Crate tnorms 

T-conorm (s-norm) families for fuzzy logic and differentiable relaxations.

A t-conorm S: [0, 1]^2 -> [0, 1] generalizes logical OR. It is:

• Commutative: S(a, b) = S(b, a)
• Associative: S(S(a, b), c) = S(a, S(b, c))
• Monotone: a1 <= a2 => S(a1, b) <= S(a2, b)
• Bounded: S(a, 0) = a (0 is identity)

Different families provide different “softness” of the OR operation, useful in differentiable relaxations and probabilistic reasoning.

Named families

The constants GODEL, PRODUCT, and LUKASIEWICZ name the three common fuzzy-logic families. LogicFamily adds the corresponding residuum for implication:

assert_eq!(tnorms::tnorm(tnorms::GODEL, 0.4, 0.7), 0.4);
assert!((tnorms::tnorm(tnorms::PRODUCT, 0.4, 0.7) - 0.28).abs() < 1e-12);
assert!((tnorms::tnorm(tnorms::LUKASIEWICZ, 0.4, 0.7) - 0.1).abs() < 1e-12);
assert_eq!(tnorms::LogicFamily::Godel.residuum(0.7, 0.4), 0.4);
assert_eq!(tnorms::LogicFamily::Product.tnorm_gradient(0.4, 0.7).da, 0.7);

TruthAlgebra exposes the same three families as zero-sized marker types for crates that choose the logic at the type level:

use tnorms::{Product, TruthAlgebra};

assert!((Product::tnorm_f32(0.4, 0.7) - 0.28).abs() < 1e-6);

The generic TConormFamily catalog is an aggregation catalog. residuum is available for families with a known left-continuous path; LogicFamily remains the standard fuzzy-logic surface where the t-norm and residuum form the adjoint pair by construction.

Catalog

Family	Formula S(a,b)	Parameter	Behavior
Maximum	max(a,b)	none	Hard OR
Probabilistic	a+b-ab	none	Independent events
Bounded (Lukasiewicz)	min(1, a+b)	none	Saturating sum
Einstein	(a+b)/(1+ab)	none	Smooth OR
Hamacher	(a+b-2ab)/(1-ab)	none	Aggressive
Yager	min(1, (a^p + b^p)^(1/p))	p >= 1	Lp norm
Frank	1 - log_s(1+(s^(1-a)-1)(s^(1-b)-1)/(s-1))	s > 0	Interpolates max<->bounded
Dombi	1/(1+((1/a-1)^p + (1/b-1)^p)^(-1/p))	p > 0	Power-based
Schweizer-Sklar	dual of (x^p+y^p-1)^(1/p)	p	spans min/product/Łukasiewicz/drastic
Sugeno-Weber	dual of max(0,(x+y-1+pxy)/(1+p))	p >= -1	spans drastic/Łukasiewicz/product
Aczel-Alsina	dual of exp(-(((-ln x)^p+(-ln y)^p)^(1/p)))	p > 0	product-to-min family
Mayor-Torrens	ordinal-sum dual	p in [0, 1]	min-to-Łukasiewicz ordinal sum
Drastic	drastic sum	none	discontinuous limit
Nilpotent minimum	nilpotent-minimum dual	none	left-continuous threshold

Based on: Petersen et al., gendr – Generalized Differentiable Rendering. The t-conorm catalog provides the relaxation families used for differentiable logical operations and soft aggregation.

Structs

BinaryGradient

Partial derivatives of a binary operation.

Godel

Godel truth algebra: minimum t-norm, maximum t-conorm, and Godel residuum.

Lukasiewicz

Lukasiewicz truth algebra: bounded-sum t-norm, bounded t-conorm, and Lukasiewicz residuum.

OrdinalSegment

One interval of an ordinal-sum t-norm or t-conorm.

OrdinalSum

Ordinal sum over disjoint sub-intervals of [0, 1].

Product

Product truth algebra: product t-norm, probabilistic t-conorm, and Goguen residuum.

Enums

CopulaCompatibility

Whether a t-norm family is known to satisfy the copula constraints.

GeneratorFamily

Archimedean additive-generator family.

ImplicationFamily

Fuzzy implication family.

LogicFamily

Standard t-norm fuzzy logic families with residual implication.

MixedRegion

Mixed-region behavior for representable uninorms.

NegationFamily

Fuzzy negation family.

TConormFamily

T-conorm family selection.

Constants

GODEL

Godel family: min(a, b) t-norm and max(a, b) t-conorm.

LUKASIEWICZ

Lukasiewicz family: max(0, a + b - 1) t-norm and min(1, a + b) t-conorm.

PRODUCT

Product family: a * b t-norm and a + b - a * b t-conorm.

Traits

TruthAlgebra

A standard residuated truth algebra on degrees in [0, 1].

Functions

implication

Evaluate a fuzzy implication a -> b.

log_product

Sum of logs for product t-norm aggregation.

negation

Evaluate a fuzzy negation.

owa

Ordered weighted averaging over values.

power_mean

Generalized power mean over values in [0, 1].

power_mean_weighted

Weighted generalized power mean over values in [0, 1].

product_from_log

Convert a log-space product back to a degree in [0, 1].

representable_uninorm

Evaluate a representable uninorm with neutral point e.

residuum

Evaluate the residuum a -> b for a t-norm family when available.

residuum_numeric

Numerically approximate the residuum as sup { c : T(a, c) <= b }.

tconorm

Evaluate a t-conorm S(a, b) for the given family.

tconorm_fold

Aggregate multiple values using a t-conorm (generalized multi-way OR).

tconorm_generic

Evaluate a t-conorm S(a, b) for the given family and float type.

tconorm_gradient

Evaluate the t-conorm gradient with respect to (a, b).

tnorm

The dual t-norm T(a,b) = 1 - S(1-a, 1-b) for a given t-conorm family.

tnorm_fold

Aggregate multiple values using a t-norm (generalized multi-way AND).

tnorm_generic

Evaluate the standard-negation dual t-norm for a generic float type.

tnorm_gradient

Evaluate the t-norm gradient with respect to (a, b).

Type Aliases

Family

Short alias for selecting a t-norm/t-conorm family.