znumbers 0.2.1

Z-numbers arithmetic library (Lotfi Zadeh’s Z-numbers, with 4 basic operations)

znumbers — Z-Number Arithmetic and Linear System Solver in Python

This package implements arithmetic operations and linear system solvers for "Z-numbers", introduced by Lotfi A. Zadeh as a powerful model for representing both uncertainty and reliability in information.

A Z-number is defined as:

Z = (A, B)

where:

• A is a fuzzy restriction (Triangular Fuzzy Number, TFN)
• B is the reliability of the restriction (also TFN)

This library is intended for research, education, and experimental modeling in fuzzy systems and uncertainty-based decision making.

Features

• Triangular Fuzzy Number (TFN) implementation
• Z-number arithmetic:
  • Addition
  • Subtraction
  • Multiplication
  • Division (regularized to avoid zero-support singularities)
• Reliability propagation using:
  • Linear Programming (LP)
  • Probabilistic convolution
• Linear system solvers with Z-number coefficients:
  • Regularized Gauss elimination (recommended for 2x2 and 3x3 systems)
• Fast approximate solver using only TFN parts (for large systems)

Installation

pip install znumbers
Basic Example
Creating Z-numbers
python
Копировать код
from znumbers import TFN, ZNumber

Z1 = ZNumber(TFN(2, 3, 4), TFN(0.7, 0.8, 0.9))
Z2 = ZNumber(TFN(1, 2, 3), TFN(0.6, 0.7, 0.8))

print("Z1 =", Z1)
print("Z2 =", Z2)
Arithmetic Operations
python
Копировать код
print("Z1 + Z2 =", Z1 + Z2)
print("Z1 - Z2 =", Z1 - Z2)
print("Z1 * Z2 =", Z1 * Z2)
print("Z1 / Z2 =", Z1 / Z2)
All operations propagate both the fuzzy restriction (A-part)
and the reliability (B-part).

Solving Linear Systems with Z-Numbers
Consider the system:

A x = b

where all coefficients are Z-numbers.

Example: 2x2 Z-number System
python
Копировать код
from znumbers import TFN, ZNumber, solve_gauss_regularized

Z11 = ZNumber(TFN(2,3,4), TFN(0.7,0.8,0.9))
Z12 = ZNumber(TFN(1,2,3), TFN(0.6,0.7,0.8))

Z21 = ZNumber(TFN(3,4,5), TFN(0.7,0.8,0.9))
Z22 = ZNumber(TFN(2,3,4), TFN(0.6,0.7,0.8))

b1 = ZNumber(TFN(5,6,7), TFN(0.7,0.8,0.9))
b2 = ZNumber(TFN(6,7,8), TFN(0.6,0.7,0.8))

A = [[Z11, Z12],
     [Z21, Z22]]
b = [b1, b2]

x = solve_gauss_regularized(A, b)

print("x1 =", x[0])
print("x2 =", x[1])
The solution is also returned as Z-numbers:

Z(A = (a, b, c), B = (a, b, c))

Fast Approximate Solver (TFN Only)
For large systems, full Z-number computation may be very slow.
A fast solver is provided using only the fuzzy restriction parts.

python
Копировать код
from znumbers import solve_tfn_system

x = solve_tfn_system(A, b)
This method returns TFN solutions and does not propagate reliability.

Mathematical Background
Reliability propagation is performed using linear programming (LP)
to obtain probability distributions over fuzzy supports.

Convolution of fuzzy supports is used to combine probabilities.

Division is regularized by shifting TFN supports when zero is included,
which ensures numerical stability.

Regularized Gauss elimination allows solving small Z-number systems
while preserving reliability information.

Performance Notes
Z-number arithmetic is computationally expensive due to:

repeated LP optimization

convolution of probability distributions

Recommended usage:

Full Z-number solver: small systems (2x2, 3x3)

TFN-only solver: large systems or fast approximation

Citation
If you use this package in academic work, please cite:

Lotfi A. Zadeh,
"Z-numbers — a new paradigm for representing uncertainty",
Information Sciences, 181(3), 2011.

## Author
Dilmurod Yusupov  
Uzbekistan  
Affiliation: Cyber University  
Email: [e-mail]  
Research interests: fuzzy logic, Z-numbers, uncertainty modeling, numerical methods