WEBVTT 00:00:00.250 --> 00:00:04.935 The manuscript suggests designing observations for a purpose as further work. 00:00:05.461 --> 00:00:09.607 This extension gives a small, explicitly stipulated example. 00:00:10.119 --> 00:00:15.099 We will not claim a new algorithm, complexity classification or real-world 00:00:15.110 --> 00:00:18.130 cost model from this four-state calculation. 00:00:19.161 --> 00:00:23.686 Our original cards use X for dot and Y for stripe: A 00:00:23.740 --> 00:00:27.785 has only a dot, B both, C only a stripe, and D 00:00:27.936 --> 00:00:28.398 neither. 00:00:28.968 --> 00:00:32.324 Their two-bit annotations follow X,Y order. 00:00:32.878 --> 00:00:37.283 A selected signature lists answers to the chosen test slots. 00:00:37.937 --> 00:00:41.952 Positive combinations of test regions generate the opens. 00:00:42.632 --> 00:00:46.917 Choosing a test makes it available; it does not assert its answer. 00:00:47.484 --> 00:00:51.891 Two states are distinguished here when those selected signatures differ. 00:00:52.919 --> 00:00:58.040 One goal is to distinguish designated valuations by different signatures. 00:00:58.669 --> 00:01:03.020 Another is to make a target subset open so membership can be 00:01:03.020 --> 00:01:04.051 certified. 00:01:04.424 --> 00:01:08.645 These goals need not select the same tests, even on the same 00:01:08.655 --> 00:01:09.359 carrier. 00:01:10.346 --> 00:01:15.783 Allow only three candidate tests: X, Y and their conjunction. 00:01:16.437 --> 00:01:21.372 Assign acquisition costs one, three and two respectively. 00:01:21.833 --> 00:01:26.445 These invented planning costs are separate from the coordinate weights used in 00:01:26.445 --> 00:01:27.877 the distance lectures. 00:01:29.002 --> 00:01:32.902 With three candidates there are only eight menus to inspect. 00:01:33.412 --> 00:01:37.619 Exhaustive enumeration solves this tiny example. 00:01:38.038 --> 00:01:42.345 It does not prove that a particular procedure scales well or is 00:01:42.366 --> 00:01:45.677 optimal for a general observation-design problem. 00:01:47.276 --> 00:01:51.161 X differs at A and D, while Y is false at both 00:01:51.182 --> 00:01:53.478 and the conjunction is false at both. 00:01:54.056 --> 00:01:57.507 The single X test therefore distinguishes the pair. 00:01:58.051 --> 00:02:03.098 Its cost is one, the least positive cost in the declared library. 00:02:03.965 --> 00:02:07.023 Selecting that test changes the menu. 00:02:07.321 --> 00:02:10.880 It does not assert that X is true and does not remove 00:02:10.880 --> 00:02:11.065 D. 00:02:11.874 --> 00:02:17.973 The design objective concerns potential distinctions supported by the interface, not a 00:02:17.983 --> 00:02:20.397 premise update already performed. 00:02:21.354 --> 00:02:24.951 Now seek a menu making the singleton B certifiable. 00:02:25.462 --> 00:02:30.384 X alone cannot do this because its true region also contains A. 00:02:30.937 --> 00:02:33.292 Y alone also includes C. 00:02:34.021 --> 00:02:39.609 Distinguishing one designated pair was therefore not the same design task. 00:02:40.549 --> 00:02:43.849 The conjunction's truth region is exactly B. 00:02:44.719 --> 00:02:48.537 Selecting it alone makes B open at cost two. 00:02:49.118 --> 00:02:55.171 Selecting X and Y separately also generates B through intersection, but their 00:02:55.182 --> 00:02:59.118 total acquisition cost in our toy library is four. 00:03:00.127 --> 00:03:05.651 For a signature map, injective means that different input states always give 00:03:05.651 --> 00:03:07.507 different output signatures. 00:03:08.637 --> 00:03:13.959 To distinguish every valuation, the signature map must be injective. 00:03:14.485 --> 00:03:18.669 X and Y together achieve this at total cost four. 00:03:19.190 --> 00:03:24.637 The conjunction alone has only true and false signatures and cannot separate 00:03:24.637 --> 00:03:26.633 three of the false valuations. 00:03:27.778 --> 00:03:30.766 The other two-test menus still merge a pair. 00:03:31.349 --> 00:03:34.887 X with the conjunction cannot distinguish C from D. 00:03:35.616 --> 00:03:39.198 Y with the conjunction cannot distinguish A from D. 00:03:39.932 --> 00:03:44.982 Thus, within our declared library, X and Y is the least-cost 00:03:44.992 --> 00:03:47.096 menu giving full separation. 00:03:47.567 --> 00:03:52.282 Adding the conjunction raises cost without adding a needed distinction. 00:03:53.137 --> 00:03:58.023 In the stipulated library, compare the cheapest way to make B alone 00:03:58.023 --> 00:04:02.200 open with the cheapest way to distinguish every valuation. 00:04:02.805 --> 00:04:08.132 Why is the cheaper event-certification menu insufficient for the stronger separation 00:04:08.132 --> 00:04:08.468 goal? 00:04:30.685 --> 00:04:35.795 The conjunction alone certifies B for cost two, but A, C and 00:04:35.828 --> 00:04:37.855 D share its false signature. 00:04:38.384 --> 00:04:42.540 X and Y together cost four and distinguish all points. 00:04:43.204 --> 00:04:48.248 The optimal answer depends on the objective and the allowed test library. 00:04:49.111 --> 00:04:54.683 A useful design problem states exactly what must be distinguished or certified, 00:04:54.884 --> 00:04:59.528 which formulas may be selected and which costs are being minimized. 00:05:00.024 --> 00:05:05.785 An exhaustive toy example illustrates that formulation; it does not establish the 00:05:05.785 --> 00:05:10.602 further algorithmic or complexity results proposed by the manuscript.