is_connected

std.topology.is_connected · Level L1

Whether a finite topological space is connected, as 1 or 0: no open set other than ∅ and X has an open complement, every open set checked. A disconnected space has a witness, an open set that is also closed; a connected one needs every candidate excluded. This is topological connectedness, not graph connectivity.

clopen ⊆ {∅, X}

Signature

is_connected(tau: f64[m, 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 no proper nonempty open set with an open complement in exact rational arithmetic, on all 40 test cases.
All 40 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

—

Called by

—

sha256:31e82f729100fda063e37e21aa9dbf414d3cc5a47c66ad8cef5f7a45c1870837

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

Control handle

Symbol: Ω:std.topology.is_connected · Ω:is_connected
Pins: sha256:ba82fe7d7dbfecea575e8bdec403724a692eadf7912a09226fc6fb5392023f05this graph alone
Evidence: sha256:f7a840c5bb47b85a9a781d6f4cebb5d2ebb8d8b2c776c6047e3189404aebbce4the 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.