52,472 research outputs found
The Tur\'{a}n number and probabilistic combinatorics
In this short expository article, we describe a mathematical tool called the
probabilistic method, and illustrate its elegance and beauty through proving a
few well-known results. Particularly, we give an unconventional probabilistic
proof of a classical theorem concerning the Tur\'{a}n number .
Surprisingly, this proof cannot be found in existing literature.Comment: 5 pages; to appear in Amer. Math. Monthly 201
Formal probabilistic analysis using theorem proving
Probabilistic analysis is a tool of fundamental importance to virtually all scientists and engineers as they often have to deal with systems that exhibit random or unpredictable elements. Traditionally, computer simulation techniques are used to perform probabilistic analysis. However, they provide less accurate results and cannot handle large-scale problems due to their enormous computer processing time requirements. To overcome these limitations, this thesis proposes to perform probabilistic analysis by formally specifying the behavior of random systems in higher-order logic and use these models for verifying the intended probabilistic and statistical properties in a computer based theorem prover. The analysis carried out in this way is free from any approximation or precision issues due to the mathematical nature of the models and the inherent soundness of the theorem proving approach. The thesis mainly targets the two most essential components for this task, i.e., the higher-order-logic formalization of random variables and the ability to formally verify the probabilistic and statistical properties of these random variables within a theorem prover. We present a framework that can be used to formalize and verify any continuous random variable for which the inverse of the cumulative distribution function can be expressed in a closed mathematical form. Similarly, we provide a formalization infrastructure that allows us to formally reason about statistical properties, such as mean, variance and tail distribution bounds, for discrete random variables. In order to in illustrate the practical effectiveness of the proposed approach, we consider the probabilistic analysis of three examples: the Coupon Collector's problem, the roundoff error in a digital processor and the Stop-and-Wait protocol. All the above mentioned work is conducted using the HOL theorem prover
On a functional contraction method
Methods for proving functional limit laws are developed for sequences of
stochastic processes which allow a recursive distributional decomposition
either in time or space. Our approach is an extension of the so-called
contraction method to the space of continuous functions
endowed with uniform topology and the space of
c\`{a}dl\`{a}g functions with the Skorokhod topology. The contraction method
originated from the probabilistic analysis of algorithms and random trees where
characteristics satisfy natural distributional recurrences. It is based on
stochastic fixed-point equations, where probability metrics can be used to
obtain contraction properties and allow the application of Banach's fixed-point
theorem. We develop the use of the Zolotarev metrics on the spaces
and in this context. Applications are
given, in particular, a short proof of Donsker's functional limit theorem is
derived and recurrences arising in the probabilistic analysis of algorithms are
discussed.Comment: Published at http://dx.doi.org/10.1214/14-AOP919 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Declarative programming for agent applications
This paper introduces the execution model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higher-order functions, modal computation, probabilistic computation, and some theorem-proving capabilities. The need for these features is motivated and examples are given to illustrate the central ideas
Formal Probabilistic Analysis of a Wireless Sensor Network for Forest Fire Detection
Wireless Sensor Networks (WSNs) have been widely explored for forest fire
detection, which is considered a fatal threat throughout the world. Energy
conservation of sensor nodes is one of the biggest challenges in this context
and random scheduling is frequently applied to overcome that. The performance
analysis of these random scheduling approaches is traditionally done by
paper-and-pencil proof methods or simulation. These traditional techniques
cannot ascertain 100% accuracy, and thus are not suitable for analyzing a
safety-critical application like forest fire detection using WSNs. In this
paper, we propose to overcome this limitation by applying formal probabilistic
analysis using theorem proving to verify scheduling performance of a real-world
WSN for forest fire detection using a k-set randomized algorithm as an energy
saving mechanism. In particular, we formally verify the expected values of
coverage intensity, the upper bound on the total number of disjoint subsets,
for a given coverage intensity, and the lower bound on the total number of
nodes.Comment: In Proceedings SCSS 2012, arXiv:1307.802
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