We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and well-foundedness. Second, an algebraic reconstruction of propositional Hoare logic.

In relational semantics, the input-output semantics of a program is a relation on its set of states. We generalize this in considering elements of Kleene algebras as semantical values. In a nondeterministic context, the demonic semantics is calculated by considering the worst behavior of the program. In this paper, we concentrate on while loops. Calculating the semantics of a loop is di#cult, but showing the correctness of any candidate abstraction is much easier. For deterministic programs, Mills has described a checking method known as the while statement verification rule.

We propose an algebraic semantics for the temporal logic CTL∗ and simplify it for its sublogics CTL and LTL. We abstractly represent state and path formulas over transition systems in Boolean left quantales. These are complete lattices with a multiplication that preserves arbitrary joins in its left argument and is isotone in its right argument. Over these quantales, the semantics of CTL∗ formulas can be encoded via finite and infinite iteration operators; the CTL and LTL operators can be related to domain operators. This yields interesting new connections between representations as known from the modal µ-calculus and Kleene/ω-algebra.

Abstract. We give a compendium of algebraic calculation rules for the operations of residuation and detachment in semirings. 1

Abstract. In this paper we present an abstract representation of pointer structures in Kleene algebras and the properties of a particular selective update function. These can be used as prerequisites for the definition of in-situ pointer updates and a general framework to derive in-situ pointer algorithms from their specification.

We generalise the designs of Unifying Theories of Programming (UTP) by defining them as matrices over semirings with ideals. This clarifies the algebraic structure of designs and considerably simplifies reasoning about them, e.g., forming a Kleene and omega algebra of designs. Moreover, we prove a generalised fixpoint theorem for isotone functions on designs. We apply our framework to investigate symmetric linear recursion and its relation to tail-recursion; this substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, pre-image, convergence and noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP.

In 1996 Zhou and Hansen proposed a first-order interval logic called Neighbourhood Logic (NL) for specifying liveness and fairness of computing systems and also defining notions of real analysis in terms of expanding modalities. After that, Roy and Zhou presented a sound and relatively complete Duration Calculus as an extension of NL. We present an embedding of NL into an idempotent semiring of intervals. This embedding allows us to extend NL from single intervals to sets of intervals as well as to extend the approach to arbitrary idempotent semirings. We show that most of the required properties follow directly from Galois connections, hence we get the properties for free. As one important result we get that some of the axioms which were postulated for NL can be dropped since they are theorems in our generalisation. Furthermore, we present some possible interpretations for neighbours beyond intervals. Here we discuss for example reachability in graphs and applications to hybrid systems. At the end of the paper we add finite and infinite iteration to NL and extend idempotent semirigs to Kleene algebras and ω algebras. These extensions are useful for formulating repetitive properties and procedures like loops.