On the equivalence of state transformer semantics and predicate transformer semantics

G. Plotkin and the author [13] have worked out the equivalence between state transformer semantics and predicate transformer semantics in a domain theoretical setting for programs combining nondeterminism and probability. Works of C. Morgan and co-authors [20], Keimel, Rosenbusch and Streicher [14, 15], already go in the same direction using only discrete state spaces. In fact, Keimel and Plotkin did not restrict to probabilities or subprobabilities, but worked in an extended setting admitting positive measures that may even have infinite values. This extended setting offers technical advantages. It was the intention of the authors to cut down their results to the subprobabilistic case in a subsequent paper. A paper by J. Goubault-Larrecq [7] already goes in this direction. When preparing a first version of the follow-up paper, the author of this paper wanted to clarify for himself the basic ideas. In fact, the paper [13] is technically quite involved, and when one reaches the last section, where the equivalence of predicate and state transformer semantics is finally put together, one is quite exhausted and has difficulties to see the leading ideas. Even the referee of the paper seemed to have given up at that point. It is the aim of this paper to begin from the other end. In all the situations that the author has been dealing with, the state transformer semantics had been given by a monad T over the category DCPO of directed complete posets (= dcpos) and Scott-continuous functions (= functions preserving the partial order and suprema of directed subsets). A state transformer interprets the input-output behavior of a program by a Scott-continuous map t from the input domain X to the ’powerdomain’ TY over the output domain Y . Thus, state transformers live in the Kleisli category associated with the monad T. If there is an equivalent predicate transformer semantics, predicate transformers have to live in a category (dually) equivalent to the Kleisli category. In my experience the equivalence between state and predicate transformer semantics is based on a very simple principle derived from the continuation monad. One starts with a dcpo R of ’observations’. The elements of the function space (the exponential) R are ’observable predicates’ over the dcpo X , and maps s : R → R are ’predicate transformers’. Assigning to every dcpo X the space R X of maps φ : R → R gives rise to a monad, the ’continuation monad’. The maps t : X → RRX are ’state transformers’. It is a simple observation that there is a natural bijection between state transformers and predicate transformers (see Section 1). Monads are used in denotational semantics to model computational effects. In lots of cases they are obtained by using a dcpo R of observations carrying an additional algebraic structure. This algebraic structure carries over to the function spaces R and R X . It leads to two kinds of monads ’subordinate’ to the continuation monad. One may assign to each dcpo X firstly the dcpo MRX ⊆ R X of all Scott-continuous algebra homomorphism φ : R → R (see Section 3) and secondly the directed complete subalgebra FRX of R X generated by the projections x̂ = (f 7→ f(x)) : R → R, x ∈ X (see Section 4). ∗Fachbereich Mathematik, Technische Universität Darmstadt, 64342 Darmstadt, Germany, email: [email protected] Work supported by Deutsche Forschungsgemeinschaft (DFG).