Nonrelativizing Separations

y z Harry Buhrman Lance Fortnow Thomas Thierauf CWI Dept. of Computer Science Abt. Theoretische Informatik University of Chicago Universit¨at Ulm PO Box 94079 1100 E. 58th St. Oberer Eselsberg 1090 GB Amsterdam Chicago, IL 60637 89069 Ulm The Netherlands USA Germany

Abstract helpus decide where andhow toput oureffortsintosolving

We show that MAEXP, the exponential time version of problems in complexity theory. It is still true that virtually the Merlin-Arthurclass, does not have polynomial size cir- all of the theorems in computational complexity theory that cuits. This signiﬁcantlyimproves the previous known result have reasonable relativizations do relativize (see [For94]). due to Kannan since we furthermore show that our result But we do have a small number of exceptions that arise does not relativize. This is the ﬁrst separation result in com- from the area of interactive proofs. These results have pre- plexity theory that does not relativize. As a corollary to our viously always taken the form of collapses such as IP

separation result we also obtain that PEXP, the exponen- PSPACE [LFKN92, Sha92], MIP NEXP [BFL91] and

O O log n tial time version of PP is not in Ppol y . PCP NP [ALM 92]. In this paper we give the ﬁrst reasonable nonrel- 1 Introduction ativizing separation results. Namely, we show that

Baker, Gill and Solovay [BGS75] noticed that the re- there exist languages in MAEXP, a one-round Merlin- sults proven in computational complexity theory relativize, Arthur game [BM88] with an exponential-time veriﬁer,

i.e. the proofs go through virtually unchanged if all of the that cannot have polynomial-size circuits, in other words, pol y machines involved have access to the same information via MAEXP P . On the other hand, we then cre-

an oracle. They then developed relativized worlds where ate a relativized world where every MAEXP language has

pol y

P NP and P NP and thus argued that the current tech- polynomial-size circuits, i.e., MAEXP P . niques in complexity theory could not settle this question. Paul, Pippenger, Szemer´edi and Trotter [PPST83] show More than two decades later, relativization still plays a separation of nondeterministiclinear time from determin- in important role in complexity theory. Though no longer istic linear time where one can create a relativized world believed to be any form of an independence result, they do where these two classes coincide. However, both the sepa-

ration and the oracle heavily depend on the machine model Email: buhrman@cwi.nl. URL: http://www.cwi.nl/˜buhrman. Par- where our results are model independent. tially supported by the Dutch foundation for scientiﬁc research (NWO) by SION project 612-34-002, and by the European Union through Neu- We can strengthen our oracle to work for MIPEXP, the roCOLT ESPRIT Working Group Nr. 8556, and HC&M grant nr. languages accepted by multiple prover interactive proof ERB4050PL93-0516. systems [BGKW88] with an exponential-time veriﬁer. In y Email: [email protected]. URL: http://www.cs.uchicago.edu/˜fortnow. Work done while on leave at fact, besides having small circuits, relative to our oracle NP

CWI. Email: [email protected]. Supported in part by NSF grant MIPEXP languages can be accepted in P and in P. CCR 92-53582, the Dutch Foundationfor Scientiﬁc Research (NWO) and Since these classes are contained in MIPEXP, we get a rel- a Fulbright Scholar award.

z Email: [email protected]. URL: http://hermes.informatik.uni-ulm.de/ti/Personen/tt.html.

M x

ativized world where Let represent the number of accepting compu-

x tations of a nondeterministic Turing machine M , and

pol y

MIPEXP P P P

M x M x

R the number of rejecting computations of . A language L is in P if there exists a polynomial-time

This contrasts greatly with the situation in the unrelativized x nondeterministic Turing machine M such that for all ,

world where we have

L M x

NP x is odd

P P PSPACE

EXPSPACE

A language L is in PP if there exists a polynomial-time

M x

NEEXP nondeterministic Turing machine such that for all ,

MIPEXP

x L M x M x R where NEEXP is nondeterministic double exponential PEXP is the exponential-time version of PP, i.e., the time. polynomial-time nondeterministic Turing machine in the

Our proof that MAEXP does not lie in P pol y uses the deﬁnition of PP is nondeterministic exponential-time. result of Babai, Fortnow, Nisan and Wigderson [BFNW93]

A language L is in MA if there exists a probabilistic

that if EXP has polynomial-size circuits then EXP MA.

q n polynomial time Turing machine M and a polynomial Since this is the only part of the proof that does not rel- such that

ativize, our paper gives the ﬁrst oracle where this Babai-

q (jxj)

x L y PrM x y Fortnow-Nisan-Wigderson result does not relativize. accepts , As a corollaryto our separation result we also obtainthe and

separation between PEXP, the exponential time version of

q (jxj)

x L y Pr M x y accepts .

PP, and P pol y . Finding nonrelativizing results like these allows us to better understand the importance and limita- This corresponds to the Merlin-Arthur games due to Babai tions of relativization and gives us new ideas on how to and Moran [BM88]. prove other nonrelativizing results. We also consider multiple-prover interactive proof sys- 2 Preliminaries tems as developed by Ben-Or, Goldwasser, Kilian and

Wigderson [BGKW88]. In this model, the veriﬁer can ask

hx x i k We use 1 to be any polynomial-time com- questions of several provers that are unable to communi- putable and invertible tupling function. For a language A,

cate with each other. Babai, Fortnow and Lund [BFL91]

A we denote the characteristic function of A as . We assume the reader familiar with basic notations in show that the class, MIP, of languages provable by such

complexity theory and classes such as P and NP. We let systems is equal to NEXP.

O (1) O (1) n

n We use MAEXP and MIPEXP to represent the Merlin-

EXP DTIME and NEXP NTIME . Arthur games and multiple-prover interactive proof sys-

The class Ppol y consists of languages accepted by

k n a family of polynomial-size circuits or equivalently a tems where we allow the veriﬁer to use time for some polynomial-time Turing machine given a polynomially- ﬁxed k . In particular, the provers can send messages up to

length advice that depends only on the length of the input. this length to the veriﬁer.

pol y A

More formally, L P if there exist P and a 3 A Nonrelativizable Separation

N polynomially length bounded function h such The computational power of polynomial-size circuits,

that for all x

Ppol y , is an interesting issue. In particular, whether one

can solve all sets in NP within Ppol y is still an open ques-

x L hx hjxji A

tion, although there are strong indications that this is not

jxj

The value h is called the advice for strings of length possible: otherwise the polynomial hierarchy collapses to

j . [KL80]. 2

2 With respect to absolute separations, Kannan [Kan82] But this contradicts Theorem 3.1. 2 showed that there are sets in NEXPNP that cannot be com- The same proof works for MAEXP coMAEXP instead puted by polynomial-size circuits. of just MAEXP. Then we need the form of Kannan’s result Theorem 3.1 (Kannan) as stated in Theorem 3.1.

NP NP

pol y

NEXP coNEXP P .

pol y Corollary 3.5 MAEXP coMAEXP P . We improve Kannan’s result and show that there are sets Vereshchagin [Ver92] shows that MA is in PP. By in MA that cannot be computed by polynomial-sizecir- EXP padding analogously to the proof of Lemma 3.3 this im- cuits. We use the followingresult of Babai, Fortnow, Nisan

plies that MAEXP PEXP.

and Wigderson [BFNW93]. pol y Corollary 3.6 PEXP P .

Theorem 3.2 (BFNW)

pol y EXP P EXP MA. 4 Relativized Collapse In order to prove the separation result we will also need In this section we show that our separation result from the following lemma. the previous section does not relativize. This is the ﬁrst known example of a non-relativizing separation result. Lemma 3.3

NP NP

NP MA NEXP MAEXP. Theorem 4.1 There exists an oracle such that A

NP A

pol y Proof: Let A be a set in NEXP , and let this be MAEXP P

witnessed by an alternating Turing machine that runs in M

Proof: Let i be an enumeration of potential veriﬁers,

p(n) p

time for some polynomial . Consider the following n

i.e., probabilisticTuring machines, that run in time . Itis padded version of A:

sufﬁcient to encode just these machines: for veriﬁers that

p(jxj)

fhx i j x Ag

A use time we can use padding to reduce the language to n

a veriﬁer that uses time.

It follows that A is in NP and by assumption is in MA. We ﬁrst describe how to encode inputs of a single This however impliesthat removing the padding yields that

length. Later we will show how to combine lengths. For

2 A MAEXP.

inputsof length n, we will encode languages proven to ver- M

We are now ready to prove our separation result. M n

iﬁers 1 .

hr i xi

The strings in our oracle A will be of the form , pol y

Theorem 3.4 MAEXP P .

5n n

i n x

where r , and . Consider the pol y Proof: Assume to the contrary that MAEXP P . following requirement:

Since EXP MAEXP it follows that

A 2

x y Pr M R y ix

r accepts

pol y

EXP P

hr i xi A

and we can apply Theorem 3.2. Hence, we have i Our construction of A will guarantee that for all , one of

the following two conditions is fulﬁlled:

EXP MA

x NP (i) either for some n and ,

Since NP EXP, we get that

NP A

Pr M x y max accepts

NP MA i

By Lemma 3.3 this yields

r (ii) or, for all n, there is some such that for all

NP n

pol y x R ix NEXP MAEXP P , r holds.

hi xi A

Note that this sufﬁces to prove the theorem. The r Case 1: tuple is not forced. Let be the value A

will act as the advice for strings of length n. To accept of right before the last stage of the construction. Since

M hi xi B S y r

a language in MAEXP with veriﬁer i , a polynomial-time is notforced, thereis no set and that causes

A B

x y A M A A B

machine on input x just needs to ask the oracle about to accept. But the ﬁnal does equal

hr i xi B S M x y

. for some r . So still will not accept and

S fhr i xi j i n x g

Let r . Dur- remains properly encoded. r

ing the course of the construction we will mark some

i xi A A

Case 2: tuple h is forced. Let be the value of

hr i xi

as frozen meaning that we guarantee that A will

i xi

at the end of the stage when h was forced. We have

hi xi

not change. We will also mark tuples as forced if we A

x y M

that some y will cause to accept. Each stage

x y y

guarantee that M accepts for some .

A S

can only add strings to in one r and there are at most

The construction works in stages. Initially, in stage n

hi xi

n stages. After the stage where was forced,

A r hi xi

, , none of the is frozen and none of the are S

we will only add strings in r that affect the probability 1

forced. A

M x y n

of accepting by at most 2 . Thus the total

62 r

Stage s. Pick the ﬁrst unfrozen . Does there exist an probability of acceptance can go down by at most

hi xi B S y

unforced , a set r , and a such that

Pr M x y

accepts

A y

Hence for the ﬁnal oracle we have that causes

x y

M to accept with probability at least 1/2. Thus we

S r

If not, then encode all stringsin r with this accord-

have

r i xi

ing to condition (ii) from above. That is, put h

max Pr M x y

A hi xi

into for all forced . Then halt the construction. 2

This r will serve as the advice. fulﬁlling either condition (i) or (ii).

hi xi B y A

Otherwise pick the ﬁrst such , and . Let We still need to make our oracle work for all input

A B hi xi r r r r r

2 n

, mark as forced and freeze . lengths. For this, we replace with 1 ,

i n r r A

where, for , i plays the role of for strings of length

x y

Consider the computation paths of M for this

i r 2

, and n is what we vary for this proof. T

new A. Assume we have paths each occurring with

p Q

equal probability. For each path let p be set of We can generalize this result.

queries made along this path. Freeze all of the r such Theorem 4.2 There exists a relativized world where that

T NP

jfp j S Q gj

pol y p

r (1) MIPEXP P P P

and go to the next stage. Proof Sketch: To get MIPEXP, we just replace the single

prover message y in the above proof by the strategy of the We need to show that the construction will terminate provers.

and that either condition (i) or (ii) is fulﬁlled. A

M x

To get P we encode accept if there are an odd

Since the sets r are disjoint and there are at most

hr i xi A

number of r such that is in . n

3 NP

queries on each path, by Equation (1) at most

To get P we pick our r in lexicographical order and A

n NP

n r ’s can be frozen in each stage. There are tuples so at

then ﬁnd r in P by using binary search to ﬁnd the

3n n 5n

n r

most ’s are frozen. Since we have

r i xi A r

largest tuple h in . We then use as our advice.

r hi xi

’s, we will run out of tuples before we run out 2

of r ’s. i

Now we argue that for each input x and machine we 5 Conclusion

r i xi r properly encoded h using the from the construction We give the ﬁrst reasonable nonrelativizing separation above. We have two cases. showing that MAEXP is not computable by polynomial-size

4 circuits. We believe our techniques give us a foot in the [KL80] R. Karp and R. Lipton. Some connections door that may open to many other exciting separations. between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Sym- References posium on the Theory of Computing, pages [ALM+ 92] S. Arora, C. Lund, R. Motwani, M. Sudan, 302–309. ACM, New York, 1980. and M. Szegedy. Proof veriﬁcation and hard- ness of approximation problems. In Proceed- [LFKN92] C. Lund, L. Fortnow, H. Karloff, and ings of the 33rd IEEE Symposium on Foun- N. Nisan. Algebraic methods for interac- dations of Computer Science, pages 14–23. tive proof systems. Journal of the ACM, IEEE, New York, 1992. 39(4):859–868, 1992. [PPST83] W. Paul, N. Pippenger, E. Szemer´edi, and [BFL91] L. Babai, L. Fortnow, and C. Lund. Non- W. Trotter. On determinism versus non- deterministicexponential time has two-prover determinism and related problems. In Pro- interactive protocols. Computational Com- ceedings of the 24th IEEE Symposium on plexity, 1(1):3–40, 1991. Foundations of Computer Science, pages [BFNW93] L. Babai, L. Fortnow, N. Nisan, and 429–438. IEEE, New York, 1983. A. Wigderson. BPP has subexponential sim- [Sha92] A. Shamir. IP = PSPACE. Journal of the ulations unless EXPTIME has publishable ACM, 39(4):869–877, 1992. proofs. Computational Complexity, 3:307– 318, 1993. [Ver92] N. Vereshchagin. On the power of PP. In Pro- ceedings of the 7th IEEE Structure in Com- [BGKW88] M. Ben-Or, S. Goldwasser, J. Kilian, and plexity Theory Conference, pages 138–143. A. Wigderson. Multi-prover interactive IEEE, New York, 1992. proofs: How to remove intractability assump- tions. In Proceedings of the 20th ACM Sym- posium on the Theory of Computing, pages 113–131. ACM, New York, 1988.

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