WEBVTT

1
00:00:00.000 --> 00:00:06.400
This overview explains how precise Boolean
inputs determine a reproducible decision.

2
00:00:06.400 --> 00:00:12.333
An explicit rule lets us trace disagreements
to a particular input and operation.

3
00:00:12.633 --> 00:00:17.267
Let A mean a valid pass and B mean access is enabled.

4
00:00:17.267 --> 00:00:20.567
The first branch, P, requires both.

5
00:00:20.567 --> 00:00:26.233
With A one and B zero, P is zero because enabled access is missing.

6
00:00:26.233 --> 00:00:29.033
With both one, P is one.

7
00:00:29.033 --> 00:00:34.600
The branch already has a complete meaning before we combine it with anything else.

8
00:00:34.600 --> 00:00:40.633
The second branch accepts either of two alternative conditions, C and D.

9
00:00:40.633 --> 00:00:45.467
This is inclusive or: one true condition is enough, and two true

10
00:00:45.467 --> 00:00:47.733
conditions still give one.

11
00:00:47.733 --> 00:00:53.233
In this teaching rule, these alternatives form a different branch; they are not extra

12
00:00:53.233 --> 00:00:56.567
requirements added to P.

13
00:00:56.567 --> 00:01:01.033
The final rule takes exclusive-or of the branch results.

14
00:01:01.033 --> 00:01:04.433
It accepts exactly one true branch.

15
00:01:04.433 --> 00:01:09.133
Neither branch gives zero; either branch alone gives one; both

16
00:01:09.133 --> 00:01:11.700
branches together give zero.

17
00:01:11.700 --> 00:01:15.767
This is a deliberately chosen rule, not the usual policy that grants

18
00:01:15.767 --> 00:01:20.329
access whenever any route succeeds.

19
00:01:20.329 --> 00:01:20.333
access whenever any route succeeds.

20
00:01:20.333 --> 00:01:24.400
Give every input a precise true-or-false meaning.

21
00:01:24.400 --> 00:01:28.933
A complete assignment then determines one output under the rule.

22
00:01:28.933 --> 00:01:32.300
First compute A AND B and C OR D.

23
00:01:32.300 --> 00:01:36.467
Our function takes the exclusive-or of those two results.

24
00:01:36.467 --> 00:01:41.067
At 1011, A AND B is zero and C OR D is one.

25
00:01:41.067 --> 00:01:44.433
Zero exclusive-or one is one.

26
00:01:44.433 --> 00:01:48.000
This traces the decision from its inputs.

27
00:01:48.000 --> 00:01:50.600
Change only B to one.

28
00:01:50.600 --> 00:01:56.633
Both inner branches now return one, so exclusive-or returns zero.

29
00:01:56.633 --> 00:02:04.733
Starting again at 1011, changing D to zero
leaves the answer at one because C is still true.

30
00:02:04.733 --> 00:02:09.867
A changed input does not change the output in every context.

31
00:02:09.867 --> 00:02:17.188
A matrix, program, and packed vector must agree
on every assignment in their declared universe.

32
00:02:17.188 --> 00:02:17.200
A matrix, program, and packed vector must agree
on every assignment in their declared universe.

33
00:02:17.500 --> 00:02:23.933
Work a second complete case with A 1, B 1, C 1, D 0.

34
00:02:23.933 --> 00:02:27.633
Keep this assignment fixed throughout the calculation.

35
00:02:27.633 --> 00:02:32.167
Each symbol retains the meaning already introduced; a new value does not

36
00:02:32.167 --> 00:02:35.633
create a new logical rule.

37
00:02:35.633 --> 00:02:39.600
Both pass requirements hold, so P is one.

38
00:02:39.600 --> 00:02:46.133
The first alternative condition holds, so Q is also one even though D is zero.

39
00:02:46.133 --> 00:02:51.733
The outer exclusive-or compares the two branch outputs and returns zero.

40
00:02:51.733 --> 00:02:57.000
It accepts exactly one true branch, not one or more.

41
00:02:57.000 --> 00:03:00.600
This case establishes a specific contrast.

42
00:03:00.600 --> 00:03:03.533
Inclusive OR and XOR differ.

43
00:03:03.533 --> 00:03:06.567
The outer gate determines the final rule.

44
00:03:06.567 --> 00:03:11.067
Keep the input, intermediate values, and conclusion together when comparing

45
00:03:11.067 --> 00:03:13.333
it with the previous case.

46
00:03:13.333 --> 00:03:19.754
The calculated answer belongs to this complete assignment under the stated rule.

47
00:03:19.754 --> 00:03:19.767
The calculated answer belongs to this complete assignment under the stated rule.

48
00:03:20.067 --> 00:03:26.133
For one zero one one, first compute P as one and zero, giving zero.

49
00:03:26.133 --> 00:03:29.800
Compute Q as one or one, giving one.

50
00:03:29.800 --> 00:03:35.033
The final exclusive-or is zero with one, so the decision is one.

51
00:03:35.033 --> 00:03:38.800
Keeping the intermediate values visible prevents us from combining the four

52
00:03:38.800 --> 00:03:41.667
inputs in the wrong order.

53
00:03:41.667 --> 00:03:44.000
Change only B to one.

54
00:03:44.000 --> 00:03:46.933
P becomes one while Q stays one.

55
00:03:46.933 --> 00:03:51.167
Now both branches hold, so exclusive-or returns zero.

56
00:03:51.167 --> 00:03:54.667
This flip is a direct consequence of the outer gate.

57
00:03:54.667 --> 00:03:58.800
It is not an error in the inner or operation.

58
00:03:58.800 --> 00:04:02.800
Reset to the first request and change D to zero.

59
00:04:02.800 --> 00:04:08.533
C is still one, so Q remains one and the final answer is unchanged.

60
00:04:08.533 --> 00:04:14.033
In the different context one zero zero zero, changing D to one changes Q

61
00:04:14.033 --> 00:04:15.933
and the final answer.

62
00:04:15.933 --> 00:04:22.300
An input can matter to the function even when one particular change is masked.

63
00:04:22.300 --> 00:04:22.300
An input can matter to the function even when one particular change is masked.

64
00:04:22.300 --> 00:04:26.533
Keep input meanings and the rule precise.

65
00:04:26.533 --> 00:04:33.200
Every representation must preserve
the same assignment-to-output mapping.