2018-06-25T09:45:55Z
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oai:digital.csic.es:10261/139265
2017-12-18T13:25:47Z
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com_10261_4
col_10261_313
The logic of distributive bilattices
Bou, Felix
Rivieccio, Umberto
European Commission
Ministerio de Educación y Ciencia (España)
Generalitat de Catalunya
Non-protoalgebraic logic
Bilattice
Reduced model
Many-valued logic
Abstract algebraic logic
Algebraizable Gentzen system
Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265-316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25-63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap-Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron's logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron's semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú (1995, Algebraizable Gentzen Systems and the Deduction of Theorem for Gentzen Systems) (even if the logic itself is not algebraizable). We also prove some purely algebraic results concerning bilattices, for instance that the variety of (unbounded) distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron (1996, Math. Struct. Comput. Sci., 6, 287-299) stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices (of which distributive bilattices are a proper subclass). © The Author 2010. Published by Oxford University Press. All rights reserved.
2016-10-24T14:26:53Z
2016-10-24T14:26:53Z
2011
2016-10-24T14:26:54Z
Artículo
Logic Journal of the IGPL 19: 183- 216 (2011)
http://hdl.handle.net/10261/139265
10.1093/jigpal/jzq041
http://dx.doi.org/10.13039/501100000780
http://dx.doi.org/10.13039/501100002809
eng
Postprint
Sí
openAccess
Oxford University Press