Predicate Transformers and Higher Order Logic

Predicate transformers are formalized in higher order logic. This gives a basis for mechanized reasoning about total correctness and refinement of programs. The notions of program variables and logical variables are explicated in the formalization. We show how to describe common program constructs, such as assignment statements, sequential and conditional composition, iteration, recursion, blocks and procedures with parameters, axe described as predicate transformers in this framework. We also describe some specification oriented constructs, such as assert statements, guards and nondeterministic assignments. The monotonicity of these constructs over the lattice of predicates is proved, as well as the monotonicity of the statement constructors with respect to the refinement ordering on predicate transformers

Refinement Calculus of Reactive Systems

Refinement calculus is a powerful and expressive tool for reasoning about sequential programs in a compositional manner. In this paper we present an extension of refinement calculus for reactive systems. Refinement calculus is based on monotonic predicate transformers, which transform sets of post-states into sets of pre-states. To model reactive systems, we introduce monotonic property transformers, which transform sets of output traces into sets of input traces. We show how to model in this semantics refinement, sequential composition, demonic choice, and other semantic operations on reactive systems. We use primarily higher order logic to express our results, but we also show how property transformers can be defined using other formalisms more amenable to automation, such as linear temporal logic (suitable for specifications) and symbolic transition systems (suitable for implementations). Finally, we show how this framework generalizes previous work on relational interfaces so as to be able to express systems with infinite behaviors and liveness properties

A Type-Directed Negation Elimination

In the modal mu-calculus, a formula is well-formed if each recursive variable occurs underneath an even number of negations. By means of De Morgan's laws, it is easy to transform any well-formed formula into an equivalent formula without negations -- its negation normal form. Moreover, if the formula is of size n, its negation normal form of is of the same size O(n). The full modal mu-calculus and the negation normal form fragment are thus equally expressive and concise. In this paper we extend this result to the higher-order modal fixed point logic (HFL), an extension of the modal mu-calculus with higher-order recursive predicate transformers. We present a procedure that converts a formula into an equivalent formula without negations of quadratic size in the worst case and of linear size when the number of variables of the formula is fixed.Comment: In Proceedings FICS 2015, arXiv:1509.0282