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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 42 (29 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.
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
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.
Residuals and Detachments
, 2005
"... Abstract. We give a compendium of algebraic calculation rules for the operations of residuation and detachment in semirings. 1 ..."
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Cited by 2 (1 self)
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Abstract. We give a compendium of algebraic calculation rules for the operations of residuation and detachment in semirings. 1
Properties of Overwriting for Updates in Typed Kleene Algebras
"... 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 insitu pointer updates and a general framework to derive insitu pointer algo ..."
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Cited by 1 (1 self)
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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 insitu pointer updates and a general framework to derive insitu pointer algorithms from their specification.
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.
Semiring Neighbours
, 2005
"... In 1996 Zhou and Hansen proposed a firstorder 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 Du ..."
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In 1996 Zhou and Hansen proposed a firstorder 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.
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: