Domain Semirings United

Fahrenberg, Uli; Johansen, Christian; Struth, Georg; Ziemiański, Krzysztof

doi:10.14232/actacyb.291111

Cited by 6 publications

(26 citation statements)

References 3 publications

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“…Given that iposets form a partial monoid (a category) under gluing composition, it is not obvious how their algebraic structure lifts to the power set level, where nondeterminism is modelled. Recent work [6,9,10] provides generic tools to deal with these issues.…”

Section: Discussionmentioning

confidence: 99%

Posets with Interfaces for Concurrent Kleene Algebra

Fahrenberg¹,

Johansen²,

Struth³

et al. 2021

Preprint

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We introduce posets with interfaces (iposets) and generalise the serial composition of posets to a new gluing composition of iposets. In partial order semantics of concurrency, this amounts to designate events that continue their execution across components. Alternatively, in terms of decomposing concurrent systems, it allows cutting through some events, whereas serial composition may cut through edges only.We show that iposets under gluing composition form a category, extending the monoid of posets under serial composition, and a 2-category when enriched with a subsumption order and a suitable parallel composition as a lax tensor. This generalises the interchange monoids used in concurrent Kleene algebra.We also consider gp-iposets, which are generated from singletons by finitary gluing and parallel compositions. We show that the class includes the seriesparallel posets as well as the interval orders, which are also well studied in concurrency theory. Finally, we show that not all posets are gp-iposets, exposing several posets that cannot occur as induced substructures of gp-iposets.

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Section: Discussionmentioning

confidence: 99%

Posets with Interfaces for Concurrent Kleene Algebra

Fahrenberg¹,

Johansen²,

Struth³

et al. 2021

Preprint

Self Cite

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“…Most formalisms for non-interleaving concurrency have a notion of events: unique occurrences of actions in space and time. HDAs, on the other hand, do not have a well-defined notion of event [9,36]. Going back to the example in Figure 1, we see that the Petri nets on each side of the figure have two events each, induced by their transitions and labeled a and b, respectively.…”

Section: Overviewmentioning

confidence: 99%

“…The function ε distinguishes events that have not yet started (labelled by 0) from those that have finished (labelled by 1) and those that are executing (labelled by ). This notation is inspired by Chu spaces [35]; see also [9] for the relation between HDAs and Chu spaces. For every morphism ( f , ε), the isomorphism f : S → ε −1 ( ) ⊆ T is unique; the map f is therefore determined by ε.…”

Section: Precube Categoriesmentioning

confidence: 99%

“…If X is a precubical subset of S , with an embedding f : X → S , then X is also event consistent. Such precubical subsets of standard cubes are called sculptures in [9], where it is shown that they correspond to Chu spaces over {0, , 1} [35].…”

Section: Labelings and Eventsmentioning

confidence: 99%

“…Example 16 shows that precubical sets may be non-selfintersecting, but not event consistent; similarly, event consistency does not imply the non-selfintersecting property, see [9].…”

Section: Labelings and Eventsmentioning

confidence: 99%

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Languages of Higher-Dimensional Automata

Fahrenberg¹,

Johansen²,

Struth³

et al. 2021

Preprint

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We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.

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Catoids and modal convolution algebras

et al. 2023

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We show how modal quantales arise as convolution algebras $$Q^X$$ Q X of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and $$Q^X$$ Q X . The catoids introduced generalise Schweizer and Sklar’s function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers.

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Domain Semirings United

Cited by 6 publications

References 3 publications

Posets with Interfaces for Concurrent Kleene Algebra

Posets with Interfaces for Concurrent Kleene Algebra

Languages of Higher-Dimensional Automata

Catoids and modal convolution algebras

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