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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 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 = 0for 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
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
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 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): 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 subtermbi(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 (t1) 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)
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.
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
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 = ¬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).
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.
