Logica II per Informatica
This year I will teach the course “Logica II” for the M.sc. degree in Computer Science. I think I will follow Moore’s modified method for the course (see also the wikipedia entry for the original method).
There are some freely available notes, looking promising, I will use during the course.
Other reference (standard books) that may be used as sources of inspiration during the course are:
- Mendelson, E. Introduction to Mathematical Logic. Chapman & Hall 2009.
- Mundici, D. Logica – Metodo breve. Springer Verlag 2011.
- Asperti, A. and Ciabattoni, A. Logica a Informatica. McGraw-Hill 2003
Further information will follow on this website.
ManyVal12 in Salerno 4-7 July
I am glad to announce that the 2012 edition of the conference series ManyVal will be hosted in Salerno, on the occasion of Antonio Di Nola $65^{th}$ birthday. Further information on the website of the conference.
Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras
We prove that the unification type of Lukasiewicz infinite-valued propositional logic and of its equivalent algebraic semantics, the variety of MV-algebras,is nullary. The proof rests upon Ghilardi’s algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra – a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.
Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras
The dual adjunction between MV-algebras and Tychonoff spaces
We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to aduality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.
The dual adjunction between MV-algebras and Tychonoff spaces
Representation of MV-algebras by regular ultrapowers of [0,1]
We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal \alpha, of a single MV-algebra in which all infinite MV-algebras of cardinality at most $\alpha$ embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of “canonical” set of values for the functional representation.
Representation of MV-algebras by regular ultrapowers of [0,1]
