mathematics

McCarthy 91 function

McCarthy 91 function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand McCarthy 91 function rather than just read about it. In short: 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 ) = { n − 10 , if n > 100 M ( M ( n + 11 ) ) , if n ≤ 100 {\displaystyle M(n)={\begin{cases}n-10,&{\mbox{if }}n>100{\mbox{ }}\\M(M(n+11)),&{\mbox{if }}n\leq 100{\mbox{ }}\end{cases}}} The results of evaluating th…

Key takeaways

McCarthy 91 function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect McCarthy 91 function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of McCarthy 91 function from memory before moving on to harder problems.

Reference excerpt

M ( n ) = { n − 10 , if n > 100 M ( M ( n + 11 ) ) , if n ≤ 100 {\displaystyle M(n)={\begin{cases}n-10,&{\mbox{if }}n>100{\mbox{ }}\\M(M(n+11)),&{\mbox{if }}n\leq 100{\mbox{ }}\end{cases}}}

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 being nested-recursive (contrasted with single recursion, such as defining f ( n ) {\displaystyle f(n)} by means of f ( n − 1 ) {\displaystyle 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. It is easier to reason about tail-recursive control flow, this is an equivalent (extensionally equal) definition:

M t ( n ) = M t ′ ( n , 1 ) {\displaystyle M_{t}(n)=M_{t}'(n,1)}

M t ′ ( n , c ) = { n , if c = 0 M t ′ ( n − 10 , c − 1 ) , if n > 100 and c ≠ 0 M t ′ ( n + 11 , c + 1 ) , if n ≤ 100 and c ≠ 0 {\displaystyle M_{t}'(n,c)={\begin{cases}n,&{\mbox{if }}c=0\\M_{t}'(n-10,c-1),&{\mbox{if }}n>100{\mbox{ and }}c\neq 0\\M_{t}'(n+11,c+1),&{\mbox{if }}n\leq 100{\mbox{ and }}c\neq 0\end{cases}}}

As one of the examples used to demonstrate such reasoning, Manna's book includes a tail-recursive algorithm equivalent to the nested-recursive 91 function. Many of the papers that report an "automated verification" (or termination proof) of the 91 function only handle the tail-recursive version. This is an equivalent mutually tail-recursive definition:

M m t ( n ) = M m t ′ ( n , 0 ) {\displaystyle M_{mt}(n)=M_{mt}'(n,0)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with McCarthy 91 function

Start with the simplest possible case. Write down what McCarthy 91 function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to McCarthy 91 function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about McCarthy 91 function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of McCarthy 91 function

In research: McCarthy 91 function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses McCarthy 91 function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: McCarthy 91 function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Formal methods, Recurrence relations, so understanding it makes those chapters shorter.
In everyday life: Look for McCarthy 91 function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study McCarthy 91 function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what McCarthy 91 function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain McCarthy 91 function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is McCarthy 91 function in simple terms?

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 ) = { n − 10 , if n > 100 M ( M ( n + 11 ) ) , if n ≤ 100 {\displaystyle M(n)={\begin{cases}…

Why does McCarthy 91 function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study McCarthy 91 function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on McCarthy 91 function.