Portland, ME 04103, USA IL 60637, USA. Abstract. Buhrman and Torenvliet created an oracle relative to which

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1 Beyond P NP = NEXP Stephen A. Fenner 1? and Lance J. Fortnow 2?? 1 University of Southern Maine, Department of Computer Science 96 Falmouth St., Portland, ME 04103, USA fenner@usm.maine.edu, Fax: (207) University of Chicago, Department of Computer Science 1100 E. 58th St., Chicago, IL 60637, USA fortnow@cs.uchicago.edu, Fax: (312) Abstract. Buhrman and Torenvliet created an oracle relative to which P NP = NEXP and thus P NP = P NEXP. Their proof uses a delicate nite injury argument that leads to a nonrecursive oracle. We simplify their proof removing the injury to create a recursive oracle making P NP = NEXP. In addition, in our construction we can make P = UP = NP \ conp. This leads to the curious situation where LOW(NP) = P but LOW(P NP ) = NEXP, and the complete p m- degree for P NP collapses to a p-isomorphism type. 1 Introduction In 1978, Seiferas, Fischer and Meyer [SFM78] showed a very strong separation theorem for nondeterministic time: For time constructible t 1 (n) and t 2 (n), if t 1 (n + 1) = o(t 2 (n)) then NTIME(t 1 (n)) does not contain NTIME(t 2 (n)). Thus we have a huge gap between nondeterministic polynomial time (NP) and nondeterministic exponential time (NEXP). We would also expect then a separation between P NP and P NEXP. Indeed, we have some evidence for that direction: Mocas [Moc94] (improving upon work of Fu, Li an Zhong [FLZ94]) showed that for any xed c, NEXP is not contained in P NP if the polynomial-time machine can only ask n c queries to the NP oracle. Recently, Buhrman and Torenvliet [BT94] showed the existence of an oracle relative to which P NP = NEXP and thus P NP = P NEXP. This says that we can not prove a separation between P NP and P NEXP with relativizable techniques. They prove this result using a dicult nite injury argument where injuries may cascade through the oracle by more than any recursive bound. Their \construction" thus does not lead to a recursive oracle. In this paper, we simplify the Buhrman and Torenvliet proof. We use their proof as a starting block but remove the nite injury and replace it with an exponential (and thus recursive) search. Thus we end up with a recursive oracle relative to which P NP = NEXP.? Partially Supported by NSF Grant CCR ?? Partially Supported by NSF Grant CCR

2 We use a tree-based construction (see [Soa87, Chapter XIV]) where if we can get an injury from one leaf in a tree we can then throw away that leaf and start using the next one. We arrange our tree so that we have more leaves than injuries. Also, we can combine our construction with techniques developed by Racko [Rac82], Hartmanis and Hemachandra [HH91] and Blum and Impagliazzo [BI87] to also get P = UP = NP \ conp while also having P NP = NEXP relative to a recursive oracle. We eliminate machines that do not categorically accept on at most one path by forcing the acceptance on one of the leaves of our tree and then using a future leaf for encoding. The relativized world we create has two interesting properties. The rst is about low sets. The low sets of a class C consist of those oracles A such that C A = C. Essentially the low sets give no help to the class C. Relative to the oracle we construct, we get a surprising relationship between the low sets for NP and the low sets for P NP : The rst consists of exactly those languages in P while the second consists of those languages in NEXP. The second property of our relativized world is that the complete p m-degree for P NP collapses to a p-isomorphism type. That is, all P NP -complete sets under polynomial-time many-one reductions are polynomial-time isomorphic. This advances a program for getting complete degrees to collapse: if C NEXP is a class with complete sets, then one can make all C-complete sets p-isomorphic in a relativized world by setting P = UP and C = EXP. Homer and Selman [HS92] were able to do this for C = p 2, and our present oracle works for C = P NP. An oracle for C = NP would arm the Isomorphism Conjecture [BH77]. Even though oracles making the Conjecture true are known [FFK92], it is still an interesting open question whether there is an oracle relative to which P = UP and NP = EXP. 2 Notation and Denitions We let = f0; 1g and identify with the natural numbers 0; 1; 2; : : : in the usual way. We will sometimes use the divider symbol, \#", and in fact, our oracles will always be languages over the alphabet f0; 1; #g. Inputs to machines will always be taken as strings in, however. For any n 0, we let n (resp. f0; 1; #g n ) be the set of strings in (resp. f0; 1; #g ) of length n. We order strings in n or f0; 1; #g n lexicographically according to the order # < 0 < 1 on the symbols. We say \lex less than" for \lexicographically less than." We say a string w extends a string v if v is a prex of w. We will also deal with partial characteristic functions f0; 1; #g! f0; 1g with nite domain, which usually represent the portion of some oracle we are committed to so far. (We will build our oracles by nite extension at each stage.) We use lower-case Greek letters to denote these nite functions. We identify a language A over f0; 1; #g with its (total) characteristic function f0; 1; #g! f0; 1g. When dealing with partial functions, we use the relation \extends" in the usual functional sense.

3 Our notation for machines is standard. See, for example, [HU79]. The only peculiarity in the paper is when a nondeterministic oracle Turing machine M accesses a partial characteristic function as its oracle. We say that M (x) has an accepting path p if and only if p is an accepting path of M on input x where all queries along p are in domain() and are answered according to. M (x) accepts if it has some accepting path (even if other paths make queries outside domain()). 3 Main Results Theorem1. There is a recursive oracle A such that P NPA = NEXP A. Proof. Fix a nondeterministic oracle machine M that accepts some NEXP B - complete set for all oracles B. We may assume without loss of generality that M asks at most 2 n queries along any path, where n is the length of the input. We dene a relativized language L B as follows: consider any x 2 of length n. Let C x be the set of all strings of the form w 0 #w 1 # #w n #x#y; such that jw i j = i + 1 for all i n, and jyj = 2n + 3. (We identify each w i with a number in the range from 0 to 2 i+1? 1 in the usual way.) For any oracle B, we dene x 2 L B if and only if C x \ B 6= ; and the rightmost bit of the lexicographically maximum element of C x \ B is 1. We call C x the coding region of x. We can determine whether a given x is in L B by binary search, asking questions of the form, \Is there an element of C x \ B which is lex no less than z?" These questions are in NP B and we only need ask polynomially many of them, so it follows that L B 2 P NPB for all B. Our goal, then, is to construct a recursive A such that L A = L(M A ). We construct A in stages 0; 1; : : :. Currently, all the work will be done in the even stages; the odd stages will be used later in the proof of Theorem 2. First some notation: if and are nite characteristic functions and w is a string, then we say that is a w-extension of if extends and for all strings z 62 domain(), if (z) = 1 then z extends w. The nite function, below, represents the committed portion of the oracle so far. Initially, before stage 0, is the empty function, and grows by nite extension throughout the construction, so that, for any z, once (z) is dened it never changes. The numbers m i will always be identied with binary strings of length i + 1. From now on throughout the proof, w stands for the string m 0 #m 1 # #m 2n. Stage 2n / ; m 0 ; m 1 ; : : : ; m 2n?1 are given. / Phase 1: Forcing Accepting Computations Let m 2n = 0. Let S = n.

4 Do the following repeatedly until no such x exists: Find some x 2 S such that there is a w-extension of making M (x) accept. If x exists, then (i) x such a, (ii) let p be some accepting path of M (x), (iii) remove x from S, (iv) extend to include the query answers along p, and (v) increment m 2n. Phase 2: Coding For each x of length n (order is unimportant): Let y x be the lex least y 2 2n+2 such that neither w#x#y0 nor w#x#y1 is in domain(). [We show later that y x always exists.] If M (x) accepts, set (w#x#y x 1) = 1. Otherwise, set (w#x#y x 0) = 1. Phase 3: Cleanup Set (z) = 0 for all z 2 S x2 n C x? domain(). Set (z) = 0 for all z 2 f0; 1; #g n? domain(). End stage 2n. Stage 2n + 1 Let m 2n+1 = 0. End stage 2n + 1. Note that in the Forcing phase, m 2n is incremented at most 2 n times, so there is no possibility of overow. In the Coding phase, for each x, the unique string z for which (z) is set to 1 we will call x's coding string. Dene the oracle A f0; 1; #g such that w 2 A if and only if (w) is set to 1 at some stage. The three claims below nish the proof. Claim 1. A is recursive. Proof. The stage-by-stage construction is recursive, is nite at every stage, and every z 2 f0; 1; #g enters domain() at some stage. ut Claim 1 Claim 2. The string y x exists for all x 2. Proof. Fix x of length n. It suces to show that at the start of the Coding phase of Stage 2n, jdomain() \ C x j < 2 2n+2. Note that all strings in C x are strictly longer than n, and all the other coding regions are disjoint from C x. This means that we can ignore the elements entering domain() in the Coding and Cleanup phases of stages 0 through 2n? 1. That is, the strings contributing to jdomain() \ C x j must have entered domain() during Phase 1 of these stages. Taken together, these phases add at most 2 n queries to domain() for each x 0 with jx 0 j n. This makes at most 2 n (2 n+1? 1) < 2 2n+2 elements in all. (The overkill in the bound will be useful in the proof of Theorem 2.) ut Claim 2

5 Claim 3. For all x of length n, x 2 L A if and only if M A (x) accepts. Proof. Let ; m 0 ; : : : ; m 2n be as at the end of phase 1 of stage 2n. We need to show two things: (1) the coding string of x is the lex maximum element of C x \ A, and (2) the accept/reject behavior of M (x) at the Coding phase is not disturbed by any later changes to, i.e., M (x) = M A (x). No strings enter A in the Cleanup phases of any stage. The coding strings for all x 0 6= x are not in C x, and hence do not aect (1). Also, the Cleanup phase of stage 2n ensures that no strings enter C x \ A after stage 2n. What is left is the Forcing phases of stages 0; : : : ; 2n. If a string enters A during the Forcing phase of some stage n 0 2n, then it must be some query extending m 0 # m n0?1#m made along some accepting path of M, for some m < m n 0. This is because all queries not extending this prex were answered \no" and kept out of A (except if they were already in domain()), and m was incremented immediately afterwards. Thus any strings in C x that enter A are lex less than x's coding string. This shows (1). If M (x) accepts, then M A (x) accepts because A extends. Suppose M (x) rejects. Since the loop in phase 1 has ended, there is no w-extension of such that M (x) accepts. It is clear by the construction that all strings put into A beyond this point in the construction have w as a prex. Therefore, M A (x) must still reject. ut Claim 3 ut Theorem 1 We can now esh out the odd stages of the construction above to get P A = UP A and P A = NP A \ conp A as well. For simplicity, we only show how to get P A = UP A here. Also getting P A = NP A \ conp A is accomplished in an entirely similar way, which we describe briey afterwards. Theorem2. There is a recursive oracle A such that P NPA P A = UP A. = NEXP A and Proof. We build A in stages just as in the proof of Theorem 1. In fact, the even-numbered stages are exactly as they were before, but we now use the oddnumbered stages to force NP machines to be illegal as UP machines (i.e., to have more than one accepting path on some input). This is done very much like the forcing of acceptances of M in the even stages. Fix a standard enumeration N 1 ; N 2 ; : : : of all ptime nondeterministic oracle TMs, where N i runs in time n i. The set-up for the construction is identical with that of Theorem 1. In stage 2n + 1 we always use w to denote the string m 0 #m 1 # #m 2n+1. Stage 2n This is identical with stage 2n of Theorem 1. End stage 2n.

6 Stage 2n + 1 Let m 2n+1 = 0. For i = 1 to n: Find the least string y with jyj 2 n=i such that there is a w-extension of where N i (y) has at least two accepting paths p 1 and p 2. If y,, p 1, and p 2 exist, then: (i) extend to include the query answers along p 1 and p 2, (ii) increment m 2n+1. End stage 2n + 1. The oracle A is dened as before, and it is clearly recursive. The proof that P NPA = NEXP A requires little change from the previous proof: Since the odd stages closely resemble the Forcing phases of the even stages, the proof of Claim 3 still applies in the present case. We only need check that Claim 2 is still valid by bounding jdomain() \ C x j at the start of the Coding phase of stage 2n. At most 2 n (2 n+1?1) strings are contributed by the even stages, and each odd stage contributes at most n2jyj i n2 n+1 strings. There are n odd stages before stage 2n, so we have jdomain() \ C x j 2 n (2 n+1? 1) + n 2 2 n+1 < 2 2n+2 : Therefore, Claim 2 holds. It remains to show that P A = UP A. First, it is important to note that this entire proof, including the construction, can be recast relative to an oracle that makes P = PSPACE, e.g., some PSPACE-complete set H. This recasting clearly alters the denition of A, but not the facts we establish. We will therefore assume for the rest of the proof that P = PSPACE (unrelativized). We can discharge this assumption by recasting everything relative to H. Let i be such that Ni A is a UP A machine. We describe a P A procedure to compute L(Ni A ). For input x, let n = i dlog jxje and run the construction through stage 2n + 1, obtaining an and w = m 0 # m 2n+1 at the end of this stage. Note that and w can be computed and stored using polynomial space. 3 Hence and w can be computed in time polynomial in jxj. Moreover, by the construction and the fact that Ni A is a UP A machine, N i (x) has at most one accepting path for every which is a w-extension of. We can now adapt a standard technique developed and used by Racko [Rac82], Hartmanis & Hemachandra [HH91], and Blum & Impagliazzo [BI87] to compute Ni A(x). ALGORITHM FOR Ni A(x). Compute and w as above. Repeat the following jxj i times: If there is a w-extension of where N i (x) accepts, then: 3 Actually, jdomain()j is superpolynomial by virtue of the Cleanup phases. We can compress, however, by only retaining the single coding string, and omitting the other strings added in the Cleanup phase.

7 Let p be some accepting path of N i (x). Extend to include all queries along p answered according to A. Accept i Ni (x) accepts. END OF ALGORITHM. The algorithm can clearly be run using polynomial space, and hence polynomial time by our assumptions. Note that throughout the algorithm, agrees with A. Thus if Ni A (x) rejects, then the algorithm rejects. Suppose Ni A(x) accepts along the unique path p?. At the start of any given loop iteration, suppose that domain() does not include all the queries made along p?. Some and p must be found in the iteration, since A is itself a w-extension of by our construction. If p = p?, then the algorithm will clearly accept. If p 6= p?, then they must share a common query that is answered dierently along each path; otherwise, there is a single w- extension of yielding two distinct accepting paths for N i (x), which violates our construction. If q is such a common query, then q 62 domain() before the iteration but enters domain() during the iteration. Thus, the number of queries along p? not in domain() decreases by one on each iteration. Since there are at most jxj i queries along p?, by the end of the loop includes all such queries, and the algorithm accepts. ut To get P A = NP A \conp A, we can easily interleave P A = UP A stages with P A = NP A \ conp A stages. In the latter stages, which are handled similarly, we consider each Ni X as a proper NP X \ conp X machine if, for each input, it has at least one accepting path, and either all accepting paths start with a left branch or they all start with a right branch. Ni X accepts an input i it has an accepting path starting with a left branch. In the P A = NP A \ conp A stages of the construction we look for w-extensions that force both types of accepting path for the same input. In corresponding P A algorithm, the main dierence is that we now search for pairs of w-extensions that yield both types of accepting path. We thus have the following theorem: Theorem3. There is a recursive oracle A such that P NPA P A = UP A = NP A \ conp A. = NEXP A and 3.1 Lowness and Collapsing Degrees For any relativizable class C, we let LOW(C) denote the class of all C-low sets, i.e., the class of all B such that C B = C. This concept itself relativizes, i.e., we say a set B is low for C relative to an oracle A if C AB = C A. Since the classes NP and P NP are \close" to each other, one might expect that LOW(NP) should likewise be close to LOW(P NP ), but instead we get the following curious lowness property relative to the oracle A of Theorem 3: Corollary4. Relative to the set A of Theorem 3, LOW(NP) = P and LOW(P NP ) = NEXP:

8 Proof. It is well-known (see [Sel79]) that LOW(NP) = NP \ conp, and the proof relativizes; thus, LOW(NP) = P relative to A. Moreover, relative to A we have (P NP ) NEXP = (P NP ) PNP PH NEXP = P NP ; and thus NEXP LOW(P NP ) P NP = NEXP. Homer and Selman [HS92] produced an oracle relative to which P = UP and p = EXP, and thus the complete 2 p m -degree for p 2 collapses. The oracle A of Theorem 3 brings the collapse further down in the polynomial hierarchy. Corollary 5. Relative to the set A of Theorem 3, all p m-complete sets for PNP are polynomial-time isomorphic. Proof. Berman [Ber77] showed via a relativizable proof that the complete p m - degree for EXP collapses to a 1-li-degree (one-to-one, length-increasing m-reductions). By results in [GS88, KLD87], all 1-li-degrees collapse if and only if P = UP, again by a relativizable proof. Relative to A, we have P = UP and P NP = EXP. ut ut 4 Open Questions Of course we would like to see the P NP = NEXP question answered in the unrelativized world. We conjecture that the classes are in fact dierent but such a proof will require vastly new techniques. In the relativized setting, we would like to see an oracle relative to which P NP = NEXP but P = BPP. We think a proof similar to the proof of Theorem 2 may work but trying to force all BPP machines to be categorical may put too many strings in the oracle. Is there an oracle relative to which P = UP and NP = EXP? 5 Acknowledgment We would like to thank Harry Buhrman for Corollary 5. References [Ber77] [BH77] [BI87] L. Berman. Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, L. Berman and J. Hartmanis. On isomorphism and density of NP and other complete sets. SIAM Journal on Computing, 1:305{322, M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages 118{126, New York, IEEE.

9 [BT94] H. Buhrman and L. Torenvliet. On the cutting edge of relativization: The resource bounded injury method. In Proceedings of the 21st International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science. Springer-Verlag, To appear. [FFK92] S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 30{39, To appear in SIAM J. Comp. [FLZ94] B. Fu, H. Li, and Y. Zhong. An application of the translational method. Mathematical Systems Theory, 27:183{186, [GS88] J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17:309{335, [HH91] J. Hartmanis and L. Hemachandra. One-way functions and the nonisomorphism of NP-complete sets. Theoretical Computer Science, 81(1):155{163, [HS92] S. Homer and A. Selman. Oracles for structural properties: The isomorphism problem and public-key cryptography. Journal of Computer and System Sciences, 44(2):287{301, [HU79] J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading, Mass., [KLD87] K. Ko, T. Long, and D. Du. A note on one-way functions and polynomialtime isomorphisms. Theoretical Computer Science, 47:263{276, [Moc94] S. Mocas. Using bounded query classes to separate classes in the exponential time hierarchy from classes in PH. In Proceedings of the 9th IEEE Structure in Complexity Theory Conference, pages 53{58. IEEE, New York, [Rac82] C. Racko. Relativized questions involving probabilistic algorithms. Journal of the ACM, 29(1):261{268, [Sel79] A. Selman. P-selective sets, tally languages, and the behavior of polynomialtime reducibilities on NP. Mathematical Systems Theory, 13:55{65, [SFM78] J. Seiferas, M. Fischer, and A. Meyer. Separating nondeterministic time complexity classes. Journal of the ACM, 25(1):146{167, [Soa87] R. I. Soare. Recursively Enumerable Sets and Degrees. Springer-Verlag, Berlin, This article was processed using the LaT E X macro package with LLNCS style