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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
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.
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.
Modal design algebra
 PROC. FIRST INTERNATIONAL SYMPOSIUM ON UNIFYING THEORIES OF PROGRAMMING. LNCS
"... We give an algebraic model of (H3) designs based on a variant of modal semirings, hence generalising the original relational model. This makes the theory applicable to a wider class of settings, e.g., to algebras of sets of traces. Moreover, we set up the connection with the weakly and strongly demo ..."
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Cited by 2 (1 self)
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We give an algebraic model of (H3) designs based on a variant of modal semirings, hence generalising the original relational model. This makes the theory applicable to a wider class of settings, e.g., to algebras of sets of traces. Moreover, we set up the connection with the weakly and strongly demonic semantics of programs as discussed by a number of authors. This is done using commands (a, t) where a corresponds to the transition relation of a program and the condition t characterises the input states from which termination is guaranteed. The commands form not only a semiring but even a Kleene and omega algebra. This is used to calculate closed expressions for the least and greatest fixed point semantics of the demonic while loop.
Normal Design Algebra
"... 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 g ..."
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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 tailrecursion; this substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, preimage, convergence and noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP.
The Journal of Logic and Algebraic Programming
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: