Distribution functions of probabilistic automata
Each probabilistic automaton M over an alphabet A defines a probability measure Prob sub(M) on the set of all finite and infinite words over A. We can identify a k letter alphabet A with the set {0, 1,..., k-1}, and, hence, we can consider every finite or infinite word w over A as a radix k expansion of a real number X(w) in the interval [0, 1]. This makes X(w) a random variable and the distribution function of M is defined as usual: F(x) := Prob sub(M) { w: X(w) < x }. Utilizing the fixed-point semantics (denotational semantics), extended to probabilistic computations, we investigate the distribution functions of probabilistic automata in detail. Automata with continuous distribution functions are characterized. By a new, and much more easier method, it is shown that the distribution function F(x) is an analytic function if it is a polynomial. Finally, answering a question posed by D. Knuth and A. Yao, we show that a polynomial distribution function F(x) on [0, 1] can be generated by a prob abilistic automaton iff all the roots of F'(x) = 0 in this interval, if any, are rational numbers. For this, we define two dynamical systems on the set of polynomial distributions and study attracting fixed points of random composition of these two systems.