Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: - Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). - FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).

We consider the problem of decoding corrupted error correcting codes with NC0[⊕] circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability Ω(ε2) even if a (1/2 − ε)-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure- versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate “poor man’s cat states” by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state |ψ_1⟩ ⊗ … ⊗ |ψ_n⟩ where each qubit is in one of two orthogonal states |ψ⟩ or |ψ^⟂⟩, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ(1/√n). Under the promise that at least 2/3 of the input qubits are in the majority state, the fidelity increases to 1 - Θ(1/n) and approaches 1 as n increases.We also consider the more general problem of computing any symmetric and equivariant Boolean function f: {0,1}ⁿ → {0,1} in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size O(n). The time complexity of the algorithm is O(n⁴ log n) where n is the number of input qubits.

Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n^3)-time or O(n^2)-time classic algorithm can be solved by a known O(n^1.5)-time or O(n)-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro, and Speelman (arXiv:1911.05686), and by Aaronson, Chia, Lin, Wang, and Zhang (arXiv:1911.01973). In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time O(n^{2-a}), for any a>0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-b})-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems.

The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It essentially states that determining whether a CNF formula is satisfiable can not be done faster than exhaustive search over all possible assignments. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are very likely beyond current techniques. In this work, we introduce an extensive framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to what SETH is for classical computation. Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in P. As an example, we provide a conditional quantum time lower bound of Ω(n^1.5) for the Longest Common Subsequence and Edit Distance problems. We also show that the n^2 SETH-based lower bound for a recent scheme for Proofs of Useful Work carries over to the quantum setting using our framework, maintaining a quadratic gap between verifier and prover. Lastly, we show that the assumptions in our framework can not be simplified further with relativizing proof techniques, as they are false in relativized worlds.

We bound separations between the entangled and classical values for several classes of nonlocal $t$-player games. Our motivating question is whether there is a family of $t$-player XOR games for which the entangled bias is $1$ but for which the classical bias goes down to $0$, for fixed $t$. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-$m$ games and show that their classical value is always at least $\frac{1}{m} + \frac{m-1}{m} t^{1-t}$. Secondly, for free XOR games, in which the input distribution is of product form, we show $β(G) \geq β^*(G)^{2^t}$ where $β(G)$ and $β^*(G)$ are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is $1-ε$ then the classical value is at least $1-\mathcal{O}(\sqrt{ε\log k})$ where $k$ is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.

We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the support rank of the iterated matrix multiplication tensor. We employ Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Newman’s theorem states that we can take any public-coin communication protocol and convert it into one that uses only private randomness with but a little increase in communication complexity. We consider a reversed scenario in the context of information complexity: can we take a protocol that uses private randomness and convert it into one that only uses public randomness while preserving the information revealed to each player? We prove that the answer is yes, at least for protocols that use a bounded number of rounds. As an application, we prove new direct-sum theorems through the compression of interactive communication in the bounded-round setting. To obtain this application, we prove a new one-shot variant of the Slepian–Wolf coding theorem, interesting in its own right. Furthermore, we show that if a Reverse Newman’s Theorem can be proven in full generality, then full compression of interactive communication and fully-general direct-sum theorems will result.

We study the use of quantum entanglement in the zero-error source-channel coding problem. Here, Alice and Bob are connected by a noisy classical one-way channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice's input while using the channel as little as possible. In the zero-error regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lovász theta number, a graph parameter defined by a semidefinite program, gives the best efficiently computable upper bound on the Shannon capacity and it also upper bounds its entanglement-assisted counterpart. At the same time, it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here, we partially extend these results to the source-coding problem and to the more general source-channel coding problem. We prove a lower bound on the rate of entanglement-assisted source-codes in terms of Szegedy's number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical source-channel codes. Our separation results use low-degree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu, and a new application of remote state preparation.

We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace.

In this work, we study position-based cryptography in the quantum setting. The aim is to use the geographical position of a party as its only credential. On the negative side, we show that if adversaries are allowed to share an arbitrarily large entangled quantum state, the task of secure position-verification is impossible. To this end, we prove the following very general result. Assume that Alice and Bob hold respectively subsystems $A$ and $B$ of a (possibly) unknown quantum state $|\psi\rangle \in {\cal H}_A \otimes {\cal H}_B$. Their goal is to calculate and share a new state $|\varphi\rangle = U|\psi\rangle$, where $U$ is a fixed unitary operation. The question that we ask is how many rounds of mutual communication are needed. It is easy to achieve such a task using two rounds of classical communication, whereas, in general, it is impossible with no communication at all. Surprisingly, in case Alice and Bob share enough entanglement to start with and we allow an arbitrarily small failure probability, we show that the same task can be done using a single round of classical communication in which Alice and Bob exchange two classical messages. Actually, we prove that a relaxed version of the task can be done with no communication at all, where the task is to compute instead a state $|\varphi'\rangle$ that coincides with $|\varphi\rangle = U|\psi\rangle$ up to local operations on $A$ and on $B$, which are determined by classical information held by Alice and Bob. The one-round scheme for the original task then follows as a simple corollary. We also show that these results generalize to more players. As a consequence, we show a generic attack that breaks any position-verification scheme. On the positive side, we show that if adversaries do not share any entangled quantum state but can compute arbitrary quantum operations, secure position-verification is achievable. Jointly, these results suggest the interesting question whether secure position-verification is possible in case of a bounded amount of entanglement. Our positive result can be interpreted as resolving this question in the simplest case, where the bound is set to zero. In models where secure position-verification is achievable, it has a number of interesting applications. For example, it enables secure communication over an insecure channel without having any preshared key, with the guarantee that only a party at a specific location can learn the content of the conversation. More generally, we show that in settings where secure position-verification is achievable, other position-based cryptographic schemes are possible as well, such as secure position-based authentication and position-based key agreement.

Newman's theorem states that we can take any public-coin communication protocol and convert it into one that uses only private randomness with only a little increase in communication complexity. We consider a reversed scenario in the context of information complexity: can we take a protocol that uses private randomness and convert it into one that only uses public randomness while preserving the information revealed to each player? We prove that the answer is yes, at least for protocols that use a bounded number of rounds. As an application, we prove new direct sum theorems through the compression of interactive communication in the bounded-round setting. Furthermore, we show that if a Reverse Newman's Theorem can be proven in full generality, then full compression of interactive communication and fully-general direct-sum theorems will result.

We define a new model of communication complexity, called the garden-hose model. Informally, the garden-hose complexity of a function f:{0,1}n x {0,1}n -> {0,1} is given by the minimal number of water pipes that need to be shared between two parties, Alice and Bob, in order for them to compute the function f as follows: Alice connects her ends of the pipes in a way that is determined solely by her input x ∈ {0,1}n and, similarly, Bob connects his ends of the pipes in a way that is determined solely by his input y ∈ {0,1}n. Alice turns on the water tap that she also connected to one of the pipes. Then, the water comes out on Alice's or Bob's side depending on the function value f(x,y).

Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work.The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPR-pairs), while we show that the winning probability of any classical strategy differs from 1/2 by at most O(log n/√n).The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here n-dimensional entanglement allows to win the game with probability 1/(log n)2, while the best winning probability without entanglement is 1/n. This near-linear ratio ("Bell inequality violation'') is near-optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.

The genetic code is known to have a high level of error robustness and has been shown to be very error robust compared to randomly selected codes, but to be significantly less error robust than a certain code found by a heuristic algorithm. We formulate this optimization problem as a Quadratic Assignment Problem and use this to formally verify that the code found by the heuristic algorithm is the global optimum. We also argue that it is strongly misleading to compare the genetic code only with codes sampled from the fixed block model, because the real code space is orders of magnitude larger. We thus enlarge the space from which random codes can be sampled from approximately 2.433 × 10(18) codes to approximately 5.908 × 10(45) codes. We do this by leaving the fixed block model, and using the wobble rules to formulate the characteristics acceptable for a genetic code. By relaxing more constraints, three larger spaces are also constructed. Using a modified error function, the genetic code is found to be more error robust compared to a background of randomly generated codes with increasing space size. We point out that these results do not necessarily imply that the code was optimized during evolution for error minimization, but that other mechanisms could be the reason for this error robustness.

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K has high Kolmogorov complexity. As a partial converse to our claim we show that strings of very high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.

The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic polynomial-time we mean on every input producing one of the possible values of the function on that input. We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time hierarchy is infinite. We also show that relative to this same oracle, P≠UP and TFNP NP functions are not computable in polynomial-time with an NP oracle.

We study properties of non-uniform reductions and related completeness notions. We strengthen several results of Hitchcock and Pavan (ICALP (1), Lecture Notes in Computer Science, vol. 4051, pp. 465–476, Springer, 2006 ) and give a trade-off between the amount of advice needed for a reduction and its honesty on NEXP. We construct an oracle relative to which this trade-off is optimal. We show, in a more systematic study of non-uniform reductions, among other things that non-uniformity can be removed at the cost of more queries. In line with Post’s program for complexity theory (Buhrman and Torenvliet in Bulletin of the EATCS 85, pp. 41–51, 2005 ) we connect such ‘uniformization’ properties to the separation of complexity classes.

We show that hard sets S for NP must have exponential density, i.e. |S=n| ges 2nepsifor some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1-isinqueries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n1-isinqueries.

In this paper we study the individual communication complexity of the following problem. Alice receives an input string x and Bob an input string y, and Alice has to output y. For deterministic protocols it has been shown in Buhrman et al. (2004), that C(y) many bits need to be exchanged even if the actual amount of information C(y|x) is much smaller than C(y). It turns out that for randomised protocols the situation is very different. We establish randomised protocols whose communication complexity is close to the information theoretical lower bound. We furthermore initiate and obtain results about the randomised round complexity of this problem and show trade-offs between the amount of communication and the number of rounds. In order to do this we establish a general framework for studying these types of questions.

A language L has a property tester if there exists a probabilistic algorithm that given an input x queries only a small number of bits of x and distinguishes the cases as to whether x is in L and x has large Hamming distance from all y in L. We define a similar notion of quantum property testing and show that there exist languages with good quantum property testers but no good classical testers. We also show there exist languages which require a large number of queries even for quantumly testing.

We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PPcc, and a version with unrestricted bias called UPPcc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PPccsubne UPPcc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. Second, we study how small the bias of minimal-degree polynomials that sign-represent Boolean functions needs to be. We show that the worst-case bias is at worst double- exponentially small in the sign-degree (which was very recently shown to be optimal by Podolski), while the average- case bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal).

We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of thetas = (6 - 2radic2)/7 ap 45%, thereby improving on a previous bound of 50% (due to Razborov, 2004). More precisely, the circuit model for which we prove this bound contains perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, Z) and arbitrary additional one-qubit gates that are subject to depolarizing noise thetas. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for fault-tolerant quantum computation follows

Abstract A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n. h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A.

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual many-one reductions. Let ${{R_{\rm C}}}, {{R_{\rm Kt}}}, {{R_{\rm KS}}}, {{R_{\rm KT}}}$ be the sets of strings x having complexity at least $|x|/2$, according to the usual Kolmogorov complexity measure ${\mbox{\rm C}}$, Levin's time-bounded Kolmogorov complexity ${\mbox{\rm Kt}}$ [L. Levin, Inform. and Control, 61 (1984), pp. 15-37], a space-bounded Kolmogorov measure ${\mbox{\rm KS}}$, and a new time-bounded Kolmogorov complexity measure ${\mbox{\rm KT}}$, respectively. Our main results are as follows: \begin{remunerate} \item ${{R_{\rm KS}}}$ and ${{R_{\rm Kt}}}$ are complete for ${{\rm{PSPACE}}}$ and {\mbox{\rm EXP}}, respectively, under ${\mbox{\rm P/poly}}$-truth-table reductions. Similar results hold for other classes with ${{\rm{PSPACE}}}$-robust Turing complete sets. \item ${\mbox{\rm EXP}} = {\mbox{\rm NP}}^{{{R_{\rm Kt}}}}.$ \item ${{\rm{PSPACE}}} = {\mbox{\rm ZPP}}^{{{R_{\rm KS}}}} \subseteq {\mbox{\rm P}}^{{{R_{\rm C}}}}$. \item The Discrete Log, Factoring, and several lattice problems are solvable in ${\mbox{\rm BPP}}^{{{R_{\rm KT}}}}$. \end{remunerate} Our hardness result for ${{\rm{PSPACE}}}$ gives rise to fairly natural problems that are complete for ${{\rm{PSPACE}}}$ under ${\mbox{$\leq^{\rm p}_{\rm T}$}}$ reductions, but not under ${\mbox{$\leq^{\rm log}_{\rm m}$}}$ reductions. Our techniques also allow us to show that all computably enumerable sets are reducible to ${{R_{\rm C}}}$ via ${\mbox{\rm P/poly}}$-truth-table reductions. This provides the first "efficient" reduction of the halting problem to ${{R_{\rm C}}}$.

The language compression problem asks for succinct descriptions of the strings in a language A such that the strings can be efficiently recovered from their description when given a membership oracle for A. We study randomized and nondeterministic decompression schemes and investigate how close we can get to the information theoretic lower bound of $$\log {\left\| {A^{{ = n}} } \right\|}$$ for the description length of strings of length n. Using nondeterminism alone, we can achieve the information theoretic lower bound up to an additive term of $$O{\left( {{\left( {{\sqrt {\log {\left\| {A^{{ = n}} } \right\|}} } + \log n} \right)}\log n} \right)};$$ using both nondeterminism and randomness, we can make do with an excess term of $$O{\left( {\log ^{3} n} \right)}.$$ With randomness alone, we show a lower bound of $$n - \log {\left\| {A^{{ = n}} } \right\|} - O{\left( {\log n} \right)}$$ on the description length of strings in A of length n, and a lower bound of $$2 \cdot \log {\left\| {A^{{ = n}} } \right\|} - O(1)$$ on the length of any program that distinguishes a given string of length n in A from any other string. The latter lower bound is tight up to an additive term of $$O{\left( {\log n} \right)}.$$ The key ingredient for our upper bounds is the relativizable hardness versus randomness tradeoffs based on the Nisan–Wigderson pseudorandom generator construction.

We present several applications of quantum amplitude amplification for deciding whether all elements in the image of a given function are distinct, for finding an intersection of two sorted tables, and for finding a triangle in a graph. Our techniques generalize and improve those of Brassard, Hoyer, and Tapp [ACM SIGACT News, 28 (1997), pp. 14--19]. This shows that in the quantum world element distinctness is significantly easier than sorting, in contrast to the classical world.

We investigate coin-flipping protocols for multiple parties in a quantum broadcast setting: (1) we propose and motivate a definition for quantum broadcast. Our model of quantum broadcast channel is new. (2) We discovered that quantum broadcast is essentially a combination of pairwise quantum channels and a classical broadcast channel. This is a somewhat surprising conclusion, but helps us in both our lower and upper bounds. (3) We provide tight upper and lower bounds on the optimal bias /spl epsiv/ of a coin which can be flipped by k parties of which exactly g parties are honest: for any 1 /spl les/ g /spl les/ k, /spl epsiv/ = 1/2 - /spl Theta/ (g/k). Thus, as long as a constant fraction of the players are honest, they can prevent the coin from being fixed with at least a constant probability. This result stands in sharp contrast with the classical setting, where no non-trivial coin-flipping is possible when g /spl les/ k/2.

The language compression problem asks for succinct descriptions of the strings in a language A such that the strings can be efficiently recovered from their description when given a membership oracle for A. We study randomized and nondeterministic decompression schemes and investigate how close we can get to the information theoretic lower bound of log /spl par/A/sup = n//spl par/ for the description length of strings of length n. Using nondeterminism alone, we can achieve the information theoretic lower bound up to an additive term of 0((/spl radic/ /spl par/A/sup = n//spl par/ + log n)log n); using both nondeterminism and randomness, we can make do with an excess term of 0(log/sup 3/ n). With randomness alone, we show a lower bound of n - log /spl par/A/sup = n//spl par/ - 0(log n) on the description length of strings in A of length n, and a lower bound of 2/spl middot/log /spl par/A/sup = n//spl par/ - 0(1) on the length of any program that distinguishes a given string length n in A from any other string. The latter lower bound is tight up to an additive term of 0(log n). The key ingredient for our upper bounds is the relativizable hardness versus randomness trade offs based on the Nisan-Wigderson pseudorandom generator construction.

We study the robustness of complete sets for various complexity classes. A complete set A is robust if for any f(n)-dense set S /spl isin/ P, A - S is still complete, where f(n) ranges from log(n), polynomial, to subexponential. We show that robustness can be used to separate complexity classes: For every /spl les//sub m//sup p/-complete set A for EXP and any subexponential dense sets S /spl isin/ P, A - S is still Turing complete and under a reasonable hardness assumption even /spl les//sub m//sup p/-complete. For EXP and the delta levels of the exponential hierarchy we show that for every Turing complete set A and any log-dense set S /spl isin/ P, A - S is still Turing complete. There exists a 3-truth-table complete set A for EEXPSPACE, and a log-dense set S /spl isin/ P such that A - S is not Turing complete. This implies that settling this issue for EEXP will either separate P from PSPACE or PUfrom EXP. We show that the robustness results for EXP and the delta levels of the exponential hierarchy do not relativize.

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let R/sub K/, R/sub Kt/, R/sub KS/, R/sub KT/ be the sets of strings x having complexity at least |x|/2, according to the usual Kolmogorov complexity measure K, Levin's time-bounded Kolmogorov complexity Kt [27], a space-bounded Kolmogorov measure KS, and the time-bounded Kolmogorov complexity measure KT that was introduced in [4], respectively. Our main results are: 1. R/sub KS/ and R/sub Kt/ are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions. 2. EXP = NP/sup R(Kt)/. 3. PSPACE = ZPP/sup R(KS)/ /spl sube/ P/sup R(K)/. 4. The Discrete Log, Factoring, and several lattice problems are solvable in BPP/sup R(KT)/.

We study the it static membership problem: Given a set S of at most n keys drawn from a universe U of size m, store it so that queries of the form "Is u in S?" can be answered by making few accesses to the memory. We study schemes for this problem that use space close to the information theoretic lower bound of $\Omega(n\log(\frac{m}{n}))$ bits and yet answer queries by reading a small number of bits of the memory. We show that, for $\epsilon > 0$, there is a scheme that stores $O(\frac{n}{\epsilon^2}\log m)$ bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most $\epsilon$. We consider schemes that make no error for queries in S but are allowed to err with probability at most $\epsilon$ for queries not in S. We show that there exist such schemes that store $O((\frac{n}{\epsilon})^2 \log m)$ bits and answer queries using just one bitprobe. If multiple probes are allowed, then the number of bits stored can be reduced to $O(n^{1+\delta}\log m)$ for any $\delta > 0$. The schemes mentioned above are based on probabilistic constructions of set systems with small intersections. We show lower bounds that come close to our upper bounds (for a large range of n and $\epsilon$): Schemes that answer queries with just one bitprobe and error probability $\epsilon$ must use $\Omega(\frac{n}{\epsilon\log(1/\epsilon)} \log m)$ bits of storage; if the error is restricted to queries not in S, then the scheme must use $\Omega(\frac{n^2}{\epsilon^2 \log (n/\epsilon)}\log m)$ bits of storage. We also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes.

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp (1998), and imply an O(N/sup 3/4/ log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with /spl Theta/(N log N) classical complexity. We also prove a lower bound of /spl Omega/(/spl radic/N) comparisons for this problem and derive bounds for a number of related problems.

The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.

We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1} N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O ( Ts 6 ) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

We take a fresh look at CD complexity, where CD t (x) is the size of the smallest program that distinguishes x from all other strings in time t(|x|). We also look at CND complexity, a new nondeterministic variant of CD complexity, and time-bounded Kolmogorov complexity, denoted by C complexity. We show several results relating time-bounded C, CD, and CND complexity and their applications to a variety of questions in computational complexity theory, including the following: Showing how to approximate the size of a set using CD complexity without using the random string as needed in Sipser's earlier proof of a similar result. Also, we give a new simpler proof of this result of Sipser's. Improving these bounds for almost all strings, using extractors. A proof of the Valiant--Vazirani lemma directly from Sipser's earlier CD lemma. A relativized lower bound for CND complexity. Exact characterizations of equivalences between C, CD, and CND complexity. Showing that satisfying assignments of a satisfiable Boolean formula can be enumerated in time polynomial in the size of the output if and only if a unique assignment can be found quickly. This answers an open question of Papadimitriou. A new Kolmogorov complexity-based proof that BPP\subseteq\Sigma_2^p$. New Kolmogorov complexity based constructions of the following relativized worlds: There exists an infinite set in P with no sparse infinite NP subsets. EXP=NEXP but there exists a NEXP machine whose accepting paths cannot be found in exponential time. Satisfying assignments cannot be found with nonadaptive queries to SAT.

We give a new definition of resource bounded measure based on compressibility of infinite binary strings. We prove that the new definition is equivalent to the one commonly used. This new characterization offers us a different way to look at resource bounded measure, shedding more light on the meaning of measure zero results and providing one more tool to prove such results. The main contribution of the paper is the new definition and the proofs leading to the equivalence result. We then show how this new characterization can be used to prove that the class of linear autoreducible sets has p- measure 0. We also prove that the class of sets that are truth-table reducible to a p-selective set has p-measure 0 and that the class of sets that Turing reduce to a subpolynomial dense set has p-measure 0. This strengthens various results.

Let f:{0, 1}/sup n//spl times/{0, 1}/sup n//spl rarr/{0, 1}. Assume Alice has x/sub 1/, ..., x/sub k//spl isin/{0, 1}/sup n/, Bob has y/sub 1/, ..., y/sub k//spl isin/{0, 1}/sup n/, and they want to compute f(x/sub 1/, y/sub 1/)/spl middot//spl middot//spl middot/f(x/sub k/, y/sub k/) communicating as few bits as possible. The Direct Sum Conjecture of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x/sub 1/, y/sub 1/), then f(x/sub 2/, y/sub 2/), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits. Since a variant of it implies NC/sup 1//spl ne/NC/sup 2/. We consider three related problems. Enumeration: Alice and Bob output e/spl les/2/sup k/-1 elements of {0, 1}/sup k/: one of which is f(x/sub 1/, y/sub 1/)/spl middot//spl middot//spl middot/f(x/sub k/, y/sub k/). Elimination: Alice and Bob output an element of {0, 1}/sup k/ that is not f(x/sub 1/ y/sub 1/)/spl middot//spl middot//spl middot/f(x/sub k/, y/sub k/) Selection: (k=2) Alice and Bob output i/spl sim/{1,2} such that if f(x/sub 1/, y/sub 1/)=1 V f(x/sub 2/, Y/sub 2/)=1 then f(x/sub i/, y/sub i/)=1. We establish lower bounds on ELIM(f/sup k/) for particular f and connect the complexity of ELIM(f/sup k/), ENUM(k, f/sup k/), and SELECT(f/sup 2/) to the direct sum conjecture and other conjectures.

The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD(A/sup /spl les/n/) complexity of all strings x in some set A. The best known upper bound for this problem is 2log(/spl par/A/sup /spl les/n//spl par/)+O(log(n)), due to Buhrman and Fortnow. We show that the constant factor 2 in this bound is optimal. We also give new bounds for a certain kind of random sets R/spl sube/{0, 1}/sup n/, for which we show an upper bound of log (/spl par/R/sup /spl les/n//spl par/)+O(log(n)).

We study the static membership problem: Given a set S of at most n keys drawn from a universe of size m, store it so that queries of the form "Is x in S?" can be answered quickly.We study schemes for this problem that use space close to the information theoretic lower bound of 12(nlog(~)) bits and yet answer queries by reading a small number of bits of the memory.We show that there is a randomized scheme with error e that stores O(;~-logm) bits and answers queries using a single bitprobe.It is based on a family of sets with small intersections.If the error is required to be restricted to queries not in S, then we have a scheme that stores o((n) 2 logm) bits, answers queries with one bitprobe and works with probability of error less than e.We also show that better schemes with one-sided error can be obtained if more probes are allowed.We show lower bounds that come close to our upper bounds (for a large range of n and e): Schemes that answer queries with just one bitprobe and error probability e must use f~(~ log m) bits of storage; if the error is restricted to r~ 2 queries not in S, then the scheme must use ~(~ log m) bits of storage.We also consider deterministic schemes for the static membership problem and show upper and lower bounds.Ijaikumar, venkat}@tcs.tifr.

We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complexity problems where the "output" is embodied in the correlations between the outputs of the individual parties. Without entanglement, the parties must communicate to produce the required correlations; whereas, with entanglement, no communication is necessary to produce the correlations. In this sense, nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication. We show how to transform examples of nonlocality into more traditional communication complexity problems, where the output is explicitly determined by each individual party. The resulting problems require communication with or without entanglement, but the required communication is less when entanglement is available. All these results are a noteworthy contrast to the well-known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties.

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from exponential space by showing that all Turing complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic, and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.

We introduce resource-bounded betting games and propose a generalization of Lutz's resource-bounded measure in which the choice of the next string to bet on is fully adaptive. Lutz's martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong pseudorandom number generators exist, then betting games are equivalent to martingales for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz measures are important open problems: the class of polynomial-time Turing-complete languages in EXP and its superclass of polynomial-time Turing-autoreducible languages. If an EXP-martingale succeeds on either of these classes, or if betting games have the "finite union property" possessed by Lutz's measure, one obtains the nonrelativizable consequence $\mbox{BPP} \neq \mbox{EXP}$. We also show that if $\mbox{EXP} \neq \mbox{MA}$, then the polynomial-time truth-table-autoreducible languages have Lutz measure zero, whereas if $\mbox{EXP} = \mbox{BPP}$, they have measure one.

We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are polynomial time CD complexity defined by Sipser, the nondeterministic variant CND due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity CS introduced by Hartmanis. For all of these measures we define the set of random strings $\mathrm{R}^{\mathit{CD}}_t$, $\mathrm{R}^{\mathit{CND}}_t$, and $\mathrm{R}^{\mathit{CS}}_t$ as the set of strings x such that $\mathit{CD}^t(x)$, $\mathit{CND}^t(x)$, and $\mathit{CS}^s(x)$ is greater than or equal to the length of x for s and t polynomials. We show the following: $\mathrm{MA} \subseteq \mathrm{NP}^{\mathrm{R}^{\mathit{CD}}_t}$, where $\mathrm{MA}$ is the class of Merlin--Arthur games defined by Babai. $\mathrm{AM} \subseteq \mathrm{NP}^{\mathrm{R}^{\mathit{CND}}_t}$, where $\mathrm{AM}$ is the class of Arthur--Merlin games. $\mathrm{PSPACE} \subseteq \mathrm{NP}^{\mathrm{cR}^{\mathit{CS}}_s}$. In the last item $\mathrm{cR}^{\mathit{CS}}_s$ is the set of pairs $\langle x,y \rangle$ so that x is random given y. These results show that the set of random strings for various resource bounds is hard for complexity classes under nondeterministic reductions. This paper contrasts the earlier work of Buhrman and Mayordomo where they show that for polynomial time deterministic reductions the set of exponential time Kolmogorov random strings is not complete for EXP.

We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query, complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: P/sub /spl par///sup NP[2k]//spl sube/EQP/sub /spl par///sup NP[k]/; P/sub /spl par///sup NP[2k+1-2]//spl sube/EQP/sup NP[k]/; FP/sub /spl par///sup NP[2k=1-2]//spl sube/FEQP/sup NP[2k]/; FP/sub /spl par///sup NP/spl sube/FEQP(NP[Olog n)]/. For sets A that are many-one complete for PSPACE or EXP we show that Fp/sup A//spl sube/FEQP/sup A[1]/. Sets A that are many-one complete for PP have the property that FP/sub /spl par///sup A//spl sube/FEQP/sup A[1]/. In general we prove that for any set A there is a set X such that FP/sup A//spl sube/FEQP/sup X[1]/, establishing that no set is superterse in the quantum setting.

Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. Using Kolmogorov complexity for oracle construction, we obtain separation results that are much stronger than separations obtained previously even with the use of very complicated combinatorial arguments. Moreover the use of Kolmogorov arguments almost trivializes the construction itself: In particular we construct relativized worlds where: 1. NP/spl cap/CoNP/spl isin/P/poly. 2. NP has a set that is both simple and NP/spl cap/CoNP-immune. 3. CoNP has a set that is both simple and NP/spl cap/CoNP-immune. 4. /spl Pi//sub 2//sup p/ has a set that is both simple and /spl Pi//sub 2//sup p//spl cap//spl Sigma//sup 2p/-immune.

We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the number of queries of quantum search algorithms, their error probability, the size of the search space, and the number of solutions in this space. Using this, we deduce new lower and upper bounds for quantum versions of amplification problems. Next, we establish nearly optimal quantum-classical separations for the query complexity of monotone functions in the zero-error model (where our quantum zero-error model is defined so as to be robust when the quantum gates are noisy). Also, we present a communication complexity problem related to a total function for which there is a quantum-classical communication complexity gap in the zero-error model. Finally, we prove separations for monotone graph properties in the zero-error and other error models which imply that the evasiveness conjecture for such properties does not hold for quantum computers.

We introduce a search problem called “mutual search” where k agents, arbitrarily distributed over n sites, are required to locate one another by posing queries of the form “Anybody at site i ?”. We ask for the least number of queries that is necessary and sufficient. For the case of two agents using deterministic protocols, we obtain the following worst-case results: In an oblivious setting (where all pre-planned queries are executed), there is no savings: n -1 queries are required and are sufficient. In a nonoblivious setting, we can exploit the paradigm of “no news is also news” to obtain significant savings: in the synchronous case 0.586 n queries are required; in the asynchronous case 0.896 n queries suffice and a fortiori 0.536 n queries are required; for o(√n) agents using a synchronous deterministic protocol less than n queries suffice; there is a simple randomized protocol for two agents with worst-case expected 0.5 n queries and all radomized protocols require at least 0.25 n worst-case expected queries. The graph-theoretic framework we formulate for expressing and analyzing algorithms for this problem may be of independent interest.

We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs, $\Theta (n^2)$ bits are necessary and sufficient for shortest path routing. By "almost all graphs" we mean the Kolmogorov random graphs which constitute a fraction of 1 - 1/n c of all graphs on n nodes, where c > 0 is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $\Omega(n^2 \log n )$ and another model where the average case upper bound drops to $O(n \log^2 n)$. This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires $\Theta (n^3)$ bits on average. For worst-case static networks we prove an $\Omega(n^2 \log n )$ lower bound for shortest path routing and all stretch factors < 2 in some networks where free relabeling is not allowed.

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of the maximum possible Kolmogorov complexity) and (ii) a new elementary proof for the number of unlabeled graphs.

We consider the question whether two queries to SAT are as powerful as one query. We show that if P/sup NP[1]/=P/sup NP[2]/ then; locally either NP=coNP or NP has polynomial-size circuits; P/sup NP/=P/sup NP[1]/; /spl Sigma//sub 2//sup p/=UP/sup NP[1]//spl cap/RP/sup NP[1]/; PH=BPP/sup NP[1]/. Moreover we extend work of E. Hemaspaandra et al. (1997) to show that if P(/spl Sigma//sub 2//sup p/[1])=P(/spl Sigma//sub 2//sup p/[2]) then /spl Sigma//sub 2//sup p/=/spl Pi//sub 2//sup p/. We also give a relativized world where P/sup NP[1]/=P/sup NP[2]/ but NP/spl ne/coNP.

We show that MA/sub EXP/, the exponential time version of the Merlin-Arthur class, does not have polynomial size circuits. This significantly improves the previous known result due to Kannan since we furthermore show that our result does not relativize. This is the first separation result in complexity theory that does not relativize. As a corollary to our separation result we also obtain that PEXP, the exponential time version of PP is nor in P/poly.

We investigate the frequency of complete sets for various complexity classes within EXP under several polynomial-time reductions in the sense of resource bounded measure. We show that these sets are scarce: The sets that are complete under /spl les/(n/sup /spl alpha//-tt/sup -/)/sup P/ reductions for NP, the levels of the polynomial-time hierarchy, and PSPACE have p/sub 2/-measure zero for any constant /spl alpha/<1. The /spl les/(n/sup c/-T)/sup P/-complete sets for EXP have p/sub 2/-measure zero for any constant c. Assuming MA/spl ne/EXP, the /spl les//sub tt//sup P/-complete sets for EXP have p-measure zero. A key ingredient is the Small Span Theorem, which states that for any set A in EXP at least one of its lower span (i.e., the sets that reduce to A) or its upper span (i.e., the sets that A reduces to) has p/sub 2/-measure zero. Previous to our work, the theorem was only known to hold for /spl les//sub btt//sup p/-reductions. We establish it for /spl les/(n/sup 0/(1)-tt)/sup p/-reductions.

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T/sup 6/) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Theorem 1.3There exists a relativized world where we can detect unique solutions for NP problems yet P # NP.

AbotractWC present n simple and general simulation technique that transforms any black-box quantum algorithm (6 la Grover's database search nlgorithm) to a quantum communication protocol for a relntcd problem, in a way that fully exploits the quantum parallelism.This allows us to obtain new positive and negative results.The positive results are novel quantum communication protocols thnt nre built from nontrivial quantum algorithms via this simulation, These protocols, combined with (old and new) classical lower bounds, nre shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) hvoparty communication complexity models.In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separntion for the zero-error model.The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model, In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor ia It possible for the OR function itself in the exact case.This dichotomy naturally suggests a study of bounded-depth predicates (Le.those in the polynomial hierarchy) between OR and MAJORITY.We present black-box algorithms that achieve near quadratic speed up for nil such predicates.

We investigate the structure of EXP-complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. We study for various types of reductions the question of whether the set difference A-S for a hard set A and a sparse set S is still hard. We also address the question of which complete sets A can be split into sets A 1 and A 2 such that $A\equiv^P_r A_1\equiv^P_r A_2$ for reduction type r, i.e., which complete sets are mitotic. We obtain both positive and negative answers to these questions depending on the reduction type and the structure of the sparse set.

We consider the following six hypotheses: *P=NP. *SAT is truth-table reducible to a P-selective set. *SAT is truth-table reducible to a k-approximable set for some k. *FP/sub /spl par///sup NP/=FP/sup NP[log]/ *SAT is O(log n)-approximable. *Solving SAT is in P on formulae with at most one assignment. We discuss their importance and relationships among them.

The optimal space used to represent routing schemes in communication networks is established, both for worst-case static networks and on the average for all static networks. Several factors may influence the cost of representing a routing scheme for a particular network. It is therefore unavoidable that we first describe several reasonable models in which to measure this cost. Failure to do so in the past has obfuscated previous results. We show that, in most models, for almost all graphs \\Theta(n 2 ) bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of 1 \\Gamma 1=n c of all graphs on n nodes, where c 3 is an arbitrary fixed constant. In contrast, there is a model that rises the average case lower bound to \\Omega\\Gamma n 2 log n) and another model where the average case upper bound drops to O(n log 2 n). This clearly exposes the sensitivity of such bounds to the model under consi...

A language is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate exponential space from doubly exponential space by showing that all Turing complete sets for exponential space are autoreducible but there exists some Turing complete set for doubly exponential space that is not. We immediately also get a separation of logarithmic space from polynomial space. Although we already know how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's Program (E. Pos, 1944) to complexity theory. We feel such techniques may prove unknown separations in the future. In particular if we could settle the question as to whether all complete sets for doubly exponential time were autoreducible we would separate polynomial time from either logarithmic space or polynomial space. We also show several other theorems about autoreducibility.

In the long-lived renaming problem --- a generalization of the classical one-time renaming problem --- n processors with unique names ranging over a source name space f0; : : : ; S \\Gamma 1g repeatedly acquire and release unique names from a (smaller) destination name space f0; : : : ; D \\Gamma 1g. It is assumed that at most k out of n processors concurrently request or hold names. An efficient renaming protocol provides a useful front-end for protocols whose time complexity depends on the size of the name space containing the participating processes. We consider long-lived renaming in the context of asynchronous, shared-memory multiprocessing systems that provide only read and write operations. A renaming protocol is fast iff the time complexity of acquiring and releasing a name is polynomial in k and independent of n and S. We present a wait-free, read/write protocol for long-lived renaming that achieves a destination name space of size O(k 2 ) with time complexity O(k 3 ). If ...

Polynomials over finite fields are used to show that any sparse set can conjunctively reduce to a tally set. This leads to new results and to simple proofs of known results about various classes that lie between P and P/poly.