Boolean algebra and Shannon's circuit analysis | Math Foundations 260 | N J Wildberger

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The development of circuit analysis in the 20th century had strong connections to the theory of logic. In this video we discuss Huntington's general formulation of Boolean algebra as an abstract theory, and then investigate Claude Shannon's important 1938 paper which first lays out the connections between Boolean algebra and circuit analysis.

There are some surprises here, as Shannon's initial orientation was in some sense opposite to the modern point of view --- he sees the logical laws as applying to what he calls "hindrances" to flow in circuits. Nevertheless he shows that Huntington's Boolean algebra with its many interesting reduction formulas was well suited to simplify, and more generally understand, complicated circuits.

Unfortunately he appears not to have read the original paper of Boole; and this ends up affecting the development of electrical engineering in the 20th century.

Video Content:
00:00 Introduction
2:42 Edward Huntington 1904
7:41 Claude Shannon
11:24 Series and parallel
16:13 Shannon's example
20:42 Reduction rules in Boolean algebra
23:10 Exercises

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This is absolutely brilliant! Opening my eyes :) Thank you for this effort to make it public!

aimlresearch
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I was brought up in the old electronics school with the BooleAlgebra "inclusive or". Your "exclusive or" seems very much better as I like maths and algebra, polynomial long division, modular arithmetic etc. Great stuff! However, for me the crowning argument is the single solitary algebraic expression that remains at the end with AlgebraBoole.
Bravo for your insight and advocacy in this matter! I think your method will win out eventually.

rbrowne
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Great vídeo professor Wildberger!
I watched your last video about Boolean algebra.
I guess a video about information theory from Shannon it can be interesting.

yscosta
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Another great video. At 11:16 it is mentioned that Shannon did not reference Boole. However, on page 8 of the thesis, after laying out the postulates, Shannon writes:
Analogue with the Calculus of Propositions.
We are now in a position to demonstrate the equivalence of this calculus with certain elementary parts of the calculus of propositions. The algebra of logic (1), (2), (3) originated by 𝗚𝗲𝗼𝗿𝗴𝗲 𝗕𝗼𝗼𝗹𝗲, is a symbolic method of investigating logical relationships. The symbols of 𝗕𝗼𝗼𝗹𝗲𝗮𝗻 algebra admit of two logical interpretations.

iconjack
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If contrary to Shannon we adopt 1 for passing current through a close gate and 0 else, than SERIE GATES are represented by the AND PRODUCT, and PARALLEL GATES by the XOR

Igdrazil
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The circle sum symbol is usually not an alternative way of writing inclusive or (P v Q), but the exclusive or. (P v Q) ^ ~(P ^ Q) would be the way to write the exclusive or.

Thanks for this series. I was never introduced to Boole's original system, and working through it has/is deepening my understanding. I'm probably bound to continue thinking from the standard perspective, but perhaps from behind a new lense.

DingDong-fqmo
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Is it possible that Shannon used the term "hinderance" as a kind of digital analogy to "(electrical) resistance"? Zero electrical resistance in a segment means current flows there. Similarly zero hinderance, with one for the opposite (ie non zero resistance/ hinderance). This would fit engineers experience with Ohms law.

roys
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Another great video. It is great to have a fresh look at many ideas which have become so central to modern life. You do seem to claim though that the development of digital electronics might have been different if Shannon and others had used the "algebra of Boole" rather than Boolean algebra. I disagree on that particular point. While the algebra of Boole makes it easier to analyze logical statements than Boolean algebra owing to its ring structure, it is actually more complicated to implement in electronics. XOR gates require more transistors (and therefore introduce a greater propagation delay) than simple AND and OR gates. In fact, in CMOS logic, one of the most common, inverters and NAND gates are the simplest. So optimizing circuits for minimal numbers of NAND gates tends to lead to better (faster) circuits than other optimizations. In fact, we see hints of this in Shannon's paper where he shows simple implementations of AND and OR gates by parallel and series arrangements of "hindrances", but no such simple one for the XOR gate which is central to the algebra of Boole.

twwc