An Isomorphism Theorem and a Gap Theorem

Reductions in Circuit Complexity:

An Isomorphism Theorem and a Gap Theorem.

Manindra Agrawal Eric Allender

Department of Computer Science Department of Computer Science

Indian Institute of Technology Rutgers University

Kanpur 208016 P.O. Box 1179

India Piscataway, NJ 08855-117 9

USA

[email protected] [email protected]

Steven Rudich

Computer Science Department

Carnegie Mellon University

Pittsburgh, PA 15213

USA

[email protected]

Abstract

We show that al l sets that arecomplete for NP under non-uniform AC reductions

are isomorphic under non-uniform AC -computable isomorphisms. Furthermore, these

sets remain NP-complete even under non-uniform NC reductions.

More general ly, we show two theorems that hold for any complexity class C closed un-

der uniform NC -computable many-one reductions. Gap: The sets that arecomplete

0 0

for C under AC and NC reducibility coincide. Isomorphism: The sets complete for

C under AC reductions are al l isomorphic under isomorphisms computable and invert-

ible by AC circuits of depth three.

Our Gap Theorem does not hold for strongly uniform reductions: we show that there

0 1 0

are Dlogtime-uniform AC -complete sets for NC that are not Dlogtime-uniform NC -

complete.

1 Intro duction

The notion of complete sets in complexity classes provides one of the most useful to ols

currently available for classifying the complexity of computational problems. Since the

mid-1970's, one of the most durable conjectures ab out the nature of complete sets is the

Berman-Hartmanis conjecture [BH77], which states that all sets complete for NP under

This researchwas done while visiting the University of Ulm under an Alexander von Humb oldt

Fellowship.

Supp orted in part by NSF grant CCR-9509603.

p olynomial-time many-one reductions are p-isomorphic; essentially this conjecture states

that the complete sets are all merely di erent enco dings of the same set. Two sets A and

B are considered p-isomorphic if there is a 1-1 function from A onto B that is p olynomial-

time computable and p olynomial-timeinvertible. Although the isomorphism conjecture

was originally stated for the NP-complete sets, subsequentwork has considered whether the

complete sets for other complexity classes C and under other notions of reducibility collapse

to an isomorphism typ e. In this pap er, we prove such an analogue of the Berman-Hartmanis

conjecture in a very natural setting: all sets complete for NP under AC reductions are

isomorphic under AC -computable isomorphisms. Most of the results in this pap er concern

0 0 0 0

non-uniform AC and NC reductions. In all instances, AC and NC refer to non-uniform

circuits unless otherwise indicated. One ma jor ingredient of our pro of of this result is

a `gap' theorem: we show that all sets complete for NP under AC reductions are also

complete under NC reductions.

In full generality, the two main theorems of this pap er can b e stated as follows.

For any class C closed under uniform NC -computable many-one reductions:

Isomorphism Theorem: The sets complete for C under AC reductions are all isomorphic

under AC -computable isomorphisms.

0 0

Gap Theorem: The sets that are complete for C under AC and NC reducibility coincide.

The remainder of the intro duction will present a more detailed description of these

theorems and previous related work.

1.1 The Isomorphism Theorem

The Berman-Hartmanis conjecture has inspired a great deal of work in complexity theory,

and we cannot review all of the previous work here. For an excellent survey, see [KMR90].

Wedowant to call attention to two general trends this work has taken, regarding 1

one-way functions, and 2 more restrictive reducibilities.

One-way functions in the worst case sense are functions that can b e computed in

p olynomial-time, but whose inverse functions are not p olynomial-time computable. Be-

ginning with [JY85] see also [KMR95, Se92, KLD86], among others many authors have

noticed that if worst case one-way functions exist, then the Berman-Hartmanis conjecture

might not b e true. In particular, if f is one-way, nob o dy has presented a general technique

for constructing a p-isomorphism b etween SAT and f SAT, even though f SAT is clearly

NP-complete. Rogers [Ro95] do es showhow to construct such isomorphisms relativetoan

oracle. However, the fo cus of our work is on non-relativized classes. Note also that it has

b een shown in [KMR88, JPY94] that there are non-complete degrees where isomorphisms

can b e constructed; however the fo cus of our work is on complete degrees.

An even stronger notion than one-way functions is that of average case one-way func-

tions: these are p olynomial-time computable functions whose inverse can b e eciently

computed only on a negligible fraction of the range. Advances in the theoretical founda-

tions of cryptographyhave shown that average case one-way functions can b e used to con-

struct \pseudo-random" functions that are computable in p olynomial-time [HILL90 ]. For 2

an excellent distillation of the most imp ortant results in this area we recommend [Lu96].

Intuitively,iff is a p olynomial-time, random-like function, f SAT is NP-complete, but

has no apparent structure to facilitate the construction of an isomorphism to SAT. Kurtz,

Mahaney, and Royer [KMR95] are able to make this intuition technically precise in the

random oracle setting. They show that when f is a truly random function, f SAT is not

isomorphic to SATeven when f is given as an oracle. This suggests that a pseudo-random

f might similarly guarantee that no isomorphism to f SAT is p ossible. As we argue b elow,

the results in this pap er greatly undermine our con dence in this approach to resolving the

Berman-Hartmanis conjecture.

Our Isomorphism Theorem negates the ab oveintuition in two imp ortant sp ecial cases.

Firstly, it is easy to observe that thereare worst-case one-way functions in uniform NC

if there are any one-way functions at all. Thus, if the worst-case one-way functions are

suciently easy to compute, the intuition that worst-case one-way functions cause the

isomorphism conjecture to fail is incorrect.

Secondly,we prove that all sets complete for NP under AC reductions are complete

under reductions that are computable via depth two AC circuits, and these sets are all

isomorphic to SAT under isomorphisms computable and invertible byAC circuits of depth

three. But it is known that there are functions computable in AC that are average-case

hard to compute for AC circuits of depth three, and that this allows one to pro duce output

that app ears pseudorandom to AC circuits of depth three [Ni92]. Using these to ols, one

can construct one-one functions f , many of whose output bits lo ok random to depth three

circuits. Although one might b elieve that for such a function f , f SAT should app ear

random to AC circuits of depth three, there is nonetheless a reduction from SATtof SAT

computable in depth two, and an isomorphism computable and invertible in depth three.

Nevertheless, we refrain from concluding that our results indicate that the isomorphism

conjecture is true. What is true in the AC settings need not hold for the much more general

p olynomial-time settings.

There has b een previous work on versions of the isomorphism conjecture for more restric-

tive reducibilities see e.g., [Ag94, ABI93]. In fact, in [Ag94] a class of reductions, 1-NL, was

presented such that the 1-NL-complete sets in natural unrelativized complexity classes are

all 1-NL-isomorphic. However, all such reductions can b e inverted in p olynomial time. For

instance, the 1-L and 1-NL reductions in [Ag94, Ag95] and earlier work, and the rst-order

pro jections considered in [ABI93] have this prop erty. A p ossible exception is the so-called

\1-omL reducibility" considered in [Ag94], which shares the non-invertibility prop ertyof

0 0

NC and AC reductions considered here. However 1-omL reducibility is a rather contrived

reducibilityinvented solely for the purp ose of proving the \collapse" result in [Ag94], and

the pro of of that result relies heavily on the invertibility of the related 1-L and 1-NL reduc-

tions. That is not the case with the results presented in this pap er. Additionally,we show

that the sets that are complete under the reducibilities considered in [Ag94] are in fact com-

plete under NLOG-uniform pro jections, and hence are also complete under NC reductions.

Thus the complete sets considered in [Ag94] are a sub class of the sets for whichwe present

isomorphisms.

One of the ma jor goals of the work presented here is to correct a shortcoming of the

results presented in [ABI93]. In [ABI93], it is shown that, for essentially any natural

complexity class C , the sets complete for C under rst-order pro jections are all isomorphic 3

under rst-order isomorphisms. The shortcoming of [ABI93] to whichwe refer is this: the

complete degree under rst-order pro jections is properly containedinthe isomorphism typ e

of the complete sets. In order to improve the result in [ABI93] to obtain a true analog of

the Berman-Hartmanis conjecture [BH77] it would b e necessary to show that the complete

degree under rst-order reductions coincides exactly with the rst-order isomorphism typ e

of the complete sets. Since rst-order reductions are precisely the functions computable by

uniform families of AC circuits [BIS90], the result we present here can b e seen as correcting

this defect in [ABI93], except for the question of uniformity.

Note that, since rst-order pro jections are a very restricted sort of NC reduction, our

0 0

result showing that the sets complete under AC reductions are all AC -isomorphic would

b e a strict improvement of [ABI93] if not for the question of uniformity. [ABI93]works

in the Dlogtime-uniform setting; our results are known to hold only in the less-restrictive

P-uniform setting in some cases and in the non-uniform setting. We b elieve that the

result for non-uniform reducibilities is interesting in its own right, and that the technical

asp ects of the argument lend additional interest to this work.

1.2 The Gap Theorem

A curious, often observed fact is that all sets known to b e NP-complete under p olynomial-

time, many-one reductions remain NP-complete under many-one, AC reductions or, even

weaker reductions. This is interesting b ecause AC is known to compute a much smaller

class of functions than p olynomial-time. It is not known if each p olynomial-time reduction

to an NP-complete set can always b e replaced byanAC reduction. In this pap er, we

0 0

prove that such a gap in the p ower of reductions do es exist b etween AC and NC :any

set that is NP-complete under non-uniform AC reductions remains NP-complete under

0 0

non-uniform NC reductions. As is the case for p olynomial-timeversus AC , the di erence

0 0

in computational p ower b etween AC and NC is enormous. For example, each output of

0 0

an NC computable function can dep end on only nitely many inputs. Thus, NC can't

even compute an AND of all its inputs in contrast, the unb ounded fan-in AND is an

0 0 0

AC function. Nonetheless, our theorem shows that AC and NC are equivalent from the

p oint of view of p erforming NP-completeness reductions. It follows that all known NP-

0 0

complete problems are complete under NC reductions. The fact that in an NC reduction

each output bit dep ends on only nitely many of the input bits means that NC reductions

are lo cal and simple by nature. Intuitively,NC reductions corresp ond to a radically simple

form of \gadget reduction."

It is instructive to consider the standard reduction from 3SAT to CLIQUE. For each

clause i, for each of its literals L,we create a no de i; L. Place an edge b etween no des

i; L and j; M if and only if i 6= j and L is not the negation of M . The resulting graph

contains a clique of one third its size if and only if the original 3SAT formula is satis able.

We x a particular choice of enco ding a 3SAT formula with n clauses. Each literal will

b e enco ded as a 1 + dlog 3ne bit string. The rst bit indicates whether the literal is p ositive

or negative; the remaining bits index the name of the variable. Since there are n clauses

there will b e at most 3n di erentvariable names. The enco ding for a given 3SAT instance

will b e the concatenation of the enco dings of each of its literals.

This is a very simple reduction to compute; it is in AC . The reducing circuit takes a 4

3SAT formula with n clauses and pro duces outputs, one for each edge of the 3n no de

graph pro duced by the reduction. Each output bit that is asso ciated with two identical

clauses is set to 0. Each other output bit dep ends on two of the input literals; it should

output 1 if and only if the literals are not negations of each other. This is clearly in AC .

However, this is not computable in NC . Why not? Tocheckiftwo literals are negations

of one another is not p ossible without considering all of the relevant 21 + dlog 3ne bits of

input. But an NC reduction can only consider a constant numb er of the relevant input

bits.

It is an exercise to nd an NC reduction from 3SAT to CLIQUE. Using our results,

any such exercise is a corollary of our Gap Theorem.

As stated ab ove, we prove that for any class C closed under NC -computable many-one

0 0

reductions the sets that are complete for C under AC and NC reducibility coincide. This

is a gap theorem in the sense that there is a big di erence gap in the computational

0 0

power of NC and AC functions, but no di erence in their p ower to p erform completeness

reductions. This is analogous to the Boro din-Trakhtenbrot Gap Theorem [Bo72,Tr64].

Some other similar gap theorems are presented in [Ag94, Ag96], and in [Ag95]itisshown

that sometimes, the hyp othesis that a gap exists for two reducibilities can haveinteresting

consequences.

We do not know if our Isomorphism Theorem holds for Dlogtime-uniform AC isomorphisms

also known as rst-order isomorphisms. In fact, we show that our Gap Theorem fails in

0 1

the Dlogtime-uniform setting, i.e., there are Dlogtime-uniformAC -complete sets for NC

or any other natural class that are not Dlogtime-uniform NC -complete. This implies

that with our approach one cannot hop e to prove the Isomorphism Theorem for Dlogtime-

uniform AC isomorphisms.

1.3 Section Organization

Section 2 presents de nitions for the classes of reductions considered in this pap er.

Section 3 presents our results ab out sets complete under NC reductions, the gap the-

orem, the isomorphism theorem, and that it cannot b e improved to Dlogtime-uniform re-

ductions using our approach.

Section 4 presents some concluding remarks.

2 Basic De nitions and Preliminaries

We assume familiarity with the basic notions of many-one reducibility as presented, for

example, in [BDG88]. In this pap er, only many-one reductions will b e considered.

A circuit family is a set fC : n 2 Ng where each C is an acyclic circuit with n Bo olean

n n

inputs x ;:::;x as well as the constants 0 and 1 allowed as inputs and some number of

1 n

output gates y ;:::;y . fC g has size snifeach circuit C has at most sn gates; it has

1 r n n

depth dn if the length of the longest path from input to output in C is at most dn.

A family fC g is uniform if the function n 7! C is easy to compute in some sense. In

n n

this pap er, we will consider only Dlogtime-uniformity [BIS90] and P-uniformity [Al89] in

addition to non-uniform circuit families. 5

0 O 1

A function f is said to b e in AC if there is a circuit family fC g of size n and

depth O 1 consisting of unb ounded fan-in AND and OR and NOT gates such that for

each input x of length n, the output of C on input x is f x. Note that, according to this

strict de nition, a function f in AC must satisfy the restriction that jxj = jy j =jf xj =

jf y j.However, the imp osition of this restriction is an unintentional artifact of the circuit-

based de nition given ab ove, and it has the e ect of disallowing anyinteresting results

ab out the class of sets isomorphic to SAT or other complete sets, since there could b e no

AC -isomorphism b etween a set containing only even length strings and a set containing

only o dd length strings { and it is precisely this sort of indi erence to enco ding details

that motivates much of the study of isomorphisms of complete sets. Thus we allowAC -

computable functions to consist of functions computed by circuits of this sort, where some

simple convention is used to enco de inputs of di erent lengths for example, \00" denotes

zero, \01" denotes one, and \11" denotes end-of-string; other reasonable conventions yield

exactly the same class of functions. For technical reasons, we will adopt the following

sp eci c convention: each C will have n + k logn output bits for some k . The last

k log n output bits will b e viewed as a binary number r , and the output pro duced by the

circuit will b e the binary string contained in the rst r output bits. It is easy to verify

that this convention is AC -equivalent to the other convention mentioned ab ove, and for

us it has the advantage that only O log n output bits are used to enco de the length. It

is worth noting that, with this de nition, the class of Dlogtime-uniform AC -computable

functions admits many alternativecharacterizations, including expressibility in rst-order

with f+; ; g [Li94, BIS90] , the logspace-rudimentary reductions of Jones [Jo75,AG91],

logarithmic-time alternating Turing machines with O 1 alternations [BIS90] and others.

This lends additional weight to our choice of this de nition.

1 0

NC and NC are the classes of functions computed in this wayby circuit families of size

O 1

n and depth O log n or O 1, resp ectively, consisting of fan-in two AND and OR and

NOT gates. Note that for anyNC circuit family, there is some constant c such that each

output bit dep ends on at most c di erent input bits. An NC function is a projection if its

circuit family contains no AND or OR gates. The class of functions in NC was considered

previously in [Ha87]. The class of pro jections is clearly a sub class of NC and has b een

studied by many authors; consult the references in [ABI93].

For technical reasons and for simplicity of exp osition, wedo not allowanNC circuit

C to pro duce outputs of di erent lengths for di erent inputs of length n, although we do

0 1

allowAC and NC circuits to do this by following the convention mentioned ab ove. That

is, if f is computed byNC circuit family fC g where each C has sn output bits, then for

n n

all inputs x of length n, jf xj = sn. Our chief justi cation for imp osing this restriction

is that Theorem 10 shows that any set hard for NP or other complexity classes under

AC reductions using the less-restrictive convention allowing outputs of di erent lengths

is in fact hard under NC reductions using the more-restrictive convention. Thus we are

able to obtain our corollaries ab out sets complete under AC reductions without dealing

with the technical complications caused by allowing NC reductions to output strings of

di erent lengths. Also note that, even with this restriction, the NC reductions we consider

Lindell [Li94] shows only that this coincides with rst-order expressibili ty in rst order with f+; ; ;

expg, where \exp" denotes exp onentiation. However, p ersonal communication from K. Regan and S. Lindell

shows that exp onentiatio n can b e eliminated. 6

are still more general than the rst-order pro jections considered in [ABI93].

For a complexity class C ,aC -isomorphism is a bijection f such that b oth f and f

are in C . Since only many-one reductions are considered in this pap er, a \C -reduction" is

simply a function in C .

A language is in a complexity class C if its characteristic function is in C . This con-

0 1

vention allows us to avoid intro ducing additional notation suchasFAC , FNC , etc. to

distinguish b etween classes of languages and classes of functions.

The theorems we prove in this pap er hold for most complexity classes that are of interest

to theoreticians; we require only closure under certain easily-computable reductions. To

make this precise, wesay that a class of languages C is a proper complexity class if C is

closed under Dlogtime-uniform NC reductions. That is, if A is in C , and B is reducible to

A via a many-one reduction computable in NC , then B is in C . Note that most complexity

classes, suchasNP,P, PSPACE, BPP, etc. are prop er complexity classes.

In fact, insp ection of our pro ofs shows that our results hold even for any class C that is

closed under reductions computed by uniform threshold circuits of depth ve. The number

ve can probably b e reduced. We do not knowhowtoweaken the assumption to closure

under reductions computed in ACC ; it is easy to see that our results do not hold for some

classes closed under AC reductions. For instance, the sets f1g and f1; 11g are b oth hard

0 0

for AC under AC reductions, but they are not isomorphic, and they are not hard under

NC reductions.

A function is length-nondecreasing length-increasing, length-squaring if, for all x, jxj

jf xj jxj < jf xj, jxj jf xj; it is C -invertible if there is a function g 2C such that

for all x; f g f x = f x.

The following prop osition is well-known:

Prop osition 1 P=NP i every length-increasing Dlogtime-uniform NC function is P-

invertible.

Pro of. The forward direction is obvious. For the converse, assume that 3SAT is not

in P. Consider the following enco ding of a 3CNF formula with variables v ;:::;v . Note

1 n

that there are only 8n clauses on these variables that can p ossibly app ear in any 3CNF

formula. A formula can thus b e enco ded as a sequence of 8n bits, with each bit denoting

the presence or absence of the corresp onding clause. Now consider the function f de ned

by f ; ~v = ; x where x is the string of length 8n such that the ith bit of x is 1 i the

ith bit of is 0 or the ith bit of is 1 and the corresp onding clause evaluates to 1 when

the variables are set according to the assignment ~v . It is clear that f is length-increasing,

and it is not hard to see that f is computed by a Dlogtime-uniform family of NC circuits.

Note that is in 3SATi ; 1 is in the range of f , which can b e detected in p olynomial

time if f is P-invertible.

3 Main Results

3.1 Sup erpro jections

De nition 1 AnNC reduction fC g is a sup erpro jection if the circuit that results by

deleting zero or more of the output bits in each C isaprojection wherein each input bit or

n 7

its negation is mapped to some output. Stated another way, it is a superprojection if, for

each input bit x , there is an output bit whose value is completely determinedbyx . That

i i

is, this output bit is either x or :x .

i i

Note that every sup erpro jection has an inverse that is computable in AC : On input

y ,wewant to determine if there is an x such that f x= y . The AC circuit will have

a sub circuit for each n jy j since a sup erpro jection is by de nition one-one and length-

nondecreasing checking to see if there is suchanx of length n. This sub circuit will nd

the n output bits that completely determine what x must b e if suchanx exists, and then

will check to see if f x= y .

Theorem 2 For every proper complexity class C , every set hard for C under P-uniform

NC reductions is hard under P-uniform one-one, length-squaring superprojections.

Pro of. Let A b e hard for C under NC reductions. We shall show A to b e hard under

one-one length-squaring sup erpro jections in two stages.

Stage 1: In the rst stage, we show that A is also hard under length-nondecreasing sup er-

pro jections.

Takeany set B in C .We de ne a new set C ,TC -reducible to B , that is accepted by

the following pro cedure:

On input y , let y =1 0z .Ifk do es not divide jz j, then reject. Otherwise, break

z into blo cks of k consecutive bits each. Let these b e u u u :::u . Accept if

1 2 3 p

there is an i,1 i p, such that u =1 . Otherwise, reject if there is an i,

1 i p, such that u =0 . Otherwise, for each i,1 i p, lab el u as nul l if

i i

the numb er of ones in it is less than k=2; as zero if the numb er of ones in it are

between k=2 and 3k=4; and as one otherwise. Let v = if u is null, 0 if u is

i i i

zero, and 1 otherwise. Let x = v v v , and accept i x 2 B .

1 2 p

It is straightforward to see that C reduces to B via a Dlogtime-uniform NC reduction.

Therefore, C 2C by the closure prop erties of C . Since A is NC -hard for C , there exists an

NC reduction of C to A. Let this b e given by the family of circuits fD g. In what follows,

we will use the circuits D reducing C to A to construct a pro jection reducing B to C

with the prop erty that the comp osition of these two reductions is a sup erpro jection from

B to A.

Let the depth of circuits in the family fD g b e b ounded by the constant d and let

c =2 . The pro jection from B to C will map strings of size n to strings of size 4c +1+4cm

where m = O n the exact value of m will b e given later. It will map string x, jxj = n,

to the string 1 0u u :::u where each u is of size 4c, and where the string formed out of

1 2 m i

these u s as describ ed in the pro cedure de ning C isx.We show b elowhow the values of

the u s are computed. In the discussion b elow, we refer to the u sasblocks.

i i

Consider the circuit D . Set the rst 4c + 1 bits of the circuit to 1 0 and consider

4c+1+4cm

the reduced circuit with 4cm unset input bits. Each output bit of this circuit dep ends on

at most 2 input bits. Let O be a maximal set of output bits satisfying the prop erty that

1 each output bit in O dep ends on at least one input bit so there are no constant output 8

bits in O , and 2 no two output bits in O dep end on the same set of input bits. We rst

show that jO j m=2 . Supp ose not. Since any output bit of the circuit dep ends on at most

d d

2 input bits, the bits in O dep end on at most jO j2

of the circuit that is not in O can dep end only on these input bits since otherwise it would

have b een included in O .Thus, there are fewer than m input bits, out of 4cm, on which

the output of the circuit dep ends. This implies that there is one whole blo ck of input bits,

say u , that do es not a ect the output. Set all the blo cks except u to zero i.e., to a value

i i

2c 2c 4c 4c

with numb er of ones b etween 2c and 3c, e.g., 1 0 . Now, setting u to 1 or 0 keeps

the output of the circuit identical, which is a contradiction since D reduces strings

4c+1+4cm

of C to A.

Therefore, the size of O is r , for some

r m=2 : 1

We will start with this set O of output bits, and p ossibly remove some bits from O as the

construction pro ceeds.

The remaining input bits are denoted as usual with x , :::, x .We view each output

1 4cm

bit as the outcome of a b ounded truth-table evaluation on the input bits on whichit

dep ends. We need to b e fairly precise here ab out howwe asso ciate a truth table to each

output bit. Consider one of the output bits in O , and consider the fan-in two circuit of

depth d that computes this output bit. Order these input bits according to the index, with

x coming b efore x if i

i j

the value of the output, then remove that input bit and simplify the circuit computing the

0 d

output bit accordingly, and let the numb er of input bits remaining b e d 2 . The \truth

. The value of the output bit is obtained table" for this output bit has variables v ;:::;v

1 d

= c by plugging in the appropriate input bit for each v . Note that there are at most 2

di erent such truth-tables. Wecho ose some truth-table, say , that is asso ciated with a

maximal numb er of output bits i.e., at least as manyasisany other truth table.

At this p oint, remove from O all output bits that do not have as their truth table,

and let s b e the numb er of output bits that now remain in O . Clearly,

s r =c: 2

Consider the ith output bit remaining in O . Let the ordered set of input bits on which

g where d c. Consider the family of sets F = fS g where it dep ends b e fx ;x ;:::;x

i i;1 i;2

i;d

S = fj; x :1 j d g. All of the sets in F have size at most c, and note that, by

i i;j

.Thus by the the waywehave constructed our set O ,wehave that if i 6= i , then S 6= S

Sun ower Lemma of [ER60] see also [BS90, Lemma 4.1], the collection of sets F contains

a sun ower of size at least

1=c

t bs=c!c : 3

This \Sun ower" is a collection of t sets from F with the prop erty that, for all pairwise

distinct sets S ;S ;S ;S in the sun ower, S \ S = S \ S . The set that one obtains by

1 2 3 4 1 2 3 4

intersecting anytwo elements of the sun ower is called the \core" of the sun ower. We

now remove from O all of those output bits i such that S is not in this sun ower. Thus

jO j = t now. 9

Consider anytwo bits i and j that remain in O . S and S record the input bits on

i j

which output bits i and j dep end, and note that the input bits in S \ S corresp ond to

i j

exactly the same variables in the truth table that determines how i and j dep end on these

inputs. Now, set the bits in the core of the sun ower to 0-1 values such that the truth-table

do es not b ecome a constant this can always b e done, b ecause the truth table dep ends

on al l of its d input bits. So now each output bit in O dep ends only on the input bits that

are in the corresp onding p etal of the sun ower. We will pro cess each p etal of the sun ower

in turn.

Consider the rst p etal corresp onding to output bit i . Since setting the bits in the

core did not make a constant, there is some bit z in this p etal and some assignmentof

f0,1g values to the other bits in the p etal such that the output bit i dep ends only on the

value of z . Moreover, since the truth-table relating the output bits to p etals is identical for

all p etals, we obtain a corresp onding bit z in each p etal, along with an identical assignment

to the remaining bits in each p etal. There is a subtle p oint here. Although the sets in

our sun ower F have pairwise intersection equal to the core of the sun ower, and thus the

p etals are each pairwise disjoint, this says only that a tuple j; x can app ear in at most

k;j

one p etal, but it do es not say that a given input variable can app ear in at most one p etal

although it do es follow that a given input variable can app ear in at most d c di erent

p etals, each time paired with a di erentnumber j . In particular, it is certainly p ossible

that the \identical assignments to the remaining bits in each p etal" referred to ab ove will

con ict with each other. We will show b elowhow to deal with this. Call the bit z the

identi ed bit for the p etal i.

Recall that our goal is to map a string x of length n to a string of the form 1 0u u :::u

1 2 m

where each u is of size 4c, and where each u is either \null" or represents a single bit of

i i

x. Our overall approach is to map input bits of x to the identi ed bits z in the p etals of

the sun ower. When we try to do this wehave to assign values to the bits in the core of

the sun ower and to the other bits in the p etals of the sun ower; this will cause us to make

some of the blo cks u \null", and it will cause us to remove some of the p etals from O .

We will succeed if we can showhow to make this assignment and still end up with enough

p etals to enco de all of the bits of x.

Pro cess each output bit i 2 O in turn. Consider the unset bits in S . Initially, none of

the bits in S are set. When we pro cess the rst bit i in O we will set all of the bits in S

i i

except for z , including all of the bits in the core. When we pro cess the other bits in O only

the bits in the p etal will b e unset. For each of these bits other than z , set this bit to the

value discussed ab ove so that output bit i dep ends only on the value of z . This causes

at most c 1 bits to b e set. Each of these bits is in some blo ck u . Consider any such

blo ck u that contains one of these bits that has just b een set but does not contain z . Set

j i

the rest of the bits in such a blo ck u to zero. Note that this has the e ect of making blo ck

u nul l, since the length of u is 4c and we are setting at least 3c +1 > 2c variables in this

j j

blo ck to zero. Wehavenow set all blo cks containing variables in S except for the blo ck

containing z . This blo ck contains at most c 1variables that have b een set. Set the u

i j

rest of the inputs in blo ck u that is, set the variables in u other than z so that there

j j i

i i

are exactly 3c ones and c 1 zero es in the blo ck. This has the e ect of making the blo ck

dep end on the identi ed bit: it is zero when the identi ed bit is zero and one otherwise.

Thus far in pro cessing p etal i,wehave set fewer than 4c input bits at most 4c for each bit 10

other than z , and at most 4c 1 for z . Some of the input bits wehave set including some

i i

of the bits in the p etal just pro cessed may b e elements of other p etals in our sun ower.

Remove from O any output j such that its p etal contains a bit that has b een set in this

way; remove also any j such that its p etal contains the bit z just pro cessed. This causes

3 2

O to lose fewer than 4c output bits since eachofthe< 4c bits can app ear in at most c

p etals, and in the remaining sun ower, none of the bits in any p etal has b een set. Note

that the end result of pro cessing this element i 2 O is that wehave obtained an input bit

z such that the output bit i is a pro jection of z .Now rep eat the pro cess describ ed ab ove

i i

for the next bit remaining in O .

We rep eat this pro cess jxj times to obtain jxj such bits z . In order for this to b e p ossible,

3 d

it is sucient for t to b e at least 4c jxj. This gives us a b ound on m: m 2 r by 1

d d c 3

2 c s by 2 2 c t +1 c!by 3, and thus if we pick t to b e 4c jxj, it follows

00 c 00

that it is sucienttocho ose m to b e c jxj for some constant c dep ending on d. Recall

that c dep ends on d.

So our reduction of B to C will, on input x of length n, identify bits z ;:::;z and map

1 n

x to bit z , where the other bits of circuit D are set according to the pro cedure

i i 4c+1+4cm

listed ab ove or if there are any remaining bits left unset by this pro cedure, we set those bits

to zero, having the e ect of nullifying all remaining blo cks not containing one of the z s.

This reduction of B to C is just a pro jection from B , since every output bit dep ends on at

00 c

most one input bit. It maps a string of size n to one of size 4c +1+4cm with m = c n .

If wenow consider the reduction from B to A that results by comp osing the pro jection

from B to C with the reduction D ,we note that the n bits that are determined

4c+1+4cm

by the z are merely the pro jections of the input x, and the other bits are either xed

or corresp ond to output bits that were deleted from O by the foregoing pro cedure, but

nonetheless are still computed by the NC circuit. Thus the reduction is a sup erpro jection.

It is somewhat tedious to verify that this reduction can b e made P-uniform. First

observe that can b e found in logspace. Then observe that there are at most n sets

that could p ossibly b e the core of the desired sun ower; exhaustively trying each such

p ossible core in turn, and then using a greedy algorithm to nd a maximal collection of sets

containing the core and with pairwise disjoint \p etals" will eventually uncover a sun ower

of the desired size. The pro of of the Sun ower Lemma given in [BS90] shows that this

approach will succeed. Finding the desired setting of the bits in the core and p etals is easy.

Then sequentially deleting the bits from the p etals is straightforward.

Stage 2: It is clear at this p oint that the reduction of B to A describ ed ab ove is length-

nondecreasing and also 1-1 at least on strings of the same length. However, it may map

strings of two di erent lengths to the same string. To take care of this problem, we add

another stage of the construction.

Once more, we takeany set B in C . Once more, we de ne a new set E AC -reducible

to B . The de nition of E is straightforward:

E = fx10 j x 2 B k 2 Ng:

E is clearly in C and therefore there exists an NC reduction of E to A that is a length-

increasing sup erpro jection. Let this reduction b e given by the function f .We know that for

all x: jxj jf xj pjxj for some p olynomial p. De ne a function r as follows: r 0 = 1, 11

and r t +1= pr t + 1. And now de ne a reduction g of B to E as: g x= x10 where

k is the smallest numb er such that: k jxj and jxj +1+ k = r t for some t.Function g

can clearly b e computed by a pro jection circuit, and so f g is an NC reduction of B to A.

It is length-increasing b ecause g and f are b oth length-increasing. It is 1-1 also, which can

b e seen as follows: for anytwo strings x and y such that jg xj = jg y j, f g x 6= f g y

follows from the nature of f . And when jg xj > jg y j then jf g y j pjg y j= pr t

for some t

Also note that, since g is length-squaring and f is length-nondecreasing, the resulting

sup erpro jection is at least length-squaring.

Checking P-uniformity of this step is trivial.

The following corollary the non-uniform case is a trivial consequence of the foregoing.

Corollary 3 For every proper complexity class C , every set complete for C under NC reductions

is complete under one-one, length-squaring superprojections.

Corollary 4 For every proper complexity class C , every set complete for C under NC reductions

is complete under reductions computable by depth two AC circuits and invertible by depth

0 0 0

threeAC circuits. If the NC reductions are P-uniform, then so are the AC circuits.

Pro of. First note that since a sup erpro jection is an NC reduction, it can b e computed in

depth two simply by expressing each output bit in DNF or CNF form.

Next note that b ecause of the construction of Stage 2 of the pro of of Theorem 2, we know

that if A is complete for C under NC reductions, then it is complete under sup erpro jections

f of the form h g where h is a sup erpro jection when restricted to strings in the range of

g , and strings in the range of g have the form y 10 .Furthermore, for each n there is an

easily-computed m such that, for each string x of length n,iff x exists, then there

k 1 k

exist y and k such that jg y j = jy 10 j = m, f x=y , and hy 10 =x. The p oint

here is that m dep ends only on n = jxj. Now to compute f for inputs of length n,it

suces to consider the circuit computing h on inputs of length m.For inputs x of length n,

1 1 k

f x exists if and only if h x exists and is of the form y 10 for k in the correct range.

1 k

If h x exists, then there are m bits of x that directly enco de the bits of y 10 , since h is

a sup erpro jection.

1 k

Thus our circuit to compute f x rst takes the string y 10 that is available on the

input level of the circuit as determined by m bits of input x and that is a candidate

1 k

for h x. Then in depth two it computes hy 10 , and checks that all of the bits of

k 0

hy 10 and x agree. This is an AND of several NC predicates, and by expressing the

NC predicates in CNF and merging the two levels of AND gates we obtain a depth two

k 1 k

circuit pro ducing output y 10 if h x= y 10 . Since our goal is to pro duce output y and

also output jy j in the length-enco ding eld we obtain a depth three circuit by taking the

1 k

OR over all p ossible values of r = jy j of the predicate \h x= y 10 AND the last m r

bits of y 10 are in 10 ".

Corollary 5 For every proper complexity class C , al l sets complete for C under P-uniform

0 0

NC reductions are P-uniform AC -isomorphic. Furthermore, these isomorphisms arecom-

putable and invertible by P-uniform AC circuits of depth three. 12

Pro of. The main result in [ABI93], showing that all sets complete under rst-order pro jec-

tions are rst-order isomorphic, carries over also into the P-uniform setting, and the same

pro of also works for sup erpro jections. We refer the reader to [ABI93] for details, but we

sketch some of the imp ortant steps here.

Let A and B b e complete for C under NC reductions. Thus there are sup erpro jections

f and g reducing A to B and reducing B to A, resp ectively. Our goal is to construct an

isomorphism mapping A onto B .We rst construct a depth four isomorphism b etween A

and B , and then improve it to depth three. As in most other work constructing isomor-

phisms see [BH77] for example, given an input x,we will need to compute the length of

the \ancestor chain" of x, and output f x if the length of the chain is even, and output

g x if the length of the chain is o dd.

Note that if the k ancestor exists, then just as in the case k = 1 in the pro of of

Corollary 4 the bits of the k ancestor are available at the input level. Thus one can

determine in depth three if the length of the ancestor chain is exactly r . Namely, for all

th th

r ancestor. Now, the i bit of the output would b e

_ ^ _ _ ^

th th 1

r = k i bit of f x r = k i bit of g x:

k odd k even

This gives a depth six circuit, however, note that the top two levels are of fan-in two, and

therefore, can b e \pushed down" and collapsed with the b ottom two levels. This results in

a depth four circuit.

To reduce the depth further, we observe that we do not need to explicitly check for

the existence of the inverse at level two as is done in the pro of of Corollary 4. Instead,

we distribute this work to the top two levels: Let C b e the NC circuit that outputs

k;m

th 0

a sequence of ones i the k ancestor exists and has length m. Let C b e the depth two

AC circuit with top level AND gates that outputs a sequence containing at least one

th 0 th

zero i the k ancestor do es not exist. To see how to construct C , note that if the k

ancestor do es exist, then there is a k 1 ancestor z of some length m that is completely

th 1 r

determined by n and k , and the k ancestor is a string y where h z =y 10 where r

cannot b e to o large. That is, in addition to the lo cal consistency checks each bit of which

can computed in NC , the only other condition that must b e checked is to say that the

th 1 r

k ancestor do es not exist if h z = y 10 ends in to o many zeros. This can clearly b e

checked by a CNF circuit. Let D b e the circuit computing

r;l;m ~

^ ^

th 0

outputC 2 1 l bit of outputC =0:

k;m

r +1

k r

Here, m~ is a vector of O log n bits enco ding a sequence m ;:::;m of numb ers such

1 r

that m m n. Since f and g are b oth length-squaring, the sequence of lengths

i i+1

o ccurring in the ancestor chain can b e enco ded as suchavector. D is still a depth two

r;l;m ~

circuit since the AND gate at the top can b e merged with the AND gates on top of the

0 th

s and C s. The i bit of the output can now b e de ned as: C

k;m

r +1

_ _ ^ _ _ _ _ ^ _

th th 1

D i bit of f x D i bit of g x:

r;l;m ~ r;l;m ~

r even

r odd l l

m~ m~ 13

As argued b efore, this is a depth three circuit. A similar circuit computes the inverse of the

isomorphism.

Corollary 6 For every proper complexity class C , al l sets complete for C under NC reductions

areAC -isomorphic. Furthermore, these isomorphisms arecomputable and invertible by

AC circuits of depth three.

The ab ove corollary generalizes the result not only of [ABI93] to a larger class of complete

sets, but also the results of [Ag94]. To see this, we observe that the complete sets under all

the reducibilities considered in [Ag94] are also complete under NLOG-uniform pro jections.

It was shown in [Ag94] that the complete sets under 1-L, 1-NL, and 1-omL reductions are

also complete under forgetful 1-L, 1-NL, and 1-omL reductions resp ectively. A forgetful

reducibilitywas de ned there as one computed by a TM that, after scanning each bit of the

input, ends up in a con guration that dep ends only on the size of the input|in particular,

it is indep endent of the value of the bits scanned so far. It is easy to see that a forgetful TM

computes a function that is a pro jection|the output of the TM during its scan of any input

bit dep ends only on the bit and the length of the input; and thus every output bit dep ends

on at most one input bit. And since all the three reducibilities 1-L, 1-NL, and 1-omL are

sub classes of NLOG-reducibility, the corresp onding forgetful TMs compute NLOG-uniform

pro jections. Thus, these complete sets form a strict sub class of NC -complete sets as it

is straightforward to construct a set that is NC -complete for any prop er class but not

complete under even non-uniform pro jections [ABI93].

3.2 The Gap Theorem and The Isomorphism Theorem

3.2.1 Random Restrictions of AC reductions

An imp ortant to ol will b e the fact that when we randomly restrict the inputs to a circuit

0 0

family computing an AC function we obtain a circuit family computing an NC reduction.

This has b een a folklore theorem since [FSS84, Aj83]. Lemma 7 b elow is explicitly stated

in [Ar95]. The pro of can b e gleaned by suitable mo di cations of any of several alternative

presentations [BS90, Fo95, B95]. The pro of of a slightly stronger statement can b e found

in the app endix of this pap er.

De nition: For a natural number a,ana-random restriction on m variables is a

function indep endently assigning to eachvariable a value in f0; 1; g as follows: set it to 0

1=a1 1=a1

with probability1 m =2, set it to 1 with probability1 m =2, and set it

1=a1

to * with probability m .

Lemma7 For any AC reduction computed by a family of circuits fC g, there exists an

a 2 N such that, with probability 1 o1,ana-random restriction on m variables

0 1=a

transforms C into an NC -circuit with m input variables.

In the app endix of this pap er we prove a slightly stronger version of this, which to the

b est of our knowledge has b een a folklore theorem.

Lemma8 Let m = r , and let the m variables x ;:::;x be divided into r blocks of length

1 m

2a1

r where block i consists of variables x ;:::;x . Then with probability 1 o1,

ir +1 ir +r

an a-random restriction assigns * to at least three variables in each block. 14

1=a1

1=a1 2a 22a

Pro of. Each one of the m bits will have probability p = m = r = r

2a1

of getting set to *. Notice that the blo ck size is r = r=p. The probability that any

particular blo ck of size r=p gets no more than two*sisgiven by:

r=p

r=p1 2 r=p2 r=p

p1 p + p 1 p 1 p +

2 p

r r 2 r

e + re + r e

2 r

= Or e

3 r

Since there are r blo cks the probability that one of them gets fewer than 3 *'s is O r e =

o1. It follows that all blo cks get at least 3 *'s with probability1 o1.

All that is needed for the results of the following section is this easy consequence of the

preceding two lemmas.

Corollary 9 For any AC reduction computed by a family of circuits fC g, there exists an

a 2 N such that, for al l large m of the form r , thereisarestriction such that

m m

transforms C into an NC circuit, and assigns * to at least three variables in each

m m

2a1

block of length r .

3.2.2 The Gap Theorem

Theorem 10 Gap Theorem Let C be any proper complexity class. The sets hard for

0 0

C under non-uniform AC reductions are hard for C under non-uniform NC reductions.

Pro of.

Let C be any prop er complexity class, i.e., C is closed under Dlogtime-uniform NC

reductions. Let A be any set hard for C under AC -reductions. Let B be any set in C .

0 0

Clearly, B is AC -reducible to A.We seek to show that B is actually NC -reducible to A.

The pro of strategy will b e similar to the one followed in Stage 1 of the pro of of Theo-

0 0 0

rem 2. As there, we de ne a set B 2C, and use the AC -reduction C from B to A as

a starting p oint for a reduction from B to A. B will have b een chosen so that a suitable

non-uniform restriction of C will giveusanNC reduction from B to A.

to b e the set of strings accepted by the following pro cedure: We de ne B

On input y , let y =1 0z . Reject if k do es not divide jz j. Otherwise, break z

into blo cks of k consecutive bits each. Let these b e u u u u .For each i,

1 2 3 q

1 i q , let v =0ifthenumb er of ones in u equals 0 mo dulo 3; v =1if

i i i

the numb er of ones in u equals 1 mo dulo 3; and v = otherwise. Accept i

i i

v v v 2 B .

1 2 q

0 1

It is easy to see that B is Dlogtime-uniform NC reducible to B . Hence, by the de nition

0 0

of prop er complexity class, B 2C. Since A is hard for C under AC -reductions, there must

0 0

exist an AC circuit family C computing a reduction from B to A. Let d b e a b ound on

the depth of the family C . W.l.o.g. we can assume that C takes n input bits and has no

n n

more than n output bits. Recall that the nal O log n bits are used to enco de a number 15

that indicates how many of the output bits to use in the reduction. We refer to these bits

as the \length-enco der bits".

Let a b e the constant dep ending on the depth d of C from Lemma 7. Let C b e the

2a 0

family of circuits for inputs of length m =2q , where C is obtained by taking circuit

2a1

2a1 2a1 2q

C for n =1+2q + m and setting the rst 1 + 2q bits to 1 0.

By Corollary 9, for all large m, there is a restriction such that transforms C into

m m

0 2a1

an NC circuit, and assigns * to at least three variables in each blo ck of length 2q .

We will now showhow to extend to obtain a further restriction of C having only q

variables, and having the length-enco der bits set to constantvalues. We will call this new

circuit family D . This circuit family D will b e our NC reduction from B to A.

q q

EachofO log n length enco der bits dep ends only on a constantnumb er of remaining

input bits. Thus, the enco der bits dep end only on O log n blo cks. For each of these

O log n blo cks, x al l the remaining bits to constants so that the numb er of 1's in each

blo ck is 2 mo dulo 3 this is always p ossible b ecause wehave at least 3 unset bits in each

blo ck. The length-enco der bits are xed and we still have2q O log q blo cks that we

have not tamp ered with. Pick all but the rst q blo cks and also x their inputs so that the

numb er of 1's in each of them is 2 mo dulo 3. For each of the remaining q blo cks, set all but

one bit in each blo ck so that the total numb er of 1's in the blo ck is 0 mo dulo 3 again this is

p ossible since there are at least three unset bits in each blo ck. The result is a circuit with

exactly q input bits and a xed output size. That is, all of the length-enco der bits have

b een set to constantvalues by setting the bits on which they dep end. Let this value b e r .

Thus we can delete the length enco der bits and all but the rst r output bits. Call this

circuit family D . Notice that it has size p olynomial in q b ecause q is n for some >0

and C is of size p olynomial in n. Also note that D is obtained from C by restricting

n q n

2a1

attention to inputs of the form y =1 0z , where z is a string with exactly q *'s. For

any string x of length q , denote by y x the result of plugging the q bits of x into the q

p ositions in z , and note that the algorithm for B accepts y x if and only x 2 B b ecause

the algorithm for B deco des z to obtain x.

0 0

Since C reduces B to A,we see that D reduces B to A. This is the desired NC -

n q

reduction from B to A.

3.2.3 The Isomorphism Theorem

The Isomorphism Theorem is an immediate consequence of the Gap Theorem and Corollary

Theorem 11 Isomorphism Theorem Let C be any proper complexity class. Al l sets

0 0

complete for C under non-uniform AC reductions areAC -isomorphic. Furthermore the

isomorphisms arecomputable and invertible by depth threeAC circuits.

Note that this is, in some sense, a true analog of the Berman-Hartmanis conjecture,

since it presents a natural notion of computation which then yields natural notions of

reducibility and isomorphism and it shows that in this setting the complete sets coincide

with the isomorphism typ e of the standard complete set.

In the rest of this section, we include a few observations that we hop e will shed additional

light on the original Berman-Hartmanis conjecture. 16

Recall that our Gap Theorem says that, for sets A that are complete for NP under

0 0

AC reductions, all sets reducible to A under AC reductions are already reducible to A

under NC reductions. Let us now consider another typ e of p ossible \collapse" and inves-

tigate its consequences.

For the purp oses of this section, let us say that a set A is special if it has the prop erty

that the class of sets reducible to A via uniform AC -reductions is equal to the class of

sets reducible to A via reductions computed by uniform threshold circuits of depth ve.

Since AC cannot compute the MAJORITY function computable by a depth one threshold

circuit, clearly no nite set is sp ecial, and it is easy to construct many other sets that are

not sp ecial.

On the other hand, any set A that is complete for NP under AC -reductions clearly is

sp ecial, since NP is exactly the class of sets reducible to A under either form of reducibility.

Are all sets that are complete for NP under polynomial-time reductions sp ecial? If so,

then as wesketch b elow, a non-uniform version of the Berman-Hartmanis Conjecture is

true.

Let A be any set that is complete for NP under p olynomial-time reductions. We observe

rst of all that there is a set A that is b oth

1. P-isomorphic to SAT, and

2. AC -reducible to A.

Let f b e a p olynomial-time reduction from SATtoA, and let f b e computed bya

p olynomial-time machine M . Let A b e the set f< x;y;C ;:::;C >j C is an initial

1 m 1

con guration of M on input x, each C yields C via one computation step of M , C

i i+1 m

is the nal con guration of M pro ducing output y , and y 2 Ag. Since SAT is reducible

to A via a length-increasing and invertible reduction, it follows from [BH77] that SATis

P-isomorphic to A .

If in addition A is sp ecial, then by Theorem 10 there is a non-uniform NC reduction

0 0

from A to A, and by Theorem 2, there is a non-uniform sup erpro jection reducing A

to A. In particular, this implies that there is a length-increasing and invertible P/p oly

reduction from SATtoA, and hence A is P/p oly-isomorphic to SAT. Most of the non-

0 0

uniformity here can in fact b e eliminated. The NC reduction from A to A can in fact

O 1

log n

b e made DTIMEn -uniform, by noting that the pseudorandom output pro duced

by Nisan's generator [Ni92]must frequently pro duce restrictions satisfying the condition of

Corollary 9; we leave the details to the reader. It follows that, if A is sp ecial, then it is

quasip olynomial-time isomorphic to SAT.

We do not view this as strong evidence in favor of the Berman-Hartmanis Conjecture,

but we do feel that it casts the problem in a new light.

3.3 Uniform Versus Non-uniform Gap Theorems

Theorem 10 cannot b e made Dlogtime-uniform. That is, there exist Dlogtime-uniform AC -

1 0

complete sets for NC that are not Dlogtime-uniform NC -complete. It is worth mentioning

See the comments in Section 2, where the notion of a prop er complexity class is de ned, to understand

this reference to threshold circuits. 17

at this p oint that it is not entirely clear what should b e the \right" notion of uniformity for

NC circuits, since Dlogtime Turing machines can do things that cannot b e done byany

NC circuit and thus one mightwant to consider a more restrictiveversion of uniformity

when discussing uniform NC , so as not to allow the uniformity machine to overwhelm the

computation done by the NC circuit itself . However, we show here that even under this

\p owerful" notion of uniformity for NC , Theorem 10 fails to hold.

Theorem 12 For any class C closed under Dlogtime-uniform AC reductions and having

0 0

a set complete under AC reductions, thereare Dlogtime-uniform AC -complete sets for

C that are not Dlogtime-uniform NC -complete.

Pro of. The pro of uses ideas from [Ag95]. Let A be any Dlogtime-uniform AC -complete

denote the characteristic bit-vector set for C , and let L be any set in NTIMEn. Let

of L for the rst m strings of in lexicographic order. De ne a set A as:

jz j

g: A = fxz j z 2 A ^jxj = jz j^x =

The set A reduces to A via a Dlogtime-uniform AC reduction: for inputs of size 2n,

the reduction circuit computes the bit-vector this can b e done by a Dlogtime-uniform

AC circuit since L 2 NTIMEn and the size of each of the rst n strings of is at most

log n, compares it with the rst n bits of the input, and outputs the last n bits if all the

bits match, and otherwise outputs some xed string not in A.For inputs of o dd size, the

reduction circuit just outputs some xed string not in A. It is easy to see that the entire

circuit can b e made Dlogtime-uniform. Therefore, A 2C.

The set A reduces to A via a Dlogtime-uniform AC reduction: for input of size n, the

reduction circuit rst outputs and then outputs the n input bits. This circuit to o is a

0 0

Dlogtime-uniform AC circuit. Therefore, A is Dlogtime-uniform AC -complete for C .

Now, let us assume that Dlogtime-uniform AC -complete sets for C are also Dlogtime-

0 0

uniform NC -complete. Thus, A is Dlogtime-uniform NC -complete for C . By Lemma

13 b elow, A is complete under Dlogtime-uniformNC reductions for which there is a

constant c 1 such that the length of the output pro duced on input x has length at least

jxj=2 . Let fC g b e such a reduction of the set 1 to A .

n L

We will now give a deterministic pro cedure that accepts L in linear time. On input x,

n+2+c

jxj = n, the pro cedure considers the circuit C . This circuit, on input 1 ,must

n+2+c

n+2

output a string in A of size ` with ` 2 . Since the output is in A , the rst `=2 bits of

L L

`=2

it are , and therefore, the m output bit of the circuit is x where m is the p osition

n+1

of the string x in the lexicographic order. This is b ecause m 2 `=2.

The circuit C n+2+c is a Dlogtime-uniform NC circuit. Thus the following functions are

computable in time logarithmic in input size in other words, in time linear in n:

n+2+c 0 0

2 ;g;R 7! g , where g is the name of the right input to gate g in C if g

n+2+c

is a gate in C n+2+c . The output is * if g is not a gate in C n+2+c .

2 2

n+2+c 0 0

2 ;g;L 7! g , where g is the name of the left input to gate g in C n+2+c if g is

a gate in C n+2+c . The output is * if g is not a gate in C n+2+c .

2 2 18

n+2+c

2 ;g 7! T , where T is the typ e of gate that g is, i.e., T 2fAND, OR, NOT,

INPUT, 0, 1g. Note that, since C n+2+c takes input in 1 , all \INPUT" gates are

essentially constant 1 gates.

Our pro cedure to determine if x is in L rst computes the number m such that m is the

p osition of x in the lexicographic order. Then it determines g , the gate that computes the

m bit of circuit C n+2+c .We've established that such an output gate must exist. Using

a reasonable notion of uniformity,we should b e able to compute the name of this output

gate. Now it computes the right and left inputs to the gate g , right and left inputs of these

two gates and so on until the entire tree for the gate g has b een computed. Since this tree

has constant height, this computation takes only linear time in n. Finally, the pro cedure

computes the value output by the gate g . This can b e done b ecause it knows what sort

of gates are in the tree of g , and that all of the inputs are either constants or, if they are

circuit inputs, they are set to 1. The pro cedure accepts i this bit is 1.

Thus, L 2 DTIMEn. Since L was an arbitrary language in NTIMEn, wehave shown

that NTIMEn = DTIMEn, whichisacontradiction, since it is known that DTIMEn

is prop erly contained in NTIMEn [PPS83].

The pro of is now complete, except for the following technical lemma.

Lemma13 For any class C closed under Dlogtime-uniform AC reductions, al l sets com-

0 0

plete under Dlogtime-uniform NC reductions arecomplete under Dlogtime-uniform NC reductions

for which thereisaconstant d 1 such that the output produced on input x has length at

least jxj=d.

Pro of. Let A b e hard for C , let B be any set in C , and let D b e the set f0x j x 2 B g[1 . D is

0 0

Dlogtime-uniform AC -reducible to B , and thus there is a Dlogtime-uniform NC reduction

fC g reducing D to A.We will show that the NC reduction from B to A obtained by

comp osing the obvious pro jection from B to D with fC g satis es the conditions of the

lemma.

Consider the circuit C .Ifwe set the rst input bit to 1, note that each remaining input

bit must in uence at least one output bit, b ecause if all bits are set to 1, C must pro duce

an elementof A as output, but if any bit is set to 0, then the output pro duced is not in A.

The b ound follows b ecause there is some constant d such that each output bit dep ends on

at most d input bits.

4 Conclusions

In closing, let us summarize our results. Berman and Hartmanis conjectured in [BH77] that

all sets complete for NP under p oly-time many-one reductions are P-isomorphic. Following

the lead of [ABI93]wehave considered the analogous question, where p olynomial-time re-

ductions and isomorphisms are replaced byAC -computable reductions and isomorphisms.

0 0

projections are AC -isomorphic. In [ABI93] it was shown that all sets complete under AC

Wehave improved that result to show that all sets complete under NC reductions are

AC -isomorphic. To give some indication of the nature of this improvement, note that 1 19

pro jections are a very simple sort of NC reduction, and 2 pro jections are easily invert-

0 0

ible in AC , whereas NC reductions are not invertible in p olynomial time unless P=NP.

Invertibility is relevant here, since the likely existence of non-invertible p oly-time reduc-

tions is one of the main considerations leading many researchers to conjecture that the

Berman-Hartmanis conjecture is false [JY85]. We use our results ab out NC -reducibility,

sup erpro jections, and an inherent gap in the p ower of reductions to prove a true analog of

the Berman-Hartmanis conjecture. That is, the sets complete under AC reductions are

all AC -isomorphic. Finally,we show that our approach cannot yield the analog of the

conjecture for Dlogtime-uniform AC reductions.

We esp ecially call attention to the following problems:

1. Assuming the existence of a function that is one-wayinavery strong average case

sense, is it p ossible to construct a counter-example to the original Berman-Hartmanis

conjecture?

2. Is there any natural class C larger than P such that there is a set hard for C under

p olynomial-time many-one reductions that is not hard under non-uniform sup erpro-

jections?

3. Are there any natural classes C larger than P such that the classes of sets hard for

C under a p olynomial-time many-one reductions, and b uniform AC reductions,

di er? Wehave already shown that uniform NC reductions do yield a strictly smaller

class of complete sets.

Note in this regard that [Ar95] shows a there is a p oly-time reduction f :PARITY

Clique such that x 2 PARITY implies f xhasavery large clique, and x 62

PARITY implies f x has only very small cliques, and b no AC reduction can have

this prop erty. Nonetheless, there is no version of the Clique problem or any other NP-

complete problem that is currently known not to b e complete under AC many-one

reductions.

4. Is there any class C such that Dlogtime-uniform AC -complete sets for C are all

Dlogtime-uniform AC -isomorphic?

Acknowledgments

Weacknowledge helpful conversations with M. Ajtai, S. Arora, R. Boppana, O. Goldreich,

M. Ogihara, D. van Melkeb eek, R. Pruim, D. Sivakumar, and M. Saks.

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5 App endix

In this app endix, we give a pro of of Lemma 7. Although there are several p eople in the

community who were already aware that established techniques can b e used to prove this

lemma, wehave b een unable to nd a published statement of this fact, and hence we provide

the pro of for completeness.

In an earlier version of this work [AA96], the argument that was presented required a

stronger lemma, stated in terms of \blo ckwise" restrictions. Since this stronger version of

the lemma may b e useful in some other applications, wehavechosen to include the pro of

of this stronger lemma in the app endix. Note that wehave not tried to obtain the b est

constants in this pro of. Instead our goal was to make the pro of as simple as p ossible.

Let b 2 N. A b-blo ck restriction on n variables is an assignment to the variables

x ;:::;x with the prop erty that for all j 0 either al l of the variables x ;:::;x are

1 n jb+1 jb+b

given values, or none of them are. The variables x ;:::;x together constitute the

jb+1 jb+b

j -th b-blo ck. If n is not a multiple of b, then the nal blo ckmayhave length less than b.

For a; b 2 N,an a-random b-blo ck restriction on n variables is a b-blo ck restriction

chosen according to the following pro cess: For blo ckof b variables in turn, with probability

1=a1 b

n=b set all of the variables in that blo ck to *, and for any v 2f0; 1g , set the variables

1=a1 b

in the blo ck to the vector v with probability1 n=b =2 . Note that the a-random

restrictions used in Section 3.2 are 1-blo ck restrictions.

Wenow restate Lemma 7 using the more general notion of blo ck restriction.

Lemma7 For any AC reduction computed by a family of circuits fC g, there exists an

a 2 N such that for any constant b, with probability 1 o1,an a-random b-block restriction

0 1=a

on n variables transforms C into an NC -circuit with n b-blocks of input variables.

a n

Pro of. Without loss of generality, the circuit family fC g is

of depth k 2

leveled meaning that the circuit has n inputs x ;:::;x , and n negated inputs

1 n

:x ;:::;:x , and that these inputs feed into AND gates or OR gates, and that

1 n

AND gates feed into OR gates, and vice-versa

of b ottom fan-in 1 meaning that the gates that x ;:::;x , and :x ;:::;:x feed

1 n 1 n

into have fan-in only 1.

The constant a is chosen to b e 5k 1.

To simplify notation, we will assume throughout the pro of that n is a multiple of b. Let

m = n=b.

By a simple application of the Cherno b ound see, e.g., [AS92], with probability

1=a 1=a

1 o1, the number of b-blo cks that are set to * is b etween m and 2m .Thus it is

2 23

sucient to show that, with probability1 o1, C is transformed into an NC -circuit,

given that the numb er of blo cks that are set to * lies within this range.

1=a b

Since we will assume that at least m 2m blo cks are set to values in f0; 1g , it will

b e convenient to consider the following pro cess for picking the random restriction . One-

by-one, a blo ckispicked and set to a value in f0; 1g with all unset blo cks b eing equally

likely, and all values in f0; 1g equally likely after the blo ckispicked. We will see that after

1=a

m 2m blo cks have b een set in this manner, the circuit C will b e transformed into an

NC -circuit with probability1 o1. This pro cess will actually pro ceed in k 1 stages,

where the rst k 2 stages each decrease the depth of the circuit by one with probability

1 o1 this is guaranteed by Claim 15, and the nal stage changes the depth two circuit

into an NC -circuit with probability1 o1 this is guaranteed by Claim 16. Setting any

additional blo cks will not alter the fact that the circuit is transformed into an NC -circuit.

Thus, given that we are assuming that the numb er of blo cks that are set to * lies within

the acceptable range, this will prove the lemma.

1=4bk 1

k 2=k 1

More formally, let f m= m , and let r m; k = . Claim 15 shows

k 2

that we can take a depth k circuit on m blo cks, having b ottom-fan-in r m; k , and by setting

some of the blo cks randomly, obtain a circuit that is equivalent to a depth k 1 circuit

with f m blo cks, having b ottom fan-in r f m;k 1. For i

k k

i i i+1 1 i

m= m. Note that f m= f f m= f m, and f as follows: f f

k k

k k 1 k k k

k i+1=k 1 i

m . Also note that r f m;k i= b r m; k .

k 2

1=k 1

m= m Applying Claim 15 k 2 times, we obtain a circuit with depth 2, on f

k 2

k 2 1=4bk 1

blo cks, having b ottom fan-in r f m; 2 = b r m; k = m .Nowwe apply

1=4k 1 1=a

Claim 16 and obtain a circuit on m = ! n blo cks with b ottom fan-in O 1.

Before we can present the pro ofs of Claims 15 and 16, we will need some additional

notation.

Let R denote the set of all b-blo ck restrictions to the variables x ;:::;x that have

1 n

n;b

exactly ` blo cks of b variables unset. It will usually b e the case that n and b are understo o d

from context, and thus we will usually delete the subscripts.

Note that

` n`b

jR j = 2 :

Thus there is a constant c such that, for any 2 R ,

n;b

K jn; `; b c + n `b + log

Here, we are assuming some xed system for enco ding restrictions; the particulars are

irrelevant. We are assuming that the reader is familiar with Kolmogorov complexity.For

details, see [LV93]. Note also that, for s>0,

m `

` `s sb

jR j=jR j =2

2 `

In our applications, s will b e much larger than b, and n=b will b e much larger than `, and thus

`s `

R is much smaller than R . In particular, although at rst it may seem counterintuitive, 24

a Kolmogorov-random element 2 R has greater Kolmogorov complexity than any element

0 `s 0

of R even if is the result of assigning values to some of the unset variables of .

This simple fact is a key observation underlying the pro of of the switching lemma.

In the following, we will not makeany meaningful distinction b etween a circuit C and

the bit string that describ es an enco ding of C .Thus given any circuit C of size t,any

n n

gate in the circuit can b e describ ed using an additional log t bits, and thus a pair C ;i

is a description of the function computed by gate i of C . If gate i is on level twoof C

n n

and thus by our assumptions it is an OR of ANDs, where the ANDs are connected to input

literals, then the pair C ;i immediately gives a DNF formula F where this formula lists

n n;i

the AND gates feeding into gate i listing them in the order in which they app ear in C

where each AND gate is presented by the literals feeding into that gate where the ordering

on the variables imp oses an order on the literals.

Given any DNF formula F and restriction , F j is the formula that results by 1

deleting from F all terms i.e., conjunction of literals that are made false by , and 2

replacing each remaining term C by the term C j obtained by deleting from C all the literals

that are satis ed by . It should b e emphasized that this is a syntactic op eration, and that

F j is a syntactic ob ject. A formula with an empty term denotes the constant function 1;

a formula with no terms denotes the constant function 0.

Given any DNF formula F ,we follow [B95] and de ne the canonical decision tree for F

denoted T F as follows.

1. If F has no terms, then T F is a single leaf lab elled 0.

2. If the rst term in F is empty, then T F is a single leaf lab elled 1.

0 0

3. If the rst term C of F is not empty, then let F b e the rest of F i.e., F = C _ F .

1 1

The decision tree T F b egins by querying al l of the variables that app ear in any b-

blo ck containing a variable in C . That is, if C has r variables, the tree T F b egins

1 1

0 0

with a complete binary tree on the r b rb variables in the r blo cks that contain

variables app earing in C . Each leaf v of this complete binary tree is reached by

some path lab elled by a restriction recording an assignment to the variables in C .

Each suchnode v is the ro ot of a subtree in T F given by T F j . For any that

satis es C , T F j is a single no de lab elled 1. For all other , T F j = T F j .

For a restriction ; let Dom denote f0; 1g; i.e., the variables set by : For

S fx ;:::;x g; let j denote the restriction

1 n S

; x 2= S ;

j x =

S i

x x 2 S

i i

For a Bo olean expression F; we will sometimes write F for F j :

Following [B95], we will show that for any Kolmogorov-random and for any DNF

formula F , the heightofT F j denoted jT F j j is small.

It is easy to observe that if f is a Bo olean function having a decision tree with height

b ounded by s, then it has a DNF formula with term size b ounded by s. The disjuncts

consist of the conjunctions, over all paths of the tree ending in 1, of the literals queried

along those paths. Similarly, suchanf has a CNF formula with term size b ounded by s 25

since :f clearly has a small decision tree, and hence a DNF formula with small term size.

Note that saying that jT F j is small is a much stronger statement than merely saying that

the function computed by F has a decision tree with small height; this is b ecause T F may

well b e a very inecient decision tree.

Lemma14 Hastad Switching lemma Thereisaconstant c such that, for any DNF

formula F in n variables with terms of length at most r , and for any b and for any 0

` n=b, and for any 2 R ,if

n;b

n=b

K jF ; n; `; s; b >c + n `b + sb log16r + log

` s

then jT F j j

At this p oint, we can present the pro ofs of the Claims 15 and 16.

Claim 15 Let C be a leveled circuit on m blocks with depth k 3 and with bottom fan-in

r = r m; k , and having no more than 2 gates at levels 2 and above. Then for al l large

n, with probability 1 o1,a random b-block assignment leaving f m blocks unset has

a k

the property that C j is computed by a depth k 1 circuit with bottom fan-in r f m;k 1

r f m;k 1

and with no more than 2 gates at level 2 and above.

Pro of. of Claim 15 Assume without loss of generality that the b ottom level gates are

AND gates. The other case follows by DeMorgan's laws.

Let s = r = r m; k and let ` = f m. Note that for a randomly chosen in 2

R , K jC ; n; `; s; b is high. In particular, with probability1 1= log n, wehave that

n;b

K jC ; n; `;s;b n `b + log log log n.

Consider any of the gates on level twoof C , and consider the DNF formula F represented

by this gate. Before we go further, wemust argue that

n=b

K jF ; n; `; s; b >c + n `b + sb log16s + log

` s

and that we can therefore app eal to the Switching Lemma.

Since there are most 2 gates at level twoofC , it is easy to see that K jC ; n; `; s; b

K jF ; n; `; s; b+ c + s. The constant c is enough bits to say \count the number p of

5 5

gates in C at level 2 and ab ove, and use the next dlog pes bits to identify one of the OR

gates on level twoof C . Construct the DNF formula F computed by this gate, and then

use the remaining bits of information to construct from F ."

Thus K jF ; n; `; s; b K jC ; n; `; s; b c s n `b + log c s log log n:

5 5

This value is b ounded b elowby c + n `b + sb log16s + log if and only if

log c + c +s + sblog 16s + log log n:

4 5

Thus it is sucienttoshow that

m `

c + c + s + sblog 16s + log log n: log

4 5

` 26

This is equivalenttoshowing that

d b 1=s

m ` 2 `16s log n

where d =1+c + c =s. Note that d< 2 for all large n, since s = r n; k . Since there is

4 5

b 1 4b+2 1

some >0 such that `s n ,itisthus sucienttoshow that n=b ` 2 n log n,

which is clearly true for all large n: Thus wehave established that, for the chosen values of

`, s, and , the hyp othesis of the Switching Lemma is satis ed.

Thus by the Switching Lemma, each of the OR gates at level twoofC j has a decision

tree of height sb, and thus each of those OR gates can b e computed by CNF circuits

with b ottom fan-in sb. The circuit that results by substituting these CNF circuits into C j

has AND gates on levels two and three. If we merge these two levels, we obtain a circuit

having depth k 1, having ` blo cks of b variables, and with b ottom fan-in b ounded by

r `;k 1

sb = r `; k 1 and with no more than 2 gates at level two and ab ove. of Claim

Claim 16 Let n be suciently large. Let f be a function on n inputs such that jf xj n

for al l x of length n, where f is computed by a depth-two circuit C having bottom fan-in

1=4b 1=4

r = r m= m . Then with probability 1 o1,a randomly-chosen leaving m

b-blocks unset has the property that each output bit of C j hasadecision tree of height at

most sb, where s =4B .

Pro of. Assume without loss of generality that the output bits of C are ORs. The other

case follows easily.

1=4 `

Pick ` = m , and let b e randomly chosen elementofR .Thus, with probabilityat

n;b

log log n. least 1 1= log n, K jC ; n; `;s;b is greater than n `b + log

Consider any of the output gates of C , and consider the DNF formula F represented by

this gate. Before we go further, wemust argue that

n=b

K jF ; n; `; s; b >c + n `b + sb log16r + log

` s

and that we can therefore app eal to the Switching Lemma.

Since there are most n output gates in C , it is easy to see that K jC ; n; `; s; b

K jF ; n; `; s; b+ c + B log n.Thus K jF ; n; `; s; b K jC ; n; `; s; b c B log n

5 5

n `b + log c B log n log log n: This value is b ounded b elowby c + n `b +

5 4

if and only if sb log16r + log

log c + c + B log n + sblog 16r + log log n:

4 5

Thus it is sucienttoshow that

m `

log c + c +B log n + sblog 16r + log log n:

4 5

By our choice of s this is equivalent to showing that

c +c =s 1=4 b 1=s

4 5

` 2 n `16r log n :

b 27

1=4 b 3=4

Since n r ` =n , the hyp othesis of the Switching Lemma is satis ed for all large n.

Now the claim follows immediately from the Switching Lemma. of Claim 16

Thus it will suce to give a pro of of the Switching Lemma.

Pro of. of the Switching Lemma

Let F ; `; s; n; t; b e as in the hyp othesis of the lemma. We will prove the contrap os-

itive. That is, we'll show that if T F j contains a path of length at least sb, then

n=b

. K jF ; n; `; s; b c + n `b + sb log16r + log

0 `s

The strategy is to construct 2 R extending i.e., xing more variables, such

0 0

that K j ;F;`;s;b is not much bigger than K jF ; n; `; s; b. As we observed ab ove,

0 `s

K jF ; n; `; s; b is small, b ecause R is small.

Let 2 R and let be any path in T F j having length at least sb. Note that we

may view as a restriction namely the restriction that gives to the sb variables queried

along the value determined along path , and leaving the other n sb variables unset.

Wenow de ne sequences of restrictions ; ;::: and ; ;::: where each in turn

1 2 1 2 i

0 00 0 0

is decomp osed into = , where Dom = Dom . Our goal is to de ne =

i i

i i i

00 00 00 0

::: in suchaway that is easy to retrieve from .

1 2 k

1 2

Let C b e the rst term in F that is not set to 0 by . Such a term must exist, since

otherwise the heightofT F j would b e zero; by the same observation, C j is not empty.

Thus C j is the rst term of F j . Let S b e the set of variables in C j , and let b e the

1 i 1

1 1

unique restriction of the variables in S that satis es C j . Thus, S = Dom . Let P

1 i 1 1

b e the set of variables that are in b-blo cks containing an elementofS , and let P = P n S .

1 1 1

Note that, by the de nition of the canonical decision tree, P Dom , since the tree

0 00

queries all variables in each blo ck touched by a term. Let = j and = j , and let

S P

1 1 1 1

0 00

= j , so that = . Note that is a pre x of path .

1 P 1 1

1 1

If is in fact al l of , then the rst part of the construction is over. Otherwise, by

the de nition of the canonical decision tree, it must b e the case that there must b e some

C that is the rst term in F that is not set to zero by , and C j is the rst term of

i 1 i

2 2 1

F j , and it is this term that is explored next in the decision tree along path .Asabove,

let b e the unique assignment to the variables S in C j that satis es this term, let P

2 2 i 2

2 1

0 00

b e the other variables in the blo cks touched by S and let = j and = j .

2 S P

2 2 2 2

Continue in this wayuntil the entire path is pro cessed, maintaining the prop erty that

clause C is the rst term of F not falsi ed by ::: , and ::: satis es

i 1 h1 1 h1 h

C . It is imp ortant to observe also that each set S is nonempty.

i h

Note that, since the length of is at least sb,itmust touch at least sb-blo cks. A slight

problem is caused by the fact that may touch more than s blo cks. Since wewant to

be in R ,we simply consider the rst stage k such that ::: touches at least s

1 k 1 k

b-blo cks, and rede ne S to b e the initial sequence of variables in clause C up through the

k i

s blo ck touched, de ne P to b e the other variables in those blo cks, and rede ne to

k k

00 0

= j . = j , and b e the setting to those variables that do es not falsify the clause,

P S

k k

0 00 00 00

,wehave de ned the desired Note that k s. By setting equal to :::

1 2 k

1 2

elementof R .

n;b

Wenowwant to show that is easy to recover from .To do this, we de ne a sequence

di er. Each is a string of length r ;:::; , where each describ es how and

h 1 k h h

recall that r is the b ottom fan-in of the circuit C over the alphab et f0; 1; g, de ned as 28

follows. The j bit of is if either 1 clause C of the original formula F has fewer

h i

th th

than j variables, or 2 the j variable in clause C is not in S . The j bit of is0if

i h h

th th 0

agree on this variable. The j bit of is 1 if the j variable in C is in S and and

h i h h

h h

th 0

the j variable in C is in S and and disagree on this variable. Note that each

i h h h

h h

contains at least one symb ol that is not a . The total numb er of non-* symb ols is at least

s and no more than sb since the 's together touch exactly s blo cks. Let = ::: ;

h 1 k

thus is a string of length kr sr .We will pad with *'s at the end to obtain a string

of length exactly sr .

Observe that, for some constant c , K js; b; r c + sb3 + log r . This is b ecause

2 2

0 j 1 j 1 j 1

1 2

is of the form b b ::: b ::: where each b 2f0; 1g, and v bs and

1 2 v h

each j 2r since no is all *'s. Note that many of the numb ers j may b e equal to zero.

h h h

By making j = :::j = 0 and making b = ::: = b we can enco de using exactly bs

v +1 bs v bs

suchnumb ers. Thus we can enco de as a sequence of exactly bs numb ers j of exactly

dlog r e + 1 bits, and a bit string of exactly bs bits enco ding the b 's.

Nowwe claim that, for some constant c ,

n=b

: K jF ; n; `; s; b c + n `b + sb log16r + log

` s

n=b

To see this, note that can b e describ ed with c + n `b + sb + log bits. Since r can

b e obtained from F , can b e describ ed with c + sb3 + log r bits. The b ound on the

Kolmogorov complexityof now follows from an additional c bits of information, enco ding

the following instructions:

Find the rst clause C in F that is not made false by . Note that

i 1

makes C true, and the further assignments made by cannot change this.

0 th

Let b e the rst r bits of . Use to nd the elements of S . If the j

1 1 1

bit of is not *, then the j variable in C is in S .

1 i 1

Use C and S to obtain . Use and to obtain . P consists of all

i 1 1 1 1 1

00 0 0 00

of the other variables in blo cks touched by S ; is j ; = .

1 P 1

1 1 1 1

00 00

Let C b e the rst clause of F that is not made false by ::: .

i 1 2 k

Note that this can b e obtained from without knowing what is bychanging

the setting of the variables in S and P to .Asabove, compute and ,

1 1 1 2 2

00 00

and then use ::: to nd C , and continue in this way, to obtain

1 2 3 k i

3 3

;:::; .

1 k

Obtain by de ning x = if x 2 Dom ::: , and x = x

i i 1 k i i

otherwise. 29