Formalization of the Standard Uniform random variable

Continuous random variables are widely used to mathematically describe random phenomena in engineering and the physical sciences. In this paper, we present a higher-order logic formalization of the Standard Uniform random variable as the limit value of the sequence of its discrete approximations. We then show the correctness of this specification by proving the corresponding probability distribution properties within the HOL theorem prover, summarizing the proof steps. The formalized Standard Uniform random variable can be transformed to formalize other continuous random variables, such as Uniform, Exponential, Normal, etc., by using various non-uniform random number generation techniques. The formalization of these continuous random variables will enable us to perform an error free probabilistic analysis of systems within the framework of a higher-order-logic (HOL) theorem prover. For illustration purposes, we present the formalization of the Continuous Uniform random variable based on the formalized Standard Uniform random variable, and then utilize it to perform a simple probabilistic analysis of roundoff error in HOL.

[1]  Lawrence Charles Paulson,et al.  ML for the working programmer , 1991 .

[2]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[3]  Jan J. M. M. Rutten,et al.  Mathematical techniques for analyzing concurrent and probabilistic systems , 2004, CRM monograph series.

[4]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[5]  Gilles Dowek,et al.  Principles of programming languages , 1981, Prentice Hall International Series in Computer Science.

[6]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[7]  Charles M. Grinstead,et al.  Introduction to probability , 1999, Statistics for the Behavioural Sciences.

[8]  John Robert Harrison,et al.  Theorem proving with the real numbers , 1998, CPHC/BCS distinguished dissertations.

[9]  Bruce D. McCullough,et al.  Assessing the Reliability of Statistical Software: Part I , 1998 .

[10]  Radha Jagadeesan,et al.  Stochastic processes as concurrent constraint programs , 1999, POPL '99.

[11]  Sebastian Thrun,et al.  A probabilistic language based upon sampling functions , 2005, POPL '05.

[12]  P. Spreij.  Probability and Measure , 1996 .

[13]  M. Cowles.  Statistical Computing , 2004 .

[14]  Joe Hurd,et al.  Formal verification of probabilistic algorithms , 2003 .

[15]  B. Widrow.  Statistical analysis of amplitude-quantized sampled-data systems , 1961, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry.

[16]  M. Gordon,et al.  Introduction to HOL: a theorem proving environment for higher order logic , 1993 .

[17]  Kishor S. Trivedi.  Probability and Statistics with Reliability, Queuing, and Computer Science Applications , 1984 .

[18]  Andrew Chi-Chih Yao,et al.  The complexity of nonuniform random number generation , 1976 .

[19]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[20]  Christel Baier,et al.  Model-Checking Algorithms for Continuous-Time Markov Chains , 2002, IEEE Trans. Software Eng..

[21]  Joseph F. Traub,et al.  Algorithms and Complexity: New Directions and Recent Results , 1976 .

[22]  Michael J. C. Gordon,et al.  Mechanizing programming logics in higher order logic , 1989 .

[23]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[24]  Graham Birtwistle,et al.  Current Trends in Hardware Verification and Automated Theorem Proving , 1989, Springer New York.

[25]  David J. C. Mackay,et al.  Introduction to Monte Carlo Methods , 1998, Learning in Graphical Models.

[26]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[27]  Robin Milner,et al.  A Theory of Type Polymorphism in Programming , 1978, J. Comput. Syst. Sci..

[28]  L. Dworsky.  An Introduction to Probability , 2008 .

[29]  Avi Pfeffer,et al.  IBAL: A Probabilistic Rational Programming Language , 2001, IJCAI.

[30]  Lawrence C. Paulson,et al.  ML for the working programmer (2. ed.) , 1996 .