dfa_pumping_split

std.seq.dfa_pumping_split · Level L3

The pumping-lemma split of a word read by a DFA: the first repeated state among the states after 0, 1, …, L symbols gives w = xyz with x the part before the first visit, y the loop (|y| > 0), and |xy| ≤ the number of states when the word is at least that long (pigeonhole). Reading y returns the automaton to the same state, so xyⁱz is accepted exactly when w is, for every i ≥ 0: the loop is a certificate for all i, not a check of a few. Also whether a repeat exists and the loop's state.

δ*(q₀, x) = δ*(q₀, xy) ⇒ xyⁱz ∈ L ⇔ w ∈ L

Signature

dfa_pumping_split(delta: i64[q, a], start: i64[], word: i64[L]) → 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 run's states in Python, and the first repeat 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

dfa_step

Called by

—

sha256:3c3b48a35ded3a0ecee16787b22d7db3e928ef677b9e6b1e40a7e996cbd7f21f

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

Control handle

Symbol: Ω:std.seq.dfa_pumping_split · Ω:dfa_pumping_split
Pins: sha256:6a24115267b692aa3f439adc39444c34e01287c94eb4724d19d9fbd501e2f7b9this graph and the 1 it reaches through calls
Evidence: sha256:165b125e5d7d56e486b5e9be084ad80bd867a703d396a8b7829c3cee35a58a7cthe 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.