reachability

std.linalg.reachability · Level L2

Which vertices each vertex can reach in a finite directed graph, given as a 0/1 adjacency matrix: Rᵢⱼ = 1 when there is a path from i to j, of any length including 0. A Scan of reach_step, n steps from the identity. The reference runs a breadth-first search from every vertex.

Rᵢⱼ = [a path i → ⋯ → j exists]

Signature

reachability(A: f64[n, n]) → f64[n, n]

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 breadth-first search from every vertex in exact rational arithmetic, on all 40 test cases.
All 748 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

reach_step

Called by

component_labels nfa_accepts regex_node_step model_correctness

sha256:d29772af65862dca5256876fa0d43e91fd61681b22643dec2ad61a11edbb00d0

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

Control handle

Symbol: Ω:std.linalg.reachability · Ω:reachability
Pins: sha256:d3693677123fa8e8f46a80c1fa94bb5256a2a02b869bb3438c4eba6040e326dethis graph and the 1 it reaches through calls
Evidence: sha256:9ffea49b4f697bca7d775a531c45b28a885c2e4a6a3a2cc330f12a61c83d8a60the 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.