Inductive definition


In mathematical logic and computer science, a recursive definition (or inductive definition) is used to define an object in terms of itself (Aczel 1978:740ff).

A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs. For example, the factorial function n! is defined by the rules

0! = 1.
(n+1)! = (n+1)·n!.

This definition is valid for all n, because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure describing how to construct the function n!, starting from n = 0 and proceeding onwards with n = 1, n = 2, n = 3 etc..

That such a definition indeed defines a function can be proved by induction.

An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set N of natural numbers is:

  1. 1 is in N.
  2. If an element n is in N then n+1 is in N.
  3. N is the smallest set satisfying (1) and (2).

There are many sets that satisfy (1) and (2) - for example, the set {1, 1.649, 2, 2.649, 3, 3.649, ...} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members.

Properties of recursively defined functions and sets can often be proved by an induction principle that follows the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds of the natural number 0, and the property holds of n+1 whenever it holds of n, then the property holds of all natural numbers (Aczel 1978:742).

Form of recursive definitions

Most recursive definition have three foundations: a base case (basis), an inductive clause, and an extremal clause.

The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and all other cases comprising the definition must be "smaller" (closer to those base cases that terminate the recursion) in some sense. In contrast, a circular definition may have no base case, and define the value of a function in terms of that value itself, rather than on other values of the function. Such a situation would lead to an infinite regress.

Proof that recursion defines a function

Suppose one has a pair of functions f and g on the natural numbers such that f(0)=0 and f(n+1)=g(f(n)) for all n. We want to construct, for any natural number n, a subset F(n) of N^2 such that:

(1) for all natural numbers a between 0 and n: there exists a unique natural number b such that (a, b) is in F(n);

(2) (0, f(0)) is in F(n);

(3) for all natural numbers a, where a is between 0 and n-1: if (a, f(a)) is in F then (a+1, g(f(a))) is in F(n).

Clearly {(0, f(0))} satisfies (2) and (3).

Suppose these properties hold for n. Consider F(n+1)=F(n)U{(n,g(n))}. Then this set satisfies all 3 properties (1), (2), (3). By induction we have such a set for any natural number n. Let F be the set theoretic union of all the F(n), where the union is taken over all natural numbers n.

Examples of recursive definitions

Elementary functions

Addition is defined recursively based on counting

0+a=a
(1+n)+a=1+(n+a)

Multiplication is defined recursively

0 a=0
(1+n)a=a+na

Exponentiation is defined recursively

a^0=1
a^{1+n}=a a^n

Binomial coefficients are defined recursively

\binom{a}{0}=1
\binom{1+a}{1+n}=\frac{(1+a)\binom{a}{n}}{1+n}

Prime numbers

The set of prime numbers can be defined as the unique set of positive integers satisfying

  • 1 is not a prime number
  • any other positive integer is a prime number if and only if it is not divisible by any prime number smaller than itself

The primality of the integer 1 is the base case; checking the primality of any larger integer X by this definition requires knowing the primality of every integer between 1 and X, which is well defined by this definition. That last point can proved by induction on X, for which it is essential that the second clause says "if and only if"; if it had said just "if" the primality of for instance 4 would not be clear, and the further application of the second clause would be impossible.

Non-negative even numbers

The even numbers can be defined as consisting of

  • 0 is in the set E of non-negative evens (basis clause)
  • For any element x in the set E, x+2 is in E (inductive clause)
  • Nothing is in E unless it is obtained from the basis and inductive clauses (extremal clause).

Well formed formulas

It is chiefly in logic or computer programming that recursive definitions are found. For example, a well formed formula (wff) can be defined as:

  1. a symbol which stands for a proposition - like p means "Connor is a lawyer."
  2. The negation symbol, followed by a wff - like Np means "It is not true that Connor is a lawyer."
  3. Any of the four binary connectives (C, A, K, or E) followed by two wffs. The symbol K means "both are true", so Kpq may mean "Connor is a lawyer and Mary likes music."

The value of such a recursive definition is that it can be used to determine whether any particular string of symbols is "well formed".

  • Kpq is well formed, because it's K followed by the atomic wffs p and q.
  • NKpq is well formed, because it's N followed by Kpq, which is in turn a wff.
  • KNpNq is K followed by Np and Nq; and Np is a wff, etc.

See also

References

  • P. Aczel (1977), "An introduction to inductive definitions", Handbook of Mathematical Logic, J. Barwise (ed.), ISBN 0-444-86388-5
  • James L. Hein (2009), Discrete Structures, Logic, and Computability. ISBN 0-7637-7206-2
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