Buridan's Principle

Talk at Papers We Love NYC on Leslie Lamport's Buridan's Principle.

Transcript

1. Buridan’s Principle Tom Santero Papers We Love Nov 19, 2015
2. “I’m interested in the intersection of philosophy and Computer Science” -- Michael R. Bernstein
3. Philosophy Science!!!
5. The Paradox of Buridan’s Ass Part I
6. Ἀριστοτέλης 384 – 322 BC ...a man, being just as hungry as thirsty, and placed in between food and drink, must necessarily remain where he is and starve to death. -- On The Heavens
7. Jean Buridan 1295 – 1363 Should two courses be judged equal, then the will cannot break the deadlock; all it can do is suspend judgement until the circumstances change, and the right course of action is clear.
10. "Mr. Speaker, we have all heard of the animal standing in doubt between two stacks of hay, and starving to death. The like would never happen to General Cass; place the stacks a thousand miles apart, he would stand stock still midway between them, and eat them both at once, and the green grass along the line would be apt to suffer some too at the same time." -- insulting Democratic presidential candidate, Lewis Cass, in 1848 Abraham Lincoln 1809 – 1865
12. Part II
13. Given: an Ass, two equal bails of hay Proposition: If placed equidistant between each bail of hay, there exists some finite starting positions for which an Ass could possibly starve to death.
14. 1. Let x represent the starting position of our ass at time t = 0, such that x falls along the line joining two equal bails of hay at positions 0 and 1, where 0 < x < 1 [given] 2. Let A t (x) denotes the position of the ass at time t, as a function of time over x 3. t ≥ 0: A t (0) = 0; A t (1) = 1; {x ∈ ℝ | 0 < x < 1} [continuity] 4. Since A t (0) = 0 and A t (1) = 1 there must be a finite range of values of x for which 0 < A t (x) < 1 Q.E.D
15. x 0 1
16. Buridan’s Principle: A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time.
17. 1. Let x represent the starting position of our ass at time t = 0, such that x falls along the line joining two equal bails of hay at positions 0 and 1, where 0 < x < 1 [given] 2. Let A t (x) denotes the position of the ass at time t, as a function of time over x 3. t ≥ 0: A t (0) = 0; A t (1) = 1; {x ∈ ℝ | 0 < x < 1} [continuity] 4. Since A t (0) = 0 and A t (1) = 1 there must be a finite range of values of x for which 0 < A t (x) < 1 Q.E.D
18. Lamport’s Asses Part III
19. Choo Choooo! (hi Caitie, Kyle, Ryan and Jared)
22. Protip: pay attention to the direction of the arrows
25. You’re gonna get run over, you ass!
28. Continuity Part IV
29. Buridan’s Principle does not rely upon any assumption about how the decision is made; it rests only on the assumption of continuity.
30. A continuous mechanism must either forgo discreteness, permitting a continuous range of decisions, or must allow an unbounded length of time to make the decision.
31. Should we trust Lamport ?
32. Reasons to trust Lamport: - Manufacture d’horlogerie - taught us how to properly order pastries at a bakery - warned us about dangers of participating in Greek politics - the Art of War in Byzantium
35. It has been shown by Vosbury [14] and by Palais and Lamport [11] that the glitch phenomenon is inherent in arbiter circuits. These proofs are based on a continuous model of asynchronous circuits. In fact, Palais and Lamport state that a proof of this result must be based on a continuous model. We show otherwise; that is, we give a proof of the unavoidability of the glitch phenomenon that is based on a discrete circuit model.
37. The impossibility of building a bounded-time arbiter seems to be a fundamental law of physics, not a mathematical theorem. For example, Anderson and Gouda [1] proved that a bounded-time arbiter cannot be constructed from certain kinds of components, but their proof offers no insight into why the quantum- mechanical arbiter described in [9] doesn’t work. We take the nonexistence of a bounded-time arbiter as an axiom.
39. Buridan’s Law of Measurement: If x < y < z, then any measurement performed in a bounded length of time that has a nonzero probability of yielding a value in a neighborhood of x and a nonzero probability of yielding a value in a neighborhood of z must also have a nonzero probability of yielding a value in a neighborhood of y
40. Other Asses Part V
41. decisions may sometimes take longer than we expect
42. indecision considered harmful
43. there are certain circumstances where we cannot prolong choice
44. Entropy ain’t what it used to be Randomness can make it impossible deliberately to starve the ass
46. asynchronous message passing as a continuous function of our initial state over time
47. Buridan’s Reliable Failure Detector: A process cannot make a discrete decision about the state of another process based upon an input having a continuous range of values within a bounded length of time.
48. Byzantine faults as accidental starvation
49. Computability (P v NP) Topologies and Category Theory
50. Thanks!