NP Might Not Be As Easy As Detecting Unique Solutions

x Richard Beigel Harry Buhrman Lance Fortnow

Lehigh University CWI University of Chicago

Abstract Intro duction

We construct an oracle A such that Valiant and Vazirani [VV86] show the surprising power of solving

satisﬁability on formulae with at most one satisfying assignment or

A A A A

P P and NP EXP equivalently detecting unique solutions to NP problems. This relativized world has several amazing properties:

Theorem 1.1 (Valiant-Vazirani) If one could detect unique solu-

A The oracle gives the ﬁrst relativized world where one can tions to NP problems then R NP.

solve satisﬁability on formulae with at most one assignment yet P NP. The proof of Theorem 1.1 depends heavily on randomization.

They leave open whether detecting unique solutions implies P A

The oracle is the ﬁrst where NP.

A A A A

P UP NP coNP Hypothesis 1.2 If one can detect unique solutions to NP problems

then P NP. The construction gives a much simpler proof than that of Fenner, Fortnow and Kurtz of a relativized world where all the NP-complete sets are polynomial-time isomorphic. It is Theorem 1.1 relativizes. To help us gauge the difﬁculty of prov- the ﬁrst such computable oracle. ing a deterministic version of Theorem 1.1 we will show Hypothe-

sis 1.2 fails in a relativized world.

A A

A Relative to we have a collapse of EXP ZPP

PA /poly. Theorem 1.3 There exists a relativized world where we can detect unique solutions for NP problems yet P NP. We also create a different relativized world where there exists

a set L in NP that is NP-complete under reductions that make one To prove Theorem 1.3 we consider the set Q consisting of all query to L but not complete under traditional many-one reductions. formulae with an odd number of satisfying assignments. If we can This contrasts with the result of Buhrman, Spaan and Torenvliet determine membership in Q than we can detect unique solutions. showing that these two completeness notions for NEXP coincide.

This set Q also has some nice algebraic properties for our proofs.

Research done at Yale and the University of Maryland. Address: Elect Note the set Q is P-complete. Eng Comp Science, 19 Memorial Dr W Ste 2, Bethlehem PA 18015- In fact we prove a considerably stronger result.

3084, USA. Supported in part by the National Science Foundation

A A

under grants CCR-8958528, CCR-9415410 and CCR-9700417 and by Theorem 1.4 There exists an oracle A such that P P and

A A

NASA under grant NAG 52895. Email: [email protected]. NP EXP . http://www.eecs.lehigh.edu/˜beigel/

x Address: CWI, Kruislaan 413, 1098SJ Amsterdam, The Nether- Theorem 1.4 has some other important applications. Berman lands. Partially supported by the Dutch foundation for scientiﬁc re- and Hartmanis [BH77] conjectured that all NP-complete sets are

search (NWO) by SION project 612-34-002, and by the European polynomial-time isomorphic, i.e., for every pair of NP-complete B Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and sets A and there exists a polynomial-time computable and in-

HC&M grant nr. ERB4050PL93-0516. Email: buhrman@cwi.nl. B vertible bijection reducing A to . http://www.cwi.nl/˜buhrman/ Finding a relativized world where this isomorphism conjecture Research done while on leave at CWI. Address: University of Chicago, held remained open for many years. Homer and Selman [HS92]

Department of Computer Science, 1100 E. 58th. St., Chicago, IL 60637, noted that if P UP and NP EXP then the isomorphism con- USA. Supported in part by NSF grant CCR 92-53582, the Dutch Founda-

tion for Scientiﬁc Research (NWO) and a Fulbright Scholar award. Email: jecture holds. They created a relativized world where P UP and

EXP. However, even a relativized world where P UP [email protected]. http://www.cs.uchicago.edu/˜fortnow 2

NP coNP seemed much more difﬁcult to prove. Theorem 1.4 is the ﬁrst to break this barrier.

Corollary 1.5 There exists an oracle A such that

A A A A A

P UP NP coNP EXP

x L M x Fenner, Fortnow and Kurtz [FFK96] used a very different and

complicated approach to resolve the relativized isomorphism con-

x L M x jecture. Their oracle is nonconstructive and makes the polynomial- .

time hierarchy infinite. Theorem 1.4 is the first to fulfill the Homer A language L is in P if there exists a polynomial-time nonde-

and Selman approach. The proof is considerably simpler than Fen- x terministic Turing machine M such that for all ,

ner, Fortnow and Kurtz and is the ﬁrst to achieve a constructible

x L M x

oracle and a collapse of the hierarchy. is odd.

x L M x Corollary 1.6 There exists a recursive set A such that the isomor- is even.

phism conjecture holds relative to A. In addition the polynomial-

A The class EXP has the same deﬁnition as P except we allow

time hierarchy relative to A collapses to NP .

O (1)

n M to use time. Heller [Hel84] and Kurtz [Kur85] give a relativized world where We can generalize P by allowing different modula. A lan-

EXP ZPP. Theorem 1.4 gives a stronger collapse. L

guage is in Modk P if there exists a polynomial-time nondeter- x ministic Turing machine M such that for all ,

Corollary 1.7 There exists an oracle A such that

x L M x mo d k

A A

A .

pol y

EXP ZPP P

x L M x mo d k .

We can generalize Theorem 1.4 to Modk P classes on two fronts.

Without loss of generality we can replace the “ ” in the ﬁrst A

Theorem 1.8 For every prime q there is an oracle such that for

condition by “ ” (see [Bei91]). q all k not a power of

We can detect unique solutions if for every nondeterministic

A A A A A A

Turing machine M there exists a language in P such that for all

q P Mod P and Modk P NP EXP

In particular we get a relativized world where both P P and

M x x A

Mod3 P EXP.

Homer, Kurtz and Royer [HKR93] show that if L is complete

M x x A

for the class EXP under 1-truth-table reductions then L is also

M x EXP-complete under the standard many-one reducibility. Buhrman, We put no conditions on A if . Note that detecting

Spaan and Torenvliet [BST93] show that the same result holds for unique solutions is a stronger restriction than P UP. NEXP. We give a give a relativized world where this collapse does Since the famous reduction of Cook [Coo71] preserves the num- not hold for NP. ber of solutions, detecting unique solutions is equivalent to solving

satisﬁability on formulae with at most one satisfying assignment. L Theorem 1.9 There exists an oracle B and a language that is In this paper we also consider different reductions between sets. 1-tt-complete for NPB but not m-complete.

Traditionally as deﬁned by Karp [Kar72], we say a set A is com-

A C L C plete for a class C if is in and for all , there exists a

In Theorem 1.9 and Corollary 1.6 we allow the reductions to access

x L f x A function f in FP such that is in if and only if is in . the oracle. To distinguish completeness notions we often use the term m-

completeness for this Karp deﬁnition. We say A is 1-li-complete if

Relativization

f can always be a length increasing injection. Most of the proofs in computational complexity theory relativize: Cook [Coo71] uses the notion of Turing-completeness where

The results hold even if all machines involved have access to the instead of a function f we have a polynomial-time Turing machine

x L M x A

same arbitrary oracle. Thus our paper shows that the relativized M such that is in if and only if accepts. We say is A

results described in this paper cannot be proven false in the unrela- 1-tt-complete if M can make only one query to .

B A B tivized world unless one uses nonrelativizing techniques. We say two sets A and are isomorphic if m-reduces to The only reasonable use of nonrelativizing techniques so far via a polynomial-time computable function that is one-to-one, onto has been in the area of interactive proof systems [LFKN92, Sha92, and polynomial-time invertible.

BFL91, ALM+ 92]. Complexity theorists have yet to ﬁnd many

Detecting Unique Solutions other interesting applications of these techniques. At this time we know no nonrelativizing techniques to handle the questions men- tioned in this paper. Valiant and Vazirani [VV86] show how to randomly map satisﬁable For a more thorough discussion on relativization see [For94]. formula to those with unique satisfying assignments.

Lemma 3.1 There exists a probabilistic polynomial-time function

Preliminaries

f such that for all boolean formulae in variables

f We assume the reader familiar with basic notations in complexity If SAT then is never satisﬁable.

theory and classes such as P and NP.

n f We use the class R to represent probabilistic polynomial-time If SAT then with probability at least , has

exactly one assignment. computation with one-sided error. The class ZPP R coR is

probabilistic computation with zero-sided error running in expected 2

f If one takes n independent applications of for some satisﬁ- polynomial time.

able then with extremely high probability one of these outputs

O (1)

We let EXP DTIME . will have a unique assignment. Theorem 1.1 follows directly from M Let represent the number of accepting computations of a Lemma 3.1.

nondeterministic Turing machine M . Valiant and Vazirani’s construction creates random subspaces

A language L is in UP if there exists a polynomial-time nonde- of the assignments. Mulmuley, Vazirani and Vazirani [MVV87] x terministic Turing machine M such that for all , give an alternate proof looking at the maximal weighted cliques after putting random weights on the edges. Buhrman and Fort- Lemma 4.3 relativizes so Homer and Selman tried to create an or-

now [BF97] show how Lemma 3.1 follows from earlier work by acle relative to which the isomorphism conjecture holds by getting Sipser [Sip83] on Kolmogorov complexity. Gupta [Gup97] gives P UP and NP EXP. They showed the following result. a construction for Lemma 3.1 that improves the probability to a

Theorem 4.4 (Homer-Selman) There exists an oracle relativize

constant if we only require f to have an odd number of assign-

to which P UP and EXP. ments. 2

Attempts at a relativized counterexample to Hypothesis 1.2 have p

Theorem 4.4 gives the ﬁrst relativized world where all -complete 2

a long history. Rackoff [Rac82] gives a relativized world where sets are isomorphic.

P UP NP but the proof heavily uses the fact that the UP Later, Fenner, Fortnow and Kurtz [FFK96] used a very different machines must have one accepting path for all inputs. approach to settle the relativized isomorphism conjecture. An easy application of Lemma 3.1 allows one to randomly find a satisfying assignment of a formula making nonadaptive queries Theorem 4.5 (Fenner-Fortnow-Kurtz) There exists a relativized to SAT. Buhrman and Thierauf [BT96] give a relativized world world where all NP-complete sets are polynomial-time isomorphic. where this fails deterministically. Theorem 1.4 gives the first relativized counterexample to Hy- The proof of Fenner, Fortnow and Kurtz requires a complicated, pothesis 1.2. In fact Theorem 1.4 shows a stronger result. Buss nonconstructive argument using a specialized form of generic ora- and Hay [BH91] and Wagner [Wag90] show that languages com- cles with infinite conditions. Relative to their oracle the polynomial-

putable with a polynomial number of nonadaptive queries to SAT time hierarchy is inﬁnite.

A A

Since for all , UP P , Theorem 1.4 combined with

log n are equivalent to those computable with O adaptive queries to SAT. For functions the equivalence would imply we can distin- Lemma 4.3 gives us an alternative proof of Theorem 4.5. Our proof guish unique solutions (see [BFT97]). Theorem 1.4 gives a rela- is considerably simpler, constructive and collapses the polynomial-

tivized world where the converse fails. time hierarchy to NP (Corollary 1.6).

Collapsing to ZPP and Pp oly Corollary 3.2 There exists a relativized world where we can distinguish unique solutions but there is a function computable with

a polynomial number of nonadaptive queries to SAT but not with Heller [Hel84] and Kurtz [Kur85] exhibit a relativized world col-

log n O adaptive queries. lapsing EXP to ZPP.

Proof: Combine Theorem 1.4 with the fact that for all rela- Theorem 5.1 (Heller-Kurtz) There exists an oracle A such that

A A tivized worlds, if all functions computable with a polynomial num- EXP ZPP .

ber of nonadaptive queries to SAT are equivalent to those com-

If P P then by Theorem 1.1 we have that R NP. Also by

log n

putable with O adaptive queries to SAT and NP coNP

standard padding arguments P P implies EXP EXP. If 2

then P NP (see [BFT97]).

also EXP NP then we have EXP R. Since EXP is closed

under complement, we have EXP ZPP.

The Isomorphism Conjecture This whole argument relativizes. Theorem 1.4 thus gives us a

strong improvement of Theorem 5.1 giving an oracle A such that A

Berman and Hartmanis [BH77] consider whether all NP-complete A

EXP ZPP (Corollary 1.7).

B B

sets are isomorphic. B

Since for all , ZPP BPP P /poly, we also get that

A A EXP P /poly. This gives a complementary result to an

Conjecture 4.1 (Berman-Hartmanis) Every pair of NP-complete C

NP C

sets are polynomial-time isomorphic. oracle C by Heller [Hel86] showing that EXP BPP PC /poly. Berman and Hartmanis [BH77] give a powerful tool to show

that sets are isomorphic.

Pro of of Main Theorem B Lemma 4.2 (Berman-Hartmanis) Sets A and are polynomial-

In this section we prove Theorem 1.4 showing an oracle A such

A A A

time isomorphic if there exist length-increasing polynomial-time A

that P P and NP EXP .

A B B

computable and invertible injections from to and from to A

Tor´an [Tor88] constructs the ﬁrst oracle A such that NP A

. A

P which also follows from Theorem 1.4. Tarui [Tar91] gives an Berman and Hartmanis [BH77] use Lemma 4.2 to show that the alternate proof of Tor´an’s result using the high degree of the OR

known natural NP-complete sets of the time were all isomorphic. function over GF[2]. This property of the OR function also plays

Proving Conjecture 4.1 would imply that P NP since otherwise an important role in our proof. A

we would have ﬁnite NP-complete sets isomorphic to inﬁnite ones. Let M be a nondeterministic linear time Turing machine such A

The Isomorphism Conjecture has been the subject of consider- that the language L deﬁned by

able research. We recommend the surveys by Joseph and Young

A A

M w mo d [JY90] and Kurtz, Mahaney and Royer [KMR90]. w L

Berman [Ber77] showed that every m-complete set for EXP is

A A

complete via one-to-one and length-increasing reductions. Groll- is P complete for every . We assume without loss of gener-

A n ality that M makes at most queries on any computation path, mann and Selman [GS88] show that P UP is equivalent to every length-increasing polynomial-time computable injection being guesses the answers to all oracle queries and veriﬁes the answers

polynomial-time invertible. Using these results Homer and Sel- nonadaptively at the end.

A n

man [HS92] realized an implication that would imply the isomor- Let N be a deterministic machine that runs in time and for

A A K

phism conjecture. all A accepts a language that is EXP complete. w We will construct A such that for all

Lemma 4.3 (Berman-Grollman-Selman-Homer) If P UP and

A jw j

L h w i A

NP EXP then all NP-complete sets are polynomial-time isomor- w (Condition 0)

A 2

K v jv j jw j h w v i A phic. w and (Condition 1)

u u m

Condition 0 will guarantee that P P and Condition 1 will Lemma 6.3 The function OR 1 as a multivariate poly- m guarantee that NP EXP. nomial over GF[2] requires degree exactly .

We will use the terms -strings for all of the strings of the form

w j

j Proof: Every function over GF[2] has a unique representation as

w i h w v i h and -strings for the strings of the form with

a multivariate multilinear polynomial.

v j jw j A j . All other strings we immediately put in . Note that AND is just the product so by using De Morgan’s First we give some intuition for the proof. Condition 0 will be laws we can write OR as

automatically fulﬁlled by just describing how we set the -strings Y

because they force the -strings as deﬁned by Condition 0.

u u u 2

1 m

OR i

Fulﬁlling Condition 1 requires a bit more care since N

1im

can query exponentially long - and -strings. We consider each -

h w v i y

w v i

string as a variable h whose value determines whether First let us discuss how to fulﬁll condition 1 in isolation. Let

h w v i A N x

y is in . We will show that the computation can r

xv i us assume that the only variables that depends on are h for

be represented by a low-degree polynomial over these variables in some v . We have two cases.

y the ﬁeld of two elements. To encode the computation properly we r

xv i Case (1): : We just set all the variables h .

use the fact that the OR function has high degree.

r t

Case (2): x : In this case there must be a such

We will assign a polynomial z over GF[2] to all of the -

r t r x

that x . Otherwise computes the OR function on the

z z

strings and -strings . We ensure that for all

3jxj

y r

xv i

h . But the degree of is at most and by Lemma 6.3

p z A 2

1. If z then is in .

jxj

the OR function has degree . We encode using this t.

p z A

r y w z

2. If then is not in . x

w v i

However may, of course, depend on variables h for

x r

. So we must encode x in stages.

z h w v i p

First for each -string we let z be the single

x y w x

w v i

Stage : We assume all the variables h for are

w v i

variable polynomial h .

s x r

encoded. For deﬁne the polynomials s as the polynomials

We assign polynomials to the -strings recursively. Note that

r y w x

w v i

where we replace the variables h for with their

jxj x jw j

M can only query -strings with . Consider

y w x

w v i

encoded value and variables h for with 0. The only

M x

an accepting computation path of (assuming the oracle

r y

hxv i

remaining variables of s are of the form .

q q

1 m

queries are guessed correctly). Let be the queries

x x

s x b r s

For each , let s . There are two cases again.

b b b

1 m i

on this path and be the query answers with

b y

hxv i

Case (1): x : In this case just set the variables to

A b if the query was guessed in and i otherwise. Note that

zero.

m .

Case (2): x . Consider the polynomial

M x

Let P be the set of accepting computation paths of . We

jxj

x x

p z h x i

then deﬁne the polynomial z for as follows:

r b R

sx s

OR s

X Y

b p p

i q

z (1)

R i

x has degree at most

P i:1im

jxj+1 3jsj jxj+1 3jxj 4jxj+1

max

Remember that we are working over GF[2] so addition is parity.

w v i

Setting the variables h (and thus the -strings) forces the

values of z for the -strings. We have set things up properly so the

b R x

By the deﬁnition of s we have . There must be some

following lemma is straightforward.

t R t R x

vector such that x . Otherwise computes the

jxj

z h x p i

Lemma 6.1 For each -string we have z OR function over variables.

M x mo d

and Condition 0 can be satisﬁed. The polynomial We use this as our encoding.

p jxj

z has degree at most . Theorem 1.4 follows from the following lemma.

xj 2

Proof: Simple proof by induction on j . Lemma 6.4 After all the stages, Condition 1 is fulﬁlled for every r

We would like to create a polynomial x that captures the value

x N x

of N . Consider a nondeterministic machine that simu-

N x

lates by ﬁrst guessing the oracle queries and verifying them Proof: We will show by induction that after every stage r

at the end. Similar to Equation (1), we can sum up over all the paths the polynomial x is properly encoded assuming that all unencoded

jxj

r jj

of the machine. We then deﬁne x by strings are zero. After stage for all the variables used

X Y r

by x have been encoded and will no longer change.

r b p

x i q

i Note that immediately after stage we have that the value of

r s r

P i:1im s

s for all assuming the unencoded string are zero.

x x

x b r x

where the terms have similar meaning as in Equation 1. Here we Base Case ( ): If x then we encode using

x x

jxj

b r R t

x x

so Condition 1 is satisﬁed. If x then since

have m .

x x

r t b t x

, x so by using we once again satisfy Condition 1.

r N x

Lemma 6.2 The value x is exactly when accepts and

x r

Inductive Case ( ): In stage we have x properly

3jxj

otherwise. The degree of x is at most . encoded assuming all of the unencoded strings are zero. Therefore

b r r

x x

x immediately before stage . If in stage we only N

Proof: Since N is deterministic, can have at most one accept-

R t t

ing path. encode with then nothing changes. If we use , since ,

r t b r 2

q N x

x x

To bound the degree note that the queries of made by we still have x as desired.

jxj jxj 2

have length at most so the degree of q is bounded by

jxj 3jxj 2jxj

m 2 . Since this gives us a total degree of . To properly encode to fulﬁll Condition 1, we need the following lemma about the OR function.

B B

Extension of Main Theorem jxj M x and accepts an NP -complete set under unrelativized

reductions.

B B

In this section we give a proof sketch showing that for any prime Fortnow and Rogers [FR94] show that NP coNP

q A k q

there is an oracle such that for all not a power of , P PSPACE for these oracles. Thus we have a polynomial p -

A A A A

M x

Mod P and Mod P NP EXP . time computable function f such that accepts if and only

k j k j j

For all and such that divides , Modk P Mod P (see

f x f B

if M rejects. The function does not depend on . q

[Bei91]). Thus we can assume that k is a prime different from .

pjxj C

Let m be the largest tower of 2 at most . Let be the

A A A

To get an oracle such that P Modq P and NP

m C

set of strings in B of length less than . Since every string in

m C

EXP is a simple variation of the proof in Section 6: One works has length at most log , we can enumerate in polynomial-time q

over GF[ ] instead of GF[2]. with access to B by brute-force search.

z m

We will encode EXP into Mod P similarly to the way we en- We call an input x good if the number of of length such

fz g m1

coded EXP into NP in Section 6. We will add the following Con- C

that M accepts is at least . Since P PSPACE we

k q dition k for each prime .

can determine whether an input x is good in polynomial time. B

Consider the set L deﬁned as

A 2

K jfv jv j jw j hk w v i Agj mo d k

w and

B B

fx M x x g

L accepts and is good

We will fulﬁll all of the conditions k for each string by

B B B

k w a standard dovetailing through pairs . We will only encode

We will show that L is in NP , 1-tt-hard for NP but not m-

k w k log log jw j

pairs where to keep the number of condi- complete where the reductions can access the oracle. B

tions we must fulﬁll at any length low. B M L is in NP since is a nondeterministic polynomial-time

To fulﬁll condition k without interfering with the other condi- Turing machine and testing goodness is in polynomial time.

tions we need to show that the Modk function has high degree over

f x z

If a string x is not good then is good since for all ,

fz g C fz g

GF[ ]. We will use the following lemma implicitly proven by Bar- C

f x M x

M accepts exactly when rejects. B

rington, Beigel and Rudich [BBR94]. B

M L x

We create a 1-tt reduction from L to as follows: If

B B

x L f x L

Lemma 7.1 (Barrington-Beigel-Rudich) Let r be a polynomial is good then accept if otherwise accept if .

B B

x x q r

L N in binary variables 1 . Let be a prime. Suppose that We still need to show that is not m-complete for NP . Con-

satisﬁes: sider the set

B m

r x x mo d q x x

N 1 N

1 if , and

f y m y B g

S There exists a of length such that

r x x mo d q x x q

N 1 N

1 if is a power of We show how to diagonalize over a potential many-one reduc-

B B

S L

tion g from to .

N q

Then the degree of r is at least .

B m

r g

Fix m a large tower of 2. Consider putting any un-

B m B

r mo d q r x x mo d q

g B jr j m M r N

If we have and 1 encoded query made by out of . If then

x x mo d k r

m N

whenever 1 then the degree of is at least cannot query any strings of length so we can easily diagonalize.

q N . Otherwise we have two cases:

This allows us to use the same kind of argument to fulﬁll Con- Case (1): r is not good: In this case we put some unencoded

m B

m B S

dition as we used to fulﬁll Condition 1 in Section 6. string y of length that starts with in . We have in

B L

but r is not in .

y m

Case (2): r is good: Since at least strings of length

ttcompleteness for NP

C fy g

r y

make M accept there must be such an unencoded that

y B r L

Homer, Kurtz and Royer [HKR93] show that every 1-tt-complete does not start with . Putting in gives us that but

m B

S 2 set for EXP is also m-complete. If one can show that there exists a ﬁnishing the proof. 1-tt-complete set for NP that is not m-complete this would separate

NP from EXP. Theorem 1.9 shows a relativized world where this wledgments possibility holds. Ackno Buhrman, Spaan and Torenvliet [BST93] show that every 1-tt- We thank Alexis Maciel for helpful discussions and Dieter van complete set for NEXP is also m-complete. Buhrman and Fort- Melkebeek for helpful comments on the write-up. now [BF96] give a relativized world where a 1-tt-complete set for

PSPACE is not m-complete. References

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