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Nonfinite basicity of one number system with constant

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.

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Nonfinite basicity of one number system with constant

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  1. Nonfinite basicity of one number system with constant Almaz Kungozhin Kazakh National University PhD-student ACCT 2012, June 15-21

  2. Outline • History • Definitions • Known results • New definitions • Main result

  3. 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

  4. 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

  5. Negations • ”Classical” fuzzy negation ¬x = 1 - x • Godel’s negation ¬0= 1, ¬x = 0for x > 0

  6. 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

  7. 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

  8. Identity Terms 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

  9. 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

  10. Basis of identities Abasis 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): iI}- 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 subtermbi(1, 2, …, ni) to the subterm i(1, 2, …, ni) (and vice versa)

  11. 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)

  12. 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

  13. Theorem A system of identities in the number system A does not have a finite basis.

  14. 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 Examples x  y 1y  (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)

  15. 1-trivial terms A term t called A1-trivialiff any term identical to it is A1-trivially identical to it. Examples Terms x, (x), (x  y) are trivial. Terms (x  (y  x  y)), (x  y)  (x  y) are not trivial.

  16. 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= 1for any subterm inany 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

  17. 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

  18. Theorem A system of identities in the algebra A1 = [0;1], ¬, , 1, = does not have a finite basis.

  19. 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 = ¬1 Using simplification we can 1-triviallyand 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).

  20. 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.

  21. Thank You for Your Attention!

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