Thresholds and expectation-thresholds of monotone properties with small minterms

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the power set of $N$, where for $\mathcal{A} \subseteq 2^N$ $\mu_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]$. In this paper we study upward-closed families $\mathcal{A}$ for which all minimal sets in $\mathcal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\mathcal{A}) / p^k$ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables to identify in $\mathcal{A}$ either a substantial subfamily $\mathcal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using duality of linear programming. This is related to the threshold conjecture of Kahn and Kalai

Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x \in \bits^n, and $k$ parties, who have noisy versions of the source bits y^i \in \bits^n, where for all $i$ and $j$, it holds that \Pr[y^i_j = x_j] = 1 - \eps, independently for all $i$ and $j$. That is, each party sees each bit correctly with probability $1-\epsilon$, and incorrectly (flipped) with probability $\epsilon$, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on {\em balanced} functions f_i : \bits^n \to \bits such that $\Pr[f_1(y^1) = ... = f_k(y^k)]$ is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The functions $f_i$ may be thought of as an error correcting procedure for the source $x$. When $k=2,3$ no error correction is possible, as the optimal protocol is given by $f_i(x^i) = y^i_1$. On the other hand, for large values of $k$, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to $k$, $n$ and \eps. Our analysis uses tools from probability, discrete Fourier analysis, convexity and discrete symmetrization

Average-case complexity of detecting cliques

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 79-83).The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for k-CLIQUE, as this problem is known. Our results show that, in certain models of computation, solving k-CLIQUE in the average case requires Q(nk/4) resources (moreover, k/4 is tight). Here the models of computation are bounded-depth Boolean circuits and unbounded-depth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the well-studied Erdos-Renyi random graphs) are widely believed to be a source of computationally hard instances for clique problems (as Karp suggested in 1976). Our results are the first unconditional lower bounds supporting this hypothesis. For bounded-depth Boolean circuits, our average-case hardness result significantly improves the previous worst-case lower bounds of Q(nk/Poly(d)) for depth-d circuits. In particular, our lower bound of Q(nk/ 4 ) has no noticeable dependence on d for circuits of depth d ; k- log n/log log n, thus bypassing the previous "size-depth tradeoffs". As a consequence, we obtain a novel Size Hierarchy Theorem for uniform AC0 . A related application answers a longstanding open question in finite model theory (raised by Immerman in 1982): we show that the hierarchy of bounded-variable fragments of first-order logic is strict on finite ordered graphs. Additional results of this thesis characterize the average-case descriptive complexity of k-CLIQUE through the lens of first-order logic.by Benjamin Rossman.Ph.D

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Efficient and Robust Methods for Audio and Video Signal Analysis

This thesis presents my research concerning audio and video signal processing and machine learning. Specifically, the topics of my research include computationally efficient classifier compounds, automatic speech recognition (ASR), music dereverberation, video cut point detection and video classification.Computational efficacy of information retrieval based on multiple measurement modalities has been considered in this thesis. Specifically, a cascade processing framework, including a training algorithm to set its parameters has been developed for combining multiple detectors or binary classifiers in computationally efficient way. The developed cascade processing framework has been applied on video information retrieval tasks of video cut point detection and video classification. The results in video classification, compared to others found in the literature, indicate that the developed framework is capable of both accurate and computationally efficient classification. The idea of cascade processing has been additionally adapted for the ASR task. A procedure for combining multiple speech state likelihood estimation methods within an ASR framework in cascaded manner has been developed. The results obtained clearly show that without impairing the transcription accuracy the computational load of ASR can be reduced using the cascaded speech state likelihood estimation process.Additionally, this thesis presents my work on noise robustness of ASR using a nonnegative matrix factorization (NMF) -based approach. Specifically, methods for transformation of sparse NMF-features into speech state likelihoods has been explored. The results reveal that learned transformations from NMF activations to speech state likelihoods provide better ASR transcription accuracy than dictionary label -based transformations. The results, compared to others in a noisy speech recognition -challenge show that NMF-based processing is an efficient strategy for noise robustness in ASR.The thesis also presents my work on audio signal enhancement, specifically, on removing the detrimental effect of reverberation from music audio. In the work, a linear prediction -based dereverberation algorithm, which has originally been developed for speech signal enhancement, was applied for music. The results obtained show that the algorithm performs well in conjunction with music signals and indicate that dynamic compression of music does not impair the dereverberation performance

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Tools and Algorithms for the Construction and Analysis of Systems

This book is Open Access under a CC BY licence. The LNCS 11427 and 11428 proceedings set constitutes the proceedings of the 25th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019. The total of 42 full and 8 short tool demo papers presented in these volumes was carefully reviewed and selected from 164 submissions. The papers are organized in topical sections as follows: Part I: SAT and SMT, SAT solving and theorem proving; verification and analysis; model checking; tool demo; and machine learning. Part II: concurrent and distributed systems; monitoring and runtime verification; hybrid and stochastic systems; synthesis; symbolic verification; and safety and fault-tolerant systems

Safety and Reliability - Safe Societies in a Changing World

The contributions cover a wide range of methodologies and application areas for safety and reliability that contribute to safe societies in a changing world. These methodologies and applications include: - foundations of risk and reliability assessment and management - mathematical methods in reliability and safety - risk assessment - risk management - system reliability - uncertainty analysis - digitalization and big data - prognostics and system health management - occupational safety - accident and incident modeling - maintenance modeling and applications - simulation for safety and reliability analysis - dynamic risk and barrier management - organizational factors and safety culture - human factors and human reliability - resilience engineering - structural reliability - natural hazards - security - economic analysis in risk managemen