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| The '''McCarthy 91 function''' is a [[Recursion (computer science)|recursive function]], defined by the [[computer scientist]] [[John McCarthy (computer scientist)|John McCarthy]] as a test case for [[formal verification]] within [[computer science]].
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| The McCarthy 91 function is defined as
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| :<math>M(n)=\left\{\begin{matrix} n - 10, & \mbox{if }n > 100\mbox{ } \\ M(M(n+11)), & \mbox{if }n \le 100\mbox{ } \end{matrix}\right.</math> | |
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| The results of evaluating the function are given by ''M''(''n'') = 91 for all integer arguments ''n'' ≤ 100, and ''M''(''n'') = ''n'' − 10 for ''n'' ≥ 101.
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| ==History==
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| The 91 function was introduced in papers published by [[Zohar Manna]], [[Amir Pnueli]] and [[John McCarthy (computer scientist)|John McCarthy]] in 1970. These papers represented early developments towards the application of [[formal methods]] to [[formal verification|program verification]]. The 91 function was chosen for having a complex recursion pattern (contrasted with simple patterns, such as defining <math>f(n)</math> by means of <math>f(n-1)</math>). The example was popularized by Manna's book, ''Mathematical Theory of Computation'' (1974). As the field of Formal Methods advanced, this example appeared repeatedly in the research literature.
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| In particular, it is viewed as a "challenge problem" for automated program verification.
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| Often, it is easier to reason about non-recursive computation. As one of the examples used to demonstrate such reasoning, Manna's book includes a non-recursive algorithm that simulates the original (recursive) 91 function. Many of the papers that report an "automated verification" (or [[termination proof]]) of the 91 function only handle the non-recursive version.
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| A formal derivation of the non-recursive version from the recursive one was given in a 1980 article by [[Mitchell Wand]], based on the use of [[continuation]]s.
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| ==Examples==
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| Example A:
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| M(99) = M(M(110)) since 99 ≤ 100
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| = M(100) since 110 > 100
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| = M(M(111)) since 100 ≤ 100
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| = M(101) since 111 > 100
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| = 91 since 101 > 100
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| Example B:
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| M(87) = M(M(98))
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| = M(M(M(109)))
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| = M(M(99))
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| = M(M(M(110)))
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| = M(M(100))
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| = M(M(M(111)))
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| = M(M(101))
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| = M(91)
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| = M(M(102))
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| = M(92)
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| = M(M(103))
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| = M(93)
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| .... Pattern continues
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| = M(99)
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| (same as example A)
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| = 91
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| ==Code==
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| Here is an implementation of the recursive algorithm in [[Lisp programming language|Lisp]]: | |
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| <source lang="lisp"> | |
| (defun mc91 (n)
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| (cond ((<= n 100) (mc91 (mc91 (+ n 11))))
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| (t (- n 10))))
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| </source>
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| Here is an implementation of the recursive algorithm in [[Python (programming language)|Python]]:
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| <source lang="python"> | |
| def m91(n):
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| if n > 100:
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| return n - 10
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| else:
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| return m91(m91(n + 11))
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| </source> | |
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| Here is an implementation of the recursive algorithm in [[C (programming language)|C]]:
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| <source lang="c">
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| int mc91(int n)
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| {
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| if (n > 100) {
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| return n - 10;
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| } else {
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| return mc91(mc91(n+11));
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| }
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| }
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| </source>
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| Here is an implementation of the non-recursive algorithm in [[C (programming language)|C]]:
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| <source lang="c">
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| int mccarthy(int n)
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| {
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| int c;
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| for (c = 1; c != 0; ) {
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| if (n > 100) {
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| n = n - 10;
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| c--;
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| } else {
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| n = n + 11;
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| c++;
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| }
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| }
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| return n;
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| }
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| </source>
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| ==Proof== | |
| Here is a proof that the function behaves as expected.
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| For 90 ≤ ''n'' < 101,
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| M(n) = M(M(n + 11))
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| = M(n + 11 - 10), where n + 11 >= 101 since n >= 90
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| = M(n + 1)
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| So ''M''(''n'') = 91 for 90 ≤ ''n'' < 101.
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| We can use this as a base case for [[Inductive proof|induction]] on blocks of 11 numbers, like so:
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| Assume that ''M''(''n'') = 91 for ''a'' ≤ ''n'' < ''a'' + 11.
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| Then, for any ''n'' such that ''a'' - 11 ≤ ''n'' < ''a'',
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| M(n) = M(M(n + 11))
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| = M(91), by hypothesis, since a ≤ n + 11 < a + 11
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| = 91, by the base case.
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| Now by induction ''M''(''n'') = 91 for any ''n'' in such a block. There are no holes between the blocks, so ''M''(''n'') = 91 for ''n'' < 101. We can also add ''n'' = 101 as a special case.
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| == Knuth's generalization ==
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| [[Donald Knuth]] generalized the 91 function to include additional parameters. [[John Cowles]] developed a formal proof that Knuth's generalized function was total, using the [[ACL2]] theorem prover.
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| == References ==
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| * {{cite journal | author=Zohar Manna and Amir Pnueli | title=Formalization of Properties of Functional Programs | journal=Journal of the ACM | year=1970 | volume=17 | issue=3 | month=July | pages=555–569 | doi=10.1145/321592.321606}}
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| * {{cite journal | author=Zohar Manna and John McCarthy | title=Properties of programs and partial function logic | journal=Machine Intelligence | year=1970 | volume=5 }}
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| * Zohar Manna. ''Mathematical Theory of Computation.'' McGraw-Hill Book Company, New-York, 1974. Reprinted in 2003 by Dover Publications.
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| * {{cite journal | author=Mitchell Wand | title=Continuation-Based Program Transformation Strategies | journal=Journal of the ACM | year=1980 | volume=27 | issue=1 | month=January | pages=164–180 | doi=10.1145/322169.322183}}
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| * {{cite journal | author = Donald E. Knuth | title = Textbook Examples of Recursion | year = 1991 | journal = Artificial intelligence and mathematical theory of computation | arxiv = cs/9301113}}
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| * {{cite book | author = John Cowles | chapter = Knuth's generalization of McCarthy's 91 function
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| | title = Computer-Aided reasoning: ACL2 case studies | publisher = Kluwer Academic Publishers
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| | year = 2000 | pages = 283–299 | url = http://www.cs.utexas.edu/users/moore/acl2/workshop-1999/Cowles-abstract.html}}
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| {{John McCarthy navbox}}
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| [[Category:Formal methods]]
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| [[Category:Recurrence relations]]
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