On probabilistic term rewriting

Abstract

We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method-polynomial and matrix interpretations-are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by way of multidistribution reduction sequences, this way accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

“…, x d1 1 · · · x d k k . For example, if k = 2, then Inc 2,(1,2) can be defined by Inc 2, (1,2) x y z = AddPr x (1,2) (AddPr x (1,1) (AddPr x (1,1) x (1,0) )) y z, because…”

“…(In this particular case, one can easily compute the termination probability by unfolding all the functions, but we wish to find a compositional translation so that it can work in the presence of recursion.) We need to compute the probability that F (H x) x reaches (i.e., reduces to) x, which is the probability p 1 that F (H x (1) ) x (2) reaches x (1) , plus the probability p 2 that F (H x (1) ) x (2) reaches x (2) (we have added annotations to distinguish between the two occurrences of x). What information on F is required for computing it?…”

“…We can thus represent p 2 as F 1 (H 2 ), where H 2 is 1 4 , the probability that f = H x uses the first argument, i.e., the probability that H uses the second argument. Now let us consider how to represent p 1 , the probability that F (H x (1) ) x (2) reaches x (1) . We construct another function F 0 from the definition of F for this purpose.…”

“…Here, h 2 (H 1 , H 2 ) and h 1 (1) respectively represent the probabilities that G (H z (1) ) z (2) reaches z (1) , and z (2) . The first argument of h 2 (i.e., H 1 ) represents the probability that H z reaches z, and H 2 represents the probability that H z reaches its argument (the second argument of H).…”

Lambda Calculus and Probabilistic Computation

“…To syntactically represent the global evolution of a probabilistic system, we rely on the notion of multidistribution [2].…”