sigma_algebra_axioms

std.stats.sigma_algebra_axioms · Level L3

The axioms of a σ-algebra on a finite set, given as the complete list of its members (one subset per row), as 1 or 0: it contains ∅ and X, it is closed under complements, and under unions. On a finite set a countable union reduces to finitely many distinct sets, so pairwise unions suffice.

∅, X ∈ F; A ∈ F ⇒ X \ A ∈ F; A, B ∈ F ⇒ A ∪ B ∈ F

Signature

sigma_algebra_axioms(F: f64[m, n]) → f64[], f64[], f64[], 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 rows as Python frozensets in exact rational arithmetic, on all 40 test cases.
All 160 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

members_open topology_axioms

Called by

—

sha256:7710399b8c80ef5b0284fc8d334c7a9cc7a21892dbbbacf1f61c1377731d68e0

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

Control handle

Symbol: Ω:std.stats.sigma_algebra_axioms · Ω:sigma_algebra_axioms
Pins: sha256:efb27ce4239d381c632b01e2daf44e8b42838646f588460caf57e56b32c144d4this graph and the 2 it reaches through calls
Evidence: sha256:8302b12180e5787f8f7fb52faadab817fc89ac4d7110da6e7c5ebcd403a6dad7the 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.