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
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.
"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
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.
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
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
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.
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
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.
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.
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
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.
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