nonfinite basicity of one number system with constant
DESCRIPTION
Nonfinite basicity of one number system with constant. Almaz Kungozhin Kazakh National University PhD-student ACCT 2012, June 15-21. Outline. History Definitions Known results New definitions Main result. History. L. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353. - PowerPoint PPT PresentationTRANSCRIPT
Nonfinite basicity of one number system with
constant
Almaz Kungozhin
Kazakh National University
PhD-student
ACCT 2012, June 15-21
Outline
• History
• Definitions
• Known results
• New definitions
• Main result
History• L. Zadeh, Fuzzy sets, Inform. and Control
8 (1965), 338-353.
• P. Hádjek, L. Godo, F. Esteva, A complete many-valued logic with product-conjunction. Arch. Math. Logic 35 (1996) 191-208.
• A.Kungozhin, Nonfinite basicity for a certain number system, Algebra and Logic, v.51, No 1, 2012, 56-65
t-norms
• Łukasiewicz (Ł) t-norm
x ∗ y = max(0, x + y − 1)
• Gödel (G) t-norm
x ∗ y = min(x, y)
• Product t-norm
x ∗ y = x · y
Negations
• ”Classical” fuzzy negation
¬x = 1 - x
• Godel’s negation
¬0 = 1, ¬x = 0 for x > 0
A = [0;1], ¬, , =A1 = [0;1], ¬, , 1, =
where
[0, 1] is the segment of real numbers
¬(x) = 1 – x (negation)
x · y (ordinary product)
= – symbol of equality
1 – distinguished constant
Terms
0-complexity terms: x, y, .., x1, x2,...(,1)
If t, t1 are terms of complexity n, and complexity of t2 is not bigger than n, then
¬(t), (t1) (∗ t2) and (t2) (∗ t1)
are terms of complexity n + 1
IdentityTerms t1(x1, x2, …, xn) and t2(x1, x2, …, xn)
are identical in algebra
t1(x1, x2, …, xn) = t2(x1, x2, …, xn)
iff equation is satisfied in algebra for every values of variables.
Remark 1. Terms are identical iff so are their corresponding polynomials
Examples of identities
x = (x)
x y = y x
(x y) z = x (y z)
x y = y (x)
(x y) z = (y z) x
Basis of identitiesA basis in a set of identities is its subset such that
every identity turns out to be logical consequence of the basis.
(Birghoff’s completeness theorem 1935)
{bi(x1, x2, …, xni)= i(x1, x2, …, xni): iI}- basis
iff for any t = it is possible to build a chain
t t0 = t1 = ... = tk
each following term is obtained from previous by changing a subterm bi(1, 2, …, ni) to the
subterm i(1, 2, …, ni) (and vice versa)
Nurtazin conjecture (1997)
The basis of identities of the number system A = [0;1], ¬, , = is
x = (x)
x y = y x
(x y) z = x (y z)
Contrary instance
(x (y x y)) = (x y) (x y)
since
1 – x(1 – yx(1 – y)) = 1 – x + yx2 – y 2x2
(1 – xy) (1 – x(1 – y)) = (1 – xy) (1 – x+ xy) = 1 – x+ xy – xy + yx2 – y 2x2 =
= 1 – x + yx2 – y 2x2
Theorem
A system of identities in the number system A does not have a finite basis.
1-trivially identical terms
Two terms are 1-trivially identical (t 1) if they can be derived from each other by substitutions using equations (t) = t, t1 t2 = t2 t1, t1 (t2 t3) = (t1 t2) t3, t1 1 = t1, t1 1 = 1
Examplesx y 1 y (x), (x y) z 1 (y z) x
(x (y x y)) = (x y) (x y), but
(x (y x y)) 1 (x y) (x y)
1-trivial terms
A term t called A1-trivial iff any term identical to it is A1-trivially identical to it.
ExamplesTerms x, (x), (x y) are trivial.
Terms (x (y x y)), (x y) (x y) are not trivial.
Simplifying S(t)
Any A1-term can be simplified by applying the rules (t) = t, t1 1 = t1, 1 t1 = t1, t1 1 = 1, 1 t1 = 1 for any subterm in any order
The minimal term is S(t)
Remark 1. t1 t2 = t2 t1, t1 (t2 t3) = (t1 t2) t3 are not used
Remark 2. S(t) 1, or S(t) ¬1, or doesn’t contain 1’s.
Remark 3. S(t) defined correctly
Properties of S(t)
• t = S(t)
• t 1 if and only if S(t) S() (1 1, ¬1 ¬1)
• t is A1-trivial if and only if S(t) is trivial
• If S(t) is nested (then it is trivial) then t is A1-trivial
Theorem
A system of identities in the algebra
A1 = [0;1], ¬, , 1, =
does not have a finite basis.
Proof (by contradiction)
Let there is a finite basis then we add to it trivial axioms: double negation, commutative, associative lows and (if they are absent):
• x 1 = x• x ¬1 = ¬1Using simplification we can 1-trivially and
equivalently reduce this basis to a basis of identities without 1’s, and the equations x 1 = x, x ¬1 = ¬1, 1 = 1, ¬1 = ¬1. (Let maximal number of variables is lesser than n).
Series of nontrivial equations
For every even positive number n
¬(x1¬(x2… ¬(xn-1¬(xnx1¬(x2…¬(xn-1¬(xn))…) =
¬(x1x2… xn-1xn)¬(x1¬(x2…¬(xn-1¬(xn))…)
is valid in the algebra A1.
Thank You for Your Attention!