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!