{"schema_version":"dataset-v1","id":"cerny-reset-thresholds","version":1,"title":"Reset thresholds, computed","description":"Shortest reset words found by exhaustive search for two families: Černý’s automaton Cₙ (a: q ↦ q + 1 mod n; b: n − 1 ↦ 0, other states fixed), which meets the conjectured bound (n − 1)², and the one-cluster family Fₙ of Definition 10.1, whose reset threshold is 3n − 5.","license":"CC0-1.0 (computed values)","problems":["cerny-synchronization"],"statements":[{"problem":"cerny-synchronization","statement":"cerny-quadratic-bound","role":"The conjecture: every synchronizing automaton is reset within (n − 1)²."},{"problem":"cerny-synchronization","statement":"cerny-examples-attain","role":"Černý’s examples meet (n − 1)², so the bound cannot be lowered."},{"problem":"cerny-synchronization","statement":"cerny-one-cluster-sharp-family","role":"Fₙ has reset threshold 3n − 5, sharp for the one-cluster bound."}],"figure":{"family":"line-plot","title":"Reset thresholds, computed","caption":"Černý’s automata sit exactly on the conjectured bound; the one-cluster family Fₙ grows only linearly, meeting its own sharper bound 3n − 5.","x_label":"states n","y_label":"shortest reset word","series":["bound","cerny","family"],"boundary":"Computed for n ≤ 12 by exhaustive search; finite checks of two families, not evidence for the conjecture. Cₙ is defined here as usually stated; the cited source credits Černý with such examples without restating them.","statement":"cerny-quadratic-bound"},"provenance":{"method":"Breadth-first search over subsets of states (as bitmasks) from the full set to a singleton, for n = 2, …, 12; every value is checked against (n − 1)² and 3n − 5.","software":{"python":"3.12.15"},"precision":"Exact combinatorics.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"bound","label":"(n − 1)² bound","x":[2,3,4,5,6,7,8,9,10,11,12],"y":[1,4,9,16,25,36,49,64,81,100,121]},{"key":"cerny","label":"Cₙ, computed","x":[2,3,4,5,6,7,8,9,10,11,12],"y":[1,4,9,16,25,36,49,64,81,100,121]},{"key":"family","label":"Fₙ, computed","x":[4,5,6,7,8,9,10,11,12],"y":[7,10,13,16,19,22,25,28,31]}],"marks":[]},"sha256":"d6784f9d80e3f8c888658ba47b97e8d3c48cc85616b48aa454032b11eb9d52aa"}