4,669 research outputs found
Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page
Discontinuities in recurrent neural networks
This paper studies the computational power of various discontinuous
real computational models that are based on the classical analog
recurrent neural network (ARNN). This ARNN consists of finite number
of neurons; each neuron computes a polynomial net-function and a
sigmoid-like continuous activation-function.
The authors introducePostprint (published version
05201 Abstracts Collection -- Design and Analysis of Randomized and Approximation Algorithms
From 15.05.05 to 20.05.05, the Dagstuhl Seminar 05201 ``Design and Analysis of Randomized and Approximation Algorithms\u27\u27 was held
in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
HAMPI: A Solver for String Constraints
Many automatic testing, analysis, and verification techniques for programs can be effectively reduced to a constraint-generation phase followed by a constraint-solving phase. This separation of concerns often leads to more effective and maintainable tools. The increasing efficiency of off-the-shelf constraint solvers makes this approach even more compelling. However, there are few, if any, effective and sufficiently expressive off-the-shelf solvers for string constraints generated by analysis techniques for string-manipulating programs. We designed and implemented Hampi, a solver for string constraints over bounded string variables. Hampi constraints express membership in regular languages and bounded context-free languages. Hampi constraints may contain context-free-language definitions, regular-language definitions and operations, and the membership predicate. Given a set of constraints, Hampi outputs a string that satisfies all the constraints, or reports that the constraints are unsatisfiable. Hampi is expressive and efficient, and can be successfully applied to testing and analysis of real programs. Our experiments use Hampi in: static and dynamic analyses for finding SQL injection vulnerabilities in Web applications; automated bug finding in C programs using systematic testing; and compare Hampi with another string solver. Hampi's source code, documentation, and the experimental data are available at http://people.csail.mit.edu/akiezun/hampi
NMR Quantum Computation
In this article I will describe how NMR techniques may be used to build
simple quantum information processing devices, such as small quantum computers,
and show how these techniques are related to more conventional NMR experiments.Comment: Pedagogical mini review of NMR QC aimed at NMR folk. Commissioned by
Progress in NMR Spectroscopy (in press). 30 pages RevTex including 15 figures
(4 low quality postscript images
Relativization and Interactive Proof Systems in Parameterized Complexity Theory
We introduce some classical complexity-theoretic techniques to Parameterized Complexity. First, we study relativization for the machine models that were used by Chen, Flum, and Grohe (2005) to characterize a number of parameterized complexity classes. Here we obtain a new and non-trivial characterization of the A-Hierarchy in terms of oracle machines, and parameterize a famous result of Baker, Gill, and Solovay (1975), by proving that, relative to specific oracles, FPT and A[1] can either coincide or differ (a similar statement holds for FPT and W[P]). Second, we initiate the study of interactive proof systems in the parameterized setting, and show that every problem in the class AW[SAT] has a proof system with "short" interactions, in the sense that the number of rounds is upper-bounded in terms of the parameter value alone
Pseudorandom sequence generation using binary cellular automata
Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.Random numbers are an integral part of many applications from computer simulations,
gaming, security protocols to the practices of applied mathematics and physics. As
randomness plays more critical roles, cheap and fast generation methods are becoming a
point of interest for both scientific and technological use.
Cellular Automata (CA) is a class of functions which attracts attention mostly due to the
potential it holds in modeling complex phenomena in nature along with its discreteness
and simplicity. Several studies are available in the literature expressing its potentiality
for generating randomness and presenting its advantages over commonly used random
number generators.
Most of the researches in the CA field focus on one-dimensional 3-input CA rules. In
this study, we perform an exhaustive search over the set of 5-input CA to find out the
rules with high randomness quality. As the measure of quality, the outcomes of NIST
Statistical Test Suite are used.
Since the set of 5-input CA rules is very large (including more than 4.2 billions of rules),
they are eliminated by discarding poor-quality rules before testing.
In the literature, generally entropy is used as the elimination criterion, but we preferred
mutual information. The main motive behind that choice is to find out a metric for
elimination which is directly computed on the truth table of the CA rule instead of the
generated sequence. As the test results collected on 3- and 4-input CA indicate, all rules
with very good statistical performance have zero mutual information. By exploiting this
observation, we limit the set to be tested to the rules with zero mutual information. The
reasons and consequences of this choice are discussed.
In total, more than 248 millions of rules are tested. Among them, 120 rules show out-
standing performance with all attempted neighborhood schemes. Along with these tests,
one of them is subjected to a more detailed testing and test results are included.
Keywords: Cellular Automata, Pseudorandom Number Generators, Randomness TestsContents
Declaration of Authorship ii
Abstract iii
Öz iv
Acknowledgments v
List of Figures ix
List of Tables x
1 Introduction 1
2 Random Number Sequences 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Theoretical Approaches to Randomness . . . . . . . . . . . . . . . . . . . 5
2.2.1 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Computability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Random Number Generator Classification . . . . . . . . . . . . . . . . . . 7
2.3.1 Physical TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Non-Physical TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Pseudorandom Number Generators . . . . . . . . . . . . . . . . . . 10
2.3.3.1 Generic Design of Pseudorandom Number Generators . . 10
2.3.3.2 Cryptographically Secure Pseudorandom Number Gener- ators . . . . . . . . . . . . . .11
2.3.4 Hybrid Random Number Generators . . . . . . . . . . . . . . . . . 13
2.4 A Comparison between True and Pseudo RNGs . . . . . . . . . . . . . . . 14
2.5 General Requirements on Random Number Sequences . . . . . . . . . . . 14
2.6 Evaluation Criteria of PRNGs . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Statistical Test Suites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 NIST Test Suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.1 Hypothetical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.2 Tests in NIST Test Suite . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.2.1 Frequency Test . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.2.2 Block Frequency Test . . . . . . . . . . . . . . . . . . . . 20
2.8.2.3 Runs Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8.2.4 Longest Run of Ones in a Block . . . . . . . . . . . . . . 21
2.8.2.5 Binary Matrix Rank Test . . . . . . . . . . . . . . . . . . 21
2.8.2.6 Spectral Test . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.2.7 Non-overlapping Template Matching Test . . . . . . . . . 22
2.8.2.8 Overlapping Template Matching Test . . . . . . . . . . . 22
2.8.2.9 Universal Statistical Test . . . . . . . . . . . . . . . . . . 23
2.8.2.10 Linear Complexity Test . . . . . . . . . . . . . . . . . . . 23
2.8.2.11 Serial Test . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8.2.12 Approximate Entropy Test . . . . . . . . . . . . . . . . . 24
2.8.2.13 Cumulative Sums Test . . . . . . . . . . . . . . . . . . . . 24
2.8.2.14 Random Excursions Test . . . . . . . . . . . . . . . . . . 24
2.8.2.15 Random Excursions Variant Test . . . . . . . . . . . . . . 25
3 Cellular Automata 26 3.1 History of Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . .26
3.1.1 von Neumann’s Work . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Conway’s Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.3 Wolfram’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Cellular Automata and the Definitive Parameters . . . . . . . . . . . . . . 31
3.2.1 Lattice Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Cell Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Guiding Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Neighborhood Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 A Formal Definition of Cellular Automata . . . . . . . . . . . . . . . . . . 37
3.4 Elementary Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Rule Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Producing Randomness via Cellular Automata . . . . . . . . . . . . . . . 42
3.6.1 CA-Based PRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.2 Balancedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.3 Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Test Results 47 4.1 Output of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Testing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Interpretation of the Test Results . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Rate of success over all trials . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Distribution of P-values . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Testing over a big space of functions . . . . . . . . . . . . . . . . . . . . . 50
4.5 Our Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Results and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6.1 Change in State Width . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6.2 Change in Neighborhood Scheme . . . . . . . . . . . . . . . . . . . 53
4.6.3 Entropy vs. Statistical Quality . . . . . . . . . . . . . . . . . . . . 58
4.6.4 Mutual Information vs. Statistical Quality . . . . . . . . . . . . . . 60
4.6.5 Entropy vs. Mutual Information . . . . . . . . . . . . . . . . . . . 62
4.6.6 Overall Test Results of 4- and 5-input CA . . . . . . . . . . . . . . 6
4.7 The simplest rule: 1435932310 . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Conclusion 74
A Test Results for Rule 30 and Rule 45 77
B 120 Rules with their Shortest Boolean Formulae 80
Bibliograph
- …