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Rudimentary Reductions Revisited 1 Introduction Rudimentary reductions revisited y Vivek Gore Eric Allender Department of Computer Science Department of Computer Science Rutgers University Rutgers University New Brunswick, NJ 08903 New Brunswick, NJ 08903 April 23, 1997 Abstract: We show that log-b ounded rudimentary reductions de ned and studied by Jones in 0 1975 characterize Dlogtime-uniform AC . 1 Intro duction In the rst part of the oft-cited pap er [Jon75], Jones intro duced logspace reductions as a to ol for study- ing the relative complexity of problems in P. In the second part of [Jon75], Jones intro duced a re- stricted version of logspace reducibility, called log-b ounded rudimentary reductions. The motivation for intro ducing this restriction came from 1 the desire to have a to ol for talking ab out the structure of very small complexity classes such as DSPACElog n, and 2 interest in generalizing the notion of \rudimentary relations," which at that time was the ob ject of a considerable amount of attention [Smu61, Sal73,Wra78, Wil79, PD80]. In [Jon75], Jones went on to showthatanumb er of problems were complete for various complexity classes under log-b ounded rudimentary reductions. Although logspace reducibility has proved to b e a very useful notion in complexity theory in the intervening years, log-b ounded rudimentary reducibility has b een mentioned explicitly only seldom in subsequentwork { although, as we demonstrate, it has b een considered implicitly many times, under di erent names. A great deal of insight ab out the complexityofvarious problems has b een gained by the study of uniform circuit complexity; many complexity classes have b een characterized in terms of the compu- tational p ower of families of circuits fC : n 2 Ng where C takes inputs of length n, byvarying n n the typ es of gates and allowable fan-in on the circuits, and imp osing a \uniformity condition" which sp eci es that the de nition of C in terms of n b e \easy" in some sense. Unfortunately, the question of n Supp orted in part by NSF grant CCR-9000045. y Supp orted in part by a DIMACS research assistantship. DIMACS is a co op erative pro ject of Rutgers University, Princeton University,AT&T Bell Lab oratories and Bellcore. DIMACS is an NSF Science and Technology Center, funded under contract STC-88-09648; and also receives supp ort from the New Jersey Commission on Science and Technology. 1 which uniformity condition to use has remained somewhat controversial. For example, although much work on uniform circuit complexity uses logspace uniformity, it is not clear that this is appropriate when 0 de ning sub classes of DSPACElog n, suchasAC , the class of languages accepted by p olynomial-size, constant-depth circuits of unb ounded fan-in AND and OR gates. 0 Comp eting de nitions of \uniform AC "were prop osed by Immerman [Imm87, Imm89] and Buss [Bus87]. Then it was shown by Barrington, Immerman, and Straubing, that these de nitions in fact 0 coincide [BIS90]; they show that Dlogtime-uniform AC may b e de ned alternatively in terms of rst- order logic, O 1 time on a CRAM, inductive de nitions with O 1 inductive depth, and Sipser's log-time 0 hierarchy [Sip83]. Other characterizations of Dlogtime-uniform AC are found in [Clo90]. Since some of these notions were de ned entirely indep endently of each other and indep endently of considerations of uniform circuit complexity, the fact that these notions coincide is taken as evidence for the \correctness" 0 of the Dlogtime-uniformity condition when cho osing a de nition of uniform AC . In this pap er, we add to this b o dy of evidence by showing that Jones' logspace rudimentary reductions 0 provide yet another characterization of Dlogtime-uniform AC . Throughout the rest of this pap er, 0 0 all references to AC will denote Dlogtime-uniformAC . 2 Preliminaries De nition1 The following de nitions are due to Jones [Jon75]. th Let b e an alphab et. The ternary predicate x; i; a is true i a is the i symbol of x where x 2 . For a p ositiveinteger c and a space b ound S , de ne the predicates P , P as follows: c;S c;S P x; u; v ; w , uv 6= w ^juv w jc S jxj. P x; u; v ; w , uv = w ^juv w j c S jxj, c;S c;S The class of S -b ounded rudimentary predicates, RUD is the class of predicates built from S P for every c via logical connectives, explicit transformations, and S -b ounded ; :;P , and c;S c;S existential and universal quanti cation. See [Jon75] for a complete de nition. De nition2 A function f : ! is S -b ounded rudimentary i the predicate R de ned by Rx; i; a , f x;i;aisinRUD . S th That is, f is S -b ounded rudimentary i the predicate \a is the i symbol of f x" is in RUD . The S cS jxj fact that jij c S jxj for some c and all x implies that jf xj2 . Note that, as de ned ab ove, RUD is a set of predicates;however we can just as easily view RUD as S S a set of languages. A language L will b e said to b e in RUD i the characteristic function of L is S - S b ounded rudimentary. Alternatively, each predicate Q 2 RUD over alphab et de nes a language L S 2 over alphab et [f; g where \," is any symb ol not in de ned by L = fx; y ;:::;y : Qx; y ;:::;y g. 1 m 1 m These alternative de nitions are easily seen to b e equivalent. For more formal arguments along these lines, see [Wra78]. In this pap er as in [Jon75] we are interested primarily in RUD , the class obtained when the space log 0 b ound S is logarithmic. In Section 3, we show that RUD =AC . log The pro ofs in this pap er will not make use of the circuit formalism;thus detailed de nitions of the Dlogtime-uniformity conditions are omitted here. They may b e found in [BIS90, BCGR90]. Instead, 0 we will use twocharacterizations of AC that were shown to b e equivalent in [BIS90] see also [BCGR90], one in terms of alternating Turing machines, and one in terms of rst-order logic. We use the mo del of alternating Turing machine used by Ruzzo [Ruz81] in which access to the input is provided via a sp ecial tap e whichwe call the input address tape onto which an address i maybe th written in binary, following which in unit time the i input symbol is available. This convention is necessary to allow machines with sublinear running times to have access to all of the input. Consult [BIS90, BCGR90] for examples illustrating how these machines compute. Now let timelog n denote the class of languages accepted by alternating Turing machines running k in O log n time, b eginning in an existential con guration, and making at most k 1 alternations b etween existential and universal con gurations on any computation path. Let LH the logtime hierarchyof S [Sip83] denote timelog n. k k Immerman [Imm87, Imm89] has studied the expressivepower of rst-order logic in de ning languages; he calls the resulting class FO. Due to space limitations, we refer the reader to [Imm87, Imm89] for the relevant de nitions. 0 Theorem 3 [BIS90] AC =LH=FO. Two minor complications arise from the statement of Theorem 3: 0 1. Alternating Turing machines can accept input over any alphab et , whereas FO and AC are classes of languages over the alphab et f0; 1g.Thus we simply assume a binary enco ding of any other alphab et. Alternatively, the FO and circuit frameworks can b e mo di ed to allow arbitrary alphab ets. 0 2. It is clear what it means for a function to b e computed byanAC circuit, but it is less clear how a function can b e sp eci ed in the LH or FO settings. To resolve this diculty,we use a standard 1 technique that has b een used since at least [Jon75]. For a function f with p olynomial growth rate , def th let A c; i; z = the i symbol of f z isc. Then wesay that f is in LH or FOifA is in LH f f 1 That is, there is some p olynomial p such that for all x; jf xj pjxj. 3 0 or FO. It has b een noticed by many authors e.g., [BIS90] that f is computed byAC circuits 0 2 i A 2 AC . f 0 0 Note that the comp osition of two functions in AC is also a function in AC . 0 3 Equivalence of AC and RUD log 0 Theorem 4 AC =RUD . log Pro of. This containment is mentioned in Prop osition 5 of [Vol84] where the converse containmentis left as an op en question. We present a complete pro of here. The pro of pro ceeds by induction on the de nition of RUD , showing that each language in RUD log log is in LH. The input to an ATM is assumed to have b oth ends marked by the sp ecial symbol $. Consider the inductive de nition of RUD : log 1. x; i; a and : x; i; a are in RUD . log Let L = fx; i; a : x; i; ag. Consider an alternating TM M for L: On input $x; i; a$, M existentially guesses the p ositions of all the commas there are only a constantnumber of them.
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