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Heron’s method using integer arithmetic
Suppose we are given a positive rational number , whose square root we would like to compute. To be more precise, we would like to compute an approximation of . Let be an approximation of . Hence, we...
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Generalizing Heron’s method
Suppose we are given a positive rational number . We would like to compute an approximation of the -th root of , which we denote by . If , we can use Heron’s method to approximate , as we saw a few...
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Composing an endofunction with itself
Suppose we are given an endofunction . Let us introduce a new function , where , which we define recursively as follows where is the identity function on , i.e., for all . In Haskell, it is easy to...
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Function composition via folding
Let us suppose that we are given a list of functions. We would like to compose the functions in this list (if they are composable, of course) in the order in which they are arranged in the list, i.e.,...
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Sipser on P versus NP
On October 3, 2006 Michael Sipser was at Harvard to give the following lecture on the P versus NP problem: This talk is arguably the very best introduction to computational complexity I have ever come...
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Generating all words over an alphabet
An alphabet is a finite set of symbols. A word of length over an alphabet is a sequence of symbols from , i.e., Alternatively, we can view a word of length over the alphabet as an -tuple (i.e., an...
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Deterministic Finite Automata in Haskell
Consider the deterministic finite automaton [1] illustrated below [ state diagram courtesy of Michael Sipser [1] ] Henceforth, we will call this automaton . Note that has three states, labeled , , and...
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Freeman Dyson on Wittgenstein
[ source ] Freeman Dyson on Ludwig Wittgenstein: Wittgenstein, unlike Heidegger, did not establish an ism. He wrote very little, and everything that he wrote was simple and clear. The only book that...
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My implementation of (!!)
In Haskell we can easily create a list and then access its elements using the (!!) function, which is defined in the Prelude. Here is a very brief GHCi session: Prelude> let xs = [7,8,9] Prelude>...
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Gell-Mann on learning
Murray Gell-Mann on why being in awe inhibits learning: I said I’d rather be poor or die than be an engineer because I would be no good at it. If I designed something it would fall down. When I was...
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