equivalence_classes

std.sets.equivalence_classes · Level L2

Whether a relation on a finite set is an equivalence relation (reflexive, symmetric and transitive), as 1 or 0, and its classes: each a labelled by the smallest element of [a] = {x : a R x}, which is the canonical projection onto the quotient set X/R; and the number of classes. Two classes are equal or disjoint and together cover X. The labels mean classes only when the relation is an equivalence.

[a] = {x : aRx}; X/R = {[a] : a ∈ X}

Signature

equivalence_classes(R: f64[n, n]) → f64[], f64[n], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the smallest related element of each a in exact rational arithmetic, on all 40 test cases.
All 223 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

relation_properties

Called by

—

sha256:eaac59502041aeb28c813c34c30b8c11b55252e64e3f7aa373dfbeabffb2fc8b

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.sets.equivalence_classes · Ω:equivalence_classes
Pins: sha256:53ad4f19900f9584599f2d60add600b5c3ebcba9a3c4748efaedbcc0edb9b7b1this graph and the 1 it reaches through calls
Evidence: sha256:8167229a4f7d46c4f6c79b99ca97f065751a6acb833fc216f02fd7b5ab715995the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.