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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 $cat(xa,yb) ::= cat(xa,y)b$.