23/03/2016 – Ana Cláudia Golzio

Non-deterministic Matrices: Theory and Applications to Algebraic Semantic.
Ana Claudia Golzio*
*Ph.D. Program in Logic – IFCH/CLE in Unicamp.
Supevisor: Marcelo Esteban Coniglio**
**CLE – Unicamp.

Non-deterministic Matrices (Nmatrices) are a generalization of the usual concept of multiple-valued matrix. However, in Nmatrices, the truth value of a given complex formula may be chosen non-deterministically from a certain nonempty set of options, while the usual logical matrices (two-valued or many-valued) the truth value of a complex formula is determined solely by the truth value of its subformulas. That is, Nmatrices are based on non-deterministic algebras (also known as multialgebras), in contrast with the usual logical matrices which are based on standard algebras.
The notion of non-deterministic algebras was introduced in Computer Science in order to deal with non-determinism. For instance, non-deterministic algebras were proposed to recognize terms from absolutely free algebras. An important utility of Nmatrices is the fact that they serve as finitary semantics, and therefore decidable for many logics which have only one infinite matrix characteristic as semantics and it is not a good decision procedure for these logics.
In this talk, I will start speaking about multialgebras historical aspects and presenting a study of Nmatrices (and the underlying non-deterministic algebras) from the point of view of universal algebra, that is some concepts like products, subalgebras, congruences, quotient algebra and ultra products.
One of the main goals of such algebraic approach to Nmatrices is the possibility of obtain a kind of algebrization, therefore, I will present and use the structures based on the hyperstructures of kind that multialgebras/hyperalgebras, called swap structures and swap Lindenbaum-Tarski structures, such as semantics for some modal logics and some paraconsistent logics.

References
Avron, A. and Lev, I. Non-deterministic multiple-valued structures, Journal of Logic and Computation, 15, p. 241-261, 2005.
Brunovský, P. O zovšeobecnených algebraických systémoch. Acta Facultatis rerum naturalium universitatis comenianae mathematica, v. 3, n. 1, p. 41-54, 1958.
Carnielli, W. and Coniglio, M. E. Swap structures for LFIs. CLE e-Prints, v. 14, n. 1, p. 1-42, 2014.
Coniglio, M. E., Cerro, L. F. Del, Peron, N. M. Finite non-deterministic semantics for some modal systems. Journal of Applied Non-Classical Logics, v. 25, n. 1, p. 20-45, 2015.
Marty, F. Sur une généralisation de la notion de groupe. Proceedings of 8th Congress Math. Scandinaves, p. 45-49, 1934.
Pickert, G. Bemerkungen zum homomorphiebegriff. Mathematische Zeitschrift, Springer-Verlag, v. 53, n. 4, p. 375-386, 1950.

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