A (possibly infinite) multiset over a set is a mapping . The union of a countably many multisets is defined by
which forms a multiset if and only if is finite for every . The sum of a multiset with respect to is defined by
We use set-like notations for multisets: denotes the empty multiset , is the multiset with , and is its special case where is finite.
Weighted Abstract Reduction Systems
A weighted ARS over state space is a ternary relation . We write meaning . We define the ARS induced by at cost , by the following inference rules:
We say is a normal form with respect to if there exists no with . The set of normal forms with respect to is denoted by . The potential of with respect to is defined by . The weighted ARS is called strongly bounded on , , if for every , there exists such that implies ; This is equivalent to saying that for every .
Weighted Probabilistic Abstract Reduction Systems
A multidistribution on a set is a multiset of pairs of and , written , satisfying . We denote the set of multidistributions on by . Multidistributions are closed under convex multiset unions for every finite or countable infinite index set
and probabilitieswith , where scalar multiplication is defined by for .
The restriction of a multidistribution to a set is defined by . For a function , we denote by its homomorphic extension defined by
For , we define the expectation of a function as . Notice that .
A weighted probabilistic ARS over is a set . As before, we may write for . We define the weighted ARS over induced by as follows:
For a weighted probabilistic ARS , let us define . A weighted probabilistic ARS over is strongly bounded on a set if , i.e., for all .
2 Probabilistic While
We consider an imperative language in the spirit of Dijkstra’s Guarded Command Language [Dijkstra75], endowed with primitives for sampling from discrete distributions as well as non-deterministic and probabilistic choice. Let denote a finite set of integer-valued variables . We denote by the set of stores, that associate variables with their integer contents. The syntax of program commands over is given by the following grammar.