Title:

Formal verification of probabilistic algorithms

We begin with an extensive foundational development of probability, creating a higherorder logic formalization of mathematical measure theory. This allows the definition of the probability space we use to model a random bit generator, which informally is a stream of coinflips, or technically an infinite sequence of IID Bernoulli( 1/2 ) random variables. Probabilistic programs are modified using the statetransformer monad familiar from functional programming, where the random bit generator is passed around in the computation. Functions remove random bits from the generator to perform their calculation, and then pass back the changed random bit generator with the result. Our probability space modelling the random bit generator allows us to give precise probabilistic specifications of such programs, and then verify them in the theorem prover. We also develop technical support designed to expedite verification: probabilistic quantifiers; a compositional property subsuming measurability and independence; a probabilistic while loop together with a formal concept of termination with probability 1. We also introduce a technique for reducing properties of a probabilistic while loop to properties of programs that are guaranteed to terminate: these can then be established using induction and standard methods of program correctness. We demonstrate the formal framework with some example probabilistic programs, samples algorithms for four probability distributions; some optimal procedures for generating dice rolls from coin flips; the symmetric simple random walk. In addition, we verify the MillerRabin primality test, a wellknown and commercially used probabilistic algorithm. Our fundamental perspective allows us to define a version with strong properties, which we can execute in the logic to prove compositeness of numbers.
