Kleene Algebra with Tests (KAT) provides a framework for algebraic equational reasoning about imperative programs. The recent variant Guarded KAT (GKAT) allows to reason on non-probabilistic properties of probabilistic programs. Here we introduce an extension of this framework called approximate GKAT (aGKAT), which equips GKAT with a partially ordered monoid (real numbers) enabling to express satisfaction of (deterministic) properties except with a probability up to a certain bound. This allows to represent in equational reasoning "à la KAT" proofs of probabilistic programs based on the union bound, a technique from basic probability theory. We show how a propositional variant of approximate Hoare Logic (aHL), a program logic for union bound, can be soundly encoded in our system aGKAT. We then illustrate the use of aGKAT with an example of accuracy analysis from the field of differential privacy.

Abstract This work is devoted to formal reasoning on relational properties of probabilistic imperative programs. Relational properties are properties which relate the execution of two programs (possibly the same one) on two initial memories. We aim at extending the algebraic approach of Kleene Algebras with Tests (KAT) to relational properties of probabilistic programs. For that we consider the approach of Guarded Kleene Algebras with Tests (GKAT), which can be used for representing probabilistic programs, and define a relational version of it, called Bi-guarded Kleene Algebras with Tests (BiGKAT) together with a semantics. We show that the setting of BiGKAT is expressive enough to encode a finitary version of probabilistic Relational Hoare Logic (pRHL) (without the While rule), a program logic that has been introduced in the literature for the verification of relational properties of probabilistic programs. We also discuss the additional expressivity brought by BiGKAT.

Abstract Guarded Kleene Algebra with Tests (GKAT) was presented as a fragment of KAT to abstract imperative programming languages, where only if-then-else and while-do statements are allowed in the language. The loss of expressiveness is, nevertheless, compensated by a clear advantage over KAT: it allows almost linear decidability of program equivalence. In this work, we give the first step to optimizing the complexity of dynamic logic equivalence, which is EXPTIME-complete in propositional dynamic logic (PDL). First, and based on strict deterministic PDL, we present guarded propositional dynamic logic (GPDL), a fragment of PDL where programs correspond to GKAT terms. It comes embedded, as expected, with a semantics over relational models and a sound axiomatisation. Then, we present a Natural Deduction system for GPDL, proving its soundness and completeness, concerning the axiomatisation. Based on Smolka et al. (2019, Proceedings of the ACM Programming Language 4), we obtain the main result of this work—the equivalence of GPDL programs can be established in almost linear time.

Abstract This paper introduces a class of automata and associated languages, suitable to model a computational paradigm of fuzzy systems, in which both vagueness and simultaneity are taken as first-class citizens. This requires a weighted semantics for transitions and a precise notion of a synchronous product to enforce the simultaneous occurrence of actions. The usual relationships between automata and languages are revisited in this setting, including a specific Kleene theorem.

Fuzziness, as a way to express imprecision, or uncertainty, in computation is an important feature in a number of current application scenarios: from hybrid systems interfacing with sensor networks with error boundaries, to knowledge bases collecting data from often non-coincident human experts. Their abstraction in e.g. fuzzy transition systems led to a number of mathematical structures to model this sort of systems and reason about them. This paper adds two more elements to this family: two modal logics, framed as institutions, to reason about fuzzy transition systems and the corresponding processes. This paves the way to the development, in the second part of the paper, of an associated theory of structured specification for fuzzy computational systems.