Lecture 20: Inductive definitions

• Reading: MCS 7,7.1

• We started by briefly reviewing hashing; I have added this discussion to the previous lecture's notes

• Inductively defined sets
  • BNF notation
  • examples: lists, trees, \(\mathbb{N}\), logical formulae
  • Strings (\(Σ^*\))
• Inductively defined functions
  • examples: length, concatenation

 

• Review Exercises:
  • Give inductive definitions for the following sets: \(\mathbb{N}\); the set of strings with alphabet \(Σ\); the set of binary trees; the set of arithmetic expressions formed using addition, multiplication, exponentiation
  • Give inductive definitions of the length of a string, the concatenation of two strings, the reverse of a string, the maximum element of a list of integers, the sum of two natural numbers, the product of two natural numbers, etc.
  • Let \(B\) be given by \[b \in B ::= Z \mid D~b \mid SD~b \] Let \(f : B → \mathbb{N}\) be given by \(f(Z) ::= 0\), \(f(D~b) ::= 2f(b)\) and \(f(SD~b) ::= 2f(b) + 1\). List several elements of \(B\). Compute \(f(SD~(D~(SD~Z)))\). Is \(f\) injective? Surjective?

Inductively defined sets

An inductively defined set is a set where the elements are constructed by a finite number of applications of a given set of rules.

Examples:

• the set \(\mathbb{N}\) of natural numbers is the set of elements defined by the following rules:
  1. \(Z \in \mathbb{N}\)
  2. If \(n \in \mathbb{N}\) then \(Sn \in \mathbb{N}\).

  thus the elements of \(\mathbb{N}\) are \(\{Z, SZ, SSZ, SSSZ, \dots\}\). \(S\) stands for successor. You can then define \(1\) as \(SZ\), \(2\) as \(SSZ\), and so on.

• the set \(\Sigma^{*}\) of strings with characters in \(\Sigma\) is defined by
  1. \(\epsilon \in \Sigma^*\)
  2. If \(a \in \Sigma\) and \(x \in \Sigma^{*}\) then \(xa \in \Sigma^*\).

  thus if \(\Sigma = \{0,1\}\), then the elements of \(\Sigma^*\) are \(\{ε, ε0, ε1, ε00, ε01, \dots, ε1010101, \dots\}\). we usually leave off the \(ε\) at the beginning of strings of length 1 or more.

• the set \(T\) of binary trees with integers in the nodes is given by the rules
  1. the empty tree (, written \(nil\)) is a tree
  2. if \(t_1\) and \(t_2\) are trees, then , written \(node(a,t_1,t_2)\)) is a tree.

  thus the elements of \(T\) are things like the picture to the right (click for tex), which might be written textually as \(node(3,node(0,nil,nil),node(1,node(2,nil,nil),nil))\)

BNF

Compact way of writing down inductively defined sets: BNF (Backus Naur Form)

Only the name of the set and the rules are written down; they are separated by a "::=", and the rules are separated by vertical bar (\(|\)).

Examples (from above):

• \(n \in \mathbb{N} ::= 0 \mid Sn\)

• \(x \in \Sigma^* ::= \epsilon \mid xa\) where \(a \in \Sigma\)

• \(t \in T ::= nil \mid node(a,t_1,t_2)\) where \(a \in Z\)

• (basic mathematical expresssions) \[\begin{aligned}e \in E &::= n \mid e_1 + e_2 \mid e_1 * e_2 \mid - e \mid e_1 / e_2 \\ n \in \mathbb{Z}\end{aligned}\]

Here, the variables to the left of the \(\in\) indicate metavariables. When the same characters appear in the rules on the right-hand side of the \(::=\), they indicate an arbitrary element of the set being defined. For example, the \(e_1\) and \(e_2\) in the \(e_1 + e_2\) rule could be arbitrary elements of the set \(E\), but \(+\) is just the symbol \(+\).

Inductively defined functions

If \(X\) is an inductively defined set, you can define a function from \(X\) to \(Y\) by defining the function on each of the types of elements of \(X\); i.e. for each of the rules. In the inductive rules (i.e. the ones containing the metavariable being defined), you can assume the function is already defined on the subterms.

Examples:

• \(add2 : \mathbb{N} → \mathbb{N}\) is given by \(add2(0) ::= SS0\) and \(add2 (Sn) ::= S(add2(n))\).

• \(plus : \mathbb{N} \times \mathbb{N} → \mathbb{N}\) given by \(plus (0,n) ::= n\) and \(plus (Sn, n') ::= S(plus(n,n'))\). Note that we don't need to use induction on both of the inputs.

• \(len : Σ^* → \mathbb{N}\) is given by \(len(ε) ::= 0\) and \(len(xa) ::= 1 + len(x)\).

• \(cat : Σ^* \times Σ^* → Σ^*\) is given by \(cat(ε,ε) ::= ε\), \(cat(xa,ε) ::= xa\), \(cat(ε,xa) ::= xa\) and \(cat(xa,yb) ::= cat(xa,y)b\).