique Existing techniques have b een unable to showeven an NP complete problem to b e outside of AC C Between AC C and NC is another class which has excited considerable interest A threshold gate counts its Bo olean inputs which are and compares the total with some predetermined numb er to determine its output This generalizes unb ounded fanin AND threshold indegree OR threshold and ma jority threshold half the indegree gates but any threshold gate can b e built out of these three basic typ es TC is the class of problems solvable by families of circuits of unb ounded fanin threshold gates where circuit depth is b ounded by a constant and circuit size by a p olynomial in the input size As individual threshold gates can b e simulated in NC TC NC also it is fairly easy to see that AC C TC Considerable recentwork has dealt with TC someofitmotivated by analogies with neural computing PSHMPSTRe Uniformity In their nonuniform versions these circuit complexity classes contain problems which are not computable at all in the ordinary sense eg any unary language is in AC To compute with a circuit family wemust b e able to construct the circuit for each input size Wemay lo osely dene a uniform circuit family as one in which the b ehavior on all inputs of any size is sp ecied by a single nite bit string A weak uniformity condition would b e to allow this string to b e the description of a Turing machine which on input n pro duces the circuit C Since we are concerned with complexity and not just computabilitya better n denition places resource restrictions on the Turing machine A circuit family hC C i is P uniform if the circuit C can b e constructed from n n in time p olynomial in n It is Luniform if C can b e constructed using space O log n n These denitions suce to prove the classical result that uniform circuits of p olynomial size are equivalent in computing p ower to Turing machines using p olynomial time In fact the same pro of shows the result either for P uniform or Luniform circuits showing these two classes equivalent to each other There is a sense in whichtheLuniform version of this result is more satisfying than the P uniform one In the latter case if webelieve P Lwe know that wehave isolated an imp ortant fact ab out circuits and machines That is the p olynomialsize circuits were able to simulate the p olynomialtime machines on their own without the p otential help of a p olynomialtime machine used to construct them As we study a given circuit complexityclasswewould like to use a uniformity condition which separates these two sources of computational p ower That is wewant to allow strictly less p ower to construct the circuits than the circuits themselves p osess In this pap er we explore a variety of conditions suitable for the study of the classes AC AC C TC and NC unifor There is a reasonable though complicated notion of NC uniformityU E mity due to Ruzzo Ru see also Co which has the consequence that NC uniform NC is equivalent to alternating logarithmic time mo dulo appropriate conventions on the alternating Turing machines This denition is based not on constructing the circuit but on answering certain classes of questions ab out it in alternating logarithmic time Most pro ofs involving P orLuniform circuit families go through under this denition making it a go o d to ol for studying complexity classes ab ove NC But to go within NC itself we will require new notions still more restrictive We note that one may still sp eak of say P uniform NC and that this may b e a class of considerable interest At least by analogy it represents problems for whicha very fast chip could b e manufactured by a sequential pro cess in reasonable time Many natural problems have b een shown to b e in P uniform NC and are not known to b e in more uniform versions BCHRe Recentwork by Allender Al shows P uniform NC to b e a fairly robust class with a numb er of equivalent denitions Summary of Results In this pap er we consider three candidates for a suitable notion of uniformity within the class NC Each is based on a sub class of NC the computational p ower used in sp ecifying the circuits in a family is limited to this sub class In Section wemake precise the denitions of the circuit classes already mentioned and the ways in which circuits are sp ecied The rst notion is based on Immermans theory of expressibility as a complexity measure ImaImb The basic complexity class in this scheme is the class FO of languages which can b e expressed by rstorder formulas in a certain formal system to b e explained in detail in Section First order formulas can b e evaluated by AC circuits of a particularly regular form so that FO is a uniform version of AC In Section we will examine classes dened by rstorder formulas which include new typ es of quantiers giving uniform versions of AC C TC andNC Whereas ordinary quantiers express whether an instance or all instances of the quantied variable are satis fying the new quantiers will p erform some other function on the sequence of truth values given by the sequence of instances For example we will dene quantiers which can count the satisfying instances mo dulo some constant determine whether the ma jority of the in stances are satisfying or even interpret the truth values as elements of some nite group and multiply them In fact we will dene quantiers for any function meeting a certain technical condition that of b eing monoidal The expressibilityscheme