Semantics of sub-probabilistic programs

The aim of this paper is to extend the probabilistic choice in probabilistic programs to sub-probabilistic choice, i.e., of the form (p)P ^⋈(q)Q where p + q ⩽ 1. It means that program P is executed with probability p and program Q is executed with probability q. Then, starting from an initial state, the execution of a sub-probabilistic program results in a sub-probability distribution. This paper presents two equivalent semantics for a sub-probabilistic while-programming language. One of these interprets programs as sub-probabilistic distributions on state spaces via denotational semantics. The other interprets programs as bounded expectation transformers via wp-semantics. This paper proposes an axiomatic systems for total logic, and proves its soundness and completeness in a classical pattern on the structure of programs.