Sets with small generalized Kolmogorov complexity

Summary We study the class of sets with small generalized Kolmogorov complexity. The following results are established: 1. A set has small generalized Kolmogorov complexity if and only if it is semi-isomorphic to a tally set. 2. The class of sets with small generalized Kolmogorov complexity is properly included in the class of self-p-printable sets. 3. The class of self-p-printable sets is properly included in the class of sets with selfproducible circuits. 4. A set S has self-producible circuits if and only if there is a tally set T such that P(T)=P(S). 5. If a set S has self-producible circuits, then NP(S)=NP _B(S), where NP _B( ) is the restriction of NP( ) studied by Book, Long, and Selman [4]. 6. If a set S is such that NP(S) =NP _B(S), then NP(S) P(SSAT).     

Efficient Scheduling of Strict Multithreaded Computations

In this paper we study the problem of efficiently scheduling a wide class of multithreaded computations, called strict; that is, computations in which all dependencies from a thread go to the thread's ancestors in the computation tree. Strict multithreaded computations allow the limited use of synchronization primitives. We present the first fully distributed scheduling algorithm which applies to any strict multithreaded computation. The algorithm is asynchronous, on-line, and follows the work-stealing paradigm. We prove that our algorithm is efficient not only in terms of its memory requirements and its execution time, but also in terms of its communication complexity. Our analysis applies to both shared and distributed memory machines. More specifically, the expected execution time of our algorithm is O(T ₁ /P + hT _∈fty ) , where T ₁ is the minimum serial execution time, T _∈fty is the minimum execution time with an infinite number of processors, P is the number of processors, and h is the maximum ``distance' in the computation tree between any two threads that need to communicate. Furthermore, the total space required during the execution is O(S ₁ P) , where S ₁ is the space required by a serial computer to execute the computation, while the expected communication cost incurred by our algorithm is O(PhT _∈fty (1+n _d ) S _max ) , where n _d is the maximum number of dependencies entering any thread and S _max is the largest amount of storage needed for the execution of any specific thread of the computation. Our communication complexity bound is the first nontrivial bound ever proved for the model of strict parallel programming. Received January 20, 1999, and in revised form March 26, 1999, and November 30, 1999.     

A Tight Upper Bound on Kolmogorov Complexity and Uniformly Optimal Prediction

This paper links the concepts of Kolmogorov complexity (in complexity theory) and Hausdorff dimension (in fractal geometry) for a class of recursive (computable) ω -languages. It is shown that the complexity of an infinite string contained in a Σ ₂ -definable set of strings is upper bounded by the Hausdorff dimension of this set and that this upper bound is tight. Moreover, we show that there are computable gambling strategies guaranteeing a uniform prediction quality arbitrarily close to the optimal one estimated by Hausdorff dimension and Kolmogorov complexity provided the gambler's adversary plays according to a sequence chosen from a Σ ₂ -definable set of strings. We provide also examples which give evidence that our results do not extend further in the arithmetical hierarchy. Received February 1995, and in revised form February 1997, and in final form October 1997.     

Most Sequences Are Stochastic

The central problem in machine learning (and statistics) is the problem of predicting future events x_n+1 based on past observations x₁x₂…x_n, where n=1, 2…. The main goal is to find a method of prediction that minimizes the total loss suffered on a sequence x₁x₂…x_n+1 for n=1, 2…. We say that a data sequence is stochastic if there exists a simply described prediction algorithm whose performance is close to the best possible one. This optimal performance is defined in terms of Vovk's predictive complexity, which is a generalization of the notion of Kolmogorov complexity. Predictive complexity gives a limit on the predictive performance of simply described prediction algorithms. In this paper we argue that data sequences normally occurring in the real world are stochastic; more formally, we prove that Levin's a priori semimeasure of nonstochastic sequences is small.     

Suboptimal Measures of Predictive Complexity for Absolute Loss Function

The problem of existence of predictive complexity for the absolute loss game is studied. Predictive complexity is a generalization of Kolmogorov complexity which bounds the ability of any algorithm to predict elements of a sequence of outcomes. For perfectly mixable loss functions (logarithmic and squared difference are among them) predictive complexity is defined like Kolmogorov complexity to within an additive constant. The absolute loss function is not perfectly mixable, and the question of existence of the corresponding predictive complexity, which is defined to within an additive constant, is open. We prove that in the case of the absolute loss game the predictive complexity can be defined to within an additive term O( ), where n is the length of a sequence of outcomes. We prove also that in some restricted settings this bound cannot be improved.     

On relativizations with restricted number of accesses to the oracle set

Conflicting relativizations, also known as Baker-Gill-Solovay phenomena, are common place in complexity theory. However, Book, Long, and Selman (SIAM J. Comput.,13 (1984), 461–487) have shown thatP(A) = NP _B (A) for all oraclesA if and only ifP = NP, whereNP _B (A) denotes the class of languages accepted by nondeterministic machines which possess only a polynomial number of query strings in the computation tree. It is shown in this paper that any superpolynomial bound on the number of queries already yields the BGS phenomenon. Similar results hold for theNP = co-NP andNPC = NP questions. A second objective of this paper is to point out a technique of Hartmanis with which to achieve oracle constructions by using the Kolmogorov complexity. This relatively new technique seems to be adequate for obtaining separation results for complexity classes which are not enumerable.     

Lower bounds for the majority communication complexity of various graph accessibility problems

We investigate the probabilistic communication complexity (more exactly, the majority communication complexity), of the graph accessibility problem (GAP) and its counting versions MOD _k -GAP,k ≥ 2. Due to arguments concerning matrix variation ranks and certain projection reductions, we prove that, for any partition of the input variables, GAP and MOD _m -GAP have majority communication complexity Ω,(n), wheren denotes the number of nodes of the graph under consideration.     

Birkhoff's ergodic theorem and the piecewise-constant maximum entropy method for Frobenius–Perron operators

Let (X, Σ, σ) be a finite measure space and S: X→X be a nonsingular transformation such that the corresponding Frobenius–Perron operator P _S : L ¹(X)→L ¹(X) has a stationary density f*. We propose a piecewise-constant maximum entropy method for the numerical recovery of f* and give its relation to the classic Birkhoff's individual ergodic theorem. An advantage of the piecewise-constant method over the current maximum entropy method based on polynomial basis functions is that a nonlinear system of equations is not needed for solving the related moment problem. Numerical results are given for several one dimensional test mappings.     

Independence and k-domination in graphs

A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbours in S. The k-domination number γ _k (G) is the minimum cardinality of a k-dominating set of G, and α(G) denotes the cardinality of a maximum independent set of G. Brook's well-known bound for the chromatic number χ and the inequality α(G)≥n(G)/χ(G) for a graph G imply that α(G)≥n(G)/Δ(G) when G is non-regular and α(G)≥n(G)/(Δ(G)+1) otherwise. In this paper, we present a new proof of this property and derive some bounds on γ _k (G). In particular, we show that, if G is connected with δ(G)≥k then γ _k (G)≤(Δ(G)?1)α(G) with the exception of G being a cycle of odd length or the complete graph of order k+1. Finally, we characterize the connected non-regular graphs G satisfying equality in these bounds and present a conjecture for the regular case.     

Transformation rules for CNOT-based quantum circuits and their applications

This paper introduces a simple but nontrivial set of local transformation rules for designingControl-NOT(CNOT)-based combinatorial circuits. We also provide a proof that the rule set iscomplete, namely, for any two equivalent circuits,S ₁ andS ₂, there is a sequence of transformations, each of them in the rule set, which changesS ₁ toS ₂. Two applications of the rule set are also presented. One is to simulate Resolution with only polynomial overhead by the rule set. Therefore we can conclude that the rule set is reasonably powerful. The other is to reduce the cost of CNOT-based circuits by using the transformations in the rule set. This implies that the rule set might be used for the practical circuit design. Currently Graduate School of Information Science, Nara Institute of Science and Technology Kazuo Iwama, Ph.D.: Professor of Informatics, Kyoto University, Kyoto 606-8501, Japan. Received BE, ME, and Ph.D. degrees in Electrical Engineering from Kyoto University in 1978, 1980 and 1985, respectively. His research interests include algorithms, complexity theory and quantum computation. Editorial board of Information Processing Letters and Parallel Computing. Council Member of European Association for Theoretical Computer Science (EATCS). Shigeru Yamashita, Ph.D.: Associate Professor of Graduate School of Information Science, Nara Instutute of Science and Technology, Nara 630-0192, Japan. He received his B.E., M.E. and Ph.D. degrees in information science from Kyoto University, Kyoto, Japan, in 1993, 1995 and 2001, respectively. His research interests include new type of computer architectures and quantum computation. He received the 2000 IEEE Circuits and Systems Society Transactions on Computer-Aided Design of Integrated Circuits and Systems Best Paper Award.     

On the growth rate of words in generalized Thue-Morse sequence

The paper summarizes properties of the pattern complexity for the generalized Thue-Morse sequence T and the sequence entropy of shift X_T generated by that sequence. We give the exact value of the pattern complexity and prove that the value of sequence entropy h_∞(X_T) is always bounded by log 4. In the second part we deal with the special case of T defined by the constant sequence and we find an upper bounding on the value of h_∞(X_T). The last theorem joins the properties of the shift X_T with odometers and shows an odometer G_s where X_T is at most 4-to-1 extension of.     

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S ₁, S ₂, …, S _k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S _i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper.     

Computations on Nondeterministic Cellular Automata

This work concerns the trade-offs between the dimension and the time and space complexity of computations on nondeterministic cellular automata. We assume that the space complexity is the diameter of area in space involved in computation. It is proved that (1) every nondeterministic cellular automata (NCA) of dimensionr, computing a predicatePwith time complexityT(n) and space complexityS(n) can be simulated byr-dimensional NCA with time and space complexityO(T^1/(r+1)S^r/(r+1)) and byr+1 dimensional NCA with time and space complexityO(T^1/2+S), whereTandSare functions constructible in time, (2) for any predicatePand integerr>1 if is a fastestr-dimensional NCA computingPwith time complexityT(n) and space complexityS(n), thenT=O(S), and (3) ifT_{r, P}is the time complexity of a fastestr-dimensional NCA computing predicatePthenT_r+1,P=O((T_{r, P})^1−r/(r+1)2),T_r+1,P=O((T_{r, P})^1+2/r).Similar problems for deterministic cellular automata (CA) are discussed.     

Stability of linear infinite-dimensional systems revisited

The paper is devoted to a study of stability questions for linear infinite-dimensional discrete-time and continuous-time systems. The concepts of power stability and l ^p Instability for a linear discrete-time system x _k+1 = Ax _k (where x _k ε X, X is a Banach space, A is linear and bounded) are introduced and studied. Relationships between these concepts and the inequality r(A) < 1, where r(A) denotes the spectral radius of A, are also given. The discrete-time results are used for a simple derivation of some well-known properties of exponentially stable and L^p-stable linear continuous-time systems described by [xdot](t) = Ax(t) (A generates here a strongly continuous semigroup of linear and bounded operators on X). Some remarks on norms related to stable systems are also included.     

Kolmogorov Complexity and Information Theory. With an Interpretation in Terms of Questions and Answers

We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and wherethey are fundamentally different. We discuss and relate the basicnotions of both theories: Shannon entropy, Kolmogorov complexity, Shannon mutual informationand Kolmogorov (``algorithmic') mutual information. We explainhow universal coding may be viewed as a middle ground betweenthe two theories. We consider Shannon's rate distortion theory, whichquantifies useful (in a certain sense) information.We use the communication of information as our guiding motif, and we explain howit relates to sequential question-answer sessions.     

One-Way Functions Using Algorithmic and Classical Information Theories

We prove several results relating injective one-way functions, time-bounded conditional Kolmogorov complexity, and time-bounded conditional entropy. First we establish a connection between injective, strong and weak one-way functions and the expected value of the polynomial time-bounded Kolmogorov complexity, denoted here by?E(K ^t (x|f(x))). These results are in both directions. More precisely, conditions on?E(K ^t (x|f(x))) that imply that?f is a weak one-way function, and properties of?E(K ^t (x|f(x))) that are implied by the fact that?f is a strong one-way function. In particular, we prove a separation result: based on the concept of time-bounded Kolmogorov complexity, we find an interval in which every function?f is a necessarily weak but not a strong one-way function. Then we propose an individual approach to injective one-way functions based on Kolmogorov complexity, defining Kolmogorov one-way functions and prove some relationships between the new proposal and the classical definition of one-way functions, showing that a Kolmogorov one-way function is also a deterministic one-way function. A relationship between Kolmogorov one-way functions and the conjecture of polynomial time symmetry of information is also proved. Finally, we relate?E(K ^t (x|f(x))) and two forms of time-bounded entropy, the unpredictable entropy?H ^unp, in which ??one-wayness?? of a function can be easily expressed, and the Yao⁺ entropy, a measure based on compression/decompression schema in which only the decompressor is restricted to be time-bounded.     

When Does Eddy Viscosity Damp Subfilter Scales Sufficiently?

Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ≥Δ, where Δ denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size <Δ are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than Δ; and (b) damp any disturbances having a scale of size smaller than Δ initially. From this we deduce that the eddy viscosity ν _e has to depend on the invariants q = \frac12tr(S²)q = \frac{1}{2}\mathrm{tr}(S^{2}) and r = -\frac13tr(S³)r= -\frac{1}{3}\mathrm{tr}(S^{3}) of the (filtered) strain rate tensor S. The simplest model is then given by n_e = \frac32(D/p)² |r|/q\nu_{e} = \frac{3}{2}(\Delta/\pi)^{2} |r|/q. This model is successfully tested for a turbulent channel flow (Re  _τ =590).     

The Baire Partial Quasi-Metric Space: A?Mathematical Tool for Asymptotic Complexity Analysis in Computer Science

In 1994, S.G. Matthews introduced the notion of partial metric space in order to obtain a suitable mathematical tool for program verification (Ann. N.Y. Acad. Sci. 728:183–197, 1994). He gave an application of this new structure to parallel computing by means of a partial metric version of the celebrated Banach fixed point theorem (Theor. Comput. Sci. 151:195–205, 1995). Later on, M.P. Schellekens introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms (Electron. Notes Theor. Comput. Sci. 1:211–232, 1995). The applicability of this theory to the asymptotic complexity analysis of Divide and Conquer algorithms was also illustrated by Schellekens. In particular, he gave a new proof, based on the use of the aforenamed Banach fixed point theorem, of the well-known fact that Mergesort algorithm has optimal asymptotic average running time of computing. In this paper, motivated by the utility of partial metrics in Computer Science, we discuss whether the Matthews fixed point theorem is a suitable tool to analyze the asymptotic complexity of algorithms in the spirit of Schellekens. Specifically, we show that a slight modification of the well-known Baire partial metric on the set of all words over an alphabet constitutes an appropriate tool to carry out the asymptotic complexity analysis of algorithms via fixed point methods without the need for assuming the convergence condition inherent to the definition of the complexity space in the Schellekens framework. Finally, in order to illustrate and to validate the developed theory we apply our results to analyze the asymptotic complexity of Quicksort, Mergesort and Largesort algorithms. Concretely we retrieve through our new approach the well-known facts that the running time of computing of Quicksort (worst case behaviour), Mergesort and Largesort (average case behaviour) are in the complexity classes O(n²)\mathcal{O}(n^{2}), O(nlog₂(n))\mathcal{O}(n\log_{2}(n)) and O(2(n-1)-log₂(n))\mathcal{O}(2(n-1)-\log_{2}(n)), respectively.