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UID:7680@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20161208T103000
DTEND;TZID=Europe/Paris:20161208T113000
DTSTAMP:20241120T204756Z
URL:https://www.i2m.univ-amu.fr/evenements/the-geometry-of-logic-algebra-a
 nd-computation/
SUMMARY:Antonino Salibra (DAIS\, Universita' Ca'Foscari Venezia): The geome
 try of logic\, algebra and computation
DESCRIPTION:Antonino Salibra: A l’occasion de la soutenance de thèse de 
 Thomas Leventis l’après-midi\;\n\nWe present a research program relatin
 g Logic\, Algebra and Computation through decomposition operators\, Boolea
 n algebras of dimension n and Boolean vector spaces.\nFirst we introduce t
 he notion of dimension in universal algebra. An algebra has dimension n if
  it admits a term operation q of arity n+1 and n constants p_1\,…\,p_n s
 atisfying the fundamental properties of the n-ary if-then-else connective:
  q( p_i\, x_1\,...\,x_n) = x_i for every i=1\,...\,n. The constants p_1\,
 …\,p_n\, which act as n-ary projections\, can be\, for example\, variabl
 es in free algebras or lambda calculus\, a basis of a vector space\, the t
 ruth values of a logic\, or prime numbers in arithmetics.\nEvery algebra A
  of dimension n contains the n-dimensional Boolean algebra (nBA\, for shor
 t) of its n-central elements. An element c is n-central if it satisfies th
 e equations simultaneously satisfied by the projections p_1\,...\, p_n. Th
 is is equivalent to the requirement that the operator q(c\,-\,…\,-) is a
  decomposition operator\, so that c can decompose the algebra A as Cartesi
 an product of n more simple factors. nBAs are a dimensional generalization
  of classical Boolean algebras (= 2BA). The classical algebraic structures
  of mathematics -including groups\, rings\, fields\, Boolean algebras\, et
 c. - are only the dimension 2 case of much richer algebraic world.\nWe sho
 w a representation theorem for nBAs: every nBA is the nBA of n-central ele
 ments of a suitable Boolean vector space. As a consequence\, every algebra
  of dimension n can be linearly approximated by a Boolean vector space.\nW
 e apply this theory to Logic and Computation:\n(1) We generalize two-value
 d classical propositional logic to n-valued classical propositional logic 
 with a perfect symmetry among the n-truth values. As the Lindenbaum algebr
 a of classical logic is a Boolean algebra\, the Lindenbaum algebra of the 
 classical n-valued logic is a Boolean algebra of dimension n. A sequent ca
 lculus with n different kinds of sequents (one for each dimension) is prov
 ided.\n(2) We propose an algebraization of tabular classical and non-class
 ical logics\, based on factor varieties and decomposition operators. In pa
 rticular\, we provide a new method for determining whether a propositional
  formula is a tautology or a contradiction. This method can be automatized
  by defining a term rewriting system that enjoys confluence and strong nor
 malization. This also suggests an original notion of logical gate and circ
 uit\, where propositional variables become logical gates and logical opera
 tions are implemented by substitution. Concerning formulas with quantifier
 s\, we present a simple algorithm based on factor varieties for reducing f
 irst-order classical logic to equational logic. We achieve a completeness 
 result for first-order classical logic without requiring any additional st
 ructure.\n\n\n&nbsp\;
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Antonio_Salibra.jpg
CATEGORIES:Séminaire,Logique et Interactions
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