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Modal Kleene Algebra And Applications  A Survey
, 2004
"... Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey ..."
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Cited by 11 (5 self)
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Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey
Quantales and temporal logics
 ALGEBRAIC METHODOLOGY AND SOFTWARE TECHNOLOGY (AMAST 2006). LNCS 4019
, 2006
"... 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 ..."
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Cited by 6 (5 self)
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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 µcalculus and Kleene/ωalgebra.
Algebraic Separation Logic
, 2010
"... We present an algebraic approach to separation logic. In particular, we give an algebraic characterisation for assertions of separation logic, discuss different classes of assertions and prove abstract laws fully algebraically. After that, we use our algebraic framework to give a relational semantic ..."
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Cited by 5 (4 self)
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We present an algebraic approach to separation logic. In particular, we give an algebraic characterisation for assertions of separation logic, discuss different classes of assertions and prove abstract laws fully algebraically. After that, we use our algebraic framework to give a relational semantics of the commands of the simple programming language associated with separation logic. On this basis we prove the frame rule in an abstract and concise way. We also propose a more general version of separating conjunction which leads to a frame rule that is easier to prove. In particular, we show how to algebraically formulate the requirement that a command does not change certain variables; this is also expressed more conveniently using the generalised separating conjunction. The algebraic view does not only yield new insights on separation logic but also shortens proofs due to a point free representation. It is largely firstorder and hence enables the use of offtheshelf automated theorem provers for verifying properties at a more abstract level.
Omega Algebra, Demonic Refinement Algebra and Commands
 IN 9TH INTERNATIONAL CONFERENCE ON RELATIONAL METHODS IN COMPUTER SCIENCE AND 4TH INTERNATIONAL WORKSHOP ON APPLICATIONS OF KLEENE ALGEBRA, LECTURE
, 2006
"... Weak omega algebra and demonic refinement algebra are two ways of describing systems with finite and infinite iteration. We show that these independently introduced kinds of algebras can actually be defined in terms of each other. By defining modal operators on the underlying weak semiring, that res ..."
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Cited by 4 (3 self)
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Weak omega algebra and demonic refinement algebra are two ways of describing systems with finite and infinite iteration. We show that these independently introduced kinds of algebras can actually be defined in terms of each other. By defining modal operators on the underlying weak semiring, that result directly gives a demonic refinement algebra of commands. This yields models in which extensionality does not hold. Since in predicatetransformer models extensionality always holds, this means that the axioms of demonic refinement algebra do not characterise predicatetransformer models uniquely. The omega and the demonic refinement algebra of commands both utilise the convergence operator that is analogous to the halting predicate of modal µcalculus. We show that the convergence operator can be defined explicitly in terms of infinite iteration and domain if and only if domain coinduction for infinite iteration holds.
The linear algebra of UTP
 Mathematics of Program Construction
, 2006
"... Abstract. We show that the wellknown algebra of matrices over semirings can be used to reason conveniently about predicates and designs as used in the Unifying Theories of Programming of Hoare and He. 1 A Matrix View of UTP The Unifying Theories of Programming (UTP) developed in [1] model the termi ..."
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Cited by 2 (1 self)
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Abstract. We show that the wellknown algebra of matrices over semirings can be used to reason conveniently about predicates and designs as used in the Unifying Theories of Programming of Hoare and He. 1 A Matrix View of UTP The Unifying Theories of Programming (UTP) developed in [1] model the termination behaviour of programs using two special variables ok and ok ′ that express whether a program has been started and has terminated, respectively. Programs are identified with predicates relating the initial values v of variables with their final values v ′ ; moreover, ok and ok ′ may occur freely in predicates. However, the set of all such predicates is too general for a number of reasons not to be discussed here. Therefore, Hoare and He introduce a special class of predicates, called designs, of the form P ⊢ Q def ⇔ ok ∧ P ⇒ ok ′ ∧ Q, where ok and ok ′ are not allowed to occur in P or Q. The informal meaning is:
Algebraic Foundations of the Unifying Theories of Programming
, 2007
"... Hoare and He’s Unifying Theories of Programming take a relational view on semantics. The meaning of a nondeterministic, imperative program is described by ‘designs’ composed of two relations. They represent terminating states and relate the initial and final values of the observable variables, resp ..."
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Hoare and He’s Unifying Theories of Programming take a relational view on semantics. The meaning of a nondeterministic, imperative program is described by ‘designs’ composed of two relations. They represent terminating states and relate the initial and final values of the observable variables, respectively. Several ‘healthiness conditions’ are imposed by the theory to obtain properties found in practice. This work determines the structure of designs and modifies the theory to support nonstrict computations. It achieves these goals by identifying healthiness conditions and related axioms that involve unnecessary restrictions and subsequently removing them. The outcome provides a clear account of the algebraic foundations of the Unifying Theories of Programming. One of the results is a generalisation of designs by constructing them on semirings with ideals, structures having fewer axioms than relations. This clarifies the essential algebraic structure of designs, allows the reuse of existing mathematical theory and connects to further semantical approaches. The framework is extended by algebraic formulations of finite and infinite iteration, domain, preimage, determinacy, invariants and convergence. Calculations
Algebraic Structure of Web Services
, 2008
"... The ServiceOriented Architecture is gaining more and more attention and one way of realising it is the usage of Web Services. But which Web Services need to be invoked to reach a goal and which parameters are necessary at the beginning or are returned at the end? In this report we present an algebr ..."
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The ServiceOriented Architecture is gaining more and more attention and one way of realising it is the usage of Web Services. But which Web Services need to be invoked to reach a goal and which parameters are necessary at the beginning or are returned at the end? In this report we present an algebraic structure of Web Services in order to formally describe the Web Services and assist the users in Web Service composition. Hence, we apply relation algebra, tests, Kleene star and modal operators to characterise Web Services and Web Service Composition.
Fixing Zeno Gaps
, 2010
"... In computer science fixpoints play a crucial role. Most often least and greatest fixpoints are sufficient. However there are situations where other ones are needed. In this paper we study, on an algebraic base, a special fixpoint of the function f (x) = a · x that describes infinite iteration of an ..."
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In computer science fixpoints play a crucial role. Most often least and greatest fixpoints are sufficient. However there are situations where other ones are needed. In this paper we study, on an algebraic base, a special fixpoint of the function f (x) = a · x that describes infinite iteration of an element a. We show that the greatest fixpoint is too imprecise. Special problems arise if the iterated element contains the possibility of stepping on the spot (e.g. skip in a programming language) or if it allows Zeno behaviour. We present a construction for a fixpoint that captures these phenomena in a precise way. The theory is presented and motivated using an example from hybrid system analysis.