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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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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.
From Sequential Algebra to Kleene Algebra: Interval Modalities and Duration Calculus
, 2005
"... ..."
Dynamic Epistemic Semirings
, 2006
"... This paper proposes a semiring formulation for reasoning about an agent’s changing beliefs: a dynamic epistemic semiring (DES). A DES is a modal semiring extended with a revision operator. The revision operator is given a relational interpretation and a basic calculus is developed – based on the rev ..."
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Cited by 2 (0 self)
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This paper proposes a semiring formulation for reasoning about an agent’s changing beliefs: a dynamic epistemic semiring (DES). A DES is a modal semiring extended with a revision operator. The revision operator is given a relational interpretation and a basic calculus is developed – based on the revision operator a contraction operator is also defined. A DES only models actions on an agent’s beliefs, whereas the real dynamic epistemic semirings also introduced in this paper facilitate actions on the world as well. To allow for iterated action both structures are extended with the Kleene star.
Axiomatizability of Representable Domain Algebras
"... The family of domain algebras provide an elegant formal system for automated reasoning about programme verification. Their primary models are algebras of relations, viz. representable domain algebras. We prove that, even for the minimal signature consisting of the domain and composition operations, ..."
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The family of domain algebras provide an elegant formal system for automated reasoning about programme verification. Their primary models are algebras of relations, viz. representable domain algebras. We prove that, even for the minimal signature consisting of the domain and composition operations, the class of representable domain algebras is not finitely axiomatizable. Then we show similar results for extended similarity types of domain algebras.
Knowledge and Games in Modal Semirings
, 2007
"... Algebraic logic compacts many small steps of general logical derivation into large steps of equational reasoning. We illustrate this by representing epistemic logic and game logic in modal semirings and modal Kleene algebras. For epistemics we treat some classical examples like the wise men and mudd ..."
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Algebraic logic compacts many small steps of general logical derivation into large steps of equational reasoning. We illustrate this by representing epistemic logic and game logic in modal semirings and modal Kleene algebras. For epistemics we treat some classical examples like the wise men and muddy children puzzles; we also show how to handle knowledge update and revision algebraically. For games, we generalise the wellknown connection between game logic and dynamic logic to modal semirings and link it to predicate transformer semantics, in particular to demonic refinement algebra. The study provides evidence that modal semirings will be able to handle a wide variety of (multi)modal logics in a uniform algebraic fashion well suited to machine assistance.
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.
Synchronous Kleene AlgebraI
"... The work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus. The resulting algebraic structure is called synchronous Kleene algebra. Models are given in terms of sets of synchronous strings and finite automata accepting s ..."
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The work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus. The resulting algebraic structure is called synchronous Kleene algebra. Models are given in terms of sets of synchronous strings and finite automata accepting synchronous strings. The extension of synchronous Kleene algebra with Boolean tests is presented together with models on sets of guarded synchronous strings and the associated automata on guarded synchronous strings. Completeness w.r.t. the standard interpretations is given for each of the two new formalisms. Decidability follows from completeness. Kleene algebra with synchrony should be included in the class of true concurrency models. In this direction, a comparison with Mazurkiewicz traces is made which yields their incomparability with synchronous Kleene algebras (one cannot simulate the other). On the other hand, we isolate a class of pomsets which captures exactly synchronous Kleene algebras. We present an application to Hoarelike reasoning about parallel programs in the style of synchrony. Key words: Universal algebra, Kleene algebra, Boolean tests, synchrony, SCCS calculus,
An Algebraic Semantics for Duration Calculus
"... Abstract. We present an algebraic semantics for Duration Calculus based on semirings and quantales. Duration Calculus was originally introduced in 1991 as a powerful logic for specifying the safety of realtime systems. We embed the Duration Calculus into the theory of Boolean semirings and extend t ..."
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Abstract. We present an algebraic semantics for Duration Calculus based on semirings and quantales. Duration Calculus was originally introduced in 1991 as a powerful logic for specifying the safety of realtime systems. We embed the Duration Calculus into the theory of Boolean semirings and extend them to Kleene algebras and ωalgebras, respectively, to express finite and infinite iteration. This allows us to calculate easily with the safety requirements and to see special results of the Duration Calculus in a more general context. When formulating an algebraic semantics we also generalise parts of von Karger’s work about reactive systems, especially, the engineer’s induction. 1
— tous droits réservés — Demonic Algebra with Domain ⋆
, 2006
"... Abstract. We first recall the concept of Kleene algebra with domain (KAD). Then we explain how to use the operators of KAD to define a demonic refinement ordering and demonic operators (many of these definitions come from the literature). Then, taking the properties of the KADbased demonic operator ..."
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Abstract. We first recall the concept of Kleene algebra with domain (KAD). Then we explain how to use the operators of KAD to define a demonic refinement ordering and demonic operators (many of these definitions come from the literature). Then, taking the properties of the KADbased demonic operators as a guideline, we axiomatise an algebra that we call Demonic algebra with domain (DAD). The laws of DAD not concerning the domain operator agree with those given in the 1987 CACM paper Laws of programming by Hoare et al. Finally, we investigate the relationship between demonic algebras with domain and KADbased demonic algebras. The question is whether every DAD is isomorphic to a KADbased demonic algebra. We show that it is not the case in general. However, if a DAD D is isomorphic to a demonic algebra based on a KAD K, then it is possible to construct a KAD isomorphic to K using the operators of D. We also describe a few open problems. 1
Separability in Domain Semirings
"... First, we show with two examples that in test semirings with an incomplete test algebra a domain operation may or may not exist. Second, we show that two notions of separability in test semirings coincide, respectively, with locality of composition and with extensionality of the diamond operators in ..."
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First, we show with two examples that in test semirings with an incomplete test algebra a domain operation may or may not exist. Second, we show that two notions of separability in test semirings coincide, respectively, with locality of composition and with extensionality of the diamond operators in domain semirings. We conclude with a brief comparison of dynamic algebras and modal Kleene algebras.