Related papers: A Completeness Theorem for Probabilistic Regular E…
Kleene algebra (KA) is an important tool for reasoning about general program equivalences, with a decidable and complete equational theory. However, KA cannot always prove equivalences between specific programs. For this purpose, one adds…
We provide an extension of concurrent Kleene algebras to account for probabilistic properties. The algebra yields a unified framework containing nondeterminism, concurrency and probability and is sound with respect to the set of…
We introduce Probabilistic Guarded Kleene Algebra with Tests (ProbGKAT), an extension of GKAT that allows reasoning about uninterpreted imperative programs with probabilistic branching. We give its operational semantics in terms of special…
In the literature on Kleene algebra (KA), a number of variants have been proposed such as Kleene algebra with tests, commutative KA, bi-KA, and concurrent KA. The equational theories of some of these structures have then been studied in the…
We prove two completeness results for Kleene algebra with tests and a top element, with respect to guarded string languages and binary relations. While the equational theories of those two classes of models coincide over the signature of…
Probabilistic context-free grammars have a long-term record of use as generative models in machine learning and symbolic regression. When used for symbolic regression, they generate algebraic expressions. We define the latter as equivalence…
In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on…
Jones' rely-guarantee calculus for shared variable concurrency is extended to include probabilistic behaviours. We use an algebraic approach which combines and adapts probabilistic Kleene algebras with concurrent Kleene algebra. Soundness…
Weighted programs generalize probabilistic programs and offer a framework for specifying and encoding mathematical models by means of an algorithmic representation. Kleene algebra with tests is an algebraic formalism based on regular…
Kleene algebra axioms are complete with respect to both language models and binary relation models. In particular, two regular expressions recognise the same language if and only if they are universally equivalent in the model of binary…
We introduce a version of probabilistic Kleene algebra with angelic nondeterminism and a corresponding class of automata. Our approach implements semantics via distributions over multisets in order to overcome theoretical barriers arising…
This booklet serves as an introduction to Kleene Algebra (KA), a set of laws that can be used to study general equivalences between programs. It discusses how general programs can be modeled using regular expressions, how those expressions…
Symbolic Regression (SR) allows for the discovery of scientific equations from data. To limit the large search space of possible equations, prior knowledge has been expressed in terms of formal grammars that characterize subsets of…
We propose regular expressions to abstractly model and study properties of resource-aware computations. Inspired by nominal techniques -- as those popular in process calculi -- we extend classical regular expressions with names (to model…
Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with tests, along the lines of the coalgebraic…
We report on simulation, hierarchy, and decidability results for Practical Regular Expressions (PRE), which may include back references in addition to the standard operations union, concatenation, and star. The following results are…
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence…
We introduce a generic expression language describing behaviours of finite coalgebras over sets; besides relational systems, this covers, e.g., weighted, probabilistic, and neighbourhood-based system types. We prove a generic Kleene-type…
There are two main approach to probability, one of set-theoretic character where probability is the measure of a set, and another one of linguistic character where probability is the degree of confidence in a proposition. In this work we…
The Sigma formulas of the language of arithmetic express semidecidable relations on the natural numbers. More generally, whenever a totality of objects is regarded as incomplete, the Sigma formulas express relations that are witnessed in a…