Boolean-valued Function Research Articles

Trading information complexity for error II: The case of a large error and the external information complexity

Two problems are studied. (1) How much external or internal information cost is required to compute a Boolean-valued function with an error at most 1/2−ξ for small ξ>0? A lower bound Ω(ξ2) and an upper bound O(ξ) are established. (2) How much external information cost can be saved to compute a function with a small error ϵ>0 comparing to the case when no error is allowed? A lower bound Ω(ϵ) and an upper bound O(h(ϵ)) are established, the lower bound can be improved to Ω(h(ϵ)) for product distributions. Except the O(h(ϵ)) upper bound, the other three bounds are tight. For distribution μ that is equally distributed on (0,0) and (1,1), it is shown that ICμext(XOR,ϵ)=1−2ϵ holds for every 0≤ϵ≤1/2. This equality seems to be the first example of exact information complexity when ϵ>0.

A structure theorem for almost low-degree functions on the slice

The Fourier-Walsh expansion of a Boolean function f: {0, 1}n → {0, 1} is its unique representation as a multilinear polynomial. The Kindler-Safra theorem (2002) asserts that if in the expansion of f, the total weight on coefficients beyond degree k is very small, then f can be approximated by a Boolean-valued function depending on at most O(2k) variables. In this paper we prove a similar theorem for Boolean functions whose domain is the ‘slice’ $$\left({\matrix{{\left[n \right]} \cr {pn} \cr}} \right) = {\rm{\{}}x \in {{\rm{\{}}0,1{\rm{\}}}^n}:\sum\nolimits_i {{x_i} = pn} {\rm{\}}}$$ , where 0 ≪ p ≪ 1, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of $$f:\left({\matrix{{\left[n \right]} \cr {pn} \cr}} \right) \to {\rm{\{}}0,1{\rm{\}}}$$ , the total weight beyond degree k is at most ϵ, where ϵ =min(p, 1 − p)O(k), then f can be O(ϵ)-approximated by a degree-k Boolean function on the slice, which in turn depends on O(2k) coordinates. This proves a conjecture of Filmus, Kindler, Mossel, and Wimmer (2015). Our proof relies on hypercontractivity, along with a novel kind of a shifting procedure. In addition, we show that the approximation rate in the Kindler-Safra theorem can be improved from ϵ + exp(O(k))ϵ5/4 to ϵ + ϵ2(2 ln(1/ϵ))k/k!, which is tight in terms of the dependence on ϵ and misses at most a factor of 2O(k) in the lower-order term.

Pseudorandom generators against advised context-free languages

Pseudorandomness has played a central role in modern cryptography, finding theoretical and practical applications to various fields of computer science. A function that generates pseudorandom strings from shorter but truly random seeds is known as a pseudorandom generator. Our generators are designed to fool languages (or equivalently, Boolean-valued functions). In particular, our generator fools advised context-free languages, namely, context-free languages assisted by external information known as advice, and moreover our generator is made almost one-to-one, stretching n-bit seeds to n+1 bits. We explicitly construct such a pseudorandom generator, which is computed by a deterministic Turing machine using logarithmic space and also belongs to CFLMV(2)/n—a functional extension of the 2-conjunctive closure of CFL with the help of appropriate deterministic advice. In contrast, we show that there is no almost one-to-one pseudorandom generator against context-free languages if we demand that it should be computed by a nondeterministic pushdown automaton equipped with a write-only output tape. Our generator naturally extends known pseudorandom generators against advised regular languages. Our proof of the CFL/n-pseudorandomness of the generator is quite elementary and, in particular, one part of the proof utilizes a special feature of the behaviors of nondeterministic pushdown automata, called a swapping property, which is interesting in its own right, generalizing the swapping lemma for context-free languages.

Reduction from non-injective hidden shift problem to injective hidden shift problem

We introduce a simple tool that can be used to reduce non-injective instances of the hidden shift problem over arbitrary group to injective instances over the same group. In particular, we show that the average-case non-injective hidden shift problem admit this reduction. We show similar results for (non-injective) hidden shift problem for bent functions. We generalize the notion of influence and show how it relates to applicability of this tool for doing reductions. In particular, these results can be used to simplify the main results by Gavinsky, Roetteler, and Roland about the hidden shift problem for the Boolean-valued functions and bent functions, and also to generalize their results to non-Boolean domains (thereby answering an open question that they pose).

On perfect 2-colorings of the [formula omitted]-ary [formula omitted]-cube

A coloring of a q -ary n -dimensional cube (hypercube) is called perfect if, for every n -tuple x , the collection of the colors of the neighbors of x depends only on the color of x . A Boolean-valued function is called correlation-immune of degree n − m if it takes value 1 the same number of times for each m -dimensional face of the hypercube. Let f = χ S be a characteristic function of a subset S of hypercube. In the present paper we prove the inequality ρ ( S ) q ( cor ( f ) + 1 ) ≤ α ( S ) , where cor ( f ) is the maximum degree of the correlation immunity of f , α ( S ) is the average number of neighbors in the set S for n -tuples in the complement of a set S , and ρ ( S ) = | S | / q n is the density of the set S . Moreover, the function f is a perfect coloring if and only if we have an equality in the formula above. Also, we find a new lower bound for the cardinality of components of a perfect coloring and a 1-perfect code in the case q > 2 .

Markov-chain model of classified atomistic transition states for discrete kinetic Monte Carlo simulations

Classical harmonic transition state theory is considered and applied in discrete lattice cells with hierarchical transition levels. The scheme is then used to determine transitions that can be applied in a lattice-based kinetic Monte Carlo (KMC) atomistic simulation model. The model results in an effective reduction of KMC simulation steps by utilizing a classification scheme of transition levels for thermally activated atomistic diffusion processes. Thermally activated atomistic movements are considered as local transition events constrained in potential energy wells over certain local time periods. These processes are represented by Markov chains of multidimensional Boolean valued functions in three-dimensional lattice space. The events inhibited by the barriers under a certain level are regarded as thermal fluctuations of the canonical ensemble and accepted freely. Consequently, the fluctuating system evolution process is implemented as a Markov chain of equivalence class objects. It is shown that the process can be characterized by the acceptance of metastable local transitions. The method is applied to a problem of Au and Ag cluster growth on a rippled surface. The simulation predicts the existence of a morphology-dependent transition time limit from a local metastable to stable state for subsequent cluster growth by accretion. Excellent agreement with observed experimental results is obtained.

On the uselessness of quantum queries

Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged after the oracle responds to the queries. A familiar example is the parity of a uniformly random Boolean-valued function over { 1 , 2 , … , N } , for which N − 1 classical queries are useless. We prove that if 2 k classical queries are useless for some oracle problem, then k quantum queries are also useless. For such problems, which include classical threshold secret sharing schemes, our result also gives a new way to obtain a lower bound on the quantum query complexity, even in cases where neither the function nor the property to be determined is Boolean.

Testing Halfspaces

This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e., a function of the form $f(x)=\mathrm{sgn}(w\cdot x-\theta)$. We consider halfspaces over the continuous domain $\mathbf{R}^n$ (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube $\{-1,1\}^n$ (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are $\epsilon$-far from any halfspace using only $\mathrm{poly}(\frac{1}{\epsilon})$ queries, independent of the dimension n. Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: The first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f's degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube $\{-1,1\}^n$ (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over $\{-1,1\}^n$ poses significant additional complications and requires other ingredients. These include “cross-consistency” versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [E. Fischer, G. Kindler, D. Ron, S. Safra, and A. Samorodnitsky, Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, 2002, pp. 103–112].

INTEGRATION-FREE ANALYSIS OF NONSMOOTH LOCAL DYNAMICS IN PLANAR FILIPPOV SYSTEMS

In this paper, we present a novel method to analyze the behavior of discontinuous piecewise-smooth autonomous systems (denominated Filippov systems) in the planar neighborhood of the discontinuity boundary (DB). The method uses the evaluation of the vector fields on DB to analyze the nonsmooth local dynamics of the Filippov system without the integration of the ODE sets. The method is useful in the detection of nonsmooth bifurcations in Filippov systems. We propose a classification of the points, events and events combinations on DB. This classification is more complete in comparison with the others previously reported. Additional characteristics as flow direction and sliding stability are included explicitly. The lines and the points are characterized with didactic symbols and the exclusive conditions for their existence are based on geometric criterions. Boolean-valued functions are used to formulate the conditions of existence. Different problems are analyzed with the proposed methodology.

Characteristic point sequences in local and global bifurcation analysis of Filippov systems

We explain the set of rules behind of the LabView toolbox for bifurcation analysis of Filippov systems denominated SPTCont 1.0. This software can detect nonsmooth bifurcations in n-dimensional systems using integration-free algorithms based on the evaluation of the vector fields on the discontinuity boundary (DB). In this paper, we present the characteristic point sequences that the software detects to guarantee the existence of local and global nonsmooth bifurcations in planar Filippov systems (n = 2). These sequences can be extended to three-dimensional or higher dimension Filippov systems. Boolean-valued functions are used to formulate the conditions of existence for each point defined in the sequences. Dynamics on DB and cycles are defined in function of the set of points. The full catalog of codim 1 local and global bifurcations is used to define the characteristic point sequence when the bifurcation parameter is varied. Finally, an illustrative example is analyzed using step-by-step routines of SPTCont 1.0.

Lower Bounds for the Noisy Broadcast Problem

We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the $1984$ workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60–62]. In this model there are $n+1$ processors $P_0,P_1,\ldots,P_n$, each of which is initially given a private input bit $x_i$. The goal is for $P_0$ to learn the value of $f(x_1,\ldots,x_n)$, for some specified function f, using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176–180] gave a noise-resistant protocol that allows $P_0$ to learn the entire input with constant probability in $O(n\log\log n)$ broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of $\Omega(n \log n)$ on the depth of a tree that learns the entire input. While the above lower bound is for an n-bit function, we also show an $\Omega(n\log\log n)$ lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least $1-n^{-\alpha}$ for a constant parameter $\alpha>0$ (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of $\Omega(n\log n)$ on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) $\Omega(n)$ lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts.

Duration calculus: Logical foundations

Abstract The Duration Calculus (abbreviated DC) represents a logical approach for formal design of real-time systems, where real numbers are used to model time and Boolean valued functions over time are used to model states and events of real-time systems. Since its introduction, DC has been applied to many case studies and it has been extended in several directions. The aim of this paper is to provide a thorough presentation of the logic.

On the VC-dimension of depth four threshold circuits and the complexity of Boolean-valued functions

We consider the problem of determining the VC-dimension δ 3( h) of depth four n-input 1-output threshold circuits with h elements. Best known asymptotic lower bounds and upper bounds are proved, that is, when h → ∞, δ 3( h) is upper bounded by (( h 2 3 ) + nh)(log h)(1 + o(1)) and lower bounded by ( 1 2 )(( h 2 4 ) + nh)(log h)(1 − o(1)) . We also consider the problem of determining the complexity C 3( N)( c 3( N)) of Boolean functions defined on N-pointsets of vertices of n-dimensional hypercube (Boolean-valued functions defined on N-pointsets in R n , respectively), measured by the number of threshold elements, with which we can construct a depth four circuit to realize the functions. We also show the best known upper and lower bounds, that is, when N → ∞, C 3( N) is upper bounded by √32( N log N )(1 + o(1)) and lower bounded by √6( N log N )(1 − o(1)) and c 3( N) is upper bounded by √16( N log N )(1 + o(1)) + 4n 2 − 2n and lower bounded by √6( N log N )(1 − o(1)) + ( 9 4 )n 2 − ( 3 2 )n .

Specifying Concurrent Program Modules

article Free AccessSpecifying Concurrent Program Modules Author: Leslie Lamport Computer Science Laboratory, SRI International, 333 Ravenswood Avenue, Menlo Park, CA Computer Science Laboratory, SRI International, 333 Ravenswood Avenue, Menlo Park, CAView Profile Authors Info & Claims ACM Transactions on Programming Languages and SystemsVolume 5Issue 2April 1983 pp 190–222https://doi.org/10.1145/69624.357207Published:01 April 1983Publication History 396citation1,279DownloadsMetricsTotal Citations396Total Downloads1,279Last 12 Months82Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Error reporting and recovery in Sangrah

AbstractIn Sangrah, exception conditions of CODASYL have been replaced by fatal errors and by boolean valued functions. Fatal errors can occur either because all the preconditions necessary for a DML statement are not fulfilled or because of the state of the data base at the time when this statement is executed. A mechanism which anticipates the former situation and thereby saves the data base from corruption is presented. Another recovery scheme which protects the data base from an abnormal termination of the user program is also described. This scheme uses a differential file.