ITFRS model

ITFRS stands for Implicator/T-norm Fuzzy-Rough Sets. In frsutils, ITFRS computes lower and upper approximation degrees from a fuzzy similarity relation and class labels using a lower implicator and an upper T-norm.

Implementation contract

frsutils.core.models.ITFRS is the dense NumPy reference implementation. It expects at least two samples, a fully materialized finite fuzzy-relation matrix with values in [0, 1], a one-dimensional label vector, an upper T-norm, and a lower implicator. The low-level relation matrix may be asymmetric and need not have a unit diagonal because self-comparisons are handled explicitly. FRsutils uses relation[i, j] = R(x_i, x_j): row i is aggregated to compute the lower and upper values returned for sample x_i. Transposing an asymmetric relation can therefore change the approximation.

The public approximation API also provides exact blockwise ITFRS execution:

from frsutils import compute_approximations

result = compute_approximations(X, y, model="itfrs", engine="blockwise")

Approximation outputs

frsutils reports the following public outputs:

signed_boundary = upper_approximation - lower_approximation
positive_region = lower_approximation

boundary_region remains available as a backward-compatible name for the same signed difference. FRsutils does not clip this value, so custom component choices can produce negative values when they do not guarantee lower_approximation <= upper_approximation. Public output arrays are NumPy arrays.

Backend status

ITFRS supports:

• dense NumPy execution,
• exact blockwise NumPy execution,
• optional CuPy-backed similarity blocks,
• experimental GPU-resident blockwise approximation accumulators, with final public output converted back to NumPy arrays.