Report on the Dagstuhl - Seminar \ Structure and Complexity "

S On the Power of Randomized Branching Programs Farid Ablayev Kazan University (joint work with Marek Karpinski, Universitat Bonn) We de ne a notion of randomized branching programs in a natural way similar to the notion of randomized circuits. We present two explicit boolean functions fn : f0; 1g4n ! f0; 1g and gn : f0; 1gn ! f0; 1g such that: 1. fn can be computed by a randomized ordered read-once branching program of size polynomial in n and with a small (constant) error, 2. any nondeterministic ordered read-k-times branching program that computes fn needs exponential size (the size is (exp(n=(2k 1)))), 3. gn can be computed by a nondeterministic read-once branching program of size polynomial in n, and 4. any randomized ordered read-once branching program that computes gn with a constant error has size no less than exp(c( )n= logn). Recent Progress on the Isomorphism Conjecture Eric Allender Rutgers University http://www.cs.rutgers.edu/~allender (joint work with Manindra Agrawal and Steven Rudich) In this talk I will discuss recent progress by Manindra Agrawal, Steven Rudich, and myself, showing unexpected similarities in the structure of complete sets. We show that for any complexity class C closed under many-one 7 reductions computable in uniform TC0, the following are true: Gap: The sets that are complete for C under AC0 and NC0 reducibility coincide. Isomorphism: The sets complete for C under AC0 reductions are all isomorphic under isomorphisms computable and invertible by AC0 circuits of depth three. Our Gap Theorem does not hold for strongly uniform reductions: we show that there are Dlogtime-uniform AC0-complete sets for NC1 that are not Dlogtime-uniform NC0-complete. An important open problem is the question of whether there is any natural complexity class such that the sets complete under polynomial-time (or more powerful) reductions are not already complete under NC0 reductions. (This work extends the paper by Agrawal and myself in the 1996 Computational Complexity conference. A full paper is available at http://www.cs.rutgers.edu/~allender/publications.) Kolmogorov-Easy Circuit Expressions Jos e L. Balc azar Universitat Polit ecnica de Catalunya, Barcelona http://www-lsi.upc.es/~balqui (joint work with Harry Buhrman and Montserrat Hermo) Circuit expressions were introduced to provide a natural link between Computational Learning and certain aspects of Structural Complexity. Upper and lower bounds on the learnability of circuit expressions are known. We study here the case in which the circuit expressions are of low (timebounded) Kolmogorov complexity. We show that these are polynomial-time learnable if and only if accepting computations of accepting nondeterministic exponential-time machines can be found deterministically in exponential time. We also exactly characterize, in terms of advice classes and of doubly tally polynomial-time degrees, the sets that have such easy circuit expressions, and obtain consequences regarding their lowness. (A full paper is available at http://www-lsi.upc.es/~balqui/postscript/q-charact.ps.) 8 Looking for an Analogue of Rice's Theorem in Complexity Theory Bernd Borchert Universitat Heidelberg http://math.uni-heidelberg.de/logic/bb/bb.html (joint work with Frank Stephan, Universitat Heidelberg) Rice's Theorem says that every nontrivial semantical property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions. (A full paper is available at http://math.uni-heidelberg.de/logic/bb/papers/Rice.ps.) Six Hypotheses Harry Buhrman CWI Amsterdam (joint work with Lance Fortnow, University of Chicago) We consider the following six hypotheses: (1) P = NP, (2) SAT ptt PSEL, (3) SAT ptt K-APPROX, (4) SAT is O(logn)-APPROX, (5) FPNP[log] = FPNP j j , and (6) (1SAT; SAT) 2 P. It is known that (1) =) (2) =) (3) =) [(4)&(5)] =) (6). We show the following: 9 1. If (NP) 6= 0, then (1)|(5) are false. 2. If (NP) 6= 0 and symmetry of information w.r.t. polynomial-time CDpoly holds (i.e., (8x; y) [CDp(xy) = CDp(x)+CDp(y j x)+O(logn)]), then (1)|(6) are false. 3. Symmetry of information alone implies (1) () (5). 4. Consider the conjecture that there exists a hierarchy theorem for compressibility: For each polynomial p1 there exists a polynomial p2 such that for every n there exists a string x of length n satisfying CDp2(x) O(logn) & CDp1(x) n O(logn): The conjecture implies that (a) R P and (b) (1) () (6). The proofs use essentially an upper bound lemma for CD complexity and a construction of Zuckerman. A Variant of Immunity, Btt-Reductions, and Minimal Programs Stephen Fenner University of Southern Maine We work with a generalization of immunity called k-immunity (k 1). It is shown that if a set A is k-immune, then K 6 k-tt A, where K is the halting problem. It is also shown that MIN, the set of minimal indices for partial computable functions, is k-immune for all k, and hence K 6 btt MIN (MIN df = fe j (8i < e) [ i 6= e]g). A short history of MIN is given, and it is mentioned that MIN is not regressive, all regressive sets are either computably enumerable or k-immune for all k, and every computably enumerable non-computable T-degree contains a k-immune, non-(k+1)-immune set, for all k. (Current sources are two technical reports found on the WWW at http://www.cs.usm.maine.edu.) 10 Two Queries Lance Fortnow CWI, Amsterdam, and The University of Chicago http://www.cs.uchicago.edu/~fortnow (joint work with Harry Buhrman at CWI) Hemaspaandra, Hemaspaandra and Hempel showed that for k > 2, if P pk[2] tt = P pk[1] then pk = pk. We extend their techniques to show that if P p2 [2] tt = P p2[1] then p2 = p2. However, the techniques cannot be pushed down to k = 1. We show a relativized world where PNP[2] tt = PNP[1] but NP 6= coNP. What does happen when PNP[2] tt = PNP[1]? Kadin shows that NP coNP=poly and thus PH p3 (Yap). Beigel, Chang and Ogihara building on Chang and Kadin improve this to show that every language in the polynomial-time hierarchy can be solved by an NP query and an p2 query. Building on the techniques of the above papers we show several new collapses if PNP[2] tt = PNP[1] including: Locally either NP = coNP or NP has polynomial-size circuits. PNP = PNP[1]. p2 = UPNP[1] \ RPNP[1]. PH = BPPNP[1]. (The paper can be found at http://www.cs.uchicago.edu/~fortnow/papers.) 11 The Complexity of Deterministically Observable Finite-Horizon Markov Decision Processes Judy Goldsmith University of Kentucky http://www.cs.engr.uky.edu/~goldsmit (joint work with Martin Mundhenk, Universitat Trier, and Chris Lusena, University of Kentucky) We consider the complexity of the decision problem for di erent types of partially-observable Markov decision processes (MDPs): given an MDP, does there exist a policy with performance > 0? Lower and upper bounds on the complexity of the decision problems are shown in terms of completeness for NL, P, NP, PSPACE, EXP, NEXP or EXPSPACE, dependent on the type of the Markov decision process. For several NP-complete types, we show that they are not even polynomial-time "-approximable for any xed ", unless P = NP. These results also reveal interesting trade-o s between the power of policies, observations, and rewards. Compressibility of In nite Binary Sequences Montserrat Hermo Universitat Trier (joint work with Ricard Gavald a and Jos e L. Balc azar, Barcelona) It is known that in nite binary sequences of constant Kolmogorov complexity are exactly the recursive ones. Such a kind of statement no longer holds in the presence of resource bounds. Contrary to what intuition might suggest, there are sequences of constant, polynomial-time bounded Kolmogorov complexity that are not polynomial-time computable. This motivates the study of several resource-bounded variants in search for a characterization, similar in spirit, of the polynomial-time computable sequences. We propose a de nition, based on Kobayashi's notion of compressibility, and compare it to both the standard resource-bounded Kolmogorov complexity of in nite strings, and the uniform complexity introduced by Loveland. Some nontrivial coincidences and disagreements are proved. The resourceunbounded case is also considered. 12 The Shapes of Trees Ulrich Hertrampf Universitat Stuttgart We investigate the in uence of the types of computation trees, allowed in a leaf language description of a certain complexity class, on the computational power of this class. To this end, we introduce three potentially di erent classes Leafp(A), BalLeafp(A), and FBTLeafp(A), the classes of sets recognized using polynomial-time machines with acceptance de ned by leaf language A, and arbitrary, balanced, or full binary trees as their computation trees. For language classes C, we further de ne Leafp(C) df = SA2C Leafp(A) and analogously BalLeafp(C), and FBTLeafp(C). For A 2 N (the class of languages having a so-called neutral letter), it is easy to see that Leafp(A) = BalLeafp(A) = FBTLeafp(A). On the other hand, in [U. Hertrampf, H. Vollmer, K. W. Wagner, On Balanced Versus Unbalanced Computation Trees, Mathematical Systems Theory, 1996] it was shown that BalLeafp(DLOGTIME) = FBTLeafp(DLOGTIME) = P, but Leafp(DLOGTIME) = PPP. For the set REG of regular languages, it was shown that Leafp(REG) = BalLeafp(REG) = FBTLeafp(REG) = PSPACE (cf. [U. Hertrampf et al., On the Power of Polynomial Time Bit Reductions, Structures 1993]). Our main results are: (8L 2 REG) (9L0 2 REG) [Leafp(L) = Leafp(L0) = BalLeafp(L0) = FBTLeafp(L0)]. (8L 2 REG) (9L0 2 REG) [BalLeafp(L) = BalLeafp(L0) = FBTLeafp(L0)]. Moreover, de ning LEAFp(C) df = fLeafp(A) jA 2 Cg, and BALLEAFp(C) and FBTLEAFp(C) analogously, we obtain the existence of sets C1; C2; C3 REG such that FBTLEAFp(C1) = BALLEAFp(C1) LEAFp(C1), FBTLEAFp(C2) LEAFp(C2) BALLEAFp(C2), and LEAFp(C3) BALLEAFp(C3) FBTLEAFp(C3). If P NP, then all these inclusions are strict. (These results are from [U. Hertrampf, Regular Leaf Languages and (Non-) Regular Tree Shapes, TR 95-21, Universitat L ubeck, 1995.) 13 Mangrove Deforestation: Algorithms for Unambiguous Logspace Classes Klaus-Jorn Lange Universitat T ubingen http://www-fs.informatik.uni-tuebingen.de/~lange (joint work with Eric Allender) There are several versions of unambigouos log-space classes. We give a complete problem for the class RUSPACE(log n), making it the rst nonsyntactical class with a complete set. In the time-bounded case there are relativizations excluding the existence of complete sets. Further on, in a joint work with ERIC ALLENDER, deterministic algorithms for the membership problems of the elements in RUSPACE(log n) using O(log2 n= logn) = o(log2 n) space are constructed. As a consequence we get parallel algorithms running on rather simple machine models. (The second part has appeared as TR 96-048 at http://www.eccc.uni-trier.de; the rst part will appear in the Chicago Journal of Theoretical Computer Science.) Equivalence of Measures of Complexity Classes Jack H. Lutz Iowa State University (joint work with Josef Breutzmann) The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probabibility measure. Speci cally, for any real number > 0, any uniformly polynomial-time computable sequence ! = ( 1; 2; 3; : : :) of real numbers (biases) i 2 [ ; 1 ] and any complexity class C (such as P, NP, BPP, P/poly, PH, PSPACE, etc.) that is closed under positive polynomial-time truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent: 14 1. C has p-measure 0 (respectively, measure 0 in E, measure 0 in E2) relative to the coin-toss probability measure given by the sequence ! . 2. C has p-measure 0 (respectively, measure 0 in E, measure 0 in E2) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an e cient martingale for one probability measure into an e cient martingale for a \nearby" probability measure; (ii) the construction of a positive bias reduction, a truth-table reduction that encodes a positive, e cient, approximate simulation of one bias sequence by another; and (iii) the use of such a reduction to dilate an e cient martingale for the simulated probability measure into an e cient martingale for the simulating probability measure. Nondeterministic NC1 Computation Pierre McKenzie Universit e de Montreal http://www.iro.umontreal.ca/~mckenzie (joint work with Herv e Caussinus, Montreal, Denis Th erien, McGill, and Heribert Vollmer, W urzburg) We present and extend results from our 1996 Computational Complexity Conference paper with the same title. We de ne the counting classes #NC1, GapNC1, PNC1, and C=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic nite automata, and programs over constant integer matrices yield equivalent de nitions of the latter three classes. Alternative de nitions of nondeterministic NC1 computation lead to the open question of whether evaluating log-depth f+; g formulas over IN reduces to multiplying constant size matrices over IN. Then we adapt the leaf language concept to the level of NC1. We show how known characterizations imply ACC0 MODPH and TC0 CH. We then extend these techniques to prove that MODPH (resp., CH) contains languages which have no ACC0-type circuits (resp., TC0-type circuits) of 15 size 2g(n), provided g(n) satis es: (9 ) (8 polynomials p) [p(n)g(p(n) g(2p(n))) = o(2n )]: This in e ect matches Eric Allender's COCOON'96 lower bounds, but with an even simpler proof which does require Eric's stronger form of the time hierarchy theorem; Eric's constructive COCOON'96 lower bounds can also be deduced. Interactive Proofs with Public Coins and Small Space Bounds R udiger Reischuk Medizinische Universitat zu L ubeck (joint work with Maciej Liskiewicz) We consider interactive proof systems with a bound on the space used by the veri er. While such systems with a logarithmic space bound seem to be extremely powerful if the random coin ips are kept secret, restricting to public coins one stays within P. An alternative characterization can be given by Arthur-Merlin-Games, that means stochastic Turing machines that alternate between probabilistic (A) and existential (M) con gurations. Let AMkSpace(S) (resp. MAkSpace(S)) denote the corresponding complexity classes, where the machines start in a probabilistic (resp. existential) con guration and use at most space S. We prove for the language PATTERN df = fw1#w2# : : : wm##u##BIN(2l) j l 2 IN; u; wi 2 f0; 1g ; juj = l & 9i u = wig the following results: PATTERN 2 MA2Space(log log); PATTERN 62 AM2Space(o(log)); PATTERN 62 AM2Space(o(log)): This yields the hierarchy AM1Space(S) AM2Space(S) AM3Space(S) for any sublogarithmic space bound S. At the end we discuss how this hierarchy might be extended. 16 Characterizations of the Existence of Partial and Total One-Way Permutations Jorg Rothe Friedrich-Schiller-Universitat Jena http://www.minet.uni-jena.de/~rothe (joint work with Lane A. Hemaspaandra, University of Rochester) We study the easy certi cate classes introduced by Hemaspaandra, Rothe, and Wechsung (cf. \Easy Sets and Hard Certi cate Schemes," to appear in Acta Informatica), with regard to the question of whether or not surjective one-way functions exist. This is an important open question in cryptology. We show that the existence of partial one-way permutations can be characterized by separating P from the class of UP sets that, for all unambiguous polynomial-time Turing machines accepting them, always have easy (i.e., polynomial-time computable) certi cates. Similar results characterizing certain types of poly-one one-way functions are given. This extends the work of Grollmann and Selman (\Complexity Measures for Public-Key Cryptosystems," SIAM Journal of Computing, 1988) and Allender (\The Complexity of Sparse Sets in P," Structures 1986). By Gradel's recent results about one-way functions (\De nability on Finite Structures and the Existence of One-Way Functions," Methods of Logic in Computer Science, 1994), this is also linked to statements in nite model theory. Finally, we establish a condition necessary and su cient for the existence of total one-way permutations. Complexity at Higher Types James Royer Syracuse University http://top.cis.syr.edu/people/royer/royer.html Constable, in 1973, posed the problem of working out a computational complexity theory for functionals and operators of type-2 and higher. In particular, he wanted a good type-2 analogue of polynomial-time. Progress 17 on these questions has been slow in coming. In large part this is because understanding the dynamic complexity of type-2 computations is surprisingly tricky. In 1991 Kapron and Cook proved a machine characterization of BFF2, a particular type-2 analogue of polynomial-time and in the process introduced several lovely ideas that have been central to recent progress on Constable's problems. I will survey this work, focusing on the recent results of myself and others. I will also illustrate why sorting out complexity at type-3 and above will be an even trickier enterprise. A Decision Procedure for Unitary Linear Quantum Cellular Automata Miklos Santha Universit e Paris-Sud Linear quantum cellular automata were introduced recently as one of the models of quantum computing. A basic postulate of quantum mechanics imposes a strong constraint on any quantum machine: it has to be unitary, that is its time evolution operator has to be a unitary transformation. In this paper we give an e cient algorithm to decide if a linear quantum cellular automaton is unitary. The complexity of the algorithm is O(n 4r 3 r+1 ) = O(n4) if the automaton has a continuous neighborhood of size r. Is Testing More Complex Than Querying? Rainer Schuler Universitat Ulm In this talk we consider the complexity of generating instances of NP search problems with one of its solutions under some pre-chosen distribution. In particular, if the distribution is su ciently natural, i.e., polynomial-time samplable, then generating only instances (asking questions) is \easy" for every problem. We observe that for every problem in R (random polynomial 18 time), generating instances with solutions is easy for polynomial-time sam-plable distributions. It turns out that every problem that allows generatinginstances with solutions (certi ed instances) is contained in co-AM, whereAM is the class of Arthur-Merlin games introduced by Babai. This showsthat it is unlikely that every NP search problem allows generating certi edinstances. Nevertheless, it is still possible that natural problems not knownto be in P, like graph isomophism, fall into this class.Algebraic and Logical Characterizations ofDeterministic Linear Time ClassesThomas SchwentickUniversitat Mainzhttp://www.informatik.uni-mainz.de/PERSONEN/Schwentick.eng.htmlAn algebraic characterization of the class DLIN of functions that canbe computed in linear time by a deterministic RAM using only numbers oflinear size is given. This class was introduced by Grandjean, who showedthat it is robust and contains most computational problems that are usuallyconsidered to be solvable in deterministic linear time.The characterization is in terms of a recursion scheme for unary functions.A variation of this recursion scheme characterizes the class DLINEAR, whichallows polynomially large numbers. A second variation de nes a class thatstill contains DTIME(n), the class of functions that are computable in lineartime on a Turing machine.From these algebraic characterizations logical characterizations of DLINand DLINEAR as well as complete problems for these classes (underDTIME(n) reductions) are derived.19 An Information-Theoretic Treatment ofRandom-Self-ReducibilityMartin StraussIowa State Universityhttp://www.cs.iastate.edu/~mstrauss/homepage.html(joint work with Joan Feigenbaum, AT&T; Labs)Informally, a function f is random-self-reducible if the evaluation of f atany given instance x can be reduced in polynomial time to the evaluationof f at one or more random instances yi, each one of which is uncorrelatedwith x.Random-self-reducible functions have many applications, includingaverage-case complexity, lower bounds, cryptography, interactive proof sys-tems, and program checkers, self-testers, and self-correctors.In this paper, we initiate the study of random-self-reducibility from aninformation-theoretic point of view. Speci cally, we formally de ne the no-tion of a random-self-reduction that, with respect to a given ensemble ofdistributions, leaks a limited number of bits, i.e., produces yi's in such a man-ner that each has a limited amount of mutual information with x. We arguethat this notion is useful in studying the relationships between random-self-reducibility and other properties of interest, including self-correctability andNP-hardness. In the case of self-correctability, we show that the information-theoretic de nition of random-self-reducibility leads to somewhat di erentconclusions from those drawn by Feigenbaum, Fortnow, Laplante, andNaik [2], who used the standard de nition. In the case of NP-hardness, weuse the information-theoretic de nition to strengthen the result of Feigen-baum and Fortnow [1], who proved, using the standard de nition, that thepolynomial hierarchy collapses if an NP-hard set is random-self-reducible.(The full paper can be found athttp://www.research.att.com/library/trs/TRs/96/96.13/96.13.1.body.ps.)References[1] J. Feigenbaum and L. Fortnow. Random-self-reducibility of completesets. SIAM Journal on Computing, 22:994{1005, 1993.20 [2] J. Feigenbaum, L. Fortnow, S. Laplante, and A. Naik. On coherence,random-self-reducibility, and self-correction. In Proc. 11th Conferenceon Computational Complexity, pages 59{67. IEEE Computer SocietyPress, Los Alamitos, 1996.The Crane Beach ConjectureDenis TherienMcGill University, MontrealLet L, e 2 ; e is neutral for L if and only if, for all u; v 2 ,uv is in L if and only if uev is in L. The Crane Beach Conjecture saysthe following: If L is a language with a neutral letter, then L is in AC0if and only if L is a regular -free set. In logical terms, this means that,assuming the presence of a neutral letter, any language de nable by a rst-order sentence using arbitrary numerical predicates can also be representedby a rst-order formula using < only. Using the Furst-Saxe-Sipser/Ajtaicircuit lower bound for PARITY, we know the conjecture to be true underthe additional assumption that the language is regular. On the other hand,an independent proof of the Crane Beach Conjecture would imply the circuitlower bound.The Isomorphism Problem for One-Time-OnlyBranching Programs and Arithmetic CircuitsThomas ThieraufUniversitat UlmWe investigate the computational complexity of the isomorphism prob-lem for one-time-only branching programs (1-BPI): On input of two suchprograms, B0 and B1, decide whether there exists a permutation of the vari-ables of B1 such that it becomes equivalent to B0.21 We show that 1-BPI cannot be NP-hard, unless the polynomial hierarchycollapses to the second level. The result is extended to the isomorphismproblem for arithmetic circuits over large enough elds.Boolean Decision Trees and Structure of RelativizedComplexity ClassesNicolai VereshchaginMoscow State UniversityWe de ne analogs of the complexity classes P, NP, AM, and IP for theBoolean decision trees model. The analogs are denoted respectively by Pdt,NPdt, AMdt, and IPdt. It is known that Pdt 6= NPdt, but NPdt \ coNPdt =Pdt = BPPdt and even IPdt \ coIPdt = Pdt. The open problem is whetherIPdt = NPdt or not. Trying to prove that IPdt 6= NPdt, we de ne two classesC1 and C2 both containing all of IPdt for which we managed to prove thatNPdt 6= C1 and NPdt 6= C2 and even MAdt 6= C1 and MAdt 6= C2.Lindstrom Quanti ers in Complexity TheoryHeribert VollmerUniversitat Wurzburghttp://haegar.informatik.uni-wuerzburg.de/person/mitarbeiter/vollmer(joint work with Hans-Jorg Burtschick, Technische Universitat Berlin)We show that examinations of the expressive power of logical formulaeenriched by Lindstrom quanti ers over ordered nite structures have a well-studied complexity-theoretic counterpart: the leaf language approach to de-ne complexity classes. Model classes of formulae with Lindstrom quanti ersare nothing else than leaf language de nable sets. Along the way we tightenthe best up to now known leaf language characterization of the classes of thepolynomial time hierarchy and give a new model-theoretic characterizationof PSPACE.22 Random IsomorphismsJie WangUniversity of North Carolina at GreensboroWe extend the NP-isomorphism theory of Berman and Hartmanis in thesetting of randomized reductions for distributional NP problems. Early re-sults on isomorphisms of average-case NP-complete problems are obtainedw.r.t. deterministic reductions. We de ne, in a standard way, what it meansfor two distributional NP problems to be isomorphic under randomized reduc-tions. We then show that all the known average-case NP-complete problems,whether they are complete under deterministic or randomized reductions, areall \randomly" isomorphic, and so these problems, regardless their origins,are \random encodings" of each other.Some Combinatorial Problem from the Study ofLocally Random ReducibilityOsamu WatanabeTokyo Institute of Technologyhttp://watanabe-www.cs.titech.ac.jp/~watanabe/myhome/intro-e.htmlAbadi, Feigenbaum, and Kilian (in M. Abadi, J. Feigenbaum, and J. Kil-ian, On Hiding Information from an Oracle, Journal of Computer and Sys-tems Sciences, Vol. 39, (1989), 21{50) formally de ned the notion of \locallyrandom reducibility" and studied it from a complexity theoretic point ofview. Since then, it has been open whether every function is k-locally re-ducible to some k functions, even for some constant k 2. Based on the ideaof Yao, Fortnow and Szegedy showed that there is a Boolean function that is2-locally reducible to no pair of Boolean functions, but their argument doesnot seem to work for the question whether there is a Boolean function thatis 2-locally reducible to no pair of functions whose range is f0; 1; 2g. Here Iasked one combinatorial question, which seems to be a key for solving thisproblem.23 Query OrderGerd WechsungFriedrich-Schiller-Universitat Jenahttp://www.minet.uni-jena.de/~wechsung(joint work with Lane A. Hemaspaandra and Harald Hempel)Let PC[1]:D[1] be the class of all sets accepted by a polynomial-time oracleTuring machine asking one query to some oracle C 2 C followed by one queryto some oracle D 2 D. We ask: Does query order matter, i.e. doesPC[1]:D[1] = PD[1]:C[1]hold?We show for C;D from the Boolean hierarchy over NPPBHi[1]:BHj [1] =8><>: Rp(i+2j 1) tt(NP) if i 0(2) ^ j 1(2)Rp(i+2j) tt(NP) otherwise.From this it follows that the only nontrivial case forPBHi[1]:BHj [1] = PBHj [1]:BHi[1]is i even and j = i + 1, provided that i j. In all remaining cases we havenonequality, unless the BH, and thus also the PH, collapse.We similarly characterize oracle classes with a tree like query structureand oracles from the BH.24