Relationships Among PL, #L, and the Determinant

Relationships Among PL, L, and the Determinant

y z

Eric Allender Mitsunori Ogihara

Department of Computer Science Department of Computer Science

Hill Center, Busch Campus, P.O. Box 1179 UniversityofRochester

Piscataway, NJ 08855-117 9, USA Ro chester, NY 14627

[email protected] [email protected]

Abstract

Recent results byTo da, Vinay, Damm, and Valianthave shown that the complexity of the

determinantischaracterized by the complexity of counting the numb er of accepting compu-

tations of a nondeterministic logspace-b ounded machine. This class of functions is known as

L. By using that characterization and by establishing a few elementary closure prop erties,

we givea very simple pro of of a theorem of Jung, showing that probabilistic logspace-b ounded

PL machines lose none of their computational p ower if they are restricted to run in p olynomial

time.

We also present new results comparing and contrasting the classes of functions reducible to

PL, L, and the determinant, using various notions of reducibility.

1 Intro duction

One of the most imp ortant and in uential early results of complexity theory is the theorem of [47]

showing that the complexity of computing the p ermanentofaninteger matrix is characterized

by the complexity class P of functions that count the numb er of accepting computation paths

of a nondeterministic p olynomial-time machine. It is p erhaps surprising that well over a decade

passed b efore it was discovered that an equally-close connection exists b etween the complexityof

computing the determinant of a matrix and the class L de ned in [5] of functions that count

the numb er of accepting computation paths of a nondeterministic logspace-b ounded machine. In

order to state this connection more precisely, let us de ne GapL to b e ff : f x=g x hx for

some g and h in Lg. This de nition is by analogy to the class GapP, consisting of the di erence

of P functions, which is de ned and studied in [17, 36, 22].

A preliminary version of this pap er app eared in Pro c. 9th IEEE Structure in Complexity Theory Conference,

1994

Supp orted in part by NSF grants CCR-9204874 and CCR-9509603. Part of this work was done while on sabbatical

at Princeton University.

Supp orted in part by the JSPS under grant NSF-INT-9116781/JSPS-ENG-207, and by the NSF under grant

CCR-9002292. Work done in part while at University of Electro-Communications, Tokyo. 1

Although stated in di erentways, the results of [49, Theorem 6.5], [44, Theorem 2.1], [16], and

[48, Theorem 2] show the following fact:

Theorem 1 [44, 16, 49 , 48 ] A function f is in GapL i f is logspace many-one reducible to the

determinant.

A function f is logspace many-one reducible to the determinant if there is a function g computable

in logspace such that f x viewed as a numb er written in binary is equal to the determinantof

matrix g x. The pro of given in [44] is particularly clear and direct: To da shows that the

determinant of integer matrices is reducible to iterated matrix multiplication over the integers

using [7], which in turn is reducible to iterated matrix multiplicationover f0; 1; 1g, which in turn

is reducible to a canonical GapL-complete problem, which in turn is reducible to the determinant.

We remark that the logspace many-one reductions presented in [44], can in fact b e computed by

uniform AC circuits; we will use this fact later.

In this pap er we use Theorem 1 to giveavery simple pro of of a theorem of Jung [27], concerning

the complexity class PL. PL is de ned to b e the class of languages A for which there exists a

probabilistic Turing machine in the sense of [19]; that is, a Turing machine with access to a source

of unbiased random bits, such that on input x the machine never uses more than log jxj space, and

x 2 A if and only if the probability that the machine reaches an accepting con guration is greater

than one half. Although the de nition of PL is straightforward, it turns out that the complexity

of PL is rather dicult to analyze, in part b ecause probabilistic logspace machines can p erform

useful work after exp onentially many computation steps. An easy way to see this is to note that

the s-t connectivity problem a standard NL-complete set can b e accepted with zero error bya

PL machine that follows a randomly-chosen path from vertex s, accepting if vertex t is reached,

and otherwise trying another randomly-chosen path, and so on; if a path exists, it will eventually

b e found with probability one.

Gill showed in [19] that PL is contained in PSPACE. This was improved to DSPACElog n

in [41], but it was not until the app earance of [8] that it was even known that PL is contained

2 1

in P; [8] shows that PL is contained in NC , and this can b e improved slightly to TC the

This de nition of \functional many-one reducibili ty" is from [5]. It should b e noted that the notion of one function

b eing \many-one reducible" to another is sometimes de ned in a much less restrictiveway.For example, it is shown

in [53, 11 ] that the p ermanent of zero-one matrices is \many-one complete" for P, using a less restrictiveversion

of \many-one reductions." On the other hand, it follows from [47] that the p ermanentofinteger matrices cannot b e

complete for P or GapP using our de nition of \many-one reduction" unless P = P since the p ermanent and the

determinant are equal mo d 2. That is, the relationship b etween the determinant and L is even closer than the

relationship that is known to exist b etween the p ermanent and P. 2

class computed by logarithmic-depth threshold circuits. Similarly,PLwas not known to b e

closed under complementationuntil a complicated pro of was given in [42];amuch simpler pro of

subsequently app eared in [39]. The source of all this dicultywas the fact, already alluded to, that

probabilistic logspace machines cannot obviously b e restricted to run in p olynomial time without

loss of computational p ower. However, Jung showed in [27] that, at least in the unbounded error

mo del which de nes the class PL, the p olynomial-time restriction causes no loss of p ower. The

pro of presented in [27] is complicated; we present an easy pro of in Section 3.

Note that Theorem 1, combined with our easy pro of of Jung's theorem, give an alternative

pro of of the result of [8] concerning the complexity of PL, as well as a making closure of PL under

complement completely obvious.

Note that it would b e remarkable if a theorem analogous to Jung's theorem could also b e proved

in the bounded error case. Wehave already seen that NL can b e accepted with b ounded in fact

zero error; thus this would in some sense provide a non-uniform version of L=NL i.e., L/p oly =

NL/p oly. Furthermore, Nisan [33] has shown that b ounded-error, p olynomial-time logspace can

b e simulated in p olynomial time and space log n; hence extending Jung's theorem to b ounded

error would provide a small-space p olynomial-time algorithm for transitive closure. All of these

consequences would b e surprising. We will not discuss b ounded-error probabilistic logspace in the

remainder of the pap er; for more information on the b ounded-error classes, see [9].

Let DET denote the determinant function. Do not confuse this with the de nition of [15],

where DET is used to denote the class of functions NC -reducible to the determinant. Theorem

DET L 0 0 1

1 allows one to show that L =L ,aswell as AC DET = AC L and NC DET =

1 PP P

NC L, etc. However, no logspace-analog of the theorem P =P is known, and we susp ect

that no such analog exists. For example, although we show that any of the high-order O log n

bits of a L function can b e computed in L ,we know of no reason to susp ect that the middle or

low-order bit of a L function can b e computed in this way.

For most natural problems A, the class of things reducible to A remains the same regardless of

the notion of reducibility that is used. For instance, NL is the class of languages reducible to s-t-

connectivity under logspace many-one or Turing reductions, or under 1-L reductions or quanti er-

1 0

free pro jections or NC or AC reductions or under any of the many other low-level reductions

that have b een studied in the literature. However, DET do es not seem to have this prop erty.In

Section 6 we presentcharacterizations of the classes reducible to PL and L under various notions

The complete history of the complexity ofPLiseven more complicated. The interested reader may wish to

consult [8, p. 115], [25 ], [43 , p. 111], and [20]. The related notion of probabili sti c nite automata has also b een

studied in depth; for recent results and references, see [31, 32 ]. 3

:PL

0 PL:

of reducibility.For example, AC PL is equal to the hierarchyPL .

2 Basic Facts ab out GapL

The class L was de ned and studied in [5]; L is the class of functions f such that, for some

nondeterministic logspace-b ounded machine M , f x is the numb er of accepting computation paths

of M on x. As in [5] we restrict our de nition to those machines M that halt on all computation

paths on all inputs; clearly the running time is p olynomial. Otherwise, the numb er of accepting

computation paths can b e in nite. Let FL denote the class of functions computed in deterministic

logspace. It is noted in [5] that FL L.

The following de nitions are adapted from [17]. Given a nondeterministic Turing machine M

that halts on all computation paths on all inputs, let gap xbe[numb er of accepting paths of

M on input x] [numb er of rejecting paths of M on input x]. Let GapL b e fgap : M is a

nondeterministic logspace-b ounded Turing machineg.

Wehave de ned L and GapL functions to b e functions mapping to the integers; however, it

is convenient to assume certain conventions concerning how these values are enco ded. For instance,

if leading zeros were not written, then it would b e p ointless to talk ab out the \high-order bit" of

a L function. Thus we assume that there is some constant k such that a L function is always

represented in binary, using exactly n bits. One could devise other conventions, but they would

not a ect the theorems in anyinteresting way. GapL functions will b e enco ded similarly, where

the high-order bit records the sign.

Let PLp oly denote the class of sets accepted by PL machines with the restriction that the

machines run in p olynomial time. In the next section, we will show that this class is equal to PL.

Given function classes C and D , let CDb e the class of functions ff g : f 2C and g 2Dg.

The following elementary results relating L, GapL, and PLp oly are easy to establish via

trivial mo di cations to the pro ofs of the analogous results in [17]

Prop osition 2 GapL = L L = L FL = FL L.

Prop osition 3 The fol lowing areequivalent:

1. A 2 PLp oly.

2. 9f 2 L such that x 2 A i the high order bit of f x is 1.

3. 9f 2 GapL such that x 2 A i f x > 0. 4

Corollary 4 Pos.DET 2 PLp oly, where Pos.DET is the set of al l integer matrices with deter-

minant > 0.

Pro of. Immediate from Theorem 1 and Prop osition 3.

It is clear that PLp oly is closed under complement; the usual pro of that PP is closed under

complement carries over to this case. Thus Corollary 4 and Theorem 1 show immediately that the

log

following sets are complete for PLp oly under reductions [26]:

The set of integer matrices with determinant > 0.

The set of integer matrices with determinant 0.

fA; m : DETA >mg

fA; m : DETA mg

fA; B : DETA > DETB g

fA; B : DETA DETB g

3 A simple pro of of Jung's Theorem

FNL

In this section we de ne nondeterministic logspace many-one reducibility . We show that

FNL FNL

PLp oly is closed under , and we show that every set in PL is -reducible to a set in

m m

PLp oly, and hence that PL = PLp oly.

The class FNL of functions computable via NC circuits with oracles for NL was de ned and

studied in [4], where it was noted that FNL admits many alternativecharacterizations, including

the following:

f is in FNL i f is computed by a logspace-b ounded oracle Turing machine with an oracle

for NL.

O 1

f is in FNL i jf xj = jxj and the language fx; i; b: b 2 [f$g and the i-th symbol

of f x=bg2NL, where the presence of x; i; $ indicates that jf xj

FNL

B if there is a function f 2 FNL De nition 1 Let A and B be languages. We say that A

such that for al l x; x 2 A f x 2 B .

FNL

Theorem 5 If A B 2 PLp oly, then A 2 PLp oly.

m 5

Pro of. Let M b e a PLp oly machine accepting B , and let f b e the FNL function reducing A

to B . Let N b e the NL machine with p olynomial run time deciding memb ership in the set

fx; i; b : symbol i of f xis bg.

Let M b e a machine that, on input x, b egins simulating the b ehavior of M on input f x.

Each time the i-th symbol of f x is needed, M will probabilistically cho ose a symbol b 2 [f$g

and simulate N on input x; i; b, using coin ips to simulate the nondeterminism of N . If the

simulation of N accepts, then M will continue, using symbol b. If the simulation of N rejects, then

M will ip one more coin and halt, accepting i the outcome of the coin ip is \1".

It is easy to verify that M accepts x with probability greater than one half i x 2 A.

Theorem 6 PL = PLp oly.

Pro of. Let M b e a PL machine accepting language A, and let input x b e given. As in the pro of of

Theorem 6.4 of [19], we will construct a Markovchain mo deling the b ehavior of M on x, where state

1 of the chain is the initial con guration, state r is the unique accepting con guration, state r 1

is a rejecting, halting con guration, and there is a state of the Markovchain for each con guration

of M , so that each probability p 2f0; ; 1g records the probability of going to con guration j

i;j

from i.

Using an oracle from NL, it is easy to mo dify this Markovchain by removing all states with no

path to the accepting con guration. Any transition in the original chain to one of these removed

states is replaced by a transition to a unique rejecting state. The unique accepting and rejecting

states are made \absorbing" states, byhaving the chain lo op once it reaches either of those states.

Call this new chain B .Wehave observed that transition matrix B can b e constructed from x via

an FNL computation. Let m b e the numb er of states in B , other than the two absorbing states.

Let x b e the probability of ending in the accepting state, starting from state i of B . Observe

that

x = B + B x :

i i;m+2 i;k k

As in [19], note that this can b e rewritten as

D I X = C

where X is the column vector x ;:::;x , D is the upp er left m m submatrix of B , and C is

1 m

the column vector consisting of the top m elements of column m +2 of B .Furthermore, since B is

a Markovchain with two absorbing states, D I is invertible see, e.g., [23, Lemma I I I.4.1]. Let 6

E =2D I , and let E b e equal to E , except with column 1 replaced by2C . Then E and E

1 2 1 1 2

are integer matrices constructible from x via an FNL reduction, and D I X = C if and only

if E X = 2C . By Cramer's rule, x is equal to DETE /DETE .

1 1 2 1

That is, x is in A i x > i 2DETE - DETE > 0.

1 2 1

Since DET is in GapL, the function f taking matrices M ;M as input, such that f M ;M =

1 2 1 2

2DETM DETM is clearly also in GapL. Thus the set fM ;M :f M ;M > 0g is in

2 1 1 2 1 2

FNL

PLp oly. Wehave shown that every set in PL is -reducible to this set. The result now follows

from Theorem 5.

The pro of given ab ove makes it clear that the result \PL = PLp oly" relativizes using the

appropriate notion of \relativized PL." This was not so clear from the pro of in [27]. We will need

this fact in later sections. First, however, wemust de ne what is meantby \relativized PL."

The question of what is an appropriate or even meaningful way to provide space-b ounded

machines with an oracle has b een the sub ject of some debate. For a discussion of some of the issues

involved and a list of references, see [1]. It turns out that a simple and useful notion of relativization

is the so-called Ruzzo-Simon-Tompa relativization [39]; brie y, using this notion of relativization,

a set is in PL if there is a probabilistic logspace machine M with an oracle tap e and query states

as is usual in the de nition of oracle Turing machines along with the restriction that the machine

act deterministical ly when it is writing on its oracle tap e. Note that this forces the prop erty that

each query that can b e asked during a computation on input x is describ ed completely by x and by

M 's con guration when the query starts to b e written. In particular, only a p olynomial number of

queries can p ossibly b e asked over all of M 's computations on x.

S S

Corollary 7 For any set S , PL = PLp oly .

Pro of. In the pro of of Theorem 6, the FNL reduction to a set in PLp oly is replaced bya

FNL reduction to the same set; namely, the transitions in the Markovchain denote the transition

probabilities in the relativized computation. These can clearly b e computed using an oracle for S .

The pro of of Theorem 5 also clearly carries over to the relativized setting. That is, if A is

S S S

reducible to B 2 PLp oly via an FNL reduction, then A 2 PLp oly .

It is worth observing that a pro of essentially identical to the pro of of Theorem 5 shows the

following:

Theorem 8 PL = PL. 7

4 Closure Prop erties of GapL and PL

One wayofinterpreting the results of the previous section is that PL is the class of languages that

are reducible to the high-order bit of a L function. In this section, we will improve this, to show

that PL is the class of languages that are reducible to any of the high-order O log n bits of a L

function. Along the way,we will establish some closure prop erties of PL and of GapL.

The following theorem and its pro of are logspace-analogues of results concerning GapP that

were proved in [17].

Theorem 9 Let f be any function in GapL.

1. Let g be a function in FL. Then f g is in GapL.

jxj

f x; i is in GapL. 2.

i=0

jxj

f x; i is in GapL. 3.

i=0

f x

4. Let g be a function in FL such that g x= O 1. Then is in GapL.

g x

Pro of. The pro ofs of the rst three closure prop erties may b e taken essentially word-for-word from

[17].

For the fourth closure prop erty, observe, as in [17] that if f is the di erence of two L functions

h and h , then

! ! !

g x

f x hx+ i h x+ 1

= 1 :

g x i g x i

i=0

The result now follows from the rst three closure prop erties and from [14, Lemma 2], where it is

shown that if a is in L and b is in FL with bx= O 1, then is in GapL.

log

A logspaceconjunctive truth-table reduction of one set A to another set B denoted A B

ctt

is a logspace-computable function f x= y ;:::;y for some r such that x is in A if and only if al l

1 r

log

of the y are in B . Logspace disjunctive truth-table reductions are de ned similarly, with

dtt

\all" replaced by \at least one." For more formal de nitions, see [30].

log log

Corollary 10 PL is closed under and reductions. In particular, PL is closed under

ctt

dtt

intersection and union.

log

Pro of.We present the pro of for reductions. The pro of of the other claim is analogous. Let

ctt

log

C B via a reduction g , so that x 2 C i 8i n g x; i 2 B . Let h b e the GapL function

ctt

such that hy > 0ify 2 B , and hy < 0 otherwise. Let H x; i denote hg x; i. 8

The de nition of the following sequence of p olynomials follows the presentation in Section 2 of

[13]. De ne

P x; i =

j 2 2dc log ne +1

[H x; i 1 H x; i 2 ]

j =1

P x; i =

j 2 2dc log ne+1

[H x; i 1 H x; i 2 ]

j =1

N x; i = P x; i P x; i

D x; i = P x; i+ P x; i

N x;i

S x; i =

D x;i

Ax = S x; i n 1

i=1

A x =

c c

n n

Y X Y

D x; i N x; i D x; l

i=1 i=1

l6=i

2 c

D x; i n 1

i=1

H x; i in [13]; it is proved there that Ax S x; i is equivalent to the function that is called S

is p ositive if all of the H x; i are p ositive, and Ax is negative otherwise. Observe that A xis

p ositive if and only if Ax is, and note also that A may b e seen to b e in GapL b ecause of the

closure prop erties established ab ove. This shows that C 2 PL.

FNL FNL

Corollary 10 can b e strengthened by de ning and reductions in the obvious way.It

ctt

dtt

suces to notice that, by Theorem 5, the function H x; i= hg x; i is in GapL, if g 2 FNL. This

shows:

FNL FNL

Corollary 11 PL is closed under and reductions.

ctt

dtt

Corollary 12 Language A is in PL if and only if thereisalogspace-computable function bx=

O log jxj and a GapL function f such that [x 2 A if and only if the high-order bx-th bit of f x

is 1].

Pro of. The forward direction is obvious from the foregoing, with bx= 1.

For the converse, note that the language B = fx; y : y f xg is in PL since x; y 2 B i

the GapL function f x y 0. Thus the language C = fz ; x; y : z f x y g is also in PL,

bx1

by Corollary 10. Let u ;u ;:::;u b e the 2 strings of length bx 1, and note that x is

1 2 r

k k

n bx n bx

u 10 f x u 11 . By corollary 10, A is in PL. This pro of is essentially in A i

i i

identical to the pro of of the analogous result for time-b ounded classes, as presented in [37]. 9

5 Exact Counting in Logspace

In this section, weintro duce the class C L as the logspace-analog of the class C P.We do this

= =

in order to characterize the complexity of the class of singular matrices, which is arguably a much

more natural problem than the complete sets for PL that have b een presented ab ove. The set of

integer matrices with determinant zero is a natural complete set for C L.

First, we need to present some de nitions.

De nition 2 C L is the class of languages A for which there is a function g 2 GapL such that

x 2 A i g x= 0.

As is the case with C P [46], there are several equivalentways of de ning C L.

= =

Prop osition 13 The fol lowing areequivalent:

1. A 2 C L.

2. There is a function f 2 FL and a function g 2 GapL such that x 2 A i f x= g x.

3. There is a function f 2 FL and a function g 2 L such that x 2 A i f x= g x.

4. Thereisaprobabilistic polynomial-time logspace-bounded machine such that x 2 A i the

probability of accepting is exactly .

Pro of. 1 = 4 Let A 2 C L, and let g b e the GapL function such that x 2 A i g x= 0.

Let f and f b e L functions such that g x= f x f x. Let p b e a p olynomial, and let

1 2 1 2

M and M b e nondeterministic machines realizing f and f , resp ectively, where eachofM and

1 2 1 2 1

M ip exactly pjxj coins along every computation path. I.e., M can b e constructed from an

2 1

arbitrary machine M realizing f byhaving M simulate M until M halts using coins to simulate

1 1

M 's nondeterministic steps, and then accepting i all remaining coin ips are heads.

Let M b e the machine that ips pn + 1 coins, simulating M if the rst coin ip is heads,

3 1

and if the rst coin ip is tails simulating M and accepting i M rejects.

2 2

pjxj+1 pjxj

It is easy to verify that, out of the 2 probabilistic sequences, exactly f x+2 f x

1 2

are accepting. This number is i g x= 0.

4 = 3 and 3 = 2 are obvious.

For 2 = 1, let f and g b e the functions in FL and GapL, resp ectively, de ning language

A. Note that g f is in GapL, and x 2 A i g x f x= 0. 10

log

Theorem 14 The set of al l singular integer matrices is complete for C L under reductions.

Pro of. The pro of is immediate from Theorem 1.

More generally, the set fA; r : the determinant of matrix A is r g is complete for C L, just

as the set fA; r : the determinant of matrix A is r g is complete for PL. For analogous results

concerning C P, see [40].

Note that NL is contained in C L. This is b ecause NL is closed under complement, and

memb ership in a coNL set is equivalenttohaving zero accepting computations. Furthermore, a

pro of essentially identical to the pro of of Theorem 5 shows:

Theorem 15 C L = C L.

= =

FNL

Theorem 16 C L is closed under -reductions.

FNL FNL

and reductions. In particular, it is closed under Prop osition 17 C L is closed under

ctt

dtt

union and intersection.

FNL k

Pro of. Let A 2 C L, and let B A via reduction f so x 2 B i there is an i n such that

dtt

0 0 FNL

f x; i 2 A. Let A b e the set of all strings x; i such that f x; i 2 A. Then A A; let g be

g x; i= 0: Theorem 9 the GapL function such that x; i 2 A i g x; i = 0. Then x 2 B i

i=1

now shows that B 2 C L.

2 FNL

g x; i =0: Similarly,if B A via reduction f , then x 2 B i

i=1 ctt

It remains unknown if C L is closed under complement.

log

Closure of C L under reductions is also sucienttoshow that C L contains the class

= =

dtt

LogFew that was de ned and studied in [14]. We refer the reader to [14] for the de nition of this

class.

Theorem 18 LogFew C L.

Pro of. The de nition of LogFew makes it clear that if A 2 LogFew, then there is a logspace-

computable function f and a L function g such that x 2 A i 9i jxj g x= f x; i. Clearly,

log

then, A is reducible to a set in C L.

dtt

Wehave one more result that makes mention of C L. The motivation for this result came not

so much from any question ab out C L itself, but rather from a question ab out the relationship

between L and GapL. As these classes are de ned, it is obvious that L is prop erly contained 11

in GapL, b ecause GapL contains negatively-valued functions. However, if wecho ose any natural

binary representation of GapL and L functions such as sign-magnitude or two's complement,

etc., it is natural to ask if GapL, viewed as a set of functions from to , contains any functions

that are not in L.

Prop osition 19 If every function f : ! in GapL is also in L, then C L = NL.

0 2

Pro of. Let A 2 C L, and let g b e the GapL function that de nes A. Let g = g . That is,

0 0

x 2 A = g x = 0, and x 62 A = g x < 0.

Let f : ! N b e the function de ned on the natural numb ers so that f x is the natural

numb er whose binary representation is given by the representation of g x. All that we assume

ab out the the representation chosen for g is that zero b e represented as a string of zeros. Then it

follows that x 2 A = f x = 0, and x 62 A = f x > 0. If f 2 L, then A 2 NL.

6 Reductions to L and PL

Corollary 12 raises the obvious question of whether the low-order or middle bits of a L function can

b e computed using the p ower of PL. At present, we see no reason to b elieve that this is p ossible; it

seems plausible that L i.e., determining the low-order bit of a L function is not NC -reducible

to PL. Any attempt to investigate this question must rst make precise what it should mean to try

to compute something \using the p ower of PL."

It is a pleasing fact of complexity theory that most of the \natural" complexity classes can

b e de ned as the class of problems \reducible" to some imp ortant problem, where the de nition

do es not dep end on the particular notion of reduction that is used. For example, nondeterministic

logspace can b e de ned as the class of problems reducible to transitive closure or s t-connectivity

log 0 1

using any of the notions of reducibility that have b een considered e.g., ,AC ,NC , rst-order

pro jections, etc..

However, the class of problems reducible to the determinant and to PL seem to b e exceptions

to this rule. As an extreme example, consider the class of functions that are logspace many-one

reducible to the determinant; by Theorem 1 this is GapL. The zero-one functions i.e., languages

in this class are determined by logspace-b ounded nondeterministic Turing machines, where the

numb er of accepting and rejecting computations di er by either zero or one; there seems to b e no

reason to b elieve that all of nondeterministic logspace is contained in that class. On the other hand,

it is obvious that nondeterministic logspace can b e reduced to the determinant if less restrictive

reductions are used since all that is required is to check if a L function is non-zero. 12

When Co ok [15] considered the class of problems reducible to the determinant, he framed

1 0

his de nition using NC reducibility. Alternatively, one could consider the class of things AC -

reducible to the determinant; we show that this class corresp onds to a logspace-analog of the

counting hierarchy.We do not know if this class is equal to the class considered by Co ok, but it

1 0

follows easily from the results of this section that if NC DET is equal to AC DET, then the

L Hierarchywe de ne b elow collapses at some level.

PP PP

The counting hierarchy see [50, 46] consists of sets in the classes PP,PP ,PP , etc. Since

C P

PP is contained in C P [46], it follows that the same class of sets results if the hierarchyis

de ned in terms of C P instead of PP. One obtains a computationally-equivalent class of functions

if one considers P,P , etc. Let us now consider the analogous classes de ned in terms of L

and PL.

De nition 3 The L Hierarchy De ne LH to be L, and let LH be the class of functions

1 i+1

f such that, for some logspace-bounded nondeterministic oracle Turing machine M , and some

function g 2 LH , f x is the number of accepting computations of M on input x with oracle g .

LH . Let LH denote

The PL Hierarchy De ne PLH to be PL, and let PLH be the class of languages A such

1 i+1

that, for some logspace-boundedprobabilistic oracle Turing machine M , and some language B 2

PLH , A is the language acceptedbyM with oracle B whereacceptance is de ned using the PL

PLH . acceptance criterion. Let PLH denote

The C L Hierarchy De ne C LH to be C L, and let C LH be the class of languages A

= = 1 = = i+1

such that, for some logspace-boundedprobabilistic oracle Turing machine M , and some language

B 2 C LH , x 2 A i M with oracle B accepts x with probability exactly .Let C LH denote

= i =

C LH .

= i

In all of these de nitions, the \Ruzzo-Simon-Tompa" relativization metho d is used cf. Section 3.

We will show that PLH, C LH, and LH can b e de ned in terms of AC reductions to PL,

C L, and L, resp ectively.AnAC reduction is a uniform family of constant-depth circuits of

unb ounded-fan-in AND and OR gates, NOT gates, and \oracle" gates. If the oracle is a function

f : f0; 1g !f0; 1g , then an oracle gate has some number m of inputs x ;:::;x and some number

1 m

of outputs, representing f x ;:::;x . If the function f do es not always take inputs of length m

1 m

to outputs of the same length, then it may b e necessary to assume some sort of \end-of-string"

enco ding; our theorems do not dep end on these details. If the oracle is a language, then the oracle

gate computes the characteristic function of the language. Our results do not dep end very muchon 13

the particular uniformity condition used. Logspace-uniformity is sucient, although the theorems

also hold if more restrictive uniformity conditions are used; see [12, 10] for discussions of uniformity

issues involved in low-level circuit complexity.

It will turn out to b e useful to consider certain re nements of AC reducibility.Following [4]

0 0

see also [6], de ne AC f to b e the class of languages accepted byAC circuits with oracle gates

for f , where no path from input to output go es through more than i oracle gates. If C is a class of

functions languages, then AC C is equal to the union over all characteristic functions f in C

of AC f .

Theorem 20 For al l i, AC PL = PLH .

Pro of. It should b e stated at the outset that it has recently b een shown by Ogihara that this

entire hierarchy collapses to PL [35]. Any set A 2 PL = PLH can b e accepted with a single

0 B

oracle gate. Thus assume that PLH AC PL, and let A b e the language accepted by M

for some B in PLH and some probabilistic logspace-b ounded oracle Turing machine M . Since M

computable in makes queries using the Ruzzo-Simon-Tompa restriction, there is a list y ;:::;y

logspace from x such that all queries of M on input x come from the set fy ;:::;y g. The set

0 0 0 0

B = fx; i: y 2 B g is also in PLH AC PL. Without loss of generality, B 2 AC C , where

i i

C is the \canonical complete set" for PL: C = fY; z: Y is a transition matrix showing, for all

con gurations of a logspace-b ounded machine, the probability of getting from one con guration to

another, such that the machine accepts z g.

0 0

The AC C circuit accepting A consists of AC C circuits computing memb ership of x; i

i+1 i

in B , followed byalayer of AND and OR gates computing the transition matrix Y of M on input

x with oracle answers given by the sub circuits for B , followed by one more oracle gate, checking if

Y; xisin C .

0 0

We prove only the induction step AC PL PLH implies AC PL PLH ; the

i i+1

pro of for the basis is essentially identical.

Let A b e accepted byanAC B circuit family fC g for some B in PL. We describ e the

i+1

b ehavior of a probabilistic oracle machine M accepting A. On input x, M lo oks at circuit C .If

the output gate of C is an oracle gate, then M accepts i y is in B , where y = y y :::y is the

n 1 2 r

word input to the oracle gate. The sub circuit computing each bit y is an AC B circuit, and by

induction hyp othesis can b e simulated using an oracle from PLH .

On the other hand, if the output gate is an AND, OR, or NOT gate, then the output of the

log log

reductions to a set or circuit can b e computed by a constantnumb er of applications of

ctt

dtt 14

in PLH . But this also de nes a set in PLH ,by the closure prop erties established in the

i+1 i+1

preceding section.

0 0

Alternatively, one may observe that AC PL = PL, and that an AC PL circuit maybe

viewed as consisting of i layers of AC PL sub circuits.

3 0 C LH

. Theorem 21 For al l i, C LH AC C L L

= i =

Pro of. The rst inclusion has a pro of essentially identical to the rst inclusion proved in Theorem

20. The second inclusion is also proved by a pro of essentially identical to the pro of of Theorem 20,

but since C L is not known to b e closed under complement, we cannot \absorb" any NOT gates,

C LH

. and thus we obtain only the inclusion L

0 LH

Theorem 22 LH AC L L .

We remark that LH is de ned as a class of functions, while AC L is de ned as a class

of languages. Note however that it is quite natural to view a circuit as computing a function;

moreover, it is well-known that the set of functions de ned in this way is exactly the same as the

set of functions f such that the language fx; i; b : bit i of f xisbg is in the asso ciated language

class. It follows from the pro of b elow that LH is actually equal to the class of functions computed

byAC L circuits, with the added restriction that the output gate b e an oracle gate.

Pro of. The rst inclusion is obvious when i =1;thus assume the induction hyp othesis for

g 0

i and let f b e in L for some g 2 LH . Thus there is some f 2 L such that f x=

, where as in the pro of of the previous theorem the list of y 's contains all f x; g y ;:::;gy

i 1

p ossible queries that could b e asked on input x, and this list can b e generated in logspace from x.

0 0 0

Let g x; i= g y , and note that g is also in LH , and by induction has AC h circuits for some

i i

0 0

h 2 L. Let k 0;z= hz , and k 1;z= f z . Our AC k circuits for f consist of sub circuits

i+1

computing g y =k 0;x;i, followed by a gate computing f x= k 1;x;gy ;:::;gy at the

i 1

ro ot.

We remark that, with a little more e ort, LH functions can b e computed by circuits con-

sisting of exactly i L oracle gates, where the outputs of one gate feed directly into the gate on

the next layer.

The pro of of the second inclusion is quite similar to the pro of of the forward inclusion in the

preceding theorem.

Note that this corrects a mis-statement in the version of this pap er that app eared in Pro c. 9th IEEE Structure

in Complexity Theory Conference, 1994. It is also worth noting that it has recently b een shown that this entire

C L

hieararchy collapses to L [2]. 15

0 0

Corollary 23 LH = AC L = AC DET

PLH = AC PL

C LH = AC C L

= =

Part 2 of this corollary was previously observed by Carsten Damm and Peter Rossmanith [24,38].

PL 0

The reader should not b e alarmed by the fact that L is contained in AC PL and in fact

the containmentmayeven b e prop er, even though AC is prop erly contained in L; we refer the

reader to the discussion in [51,52].

L DET

Some of the motivation for this study came from the question of whether or not L =L

=L ,by analogy to the equalities that are known to hold for the related classes P and PP. The

L DET PL

reader should b e able to see that L and L are indeed equal and contain L , but wedonot

PL 1

see how to compute the low-order or middle bits of a L function in L or even in NC PL.

A related question is whether PLH is equal to C LH equivalently, whether PL is AC -reducible

to C L.

We close this section with an observation showing that L can b e de ned using very restricted

access to the oracle. For related observations, see [5, Prop osition 2 and Section 6].

Prop osition 24 Let C be any of the classes L, NL, PL, or L. Then the class C remains

unchanged even when the oracle Turing machines de ning the class arerestricted to make at most

one query to the oracle, and to read the answer just oncefrom left to right.

Pro of. Let an oracle Turing machine M b e given, and let the oracle b e the function g . Let the

run time of M be n , let the p ossible queries on an input x of length n b e contained in the set

l c

fhx; 1;:::;hx; n g where h is logspace-computable, and let the numb er of bits in g y be jy j .

Let c b e greater than k and c .

Then the function f de ned by

k l

n n

X X

c l c

jy j +1 jn +in

g hx; i + 2 2

j =0 i=1

is in L.

k l

Note that f x, viewed as a binary string, consists of n rep etitions of a string containing n

elds, where each eld is of length n , consisting of leading zeros, followed by a one, followed by

the enco ding of g y for some query y that could b e asked on input x. If a machine asks the query

i i

f x at the b eginning of its computation, then it can simulate M on input x,moving one blo ckto 16

the right for each step of M that is simulated, and keeping track of which cell of M 's oracle tap e

is b eing scanned at any particular moment.

7 Conclusion

By making use of recent theorems relating L and the determinant, wehave given simpli ed pro ofs

of some facts concerning PL, and wehave proved new results concerning the classes of sets reducible

to PL and to the determinant. A numb er of op en questions remain. Among them:

What closure prop erties can b e established for L and GapL? In regard to this question, it

is appropriate to mention that GapL can b e characterized as the class of functions computed

by \skew" arithmetic circuits [45]. Related results regarding arithmetic circuits and certain

sub classes of TC may b e found in [3].

Can any relationship b e established b etween PL or L and AC or logCFL?

Can anyany relationship b e established b etween PL or C L and b ounded-error probabilistic

logspace? Note in this regard that David Zuckerman and Mauricio Karchmer have recently

shown that no \black-b ox" simulation of PL can yield a b ounded-error probabilistic logspace

algorithm for PL [54].

Can anything more b e said ab out the relationship b etween PLH and NC PL or LH and

1 k k

NC L? Note that for many classes C of interest including AC ,NC , NL, L, NP [4,34],

0 1

AC C is equal to NC C .

Acknowledgments

The rst author thanks Birgit Jenner for several motivating and informative discussions. Dave Bar-

rington, Michelangelo Grigni, Sanjeev Saluja, Herib ert Vollmer, and Pierre McKenzie also provided

helpful information. We thank Carsten Damm for constructive comments on an earlier draft.

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