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), P. Hádjek, L. Godo, F. Esteva, A complete many-valued logic with product- conjunction. Arch. Math. Logic 35 (1996) 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], ¬, , =  A 1 =  [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,.., x 1, x 2,...(,1) If t, t 1 are terms of complexity n, and complexity of t 2 is not bigger than n, then ¬(t), (t 1 ) ∗ (t 2 ) and (t 2 ) ∗ (t 1 ) are terms of complexity n + 1

Identity Terms t 1 (x 1, x 2, …, x n ) and t 2 (x 1, x 2, …, x n ) are identical in algebra t 1 (x 1, x 2, …, x n ) = t 2 (x 1, x 2, …, x n ) 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 A 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) {b i (x 1, x 2, …, x ni )=  i (x 1, x 2, …, x ni ): i  I}- basis iff for any t =  it is possible to build a chain t  t 0 = t 1 =... = t k   each following term is obtained from previous by changing a subterm b i (  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 + yx 2 – y 2 x 2 (1 – xy)  (1 – x(1 – y)) = (1 – xy)  (1 – x+ xy) = 1 – x+ xy – xy + yx 2 – y 2 x 2 = = 1 – x + yx 2 – y 2 x 2

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, t 1  t 2 = t 2  t 1, t 1  (t 2  t 3 ) = (t 1  t 2 )  t 3, t 1  1 = t 1, t 1   1 =  1 Examples x  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 A 1 -trivial iff any term identical to it is A 1 -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 A 1 -term can be simplified by applying the rules  (  t) = t, t 1  1 = t 1, 1  t 1 = t 1, t 1   1 =  1,  1  t 1 =  1 for any subterm in any order The minimal term is S(t) Remark 1. t 1  t 2 = t 2  t 1, t 1  (t 2  t 3 ) = (t 1  t 2 )  t 3 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 A 1 -trivial if and only if S(t) is trivial If S(t) is nested (then it is trivial) then t is A 1 - trivial

Theorem A system of identities in the algebra A 1 =  [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-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 ¬(x 1 ¬(x 2 … ¬(x n-1 ¬(x n x 1 ¬(x 2 …¬(x n-1 ¬(x n ))…) = ¬(x 1 x 2 … x n-1 x n )¬(x 1 ¬(x 2 …¬(x n-1 ¬(x n ))…) is valid in the algebra A 1.

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