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: [email protected]. 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 NP 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 p xj 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 A 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) k 2 n fhx i j x Ag A use time we can use padding to reduce the language to n a verifier that uses time. NP 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 .