Bernhard Möller

Kleene algebra with domain

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra by simple equational axioms for a domain and a codomain operation. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of programs and state transition systems. We develop the basic calculus, present the most interesting models and discuss some related theories. We demonstrate applicability by two examples: algebraic reconstructions of Noethericity and propositional Hoare logic based on equational reasoning.

Concurrent Kleene Algebra and its Foundations

A Concurrent Kleene Algebra oers two composition operators, related by a weak version of an exchange law: when applied in a trace model of program semantics, one of them stands for sequential execution and the other for concurrent execution of program components [22]. After introducing this motivating concrete application, we investigate its abstract background in terms of a primitive independence relation between the traces. On this basis, we develop a series of richer algebras; the richest validates a proof calculus for programs similar to that of a Jones style rely/guarantee calculus. On the basis of this abstract algebra, we nally reconstruct the original trace model, using the notion of atoms from lattice theory.

Algebras of modal operators and partial correctness

Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complementarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style. This claim is supported by novel algebraic soundness and completeness proofs for Hoare logic and by connecting this formalism with an algebraic decision procedure.

An Algebra for Features and Feature Composition

Feature-Oriented Software Development (FOSD)provides a multitude of formalisms, methods, languages, and tools for building variable, customizable, and extensible software. Along different lines of research, different notions of a feature have been developed. Although these notions have similar goals, no common basis for evaluation, comparison, and integration exists. We present a feature algebra that captures the key ideas of feature orientation and provides a common ground for current and future research in this field, in which also alternative options can be explored.

Concurrent Kleene Algebra

A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partially-ordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent ones (Joness rely/guarantee calculus). This involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer.

An algebraic foundation for automatic feature-based program synthesis

Feature-Oriented Software Development provides a multitude of formalisms, methods, languages, and tools for building variable, customizable, and extensible software. Along different lines of research, different notions of a feature have been developed. Although these notions have similar goals, no common basis for evaluation, comparison, and integration exists. We present a feature algebra that captures the key ideas of feature orientation and that provides a common ground for current and future research in this field, on which also alternative options can be explored. Furthermore, our algebraic framework is meant to serve as a basis for the development of the technology of automatic feature-based program synthesis and architectural metaprogramming.

Feature algebra

Based on experience from the hardware industry, product families have entered the software development process as well, since software developers often prefer not to build a single product but rather a family of similar products that share at least one common functionality while having well-identified variabilities. Such shared commonalities, also called features, reach from common hardware parts to software artefacts such as requirements, architectural properties, components, middleware, or code. We use idempotent semirings as the basis for a feature algebra that allows a formal treatment of the above notions as well as calculations with them. In particular models of feature algebra the elements are sets of products, i.e. product families. We extend the algebra to cover product lines, refinement, product development and product classification. Finally we briefly describe a prototype implementation of one particular model.

Lazy Kleene Algebra

We propose a relaxation of Kleene algebra by giving up strictness and right-distributivity of composition. This allows the subsumption of Dijkstra’s computation calculus, Cohen’s omega algebra and von Wright’s demonic refinement algebra. Moreover, by adding domain and codomain operators we can also incorporate modal operators. Finally, it is shown that predicate transformers form lazy Kleene algebras again, the disjunctive and conjunctive ones even lazy Kleene algebras with an omega operation.

Kleene getting lazy

We propose a relaxation of Kleene algebra by giving up strictness and right-distributivity of composition. This allows the subsumption of Dijkstras computation calculus, Cohens omega algebra and von Wrights demonic refinement algebra. Moreover, by adding domain and codomain operators we can also incorporate modal operators. We show that predicate transformers form lazy Kleene algebras, the disjunctive and conjunctive ones even lazy omega Kleene algebras. We also briefly sketch two further applications: a modal lazy Kleene algebra of commands modelling total correctness and another one that abstractly characterizes sets of trajectories as used in the description of reactive and hybrid systems.

On the algebraic specification of infinite objects—ordered and continuous models of algebraic types

SummaryThe concept of algebraic types is adapted to allow axiomatic characterizations of ordered and continuous algebras; infinite objects are then limit points in the carriers of certain continuous algebras. We mainly study implicative types, i.e., types the axioms of which are conditional inequations describing partial orders. The isomorphism classes of termgenerated ordered models of an implicative type are shown to form a complete lattice under the homomorphism ordering; this includes the wellknown initiality results for equational and conditional types as special cases. For types whose axioms specify strictness of the operations, the initial models are shown to correspond to flat domains.As a special kind of continuous algebras we consider inductively generated algebras, viz. the ideal completions of term-generated ordered algebras. For an inequational type, i.e., an implicative type where all axiom premises are empty, the completion of models always yields models again, whereas for implicative types this holds only in restricted cases. One such case is provided by hierarchic types which are complete and consistent relative to their primitive parts, and which satisfy certain conditions about limit points.Examples of algebras that can be specified by such types include those of finite and infinite streams, of sets of atoms under the Egli-Milner ordering, Milners synchronisation trees, and that of a simple functional language over the natural numbers.