Rudimentary Reductions Revisited 1 Introduction

Rudimentary reductions revisited

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

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

Supp orted in part by NSF grant CCR-9000045.

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

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.

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

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

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"

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

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].

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

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 .

That is, f is S -b ounded rudimentary i the predicate \a is the i symbol of f x" is in RUD . The

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 -

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

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,

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

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

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

[Sip83] denote timelog n.

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.

Theorem 3 [BIS90] AC =LH=FO.

Two minor complications arise from the statement of Theorem 3:

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.

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

technique that has b een used since at least [Jon75]. For a function f with p olynomial growth rate ,

def

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

That is, there is some p olynomial p such that for all x; jf xj pjxj. 3

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 .

0 0

Note that the comp osition of two functions in AC is also a function in AC .

3 Equivalence of AC and RUD

log

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.

M universally veri es that that the guessed p ositions contain the appropriate markers and

the other p ositions contain 0's or 1's.

Using the marker p ositions, M calculates jxj and jij.Itwas observed in [Bus87] that this can

b e done in logarithmic time. M calculates logjxj from jxj and veri es that jijlogjxj.

M existentially guesses the bits of i and writes them on a worktap e, and then universally

checks that for each j i, the guessed value for p osition jxj +1+ j is correct.

M again uses marker p ositions to read a, and check that the i bit of x is a.

2. If c is a p ositiveinteger then b oth P x; u; v ; w and P x; u; v ; w are in RUD where

c;log c;log log

P x; u; v ; w , uv = w ^juv w j c logjxj,

c;log

P x; u; v ; w , uv 6= w ^juv w j c logjxj.

c;log

Again, for equivalence to hold, the output of the circuit must b e interpreted in sucha way as to allow an \endmarker,"

or else all functions f in AC must have jf xj = jf y j if jxj = jy j, which is clearly not the case for some functions in LH

and FO. 4

Consider an ATM M that b ehaves as follows:

On input $x; u; v ; w $,

M rst guesses and veri es the p ositions of all the markers as in the previous case, and uses

this to compute c log jxj.

Using the marker p ositions, determine whether juv w j c logjxj, and if so, read eachof

u; v ; w from the input, as in the previous case, and verify whether uv = w or uv 6= w for

P .

c;log

3. If Qx; y ;:::;y and Rx; y ;:::;y are in RUD , then so are

1 m 1 m log

Qx; y ;:::;y ^ Rx; y ;:::;y and Qx; y ;:::;y _ Rx; y ;:::;y .

1 m 1 m 1 m 1 m

This case is trivial, using universal and existential branching, resp ectively.

4. If Qx; z ;:::;z isinRUD and eachof ;:::; is either a string in or one of the variables

1 m log 1 m

y ;:::;y , then the predicate R de ned by

1 r

Rx; y ;:::;y , Qx; ;:::; isinRUD .

1 r 1 m log

Inductively, Q can b e accepted by an alternating TM M in Timelog n. Note that r and m

are constants here and ;:::; dep end only on y ;:::;y and not on x. Therefore, the conversion

1 m 1 r

of y ;:::;y into ;:::; only requires a nite amount of information that can b e supplied to

1 r 1 m

an ATM in its nite control.

Consider a function f that converts strings of the form x; y ;:::;y into x; ;:::; .Ifwe can

1 r 1 m

show that f can b e computed in LH, then we can use the fact that LH functions are closed under

comp osition to conclude the pro of for this case, using the inductivehyp othesis. That is, we need

to show that the language A = fciz : the i symbol of f z iscg is in LH. Note that, for

well-formed inputs, z will b e of the form x; y ;:::;y .

1 r

Consider an ATM M that b ehaves as follows:

On input $ciz $

M guesses all the marker p ositions in $ciz $aswell as in z itself there are a constant

numb er of these and veri es them as in the previous cases.

M uses the marker p ositions to compute jxj and jij, and rejects if jij is to o long. Otherwise,

as in the previous cases, M guesses the contents of the c, i, and y ;:::;y elds of the input,

1 r

and checks that the guesses are correct. 5

If i jxj, accept i the i bit of x is c. Otherwise, compute ;:::; and accept i c is in

1 m

p osition i jxj of ;:::; . It is easy to verify that there is ample time to p erform these

1 m

op erations.

5. If c is a p ositiveinteger, and Qx; y ;:::;y isinRUD , then the predicates R and R de ned

1 m+1 log

as follows are also in RUD

log

Rx; y ;:::;y , 9z [jz jc logjxj ^ Qx; y ;:::;y ;z]

1 m 1 m

R x; y ;:::;y , 8z [jz j c logjxj Qx; y ;:::;y ;z]

1 m 1 m

0 0

Inductively, Q is accepted by some LH machine M .We construct a machine M accepting R .On

input $x; y ;:::;y $;M starts o by getting the marker p ositions as in the previous cases. Then

1 m

it universally veri es that for all z with jz jc logjxj, M accepts $x; y ;:::;y ;z$. To see that

1 m

this is p ossible, note that M consults its input only at the end of a computation branch. As so on

0 th

as M go es into input mo de and has j written on its index tap e, M either consults the j symbol

of its input, or reads the symb ol in p osition j jx; y ;:::;y j of z .

1 m

A similar argument, using existential branching, shows that R is in LH.

To prove this inclusion, let L 2 FO. Therefore, there exists a sentence in Prenex Normal Form

=Q j Q j :::Q j j ;:::;j

1 1 2 2 m m 1 m

such that 8x

x 2 L , x j=Q j Q j :::Q j j ;:::;j

1 1 2 2 m m 1 m

where each Q is a quanti er 9 or 8.

We will show that for each such and its asso ciated there is a RUD predicate R such that

log

for each string x,

x j=Q j Q j :::Q j j ;:::;j

1 1 2 2 m m 1 m

if and only if

0 0 0

Q y Q y :::Q y R x; y ;:::;y

1 2 m 1 m

1 2 m

where each Q ranges over all binary strings of length at most d log jxje+ 1 and is the same typ e existential

or universal as Q . By the closure prop erties of RUD , and by the fact that the length of each y

i log i

is b ounded, it follows that since R is in RUD , L is also in RUD , and the pro of is complete. It

log log

remains only to construct R .

The formula is built up from the primitive predicates =;<, X , and BI T , and from the constants

1 and n. Clearly, it will suce to show that each of these primitive predicates is \equivalent" in the 6

same sense to a relation in RUD .For example, we will show that there is a relation S in RUD such

log log

that, for anynumb ers j and j , the primitive predicate j

1 2 1 2 1 2

where y and y are binary strings of length at most d log jxje + 1 representing the numb ers j and j ,

1 2 1 2

resp ectively.

The constant 1 is explicitly de nable in RUD , and Jones [Jon75] has already observed that the

log

relations u + v = w; u v = w; u = w and juj = w are all in RUD , so long as u; v ; w are constrained

log

to have length c logjxj for some c.Thus the constant n is de nable, since the relation jxj = j is

equivalentto x; j; 0 _ x; j; 1 ^: x; j +1; 0 _ x; j +1; 1. It follows easily that the following

set is in RUD : fx; y ;:::;y : for each i, jy j = d log jxje +1g. Therefore for the rest of this pro of,

log 1 m i

we will assume that all variables y refer to strings of exactly this length. This will result in a few

simpli cations.

Nowwe consider each of the primitive predicates in turn.

j = j $ 8i[ y ;i;0 , y ;i;0]

1 2 1 2

1 2 1;log 1 1;log 2

This says that the binary representation of j is uv and the binary representation j is uw such

1 2

that the most signi cant bits of v and w are 0 and 1 resp ectively. Note that the values taken by

each of the variables u; v ; w have lengths logjxj.

X j $ x; y ; 1.

1 1

BI T j ;j $ y ;y ; 1.

1 2 2 1

This completes the pro of.

We remark that standard translational techniques maynow b e used to prove the following general-

ization of Theorem 4. Note that for the case of S n=n, this is the well-known theorem of Wrathall

[Wra78].

Corollary 5 For functions S such that the binary representation of S n can b e computed from n in

time O S n, RUD = timeS n.

S k

4 Reducibilit ies

Jones de ned RUD as a rst step towards de ning a very restrictive notion of reducibility. Jones to ok

log

the familiar notion of a many-one reduction namely, f reduces A to B if x 2 A , f x 2 B , and

imp osed the restriction that f b e log-b ounded rudimentary. There was one additional restriction that

rud

Jones imp osed in de ning his logspace-rudimentary reducibility denoted ; he required that there

log 7

b e some constant c such that for all x, log jf xj = c logjxj. That is, the length of f x should b e

equal to a p olynomial in jxj, and not just b ounded by one. In [Jon75], Jones comments that this nal

restriction is stronger than the more natural restriction of simply having f have p olynomial growth rate,

rud

but explains that it \seems to b e necessary" in order to have the relation b e transitive i.e., in order

log

for the comp osition of two log-b ounded rudimentary functions to b e log-b ounded rudimentary. However

by Theorem 4 we see that for functions f with p olynomial growth rate, f is log-b ounded rudimentary

i f is in LH. Since the comp osition of two functions in LH is easily seen to b e in LH, it follows that

Jones' rationale for imp osing the stronger length requirementwas unfounded.

5 Conclusion

Wehave considered the log-b ounded rudimentary predicates de ned by Jones in [Jon75] and wehave

shown that they corresp ond exactly to the class of sets accepted by Dlogtime-uniform AC circuits.

Thus Jones was probably the rst to study this complexity class, which has lo omed large in imp ortance

in recentyears. We b elieve that this augments the already comp elling arguments of [BIS90] in favor

of using the Dlogtime-uniformity condition when studying small circuit complexity classes.

It is instructive to note some of the op en questions p osed by Jones in [Jon75]. He mentions on more

than one o ccasion the question of whether or not RUD = DSPACES n, and he explicitly considers

the p ossibility that RUD = NSPACElog n [Jon75], Theorem 23. However, the results of [Ajt83]

log

and [FSS84] refute this p ossibilityby showing that PARITY is in NSPACElog n AC . The strongest

results along this line are those of [Has87] and [Yao85], showing that for any function S such that for all

, S n=on , the PARITY language is in DSPACE1 RUD . Here, DSPACE1 denotes the class

of regular sets. However, any generalization of this result for larger functions S will require entirely

di erent techniques and will havetotake uniformityinto account in some way, since the techniques

of [Nep70] see also [Vol84] show that for each >0, NSPACElog n RUD , and it follows from

[Wra78] that RUD contains complete sets for each level of the p olynomial hierarchy.Thus the question

of whether or not RUD = DSPACEn remains an interesting op en problem. Note, however, that

an armative answer would imply that the p olynomial hierarchy is equal to PSPACE.

References

[Ajt83] M. Ajtai. -formulae on nite structures. Annals of Pure and AppliedLogic 24 1983

1{48.

[BIS90] D. A. Mix Barrington, N. Immerman, and H. Straubing. On uniformity within NC . Journal

of Computer and System Sciences 41 1990 274{306.

[Bus87] S. R. Buss. The Bo olean formula value problem is in ALOGTIME. In Proc. 19th ACM

Symposium on Theory of Computing, 1987, pp. 123{131. 8

[BCGR90] S. Buss, S. Co ok, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for

formula evaluation. Submitted, 1990.

[Clo90] P. Clote. Bounded arithmetic and computational complexity. In Proc. 5th Structurein

Complexity Theory Conference, 1990, pp. 186{199.

[FSS84] M. Furst, J. Saxe, and M. Sipser. Parity, circuits and the p olynomial time hierarchy. Math-

ematical Systems Theory 17 1984 13{27.

[Has87] J. Hastad. Computational limitations for small depth circuits . MIT Press, Cambridge, 1987.

[Imm87] N. Immerman. Expressibility as a complexity measure: Results and directions. In Proc. 2nd

Structure in Complexity Theory Conference, 1987, pp. 194{202.

[Imm89] N. Immerman. Expressibility and parallel complexity. SIAM Journal on Computing 18 1989

625{638.

[Jon75] N. D. Jones. Space-b ounded reducibility among combinatorial problems. Journal of Com-

puter and System Sciences 11 1975 68{85. Corrigendum: Journal of Computer and System

Sciences 15 1977 241.

[Nep70] V.A. Nep omnjasci . Rudimentary predicates and Turing calculations. Soviet Math. Dokl. 11

1970 1462{1465.

[PD80] J.B. Paris and C. Dimitracop oulos. Truth de nitions for formulae. In Logic and Algo-

rithmic: an International Symposium Held in Honour of Ernst Specker, Monographie no. 30

de l'Enseignement Math ematique, 1980, pp. 317{329.

[Ruz81] W.L. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences 21

1981 365{383.

[Sal73] A. Salomaa. Formal Languages . Academic Press, 1973.

[Sip83] M. Sipser. Borel sets and circuit complexity.InProc. 15th ACM Symposium on Theory of

Computing, 1983, pp. 61{69.

[Smu61] R. Smullyan. Theory of formal systems. In Annals of Math. Studies 47. Princeton University

Press, 1961.

[Vol84] H. Volger. The role of rudimentary relations in complexity theory. Bul letin de la Societ e

Math ematique de France, series 2, numb er 16 1984 pp. 41{51.

[Wil79] A. Wilkie. Applications of complexity theory to -de nability problems in arithmetic.

Lecture Notes in Mathematics vol. 834, 1979, pp. 363{369.

[Wra78] C. Wrathall. Rudimentary predicates and relative computation. SIAM Journal on Computing

7 1978 194{209.

[Yao85] A. Yao. Separating the p olynomial-time hierarchyby oracles. In Proc. 26th IEEE Symposium

on Foundations of Computer Science, 1985, pp. 1{10. 9