muflon 1.3.3

Library for martice operations

MUFLON: Matrix Utility for Intuitionistic Fuzzy Relational Equations

MUFLON is a Python library for computations with intuitionistic fuzzy values and intuitionistic fuzzy relational systems of equations.

The library treats an intuitionistic fuzzy value as a pair

$(\mu,\nu)\in L^{},\qquad L^{}={(\mu,\nu)\in[0,1]^2:\mu+\nu\le 1}$

The first coordinate $(\mu)$ is the membership degree, and the second coordinate $(\nu)$ is the non-membership degree. Every input and output pair should satisfy the condition

$\mu+\nu\le 1.$

Mathematical model

The library supports intuitionistic fuzzy relational systems of the form

$A^{\vee,{*}}_{\wedge,\diamond}\circ x=b.$

The system is decomposed into two fuzzy relational systems:

$\max_j(a^{\mu}_{ij}*x^{\mu}_j)=b^{\mu}_i,$

and

$\min_j(a^{\nu}_{ij}\diamond x^{\nu}_j)=b^{\nu}_i.$

Here () is the membership operation and $(\diamond)$ is the non-membership operation. In the most common cases, $()$ is a t-norm and $(\diamond)$ is its dual t-conorm, for example $(T_M,S_M)$, $(T_P,S_P)$, or $(T_L,S_L)$. More generally, the operations should satisfy the assumptions required for the corresponding fuzzy relational equation.

Data format

MUFLON works with CSV files in which each cell contains one intuitionistic fuzzy value.

Example:

0.3, 0.4; 0.2, 0.1; 0.5, 0.2
0.7, 0.1; 0.6, 0.2; 1.0, 0.0

The semicolon ; separates columns. The comma , separates the membership and non-membership coordinates inside a cell. Decimal numbers should use a dot.

Recommended parsing function:

from muflon.data_io import parse_ifs_csv_to_components

membership_matrix, nonmembership_matrix = parse_ifs_csv_to_components(df)

Composition of intuitionistic fuzzy matrices

For the membership component, the composition is

$$C^{\mu}{ik}=\max_j(a^{\mu}{ij}*b^{\mu}_{jk}).$$

For the non-membership component, the composition is

$$C^{\nu}{ik}=\min_j(a^{\nu}{ij}\diamond b^{\nu}_{jk}).$$

Recommended usage:

from muflon.ifs_operators import get_operator
from muflon.ifs_operations import (
    compose_component_matrices,
    combine_components_to_ifs,
    validate_l_star_condition,
)

membership_operation = get_operator("T_M")
nonmembership_operation = get_operator("S_M")

membership_result = compose_component_matrices(
    A_mu,
    B_mu,
    component_ops=[membership_operation],
    aggregation=np.max,
)

nonmembership_result = compose_component_matrices(
    A_nu,
    B_nu,
    component_ops=[nonmembership_operation],
    aggregation=np.min,
)

ifs_matrix = combine_components_to_ifs(membership_result, nonmembership_result)
is_valid, sums = validate_l_star_condition(membership_result, nonmembership_result)

The high-level function

compose_ifs_matrices(...)

computes both components and returns a matrix of intuitionistic fuzzy pairs. The result should still be checked with validate_l_star_condition().

Solving the decomposed system

For the membership system

$$\max_j(a^{\mu}_{ij}*x^{\mu}_j)=b^{\mu}_i,$$

the candidate for the greatest membership solution is computed by the induced implication:

$$u^{\mu}j=\min_i(a^{\mu}{ij}\to_{*} b^{\mu}_i).$$

For the non-membership system

$$\min_j(a^{\nu}_{ij}\diamond x^{\nu}_j)=b^{\nu}_i,$$

the corresponding component is computed by the dual induced implication:

$$z^{\nu}j=\max_i(a^{\nu}{ij}\leftarrow_\diamond b^{\nu}_i).$$

Recommended usage:

from muflon.ifs_operators import get_operator
from muflon.ifs_operations import (
    solve_component_system,
    solve_ifs_system_candidate,
    validate_l_star_condition,
)

induced_implication_mu = get_operator("IMP_T_M")
dual_induced_implication_nu = get_operator("DIMP_S_M")

greatest_membership_solution = solve_component_system(
    A_mu,
    b_mu,
    induced_implication_mu,
    aggregation=np.min,
)

least_nonmembership_solution = solve_component_system(
    A_nu,
    b_nu,
    dual_induced_implication_nu,
    aggregation=np.max,
)

is_valid, sums = validate_l_star_condition(
    greatest_membership_solution,
    least_nonmembership_solution,
)

The term least_nonmembership_solution means the least solution of the non-membership component in the usual order on $([0,1])$. Together with the greatest membership solution it forms a candidate for the greatest solution in the intuitionistic order. This candidate must satisfy the $(L^{*})$ condition.

Available operators

Meaning	Identifier
Minimum t-norm	T_M
Product t-norm	T_P
Łukasiewicz t-norm	T_L
Drastic t-norm	T_D
Fodor t-norm	T_FD
Maximum t-conorm	S_M
Probabilistic sum	S_P
Łukasiewicz t-conorm	S_L
Drastic sum	S_D
Fodor t-conorm	S_FD
Induced implication for T_M	IMP_T_M
Induced implication for T_P	IMP_T_P
Induced implication for T_L	IMP_T_L
Induced implication for T_FD	IMP_T_FD
Dual induced implication for S_M	DIMP_S_M
Dual induced implication for S_P	DIMP_S_P
Dual induced implication for S_L	DIMP_S_L

Reduced matrices and minimal component solutions

The reduced matrix is computed componentwise.

A_reduced = compute_reduced_matrix(A, x_component, b_component, operator, mode="eq")
A_binary = binarize_reduced_matrix(A_reduced)
minimal_component_solutions = find_minimal_component_solutions(
    A,
    b_component,
    A_reduced,
    dual_implication,
    operator,
    mode="eq",
)

The function find_minimal_component_solutions() concerns one component system. It does not by itself produce the full family of intuitionistic fuzzy solutions. The full intuitionistic result must also satisfy the condition $(\mu+\nu\le 1)$.Important notes

1. A joined matrix of pairs is not automatically a valid intuitionistic fuzzy matrix. It is valid only if every pair satisfies $(\mu+\nu\le 1)$.
2. The membership component should use induced implications such as IMP_T_M, IMP_T_P, or IMP_T_L.
3. The non-membership component should use dual induced implications such as DIMP_S_M, DIMP_S_P, or DIMP_S_L.
4. The non-membership operation should be described as $(\diamond)$, not simply as an arbitrary s-conorm. In standard dual cases, $(\diamond)$ is the t-conorm dual to $({*})$.
5. If a complement transformation $(1-\nu)$ is used in an algorithm, it must be explicitly implemented and documented. Otherwise, the description should not claim that such a transformation is automatic.