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Kleene Algebra with Domain
, 2003
"... 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 ..."
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Cited by 53 (32 self)
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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 wellfoundedness. Second, an algebraic reconstruction of propositional Hoare logic.
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 14 (6 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
wp is wlp
 RELATIONAL METHODS IN COMPUTER SCIENCE. LNCS 3929
, 2006
"... Using only a simple transition relation one cannot model commands that may or may not terminate in a given state. In a more general approach commands are relations enriched with termination vectors. We reconstruct this model in modal Kleene algebra. This links the recursive definition of the do od l ..."
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Cited by 8 (6 self)
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Using only a simple transition relation one cannot model commands that may or may not terminate in a given state. In a more general approach commands are relations enriched with termination vectors. We reconstruct this model in modal Kleene algebra. This links the recursive definition of the do od loop with a combination of the Kleene star and a convergence operator. Moreover, the standard wp operator coincides with the wlp operator in the modal Kleene algebra of commands. Therefore our earlier general soundness and relative completeness proof for Hoare logic in modal Kleene algebra can be reused for wp. Although the definition of the loop semantics is motivated via the standard EgliMilner ordering, the actual construction does not depend on EgliMilnerisotonicity of the constructs involved.
Kleene under a Modal Demonic Star
 JOURNAL ON LOGIC AND ALGEBRAIC PROGRAMMING, SPECIAL ISSUE ON RELATION ALGEBRA AND KLEENE ALGEBRA
, 2004
"... In relational semantics, the inputoutput 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 ..."
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Cited by 7 (5 self)
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In relational semantics, the inputoutput 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 difficult, 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. A
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.
Greedylike algorithms in Kleene algebra
 PARTICIPANTS’ PROCEEDINGS 7TH RELMICS/2ND KLEENE WORKSHOP, MALENTE, MAY 12–17, 2003
, 2003
"... This paper provides an algebraic background for the formal derivation of greedylike algorithms. Such derivations have previously been done in various frameworks including relation algebra. We propose Kleene algebra as a particularly simple alternative. Instead of converse and residuation we use mo ..."
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Cited by 2 (2 self)
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This paper provides an algebraic background for the formal derivation of greedylike algorithms. Such derivations have previously been done in various frameworks including relation algebra. We propose Kleene algebra as a particularly simple alternative. Instead of converse and residuation we use modal operators that are definable in a wide class of algebras, based on domain/codomain or image/preimage operations. By abstracting from earlier approaches we arrive at a very general theorem about the correctness of loops that covers particular forms of greedy algorithms as special cases.