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Time, Hardware, and Uniformity This is page Printer Opaque this Time Hardware and Uniformity David Mix Barrington Neil Immerman ABSTRACT We describ e three orthogonal complexity measures parallel time amount of hardware and degree of nonuniformity which together parametrize most complexity classes Weshow that the descriptive com plexity framework neatly captures these measures using three parameters quantier depth number of variables and typ e of numeric predicates re sp ectively A fairly simple picture arises in which the basic questions in complexity theory solved and unsolved can b e understo o d as ques tions ab out tradeos among these three dimensions Intro duction An initial presentation of complexity theory usually makes the implicit as sumption that problems and hence complexity classes are linearly ordered by diculty In the Chomsky Hierarchyeach new typ e of automaton can decide more languages and the Time Hierarchy Theorem tells us that adding more time allowsaTuring machine to decide more languages In deed the word complexity is often used eg in the study of algorithms to mean worstcase Turing machine running time under which problems are linearly ordered Those of us who study structural complexityknow that the situation is actually more complicated For one thing if wewant to mo del parallel com putation we need to distinguish b etween algorithms whichtakethesame amountofwork ie sequential time wecarehowmany pro cesses are op erating in parallel and howmuch parallel time is taken These two dimen sions of complexity are identiable in all the usual mo dels b o olean circuits width and depth PRAMs or other explicit parallel machines number of pro cessors and parallel time alternating Turing machines space and alternations or even deterministic Turing machines space and reversals Hongs b o ok H gives an interesting general treatment of similarity b e tween mo dels and duality b etween these two complexity measures Figure gives a twodimensional layout of some wellknown and less wellknown complexity classes To b e precise ab out our axes wehave chosen unb ounded fanin circuit depth as our measure of parallel time and circuit width as our measure of numb er of pro cesses This assumes that the circuits have b een arranged into levels and that eachedgeinto David Mix Barrington Neil Immerman O n PH PSPACE EXPTIME O log n q AC q NC q P O n AC AC NC P PSPACE O log n SC q NC PSPACE O log n LOGSPACE q NC PSPACE O NC q NC PSPACE O O O O log n n O O log n log n n width d e p t h Figure Two Dimensions of Complexity Classes a gate comes from either an input or a gate on the immediately previous level Thus the depth is the numb er of levels and the width is dened as the size of the largest level The columns of our chart represent b ounds on depth and the rows b ounds on width Blanks indicate classes where b oth are b ounded b elow p olynomiallyandthus the resulting circuits cannot access the entire input The named classes are fairly standard except for the use of a prex q to indicate a change from a p olynomial to a quasip olynomial size b ound B Thus q NC is the class of languages decidable by circuit families of p olylog depth and quasip olynomial size it is a robust class which o ccurs several times on the chart Finally question marks denote classes whichhaveno distinctive names known to the authors and ab out whichweknow nothing other than the obvious containment relations with their neighbors Of course merely sp ecifying combinatorial b ounds on the circuits in a family do es not fully sp ecify a complexity class For example any unary language even an uncomputable one has a circuit family of O size and depth which decides it In the circuit context we usually sp eak of restricting circuit families byauniformity condition wesay that the circuit must b e computable or that questions ab out it must b e answerable by resourceb ounded computation It is equally sensible to sp eak of non uniformity as a resource more of whichallows a circuit family to decide more languages This resource forms the third axis of our parametrization of complexity classes It exists in other mo dels as well advice given to Turing machines KL or precomputation in parallel machines A Ateach p oint on our twodimensional chart wehave a range of com plexity classes obtained byvarying the uniformity condition For example if b oth size and depth are p olynomially b ounded the chart indicates the class P of languages decided by p olynomialtime Turing machines This claim is true if the circuits are Puniform computable by a p olytime Turing machine or if they are DLOGTIME uniform direct connection language decidable by a randomaccess Turing machine in time O log n see BIS or any uniformity condition in b etween Howeverifweallow ourselves more than p olynomial time to compute the circuit wemaybe Time Hardware and Uniformity able to decide more languages If we allow ourselves enough extra time we can denitely do so For example if we allow more than exp onential time we can decide the unary version of the universal language for Turing machines with some sup erp olynomial time b ound On the other hand one can imagine uniformity conditions so restrictive that a general simulation of a Turing machine is imp ossible In general as wepassupward adding more nonuniformity along the third axis we pass through three regions one with to o little nonuniformity where the basic constructions relating the circuit mo del to other mo d els cannot b e carried out a second robust region where a wide range of denitions give the same class and a third region where additional non uniformity gives steadily larger classes The distinction can b e quite imp or tant as we see in the case of the class NC Asshown in BIS we can dene avery restrictive uniformity notion under whichNC b ecomes the class of regular languages If our nonuniformity resource is b etween DLOGTIME and NC itself we get a robust class equal to ALOGTIME And if weallow p olynomial time to build our circuits we can then do integer division and related problems BCH which as far as weknow we couldnt do b efore This may b e an example of where nonuniformity can replace one of the other two resources In some cases we know of limits on the p otential p ower of nonuniformity to do so at least sub ject to complexitytheoretic assump tions For example Karp and Lipton KL haveshown that no amountof nonuniformity can allowPtosimulate all of uniform NP unless the p olynomial hierarchy collapses to the second level It would b e interesting to have a parallel result for P and NC and recentwork of Ogihara Cai and Sivakumar O CS has made progress toward this One would liketo derive unlikely complexitytheoretic consequences from for example the hyp othesis that nonuniform NC contains uniform P or equivalently that there is a sparse set complete for P under NC Turing reductions Cai and Sivakumar show that if there is a sparse set complete for P under logspace uniform NC manyone reductions then P is equal to logspace uniform NC Dvan Melkeb eek vM has subsequently shown a similar result for sparse sets complete under truthtable reductions In general our techniques for proving lower b ounds on circuit complexity are combinatorial and algebraic and apply to the total ly nonuniform ver sions of the circuit classes A notable exception is the result by Allender and Gore AG that the integer p ermanent function is not in DPOLYLOGTIME uniform q ACC though for all weknowitmightbeinLOGSPACEuniform ACC There is a certain amountofoversimplication in thinking of eachof our three parameters as a single axis For one thing our nonuniformity resource is dened in terms of the complexity of languages which talk ab out the circuits so this dimension may b e as nonlinear as the whole picture However the uniformity conditions we normally consider happ en to b e linearly ordered More imp ortantly the parallel time axis is dened David Mix Barrington Neil Immerman in terms of particular primitive op erations on the data In the circuit mo del a single gate computes an AND or OR in a parallel machine the most powerful op erations are the concurrent read and concurrent write and the alternations of a Turing machine are also dened in terms of AND and OR But there is nothing sacred ab out AND and OR in each mo del wecan consider other op erations such as MAJORITY more p owerful than AND and OR or mo dular counting orthogonal to them In the circuit mo del these op erations are emb o died as new kinds of gates in Turing machines as new acceptance conditions as in the classes P or PP and in parallel machines as new global op erations such as the scan op eration on the Connection Machine Here we consider these three dimensions and the variety of op erations in the framework of descriptive complexity where we measure the complexity of a language by the syntactic resources in a particular logical formalism needed to express the prop ertyofmembershipinitIIIb We review this framework in Section b elow It has b een known for some time that two parameters in descriptive complexitynumber of variables and quantier depth corresp ond exactly to space and parallel time in either the circuit or PRAM mo dels Ib More recently along with Straubing BIS B wehave extended the framework to deal with the third dimension and with more general op erations in the context of rstorder formulas or constantdepth p olysize circuits There varying uniformity conditions corresp ond to new atomic predicates and any asso ciative op eration with an identity can b e mo deled by a new typ
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