Healthiness from Duality

Healthiness is a good old question in program logics that dates back to Dijkstra. It asks for an intrinsic characterization of those predicate transformers which arise as the (backward) interpretation of a certain class of programs. There are several results known for healthiness conditions: for deterministic programs, nondeterministic ones, probabilistic ones, etc. Building upon our previous works on so-called state-and-effect triangles, we contribute a unified categorical framework for investigating healthiness conditions. We find the framework to be centered around a dual adjunction induced by a dualizing object, together with our notion of relative Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems interesting in its own right in the context of monads, Lawvere theories and enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to LICS 201

An expectation transformer approach to predicate abstraction and data independence for probabilistic programs

In this paper we revisit the well-known technique of predicate abstraction to characterise performance attributes of system models incorporating probability. We recast the theory using expectation transformers, and identify transformer properties which correspond to abstractions that yield nevertheless exact bound on the performance of infinite state probabilistic systems. In addition, we extend the developed technique to the special case of "data independent" programs incorporating probability. Finally, we demonstrate the subtleness of the extended technique by using the PRISM model checking tool to analyse an infinite state protocol, obtaining exact bounds on its performance

A Fixpoint Semantics of Event Systems with and without Fairness Assumptions

We present a fixpoint semantics of event systems. The semantics is presented in a general framework without concerns of fairness. Soundness and completeness of rules for deriving "leads-to" properties are proved in this general framework. The general framework is instantiated to minimal progress and weak fairness assumptions and similar results are obtained. We show the power of these results by deriving sufficient conditions for "leads-to" under minimal progress proving soundness of proof obligations without reasoning over state-traces

A Recipe for State-and-Effect Triangles

In the semantics of programming languages one can view programs as state transformers, or as predicate transformers. Recently the author has introduced state-and-effect triangles which capture this situation categorically, involving an adjunction between state- and predicate-transformers. The current paper exploits a classical result in category theory, part of Jon Beck's monadicity theorem, to systematically construct such a state-and-effect triangle from an adjunction. The power of this construction is illustrated in many examples, covering many monads occurring in program semantics, including (probabilistic) power domains

Inferring Concise Specifications of APIs

Modern software relies on libraries and uses them via application programming interfaces (APIs). Correct API usage as well as many software engineering tasks are enabled when APIs have formal specifications. In this work, we analyze the implementation of each method in an API to infer a formal postcondition. Conventional wisdom is that, if one has preconditions, then one can use the strongest postcondition predicate transformer (SP) to infer postconditions. However, SP yields postconditions that are exponentially large, which makes them difficult to use, either by humans or by tools. Our key idea is an algorithm that converts such exponentially large specifications into a form that is more concise and thus more usable. This is done by leveraging the structure of the specifications that result from the use of SP. We applied our technique to infer postconditions for over 2,300 methods in seven popular Java libraries. Our technique was able to infer specifications for 75.7% of these methods, each of which was verified using an Extended Static Checker. We also found that 84.6% of resulting specifications were less than 1/4 page (20 lines) in length. Our technique was able to reduce the length of SMT proofs needed for verifying implementations by 76.7% and reduced prover execution time by 26.7%

Weakest Preconditions for Progress

Predicate transformers that map the postcondition and all intermediate conditions of a command to a precondition are introduced. They can be used to specify certain progress properties of sequential programs

A Weakest Pre-Expectation Semantics for Mixed-Sign Expectations

We present a weakest-precondition-style calculus for reasoning about the expected values (pre-expectations) of \emph{mixed-sign unbounded} random variables after execution of a probabilistic program. The semantics of a while-loop is well-defined as the limit of iteratively applying a functional to a zero-element just as in the traditional weakest pre-expectation calculus, even though a standard least fixed point argument is not applicable in this context. A striking feature of our semantics is that it is always well-defined, even if the expected values do not exist. We show that the calculus is sound, allows for compositional reasoning, and present an invariant-based approach for reasoning about pre-expectations of loops