Testing Semantics: Connecting Processes and Process Logics

Abstract

Abstract. We propose a methodology based on testing as a framework to capture the interactions of a machine represented in a denotational model and the data it manipulates. Using a duality that models machines on the one hand, and the data they manipulate on the other, testing is used to capture the interactions of each with the objects on the other side: just as the data that are input into a machine can be viewed as tests that the machine can be subjected to, the machine can be viewed as a test that can be used to distinguish data. While this approach is based on duality theories that now are common in semantics, it accomplishes much more than simply moving from one side of the duality to the other; it faithfully represents the interactions that embody what is happening as the computation proceeds. Our basic philosophy is that tests can be used as a basis for modeling interactions, as well as processes and the data on which they operate. In more abstract terms, tests can be viewed as formulas of process logics, and testing semantics connects processes and process logics, and assigns computational meanings to both.

Approximating Markov Processes by Averaging

“…[D. Pavlovic and Worrell 2006] studied testing equivalences and made the connection with process logics. [M. developed a beautiful theory of duality for labelled Markov processes which relates LMPs to C * -algebras.…”

The Duality of State and Observation in Probabilistic Transition Systems

“…Given a POMDP as in Def. 13 we define a transition function δ α , where α is a sequence of actions, inductively:…”

“…Kozen [22] established similar results in the case of probabilistic programs and probabilistic dynamic logic. More recently, there has been very interesting work by Mislove, Ouaknine, Pavlovic and Worrell [28] and by Pavlovic, Mislove and Worrell [13] on duality for labelled Markov processes (LMPs). These are like POMDPs but they do not have the notion of observation.…”

Coalgebraic Walks, in Quantum and Turing Computation

“…The above (equivalent) representations of a Turing machine via the monad construction T [n] distinguishes between the tape and the finitely many states of a machine. In contrast, for instance in [9], a Turing machine is represented as a coalgebra of the form X −→ P fin X × Γ × { , } Γ , where Γ is a set of input symbols, and , represent left and right moves. There is only one state space X, which implicitly combines both the tape and the states that steer the computation.…”

“…X is obtained via the correspondence (9) from the bihomomorphism T (X) X ⊗ T (X) X → T (X) X that one gets by abstraction from:…”