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jeremykun.wordpress.com | ||
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nickdrozd.github.io
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| | | | | The goal of the Busy Beaver contest is to find n-state k-color Turing machine programs that run for as long as possible before halting. It's basically an optimization problem: what is the longest finite computation that can squeezed out of a program of a certain length? Or from the flip-side: how much description can be packed into a program of a certain length? | |
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www.jeremykun.com
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| | | | | Decidability Versus Efficiency In the early days of computing theory, the important questions were primarily about decidability. What sorts of problems are beyond the power of a Turing machine to solve? As we saw in our last primer on Turing machines, the halting problem is such an example: it can never be solved a finite amount of time by a Turing machine. However, more recently (in the past half-century) the focus of computing theory has shifted away from possibility in favor of determining feasibility. | |
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www.yodaiken.com
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math.andrej.com
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| | | [AI summary] The discussion revolves around the nuances of proof methods in constructive mathematics, particularly the distinction between proof by contradiction and proof by negation. Key points include the definition of irrational numbers without relying on the law of excluded middle, the use of contrapositive in proofs, and the limitations of certain classical theorems like the intermediate value theorem in constructive settings. The conversation also touches on the philosophical and practical implications of these proof methods in both classical and intuitionistic logic, as well as the role of type theory and univalent foundations in modern mathematical proofs. | ||
