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use std::fmt::Debug;
use crate::test::cartesian_power;
/// Defines a monoid structure.
///
/// A monoid is a set of items along with an associative binary operation `f` and an identity element `zero`.
/// The `f` operation combines two items and the `zero` element acts as the identity for `f`.
pub fn monoid<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
zero: S, // zero is the identity element of f
) -> Result<(), &'static str> {
semigroup(items, f)?;
identity(items, f, zero)?;
Ok(())
}
/// Defines a semigroup structure.
///
/// A semigroup is a set of items along with an associative binary operation `f`.
pub fn semigroup<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
associativity(items, f)?;
Ok(())
}
/// Defines a semiring structure.
///
/// A semiring is a set of items along with two associative binary operations `f` and `g`,
/// and two identity elements `zero` and `one`.
/// f must be commutative and g must distribute over f.
/// the zero of f must also be absorbing with respect to g.
pub fn semiring<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
g: &impl Fn(S, S) -> S,
zero: S, // zero is the identity element of f
one: S, // one is the identity element of g
) -> Result<(), &'static str> {
commutative_monoid(items, f, zero.clone())?;
monoid(items, g, one.clone())?;
absorbing_element(items, g, zero)?;
distributive(items, f, g)?;
Ok(())
}
/// Defines a ring structure.
///
/// A ring is a semiring with an inverse operation on f.
pub fn ring<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
g: &impl Fn(S, S) -> S,
zero: S, // zero is the identity element of f
one: S, // one is the identity element of g
b: &impl Fn(S) -> S,
) -> Result<(), &'static str> {
semiring(items, f, g, zero.clone(), one)?;
inverse(items, f, zero, b)?;
Ok(())
}
/// Defines an integral domain structure.
///
/// An integral domain is a nonzero commutative ring with no nonzero zero divisors.
pub fn integral_domain<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
g: &impl Fn(S, S) -> S,
zero: S, // zero is the identity element of f
one: S, // one is the identity element of g
inverse_f: &impl Fn(S) -> S, /* the function to compute the inverse element of an element with respect to f */
) -> Result<(), &'static str> {
commutative_ring(items, f, g, zero.clone(), one, inverse_f)?;
no_nonzero_zero_divisors(items, g, zero)?;
Ok(())
}
/// Defines a no-nonzero-zero-divisors property.
///
/// x is a nonzero divisor if xy = 0 and y is also a nonzero element.
pub fn no_nonzero_zero_divisors<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
zero: S,
) -> Result<(), &'static str> {
for a in items {
for b in items {
if *a != zero && *b != zero {
if f(a.clone(), b.clone()) == zero {
return Err("No nonzero zero divisors check failed.");
};
if f(b.clone(), a.clone()) == zero {
return Err("No nonzero zero divisors check failed.");
};
}
}
}
Ok(())
}
/// Defines a commutative ring structure.
///
/// A commutative ring is a ring where the multiplication operation g is commutative.
pub fn commutative_ring<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S, // addition operation
g: &impl Fn(S, S) -> S, // multiplication operation
zero: S, // zero is the identity element of f
one: S, // one is the identity element of g
inverse_f: &impl Fn(S) -> S,
) -> Result<(), &'static str> {
semiring(items, f, g, zero.clone(), one)?;
inverse(items, f, zero, inverse_f)?;
commutativity(items, g)?;
Ok(())
}
/// Defines a field structure.
///
/// A field is a commutative ring where every element has a multiplicative inverse.
pub fn field<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
g: &impl Fn(S, S) -> S,
zero: S, // zero is the identity element of f
one: S, // one is the identity element of g
inverse_f: &impl Fn(S) -> S, /* inverse_f is the function that given x computes x' such that f(x,x') = zero. */
inverse_g: &impl Fn(S) -> S, /* //inverse_g is the function that given x computes x' such that g(x,x') = one. */
) -> Result<(), &'static str> {
commutative_ring(items, f, g, zero.clone(), one.clone(), inverse_f)?;
nonzero_inverse(items, g, one, zero, inverse_g)?;
Ok(())
}
/// Defines a commutative monoid structure.
///
/// A commutative monoid is a monoid where the operation f is commutative.
pub fn commutative_monoid<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
zero: S,
) -> Result<(), &'static str> {
monoid(items, f, zero)?;
commutativity(items, f)?;
Ok(())
}
/// Defines a group structure.
///
/// A group is a set of items along with an associative binary operation `f`
/// an identity element `zero`
/// and every element has an inverse element with respect to `f`
pub fn group<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
zero: S, // zero is the identity element of f
b: &impl Fn(S) -> S, /* b is the function to compute the inverse element of an element with respect to f */
) -> Result<(), &'static str> {
monoid(items, f, zero.clone())?;
inverse(items, f, zero, b)?;
Ok(())
}
/// Defines an abelian group structure.
///
/// An abelian group is a group where the operation f is commutative.
pub fn abelian_group<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
zero: S,
b: &impl Fn(S) -> S, /* b is the function to compute the inverse element of an element with respect to f */
) -> Result<(), &'static str> {
group(items, f, zero, b)?;
commutativity(items, f)?;
Ok(())
}
// Algebraic Properties
/// Defines the distributive property
///
/// a(b+c) = ab + ac
/// and (b+c)a = ba + ca
pub fn distributive<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: &impl Fn(S, S) -> S,
g: &impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
left_distributes(items, f, g)?;
right_distributes(items, f, g)?;
Ok(())
}
/// Defines the left distributive property
///
/// a(b+c) = ab + ac
pub fn left_distributes<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
g: impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
for [a, b, c] in cartesian_power(items) {
if g(a.clone(), f(b.clone(), c.clone()))
!= f(g(a.clone(), b.clone()), g(a.clone(), c.clone()))
{
return Err("Left distributive property check failed.");
}
}
Ok(())
}
/// Defines the right distributive property.
///
/// (b+c)a = ba + ca
pub fn right_distributes<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
g: impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
for [a, b, c] in cartesian_power(items) {
if g(f(b.clone(), c.clone()), a.clone())
!= f(g(b.clone(), a.clone()), g(c.clone(), a.clone()))
{
return Err("Right distributive property check failed.");
}
}
Ok(())
}
/// Defines the absorbing_element property.
///
/// An element z is absorbing if az = z and za = z for all a.
pub fn absorbing_element<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
z: S, // absorbing element (anything multiplied by z is z e.g. 0 in integers)
) -> Result<(), &'static str> {
for a in items {
// az = z
if f(a.clone(), z.clone()) != z.clone() {
return Err("Absorbing element property check failed.");
}
// za = z
if f(z.clone(), a.clone()) != z.clone() {
return Err("Absorbing element property check failed.");
}
}
Ok(())
}
/// Defines the inverse property.
///
/// An element b is the inverse of a if ab = e and ba = e for some identity element e.
pub fn inverse<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
e: S, // e is the identity element of f
b: impl Fn(S) -> S, /* b is the function to compute the inverse element of an element with respect to f */
) -> Result<(), &'static str> {
// ∃b: ab = e, ba = e
for a in items {
if f(a.clone(), b(a.clone())) != e {
return Err("Inverse check failed.");
}
if f(b(a.clone()), a.clone()) != e {
return Err("Inverse check failed.");
}
}
Ok(())
}
/// Defines the non_zero inverse property.
///
/// Every element except zero must have an inverse.
pub fn nonzero_inverse<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
e: S,
zero: S,
b: impl Fn(S) -> S,
) -> Result<(), &'static str> {
// ∃b: ab = e, ba = e
for a in items {
if *a != zero {
if f(a.clone(), b(a.clone())) != e {
return Err("Nonzero inverse check failed.");
}
if f(b(a.clone()), a.clone()) != e {
return Err("Nonzero inverse check failed.");
}
}
}
Ok(())
}
/// Defines the identity property.
///
/// An element e is the identity of f if ae = a and ea = a for all a.
pub fn identity<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
e: S,
) -> Result<(), &'static str> {
// ea = a, ae = a
for a in items {
if f(e.clone(), a.clone()) != a.clone() {
return Err("Left Identity check failed.");
}
if f(a.clone(), e.clone()) != a.clone() {
return Err("Right Identity check failed.");
}
}
Ok(())
}
/// Defines the associativity property.
///
/// a(bc) = (ab)c
pub fn associativity<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
for [a, b, c] in cartesian_power(items) {
if f(a.clone(), f(b.clone(), c.clone())) != // f(a, f(b,c)) ie a + (b + c)
f(f(a.clone(), b.clone()), c.clone())
// f(f(a,b),c) ie (a + b) + c
{
return Err("Associativity check failed.");
}
}
Ok(())
}
/// Defines the commutativity property.
///
/// xy = yx
pub fn commutativity<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
for [x, y] in cartesian_power(items) {
if f(x.clone(), y.clone()) != f(y.clone(), x.clone()) {
// a + b = b + a
return Err("Commutativity check failed.");
}
}
Ok(())
}
/// Defines the idempotency property.
///
/// xx = x
pub fn idempotency<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
) -> Result<(), &'static str> {
for x in items {
if f(x.clone(), x.clone()) != x.clone() {
return Err("Idempotency check failed.");
}
}
Ok(())
}
/// Defines the linearity property
///
/// q is linear with respect to some group operation + if q(a+b) = q(a) + q(b)
/// This is the same as q being a group homomorphism
/// As defined in the paper "DBSP: Automatic Incremental View Maintenance for Rich Query Languages"
/// Input parameters f, g, and q represent (f) the base operation of the algebraic structure for state,
/// (g) the base operation of the algebraic structure the query q outputs to
/// and (q) the query over f that we want to check for linearity (to incrementalize) respectively
pub fn linearity<S: Debug + PartialEq + Clone, R: Debug + PartialEq + Clone>(
items: &[S],
f: impl Fn(S, S) -> S,
g: impl Fn(R, R) -> R,
q: impl Fn(S) -> R,
) -> Result<(), &'static str> {
for [a, b] in cartesian_power(items) {
if q(f(a.clone(), b.clone())) != g(q(b.clone()), q(a.clone())) {
// q(f(a,b)) = f(q(a), q(b))
return Err("Linearity check failed.");
}
}
Ok(())
}
/// Defines the bilinearity property
///
/// q is bilinear with respect to + if q(a + b, c) = q(a,c) + q(b,c) and q(a,c + d) = q(a,c) + q(a,d)
/// This is the same as q being distributive over the addition operation of the three groups S, T, and R in q:S x T --> R
/// As defined in the paper "DBSP: Automatic Incremental View Maintenance for Rich Query Languages
/// Input parameters f, h, g, and q represent (f) the base operation of the algebraic structure on the left input to the query q,
/// (h) the base operation of the algebraic structure on the right input to the query q,
/// (g) the base operation of the algebraic structure the query q outputs to,
/// and (q) The query over (f,g) that we want to check for bilinearity (to incrementalize)
pub fn bilinearity<
S: Debug + PartialEq + Clone,
R: Debug + PartialEq + Clone,
T: Debug + PartialEq + Clone,
>(
items_f: &[S],
items_h: &[T],
f: impl Fn(S, S) -> S,
h: impl Fn(T, T) -> T,
g: impl Fn(R, R) -> R,
q: impl Fn(S, T) -> R,
) -> Result<(), &'static str> {
for [a, b] in cartesian_power(items_f) {
for [c, d] in cartesian_power(items_h) {
if q(f(a.clone(), b.clone()), c.clone())
!= g(q(a.clone(), c.clone()), q(b.clone(), c.clone()))
|| q(a.clone(), h(c.clone(), d.clone()))
!= g(q(a.clone(), c.clone()), q(a.clone(), d.clone()))
{
// q(a + b, c) = q(a,c) + q(b,c) AND
// q(a,c + d) = q(a,c + q(c,d)
return Err("Bilinearity check failed.");
}
}
}
Ok(())
}
// Functions for testing out whether user defined code satisfies different properties
// A list of algebraic properties of a single function that we support
// static SINGLE_FUNCTION_PROPERTIES: [(&str, fn(&[S; N], impl Fn(S, S) -> S)); 6] = [
// ("associativity", associativity),
// ("commutativity", commutativity),
// ("idempotency", idempotency),
// ("identity", identity),
// ("inverse", inverse),
// ("absorbing_element", absorbing_element)];
/// Loop through each algebraic property in SINGLE_FUNCTION_PROPERTIES and test for them.
pub fn get_single_function_properties<S: Debug + PartialEq + Clone, const N: usize>(
items: &[S; N],
f: impl Fn(S, S) -> S,
// identity element (TODO make optional parameter)
e: S,
// inverse function (TODO make optional parameter)
b: impl Fn(S) -> S,
// absorbing element (TODO make optional parameter)
z: S,
) -> Vec<String> {
// store the list of properties (strings) that are satisfied to be returned
let mut properties_satisfied: Vec<String> = Vec::new();
// TODO make this a loop through the SINGLE_FUNCTION_PROPERTIES array
if associativity(items, &f).is_ok() {
properties_satisfied.push("associativity".to_string());
}
if commutativity(items, &f).is_ok() {
properties_satisfied.push("commutativity".to_string());
}
if idempotency(items, &f).is_ok() {
properties_satisfied.push("idempotency".to_string());
}
if identity(items, &f, e.clone()).is_ok() {
properties_satisfied.push("identity".to_string());
}
if inverse(items, &f, e.clone(), b).is_ok() {
properties_satisfied.push("inverse".to_string());
}
if absorbing_element(items, &f, z).is_ok() {
properties_satisfied.push("absorbing_element".to_string());
}
properties_satisfied
}
// TODO write a function to take in a set of functions and check which pairs satisfy different pairwise properties (e.g. distributivity
// Tests
#[cfg(test)]
mod test {
use std::collections::HashSet;
use crate::algebra::*;
static TEST_ITEMS: &[u32; 14] = &[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];
static TEST_ITEMS_NONZERO: &[u32; 13] = &[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];
static TEST_MOD_PRIME_7: &[u32; 7] = &[0, 1, 2, 3, 4, 5, 6];
static TEST_BOOLS: &[bool; 2] = &[false, true];
#[test]
fn test_associativity() {
// Test that max() is associative and exponentiation isn't
assert!(associativity(TEST_ITEMS, u32::max).is_ok());
assert!(associativity(TEST_ITEMS, u32::wrapping_pow).is_err());
}
#[test]
fn test_left_distributes() {
// Test that multiplication and subtraction are left distributive a(b-c) = ab - ac.
// but exponentiation and subtraction isn't since a^(b-c) != a^b - a^c.
assert!(left_distributes(TEST_ITEMS, u32::wrapping_sub, u32::wrapping_mul).is_ok());
assert!(left_distributes(TEST_ITEMS, u32::wrapping_sub, u32::wrapping_pow).is_err());
}
#[test]
fn test_right_distributes() {
// Test that multiplication and subtraction are right distributive (b-c)a = ba - ca.
// but exponentiation and subtraction isn't since (b-c)^a != b^a - c^a.
assert!(right_distributes(TEST_ITEMS, u32::wrapping_sub, u32::wrapping_mul).is_ok());
assert!(right_distributes(TEST_ITEMS, u32::wrapping_sub, u32::wrapping_pow).is_err());
}
#[test]
fn test_nonzero_inverse() {
// Test that addition and subtraction has a nonzero inverse and that multiplication does not.
assert!(nonzero_inverse(TEST_ITEMS, u32::wrapping_add, 0, 0, |x| {
0u32.wrapping_sub(x)
})
.is_ok());
assert!(nonzero_inverse(TEST_ITEMS, u32::wrapping_sub, 0, 0, |x| {
0u32.wrapping_add(x)
})
.is_ok());
assert!(
right_distributes(TEST_ITEMS_NONZERO, u32::wrapping_div, u32::wrapping_mul).is_err()
);
}
#[test]
fn test_idempotency() {
// Test that max() is idempotent and addition is non-idempotent
assert!(idempotency(TEST_ITEMS, u32::max).is_ok());
assert!(idempotency(TEST_ITEMS, u32::wrapping_add).is_err());
}
#[test]
fn test_commutativity() {
// Test that max() is commutative and division is non-commutative
assert!(commutativity(TEST_ITEMS, u32::max).is_ok());
assert!(commutativity(TEST_ITEMS_NONZERO, u32::wrapping_div).is_err());
// Test items non-zero to avoid a divide by zero exception
}
#[test]
fn test_commutative_ring() {
// Test that (Z, +, *) is a commutative ring.
assert!(commutative_ring(
TEST_ITEMS,
&u32::wrapping_add,
&u32::wrapping_mul,
0,
1,
&|x| 0u32.wrapping_sub(x),
)
.is_ok());
// Test that (Z, +, ^) is not a commutative ring.
assert!(commutative_ring(
TEST_ITEMS,
&u32::wrapping_add,
&u32::wrapping_pow,
0,
1,
&|x| 0u32.wrapping_sub(x),
)
.is_err());
// Test that matrix multiplication is not a commutative ring.
assert!(commutative_ring(
&[[[1, 2], [3, 4]], [[5, 6], [7, 8]], [[9, 10], [11, 12]]],
&|a, b| {
[
[a[0][0] + b[0][0], a[0][1] + b[0][1]],
[a[1][0] + b[1][0], a[1][0] + b[1][1]],
]
},
&|a, b| {
[
[
a[0][0] * b[0][0] + a[0][1] * b[1][0],
a[0][0] * b[0][1] + a[0][1] * b[1][1],
],
[
a[1][0] * b[0][0] + a[1][1] * b[1][0],
a[1][0] * b[0][1] + a[1][1] * b[1][1],
],
]
},
[[0, 0], [0, 0]],
[[1, 0], [0, 1]],
&|a| {
[
[
-a[0][0] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
-a[0][1] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
],
[
-a[1][0] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
-a[1][1] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
],
]
},
)
.is_err());
}
#[test]
fn test_commutative_monoid() {
// Test that (Z, +) is commutative monoid since every abelian group is commutative monoid.
assert!(commutative_monoid(TEST_ITEMS, &u32::wrapping_add, 0).is_ok());
// Test that set of natural numbers N = {0, 1, 2, ...} is a commutative monoid under addition (identity element 0) or multiplication (identity element 1).
assert!(commutative_monoid(TEST_ITEMS, &u32::wrapping_mul, 1).is_ok());
assert!(commutative_monoid(TEST_ITEMS, &u32::wrapping_add, 0).is_ok());
// Test that ({true, false}, ∧) is a commutative monoid with identity element true.
assert!(commutative_monoid(TEST_BOOLS, &|a, b| a & b, true).is_ok()); // logical AND
// Test that (Z, -) is not a commutative monoid.
assert!(commutative_monoid(TEST_ITEMS, &u32::wrapping_sub, 0).is_err());
// Test that (N, +) is not a commutative monoid since it doesn't have an identity element (0 is missing).
assert!(commutative_monoid(TEST_ITEMS_NONZERO, &u32::wrapping_add, 1).is_err()); // Note that 1 is an arbitrary identity element in TEST_ITEMS_NONZERO since it doesn't have an identity element 0.
// Test that (Z, ^) is not a commutative monoid.
assert!(commutative_monoid(TEST_ITEMS, &u32::wrapping_pow, 3).is_err());
}
#[test]
fn test_semigroup() {
// Test that N := {1, 2, . . .} together with addition is a semigroup.
assert!(semigroup(TEST_ITEMS_NONZERO, &u32::wrapping_add).is_ok());
// Test that set of all natural numbers N = {0, 1, 2, ...} is a semigroup under addition.
assert!(semigroup(TEST_ITEMS, &u32::wrapping_add).is_ok());
// Test that set of all natural numbers N = {0, 1, 2, ...} is a semigroup under multiplication.
assert!(semigroup(TEST_ITEMS, &u32::wrapping_mul).is_ok());
// Test that ({true, false}, ∧) is a semigroup.
assert!(semigroup(TEST_BOOLS, &|a, b| a & b).is_ok()); // logical AND
// Test that matrix multiplication is a semigroup.
assert!(semigroup(
&[[[1, 2], [3, 4]], [[5, 6], [7, 8]], [[9, 10], [11, 12]]],
&|a, b| {
[
[
a[0][0] * b[0][0] + a[0][1] * b[1][0],
a[0][0] * b[0][1] + a[0][1] * b[1][1],
],
[
a[1][0] * b[0][0] + a[1][1] * b[1][0],
a[1][0] * b[0][1] + a[1][1] * b[1][1],
],
]
},
)
.is_ok());
// Test that set of all natural numbers N = {0, 1, 2, ...} is not a semigroup under exponentiation.
assert!(semigroup(TEST_ITEMS, &u32::wrapping_pow).is_err());
}
#[test]
fn test_identity() {
// Test that 0 is the identity for addition and 5 is not
assert!(identity(TEST_ITEMS, u32::wrapping_add, 0).is_ok());
assert!(identity(TEST_ITEMS, u32::wrapping_add, 5).is_err());
}
#[test]
fn test_inverse() {
// Test that subtraction is the inverse of addition and that addition is not the inverse of addition
assert!(inverse(TEST_ITEMS, u32::wrapping_add, 0, |x| 0u32.wrapping_sub(x)).is_ok());
assert!(inverse(TEST_ITEMS, u32::wrapping_add, 0, |x| 0u32.wrapping_add(x)).is_err());
}
#[test]
fn test_distributive() {
// Test that addition and multiplication are distributive and that addition and max() are not
assert!(distributive(TEST_ITEMS, &u32::wrapping_add, &u32::wrapping_mul).is_ok());
assert!(distributive(TEST_ITEMS, &u32::wrapping_add, &u32::max).is_err());
}
#[test]
fn test_linearity() {
// Test that multiplication over the (Z,+) group is linear
// but exponentiation over the (Z,+) group is not linear
assert!(
linearity(TEST_ITEMS, u32::wrapping_add, u32::wrapping_add, |x| {
u32::wrapping_mul(x, 5)
})
.is_ok()
);
assert!(
linearity(TEST_ITEMS, u32::wrapping_add, u32::wrapping_add, |x| {
u32::pow(x, 5)
})
.is_err()
);
}
#[test]
fn test_bilinearity() {
// Test that multiplication over the (Z,+) group is bilinear
// but exponentiation over the (Z,+) group is not bilinear
assert!(bilinearity(
TEST_ITEMS,
TEST_ITEMS,
u32::wrapping_add,
u32::wrapping_add,
u32::wrapping_add,
u32::wrapping_mul
)
.is_ok());
assert!(bilinearity(
TEST_ITEMS,
TEST_ITEMS,
u32::wrapping_add,
u32::wrapping_add,
u32::wrapping_add,
u32::pow
)
.is_err());
}
#[test]
fn test_group() {
// Test that (Z, +) form a group.
assert!(group(TEST_ITEMS, &u32::wrapping_add, 0, &|x| 0u32.wrapping_sub(x)).is_ok());
// Test that (Z/7Z, +) form a group.
assert!(group(TEST_MOD_PRIME_7, &modulo_add_7, 0, &modulo_sub_7).is_ok());
// Test that (Z/14Z, +) form a group.
assert!(group(TEST_ITEMS, &modulo_add_14, 0, &modulo_sub_14).is_ok());
// Test that (Z, *) do not form a group since it has no inverse.
assert!(group(TEST_ITEMS_NONZERO, &u32::wrapping_mul, 1, &|x| 1u32
.wrapping_div(x))
.is_err());
}
#[test]
fn test_abelian_group() {
// Test that (Z, +) form an abelian group.
assert!(
abelian_group(TEST_ITEMS, &u32::wrapping_add, 0, &|x| 0u32.wrapping_sub(x)).is_ok()
);
// Test that (Z/7Z, +) form an abelian group.
assert!(abelian_group(TEST_MOD_PRIME_7, &modulo_add_7, 0, &modulo_sub_7).is_ok());
// Test that (Z, *) do not form an abelian group.
assert!(
abelian_group(TEST_ITEMS_NONZERO, &u32::wrapping_mul, 1, &|x| 1u32
.wrapping_div(x))
.is_err()
);
// Test that matrix multiplication is not an abelian group.
assert!(abelian_group(
&[[[1, 2], [3, 4]], [[5, 6], [7, 8]], [[9, 10], [11, 12]]],
&|a, b| {
[
[
a[0][0] * b[0][0] + a[0][1] * b[1][0],
a[0][0] * b[0][1] + a[0][1] * b[1][1],
],
[
a[1][0] * b[0][0] + a[1][1] * b[1][0],
a[1][0] * b[0][1] + a[1][1] * b[1][1],
],
]
},
[[1, 0], [0, 1]],
&|a| {
[
[
-a[0][0] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
-a[0][1] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
],
[
-a[1][0] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
-a[1][1] / (a[0][0] * a[1][1] - a[0][1] * a[1][1]),
],
]
},
)
.is_err());
}
#[test]
fn test_monoid() {
// Test that N = {0, 1, 2, . . .} is a monoid with respect to addition
assert!(monoid(TEST_ITEMS, &u32::wrapping_add, 0).is_ok());
// Test that N+ = N − {0} and N are both monoids with respect to multiplication
assert!(monoid(TEST_ITEMS_NONZERO, &u32::wrapping_mul, 1).is_ok());
assert!(monoid(TEST_ITEMS, &u32::wrapping_mul, 1).is_ok());
// Test that the set of nxn matrix with matrix multiplication is a monoid.
assert!(monoid(
&[[[1, 2], [3, 4]], [[5, 6], [7, 8]], [[9, 10], [11, 12]]],
&|a, b| {
[
[
a[0][0] * b[0][0] + a[0][1] * b[1][0],
a[0][0] * b[0][1] + a[0][1] * b[1][1],
],
[
a[1][0] * b[0][0] + a[1][1] * b[1][0],
a[1][0] * b[0][1] + a[1][1] * b[1][1],
],
]
},
[[1, 0], [0, 1]],
)
.is_ok());
// Test that N+ = N − {0} is not a monoid with respect to addition since it doesn't have an identity element (0 is missing).
assert!(monoid(TEST_ITEMS_NONZERO, &u32::wrapping_add, 1).is_err());
}
#[test]
fn test_absorbing() {
// Test that 0 is absorbing for multiplication and 5 is not
assert!(absorbing_element(TEST_ITEMS, u32::wrapping_mul, 0).is_ok());
assert!(absorbing_element(TEST_ITEMS, u32::wrapping_mul, 5).is_err());
}
// Performs addition modulo 7, ensuring the result remains within the range of 0 to 6.
// This function is used to compute addition modulo 7 within the context of testing integral domains.
fn modulo_add_7(a: u32, b: u32) -> u32 {
u32::wrapping_add(a, b) % 7
}
// Performs addition modulo 14, ensuring the result remains within the range of 0 to 13.
// This function is used to compute addition modulo 14 within the context of testing integral domains.
fn modulo_add_14(a: u32, b: u32) -> u32 {
u32::wrapping_add(a, b) % 14
}
// Performs subtraction modulo 7, ensuring the result remains within the range of 0 to 6.
// This function is used to compute subtraction modulo 7 within the context of testing integral domains.
fn modulo_sub_7(a: u32) -> u32 {
u32::wrapping_sub(7, a) % 7
}
// Performs subtraction modulo 14, ensuring the result remains within the range of 0 to 13.
// This function is used to compute subtraction modulo 14 within the context of testing integral domains.
fn modulo_sub_14(a: u32) -> u32 {
u32::wrapping_sub(14, a) % 14
}
// Performs multiplication modulo 7, ensuring the result remains within the range of 0 to 6.
// This function is used to compute multiplication modulo 7 within the context of testing integral domains.
fn modulo_mult_7(a: u32, b: u32) -> u32 {
u32::wrapping_mul(a, b) % 7
}
// Performs multiplication modulo 14, ensuring the result remains within the range of 0 to 13.
// This function is used to compute multiplication modulo 14 within the context of testing integral domains.
fn modulo_mult_14(a: u32, b: u32) -> u32 {
u32::wrapping_mul(a, b) % 14
}
#[test]
fn test_additive_inverse_7() {
// Tests that the additive inverse of each element in the ring of integers modulo 7 is correct.
assert_eq!(0, modulo_sub_7(0));
assert_eq!(1, modulo_sub_7(6));
assert_eq!(2, modulo_sub_7(5));
assert_eq!(3, modulo_sub_7(4));
assert_eq!(4, modulo_sub_7(3));
assert_eq!(6, modulo_sub_7(1));
}
#[test]
fn test_modulo_mu14() {
// Tests that the multiplication modulo 14 is correct.
assert_eq!(0, modulo_mult_14(2, 7));
assert_eq!(3, modulo_mult_14(1, 3));
assert_eq!(2, modulo_mult_14(2, 1));
assert_eq!(3, modulo_mult_14(3, 1));
assert_eq!(4, modulo_mult_14(2, 2));
assert_eq!(6, modulo_mult_14(2, 3));
assert_eq!(9, modulo_mult_14(3, 3));
}
#[test]
fn test_modulo_mu7() {
// Tests that the multiplication modulo 7 is correct.
assert_eq!(0, modulo_mult_7(0, 0));
assert_eq!(3, modulo_mult_7(1, 3));
assert_eq!(2, modulo_mult_7(2, 1));
assert_eq!(2, modulo_mult_7(3, 3));
assert_eq!(2, modulo_mult_7(3, 3));
assert_eq!(5, modulo_mult_7(3, 4));
assert_eq!(1, modulo_mult_7(3, 5));
}
#[test]
fn test_no_nonzero_zero_divisors() {
// The ring of integer mod prime number has no nonzero zero divisors.
assert!(no_nonzero_zero_divisors(TEST_MOD_PRIME_7, &modulo_mult_7, 0).is_ok());
// The ring of integers with multiplication mod prime number has nonzero zero divisors. (e.g. 1 * 7 = 0 mod 7)
assert!(no_nonzero_zero_divisors(TEST_ITEMS, &modulo_mult_7, 0).is_err());
}
#[test]
fn test_integral_domain() {
// The ring of integers modulo a prime number is an integral domain.
assert!(integral_domain(
TEST_MOD_PRIME_7,
&modulo_add_7,
&modulo_mult_7,
0,
1,
&modulo_sub_7,
)
.is_ok());
// The ring of integers modulo a composite number is not an integral domain.
assert!(integral_domain(
TEST_ITEMS,
&modulo_add_14,
&modulo_mult_14,
0,
1,
&modulo_sub_14,
)
.is_err());
}
#[test]
fn test_field() {
// Test that GF2 (0, 1, XOR, AND) is a field and +, x, 0, 1, - is not a field (no multiplicative inverses)
// Note GF2 is the Galois Field with 2 elements.
assert!(field(
TEST_BOOLS,
&|a, b| a ^ b, // logical XOR
&|a, b| a & b, // a & b, // logical AND
false,
true,
&|x| x, // XOR(x,x) = false, the identity for XOR
&|_x| true /* AND(x,true) = true, the identity for AND. Note that the inverse doesn't need to work for the additive identity (false)
*/
)
.is_ok());
assert!(field(
TEST_ITEMS,
&u32::wrapping_add,
&u32::wrapping_mul,
0,
1,
&|x| 0u32.wrapping_sub(x),
&|x| 0u32.wrapping_sub(x) //Note there is no valid inverse function for multiplication over the integers so we just pick some function
)
.is_err());
}
#[test]
fn test_ring() {
// Test that +, x, 0, 1, - are a ring and +, x, 0, 5 are not (5 isn't a multiplicative identity)
assert!(ring(
TEST_ITEMS,
&u32::wrapping_add,
&u32::wrapping_mul,
0,
1,
&|x| 0u32.wrapping_sub(x),
)
.is_ok());
assert!(ring(
TEST_ITEMS,
&u32::wrapping_add,
&u32::wrapping_mul,
0,
5,
&|x| 0u32.wrapping_sub(x),
)
.is_err());
}
#[test]
fn test_semiring() {
// Test +, x is a semiring
assert!(semiring(TEST_ITEMS, &u32::wrapping_add, &u32::wrapping_mul, 0, 1).is_ok());
// Test boolean semiring with AND as + and OR as x
assert!(semiring(&[false, true], &|x, y| x | y, &|x, y| x & y, false, true).is_ok());
// Test min plus semiring. + is min and x is plus. Also known as the "tropical semiring"
assert!(semiring(
&[0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, f64::INFINITY],
&f64::min,
&|x, y| x + y,
f64::INFINITY,
0.0,
)
.is_ok());
// Test max plus semiring. + is max and x is plus.
assert!(semiring(
&[0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, f64::NEG_INFINITY],
&f64::max,
&|x, y| x + y,
f64::NEG_INFINITY,
0.0,
)
.is_ok());
// Test sets of strings semiring with union as + and concatenation as x
assert!(semiring(
&[
HashSet::from([]),
HashSet::from(["".to_owned()]),
HashSet::from(["a".to_owned()]),
HashSet::from(["aa".to_owned(), "bb".to_owned()]),
HashSet::from(["ab".to_owned(), "bb".to_owned(), "cc".to_owned()]),
HashSet::from(["ba".to_owned()]),
HashSet::from(["bb".to_owned()]),
],
&|x, y| x.union(&y).cloned().collect(),
&|x, y| {
let mut new_set = HashSet::new();
for a in x.iter() {
for b in y.iter() {
new_set.insert(format!("{a}{b}"));
}
}
new_set
},
HashSet::from([]),
HashSet::from(["".to_owned()]),
)
.is_ok());
}
#[test]
fn test_get_single_function_properties() {
// Test that get single function properties on addition returns associative, commutative, identity, and inverses.
let test_properties_satisfied = get_single_function_properties(
TEST_ITEMS,
u32::wrapping_add,
0,
|x| 0u32.wrapping_sub(x),
0,
);
let correct_properties = vec![
"associativity".to_string(),
"commutativity".to_string(),
"identity".to_string(),
"inverse".to_string(),
];
assert_eq!(test_properties_satisfied, correct_properties);
// Test that get single function properties on max returns associative, commutative, idempotent, identity, and absorbing element.
let test_properties_satisfied = get_single_function_properties(
&[0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, f64::INFINITY],
f64::max,
0.0,
|x| x,
f64::INFINITY,
);
let correct_properties = vec![
"associativity".to_string(),
"commutativity".to_string(),
"idempotency".to_string(),
"identity".to_string(),
"absorbing_element".to_string(),
];
assert_eq!(test_properties_satisfied, correct_properties);
// Define a function that takes in two u32s and returns the first one
let f = |x: u32, _y: u32| x;
let test_properties_satisfied =
get_single_function_properties(TEST_ITEMS, f, 0, |x| 0u32.wrapping_sub(x), 0);
let correct_properties = vec!["associativity".to_string(), "idempotency".to_string()];
assert_eq!(test_properties_satisfied, correct_properties);
}
}