Theory of computation | Functions and mappings | Computability theory | Recursion
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not that easy to devise a computable function that is not primitive recursive; some examples are shown in section below. The set of primitive recursive functions is known as PR in computational complexity theory. (Wikipedia).
Recursive Functions (Discrete Math)
This video introduces recursive formulas.
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From playlist All Things Recursive - with Math and CS Perspective
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How to use the recursive formula to evaluate the first five terms
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Foundations - Seminar 10 - Gödel's incompleteness theorem Part 2
Billy Price and Will Troiani present a series of seminars on foundations of mathematics. In this seminar Will Troiani continues with the proof of Gödel's incompleteness theorem, providing some background on general recursive functions. You can join this seminar from anywhere, on any devic
From playlist Foundations seminar
Foundations - Seminar 11 - Gödel's incompleteness theorem Part 3
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Foundations - Seminar 14 - Gödel's incompleteness theorem Part 6
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From playlist Foundations seminar
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The Most Difficult Program to Compute? - Computerphile
The story of recursion continues as Professor Brailsford explains one of the most difficult programs to compute: Ackermann's function. Professor Brailsford's programs: http://bit.ly/1nhKtW4 Follow Up Film from the Prof in response to this film: https://www.youtube.com/watch?v=uNACwX-O5l
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From playlist MIT 6.001 Structure and Interpretation, 1986
Lecture 9B | MIT 6.001 Structure and Interpretation, 1986
Explicit-control Evaluator Despite the copyright notice on the screen, this course is now offered under a Creative Commons license: BY-NC-SA. Details at http://ocw.mit.edu/terms Subtitles for this course are provided through the generous assistance of Henry Baker, Hoofar Pourzand, Heath
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How to determine the first five terms for a recursive sequence
👉 Learn all about recursive sequences. Recursive form is a way of expressing sequences apart from the explicit form. In the recursive form of defining sequences, each term of a sequence is expressed in terms of the preceding term unlike in the explicit form where each term is expressed in
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Foundations - Seminar 12 - Gödel's incompleteness theorem Part 4
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From playlist Foundations seminar