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Indeed, as we will see much later in this course, a very exciting recent line of works involves using different media for computation that would allow us to take advantage of Crab-based logic gates from the paper “Robust soldier-crab ball gate” by Gunji, Nishiyama and Adamatzky.
This is an example of an AND gate that relies on the tendency of two swarms of crabs arriving from different directions to combine to a single swarm that continues in the average of the directions.
(If \(v\) has fewer than two neighbors then we replace either \(b\) or both \(a\) and \(b\) with zero in the condition above.) Reference: NANDcirccomputedef captures the informal description above.
This might require reading the definition a second or third time, but would be crucial for the rest of this course.
The number of transistors per integrated circuits from 1959 till 1965 and a prediction that exponential growth will continue at least another decade.
Figure taken from “Cramming More Components onto Integrated Circuits”, Gordon Moore, 1965, which would be a system with two input wires \(x,y\) and one output wire \(z\), such that if we identify high voltage with “\(1\)” and low voltage with “\(0\)”, then the wire \(z\) will equal to “\(1\)” if and only if the NAND of the values of the wires \(x\) and \(y\) is \(1\) (see Reference:transistor-nand-fig).
For example, if a function \(F:\^n \rightarrow \) can be computed by a NAND program of \(s\) lines, is it possible, given an actual input \(x\in \^n\), to compute \(F(x)\) in the real world using an amount of resources that is roughly proportional to \(s\)?
In some sense, we already know that the answer to this question is Yes.
which on moving from one position to another transmit electric pulses that may cause other similar wheels to move; single or combined telegraph relays, actuated by an electromagnet and opening or closing electric circuits; combinations of these two elements;—and finally there exists the plausible and tempting possibility of using vacuum tubes” We have defined NAND programs as a model for computation, but is this model only a mathematical abstraction, or is it connected in some way to physical reality?
We have seen a program that can evaluate NAND programs, and so if we have a NAND program \(P\), we can use any computer with Python installed on it to evaluate \(P\) on inputs of our choice.
But do we really need modern computers and programming languages to run NAND programs?
While not all vertices are labeled, no two vertices get the same label.
We denote the circuit as \(C=(V, E, L)\) where \(V, E\) are the vertices and edges of the circuit, and \(L: V \rightarrow_p S\) is the (partial) one-to-one labeling function that maps vertices into the set \(S=\).