McCarthy 91 function

The McCarthy 91 function is a recursive function, defined by the computer scientist John McCarthy as a test case for formal verification within computer science.

The McCarthy 91 function is defined as

$M(n)=\left\{\begin{matrix} n - 10, & \mbox{if }n > 100\mbox{ } \\ M(M(n+11)), & \mbox{if }n \le 100\mbox{ } \end{matrix}\right.$

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 > 100. Indeed, the result of M(101) is also 91 (101 - 10 = 91). All results of M(n) after n = 101 are continually increasing by 1, e.g. M(102) = 92, M(103) = 93.

History

The 91 function was introduced in papers published by Zohar Manna, Amir Pnueli and John McCarthy in 1970. These papers represented early developments towards the application of formal methods to program verification. The 91 function was chosen for having a complex recursion pattern (contrasted with simple patterns, such as defining $f(n)$ by means of $f(n-1)$). 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. In particular, it is viewed as a "challenge problem" for automated program verification.

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.

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 continuations.

Examples

Example A:

M(99) = M(M(110)) since 99 ≤ 100
      = M(100)    since 110 > 100
      = M(M(111)) since 100 ≤ 100
      = M(101)    since 111 > 100
      = 91        since 101 > 100

Example B:

M(87) = M(M(98))
      = M(M(M(109)))
      = M(M(99))
      = M(M(M(110)))
      = M(M(100))
      = M(M(M(111)))
      = M(M(101))
      = M(91)
      = M(M(102))
      = M(92)
      = M(M(103))
      = M(93)
   .... Pattern continues
      = M(99)
     (same as example A)
      = 91

Code

Here is an implementation of the recursive algorithm in Lisp:

 (defun mc91 (n)
   (cond ((<= n 100) (mc91 (mc91 (+ n 11))))
         (t (- n 10))))

Here is an implementation of the recursive algorithm in Haskell:

  mc91 n | n > 100   = n - 10
         | otherwise = mc91 $ mc91 $ n + 11

Here is an implementation of the recursive algorithm in Python:

 def m91(n):
    if n > 100:
        return n - 10
    else:
        return m91(m91(n + 11))

Here is an implementation of the recursive algorithm in C:

 int mc91(int n)
 {
     if (n > 100) {
         return n - 10;
     } else {
         return mc91(mc91(n+11));
     }
 }

Here is an implementation of the non-recursive algorithm in C:

 int mccarthy(int n)
 {
     int c;
     for (c = 1; c != 0; ) {
         if (n > 100) {
             n = n - 10;
             c--;
         } else {
             n = n + 11;
             c++;
         }
     }
     return n;
 }

Proof

Here is a proof that the function behaves as expected.

For 90 ≤ n < 101,

M(n) = M(M(n + 11))
     = M(n + 11 - 10), where n + 11 >= 101 since n >= 90
     = M(n + 1)

So M(n) = 91 for 90 ≤ n < 101.

We can use this as a base case for induction on blocks of 11 numbers, like so:

Assume that M(n) = 91 for a ≤ n < a + 11.

Then, for any n such that a - 11 ≤ n < a,

M(n) = M(M(n + 11))
     = M(91), by hypothesis, since a ≤ n + 11 < a + 11
     = 91, by the base case.

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.

Knuth's generalization

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.

References

• Zohar Manna. Mathematical Theory of Computation. McGraw-Hill Book Company, New-York, 1974. Reprinted in 2003 by Dover Publications.