Format-Restricting Instructions (FRI)

• FRI is a set of fuzzy rule interpolation methods that generate outputs from sparse rule bases by interpolating between incomplete antecedent configurations.
• Methods like KH, MACI, and SCALEMOVE use diverse mathematical frameworks to balance computational efficiency, abnormality avoidance, and output linearity.
• The FRI Toolbox unifies these techniques, enabling systematic benchmarking and experimentation with various membership functions and multi-dimensional scenarios.

Fuzzy Rule Interpolation (FRI) methods constitute a class of inference algorithms designed to address reasoning with sparse rule bases, where classical fuzzy reasoning—such as Mamdani or Takagi-Sugeno systems—fails due to the lack of complete coverage over all possible antecedent configurations. FRI techniques construct inferred outputs from incomplete fuzzy models by interpolating (and, in some variants, extrapolating) between existing rules. The field features a diversity of mathematical frameworks, each with particular trade-offs in terms of computational complexity, ability to handle multidimensional or differently shaped fuzzy sets, and preservation of normality, convexity, and piecewise linearity in the resulting fuzzy sets.

1. Major FRI Methods and Interpolation Formalisms

Ten principal FRI methods are reviewed. Each is based on a distinct interpolation paradigm and exhibits unique behaviors in abnormality and linearity preservation:

• KH Method (Kóczy-Hirota): Linear one-step interpolation based on a ratio of distances between input and reference antecedent fuzzy sets,

$$\frac{d(A^*,A_1)}{d(A^*,A_2)}=\frac{d(B^*,B_1)}{d(B^*,B_2)}$$

where $d(\cdot,\cdot)$ is typically a Euclidean metric between fuzzy sets. It is computationally straightforward but can yield abnormal results, especially with non-uniform or multi-dimensional fuzzy sets.

• KH Stabilized: Generalizes KH by incorporating all flanking rules using nth-power distance weighting to suppress abnormality,

$$\min B^o_\alpha = \frac{\sum_{i=1}^m \inf(B^i_\alpha)/d^n(A^*,A^i)}{\sum_{i=1}^m 1/d^n(A^*,A^i)}$$

with a similar formula for the maximum.

• VKK (Vass-Kalmar-Kóczy): Uses interpolated centers and widths,

$$\text{Center}(B^*) = \frac{d(A^*,A_{i_2})\,\text{Center}(B^{i_1}) + d(A^*,A_{i_1})\,\text{Center}(B^{i_2})}{d(A_{i_1},A_{i_2})}$$

Analogous for width, provides computational efficiency but still susceptible to abnormality.

• MACI (Modified α-Cut): Vectorizes fuzzy sets (core, left, right endpoints) and interpolates them componentwise,

$$RB^* = (1-\tau_\text{core}) RB_1 + \tau_\text{core} RB_2$$

where $\tau_\text{core}$ is a parametric weight. Always yields convex, normal sets; robust for multi-dimensional cases.

• CRF (Conservation of Relative Fuzziness): Maintains fuzzy “distance ratios” between core and support widths:

$c^* = c^* \cdot \frac{d(B_1,B_2)}{d(A_1,A_2)}$

Supports multidimensional and abnormality-robust inference.

• IMUL (Improved Multidimensional α-Cut Level): Incorporates MACI/CRF principles for interpolation and fuzziness conservation, intended for high-dimensional spaces.
• GM (Generalized Method): Two-phase: uses fuzzy relations and reference point semantics,

$d(A_1,A_2) = |RP(A_2) - RP(A_1)|$

to ensure normality, convexity, and handle non-convex fuzzy sets.

• FRIPOC (POlar Cuts): Utilizes polar cut geometry and weighted means of shifted sets,

$RP_{B_i} = \sum_{j=1}^N RP(B_{ij})\,s_j$

Well-suited for both sparse and dense rules but sometimes fails to maintain piecewise linearity.

• LESFRI (Least Squares FRI): Optimizes set “breakpoints” via a weighted least-squares criterion,

$$Q' = \sum_j w_j(x_j - x^*)^2,\quad w_j = d(A_j,A^*)^n$$

Can generate new terms and supports interpolation and extrapolation.

• SCALEMOVE: Constructs the interpolated set using center of gravity (COG) based weights,

$$AREP = \frac{d(\text{REP}(A_1),\text{REP}(A^*))}{d(\text{REP}(A_1),\text{REP}(A_2))}$$

$A' = (1 - AREP) A_1 + AREP A_2$

Guarantees normality, convexity, and is naturally extensible to multiple antecedents and extrapolation.

2. FRI Toolbox Functionality

The FRI Toolbox provides a unifying MATLAB GUI and codebase to implement all listed FRI techniques. It supports:

• All major interpolation concepts (linear, distance-weighted, core-based, vectorized, least-squares, polar cut, scale-and-move, etc.)
• Multiple membership function types (triangular, trapezoidal, singleton)
• Experimentation with FIS (fuzzy inference system) and OBS (observation) files, allowing specification of rule bases and observations
• Adjustment of algorithm-specific parameters (e.g., distance exponents, $\alpha$-levels)
• Visual comparison and analysis of result fuzzy sets, including inspection of abnormality, loss of convexity, and piecewise linearity

The toolbox is intended to standardize and systematize the evaluation of FRI methods across the research community, supporting both instructional and practical application usage scenarios.

3. Benchmark Example Systematics

Seven unified benchmark examples are used to scrutinize method properties:

• Span dimensionality (single and multi-dimensional)
• Vary membership function (MF) shape: triangular, trapezoidal, singleton, mix
• Assess behavior on both regular interpolation and extrapolation tasks

Examples include pairs such as triangular-to-triangular (1D), triangular-to-trapezoidal (1D), and higher-dimensional cases with mixtures of set types. Each FRI method is applied to every example; the results are visually and mathematically compared with special attention to shape preservation and abnormality.

4. Comparative Classification of Methods

A systematic taxonomy is developed based on abnormality resilience, linearity of output, and compatibility with heterogeneous MF shapes:

Method	Multi-dim Ready	Abnormality	Linear/Convex	Extrapolation	Heterogeneous MF
KH	Yes	Yes	Linear (1D)	No	Same
KH-Stabilized	Yes	Yes	Linear (1D)	No	Same
VKK	Yes	Yes	Linear (1D)	No	Same
MACI	Yes	No	Yes	Yes	Any
CRF	Yes	No	Yes	Yes	Any
IMUL	Yes	No	Yes	Yes	Any
GM	Yes	No	Yes	Yes	Any
FRIPOC	Yes	Yes (on linearity)	Piecewise Linear	Yes	Subnormal
LESFRI	Yes	Yes (multi-dim)	Linear (usually)	Yes	Any
SCALEMOVE	Yes	No	Yes	Yes	Any

Key conclusions from the benchmark:

• MACI, IMUL, CRF, GM, SCALEMOVE: robust to abnormalities and linearity losses regardless of MF dimensionality or shape.
• KH, KH-Stabilized, LESFRI, VKK: prone to abnormal results under multi-dimensionality or with differing MF types.
• FRIPOC: may lose piecewise linearity in multi-dimensional or mixed-shape cases.
• SCALEMOVE: preserves both convexity and normality under all tested conditions.

5. Mathematical Formalisms and Abnormality Management

Representative formulas from several methods, restated:

• KH Linear Interpolation:

$$\frac{d(A^*, A_1)}{d(A^*, A_2)} = \frac{d(B^*, B_1)}{d(B^*, B_2)}$$

• KH-Stabilized (α-cuts):

$$\min B^o_\alpha = \frac{\sum_{i=1}^m \inf(B^i_\alpha)/d^n(A^*, A^i)}{\sum_{i=1}^m 1/d^n(A^*, A^i)}$$

• MACI/IMUL Vectorization:

$$RB^* = (1 - \tau_{\text{core}}) RB_1 + \tau_{\text{core}} RB_2$$

• CRF Core/Fuzziness Conservation:

$c^* = c^* \cdot \frac{d(B_1, B_2)}{d(A_1, A_2)}$

• SCALEMOVE Weight:

$$AREP = \frac{d(\text{REP}(A_1), \text{REP}(A^*))}{d(\text{REP}(A_1), \text{REP}(A_2))},\quad A' = (1-AREP)A_1 + AREP A_2$$

These mathematical mechanisms are deployed to ensure result sets maintain key properties, with some (MACI, CRF, IMUL, GM, SCALEMOVE) provably avoiding abnormality across the benchmark suite.

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In summary, the refreshed and extended FRI Toolbox offers an integrated platform for systematic evaluation and development of Fuzzy Rule Interpolation techniques. The systematized benchmarks demonstrate that robust, normal, convex, and piecewise-linear conclusions are obtainable for a wide variety of fuzzy rulebase configurations, provided advanced methods (MACI, IMUL, CRF, GM, SCALEMOVE) are selected. For more basic interpolation technologies (e.g., KH, VKK), caution is required in multi-dimensional or heterogeneous settings due to persistent abnormality risk. This quantitative taxonomy provides both a pragmatic guide for application engineers and a foundation for further theoretical work in fuzzy systems reasoning.