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A Note on Drastic Product Logic

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Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2014)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 443))

Abstract

The drastic product * D is known to be the smallest t-norm, since x * D y = 0 whenever x, y < 1. This t-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product t-norm based many-valued logics, in the sense of [7]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called S3MTL in [17]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the Δ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Universitá degli Studi di Milano, via Comelico 39/41, 20135, Milano, Italy

    Stefano Aguzzoli & Matteo Bianchi

  2. Dipartimento di Scienze Teoriche e Applicate, Universitá degli Studi dell’Insubria, via Mazzini 5, 21100, Varese, Italy

    Diego Valota

Authors
  1. Stefano Aguzzoli
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  2. Matteo Bianchi
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  3. Diego Valota
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Editor information

Editors and Affiliations

  1. University Montpellier 2, LIRMM - CNRS UMR 5506, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Anne Laurent

  2. LIRMM, UMR CNRS/Universite Montpellier II, 161 rue Ada, 34392, Montpellier cedex 5, France

    Olivier Strauss

  3. UPMC Univ. Paris 06, CNRS UMR 7606, LIP6, F-75005, Paris, France

    Bernadette Bouchon-Meunier

  4. Dept. of Information Systems, Iona College, 710 North Ave, 10801, New Rochelle, NY, USA

    Ronald R. Yager

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© 2014 Springer International Publishing Switzerland

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Aguzzoli, S., Bianchi, M., Valota, D. (2014). A Note on Drastic Product Logic. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 443. Springer, Cham. https://doi.org/10.1007/978-3-319-08855-6_37

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  • DOI: https://doi.org/10.1007/978-3-319-08855-6_37

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08854-9

  • Online ISBN: 978-3-319-08855-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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