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And... Action! – Monoid Actions and (Pre)orders

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Declarative Programming and Knowledge Management (INAP 2013, WLP 2013, WFLP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8439))

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Abstract

Orders occur naturally in many areas of computer science and mathematics. In several cases it is very simple do describe an order mathematically, but it may be cumbersome to implement in some programming language. On the other hand many order relations are defined in terms of an existential quantification. We provide a simple abstraction of such definitions using the well-known concept of monoid actions and furthermore show that in fact every order relation can be obtained from a specific monoid action.

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Notes

  1. 1.

    A reflexive and transitive relation is called preorder.

  2. 2.

    All of these can be considered functional classes in the sense of set theory.

  3. 3.

    Actually we should have written \(+ : {\mathbb {N}}\times \mathbb Z\rightarrow \mathbb Z\) for congruence with the following definition. The reason we did not is that for the three examples the monoid is better placed in the second component, whereas from a mathematical point of view it is more convenient to place it in the first one.

  4. 4.

    The cited source deals with finite state sets only, but the technique easily carries over to infinite sets as well.

  5. 5.

    In fact \(\leadsto _G\) is the reflexive-transitive closure of \(E\), cf. [9].

  6. 6.

    Clearly this is a greatly simplified approach, since not every function \( f \mathbin {::}\mu \rightarrow \alpha \rightarrow \alpha \) is a monoid action. A user has to verify that \(\mu \) is a monoid and \( f \) is a monoid action to ensure that \( preOrder \, f \) is in fact a preorder. Alternatively, a proof assistant (e.g. Coq [4]) can be used to guarantee that \( preOrder \) is applicable only once the necessary conditions have been proved.

  7. 7.

    It is simple (but lengthy) to define the integers based on the naturals and to extend the definitions of \(\oplus \) and \(\sqsubseteq _\oplus \) to allow the implementation of \(\le _\mathbb Z\).

  8. 8.

    An order \(\le \subseteq A \times A\) is called total iff for all \(x, y \in A\) it is true that \(x \le y\) or \(y \le x\).

  9. 9.

    The pattern matching in the first rule makes the rules non-overlapping.

References

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Authors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Nikita Danilenko

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  1. Nikita Danilenko
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Correspondence to Nikita Danilenko .

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Editors and Affiliations

  1. Universität Kiel, Kiel, Germany

    Michael Hanus

  2. CRACS & INESC-Porto LA, University of Porto, Porto, Portugal

    Ricardo Rocha

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Danilenko, N. (2014). And... Action! – Monoid Actions and (Pre)orders. In: Hanus, M., Rocha, R. (eds) Declarative Programming and Knowledge Management. INAP WLP WFLP 2013 2013 2013. Lecture Notes in Computer Science(), vol 8439. Springer, Cham. https://doi.org/10.1007/978-3-319-08909-6_6

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  • DOI: https://doi.org/10.1007/978-3-319-08909-6_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08908-9

  • Online ISBN: 978-3-319-08909-6

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