# Lecture 20: Inductive definitions

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

• 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.

• Prove that $$len(cat(x,y)) = len(x) + len(y)$$.

• Prove that $$len(reverse(x)) = len(x)$$.

• Use the inductive definitions of $$\mathbb{N}$$ and $$plus$$ to show that $$plus(a,b) = plus(b,a)$$.

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

## Idea behind structural induction

Consider the definition $$x \in Σ^* ::= ε \mid xa$$. I will refer to $$x ::= ε$$ as "rule 1" and $$x ::= xa$$ as "rule 2". This definition says that there are two kinds of strings: empty strings (formed using rule 1), and strings of the form $$xa$$, where $$x$$ is a smaller string (formed using rule 2); these are the only kinds of strings.

If we want to prove that property $$P$$ holds on all strings (i.e. $$∀x \in Σ^*, P(x)$$), we can do it by giving a proof for strings formed using rule 1 (let's call it proof 1), and another proof for strings formed using rule 2 (let's call it proof 2). In the second proof, we may assume that $$P(y)$$ holds.

Why can we make this assumption? Suppose we have some complicated string, like $$εabc$$, and we want to conclude $$P(εabc)$$. We build the string $$εabc$$ by snapping together smaller strings using rules 1 and 2; we can imagine building a proof of $$P(εabc)$$ by snapping together smaller proofs using proofs 1 and 2.

To show that $$εabc$$ is a string, we first use rule 1 to show that $$ε$$ is a string, then rule 2 to show that $$εa$$ is a string (this assumes that $$ε$$ is a string, but we just argued it was), and then rule 2 again to show that $$εab$$ is a string (using the fact that $$εa$$ is a string), and finally use rule 2 a third time to show that $$εabc$$ is a string.

Similarly, we can use proof 1 to show that $$P(ε)$$ holds, then use proof 2 to show that $$P(εa)$$ holds (this assumes that $$P(ε)$$ holds, but we just argued it does), and then use proof 2 again to show that $$P(εab)$$ holds (using the fact that $$P(εa)$$ holds), and finally use proof 2 a third time to show that $$P(εabc)$$ holds.

In general, any element of an inductively defined set is built up by applying the rules defining the set, so if you provide a proof for each rule, you have given a proof for every element. Before you can build a complex structure, you have to build the parts, so while building the proof that some property holds on a complex structure, you can assume that you have already proved it for the subparts.

## Structural induction step by step

In general, if an inductive set $$X$$ is defined by a set of rules (rule 1, rule 2, etc.), then we can prove $$∀x \in X, P(X)$$ by giving a separate proof of $$P(x)$$ for $$x$$ formed by each of the rules. In the cases where the rule recursively uses elements $$y_1, y_2, \dots$$ of the set being defined, we can assume $$P(y_1), P(y_2), \dots$$.

Example structures:

• $$Σ^*$$ is defined by $$x ∈ Σ^* ::= ε \mid xa$$. To prove $$∀x \in Σ^*, P(x)$$, you must prove (1) $$P(ε)$$, and (2) $$P(xa)$$; but in the proof of (2) you may assume $$P(x)$$.

• If a set $$T$$ is defined by $$t \in T ::= nil \mid node(a,t_1,t_2)$$, you must prove (1) $$P(nil)$$ and (2) $$P(node(a,t_1,t_2))$$. But, in the proof of (2) you may assume $$P(t_1)$$ and $$P(t_2)$$.

• If a set $$F$$ is defined by $$φ \in F ::= Q \mid \lnot φ \mid φ_1 \land φ_2 \mid φ_1 \lor φ_2$$, you can prove $$∀φ ∈ F, P(φ)$$ by proving (1) $$P(Q)$$, (2) $$P(\lnot φ)$$ [assuming $$P(φ)$$], (3) $$P(φ_1 \land φ_2)$$ [assuming $$P(φ_1)$$ and $$P(φ_2)$$], (4) $$P(φ_1 \lor φ_2)$$ [assuming $$P(φ_1)$$ and $$P(φ_2)$$].

## Example proofs

### lengths of strings are nonnegative

Recall $$Σ^*$$ is defined by $$x \in Σ^* ::= ε \mid xa$$ and $$len : Σ^* → \N$$ is given by $$len(ε) ::= 0$$ and $$len(xa) ::= 1 + len(x)$$.

Claim: For all $$x \in Σ^*$$, $$len(x) \geq 0$$ Proof: By induction on the structure of $$x$$. Let $$P(x)$$ be the statement "$$len(x) \geq 0$$". We must prove $$P(ε)$$, and $$P(xa)$$ assuming $$P(x)$$.

$$P(ε)$$ case: we want to show $$len(ε) \geq 0$$. Well, by definition, $$P(ε) = 0 \geq 0$$.

$$P(xa)$$ case: assume $$P(x)$$. That is, $$len(x) \geq 0$$. We wish to show $$P(xa)$$, i.e. that $$len(xa) \geq 0$$. Well, $$len(xa) = 1 + len(x) \geq 1 + 0 = 1$$.

### balanced trees of height $$k$$ have height $$2^k - 1$$

Here is another example proof by structural induction, this time using the definition of trees. We proved this in lecture 21 but it has been moved here.

Definition: We say that a tree $$t \in T$$ is balanced of height $$k$$ if either 1. $$t = nil$$ and $$k = 0$$, or 2. $$t = node(a,t_1,t_2)$$ and $$t_1$$ and $$t_2$$ are both balanced of height $$k-1$$.

Definition: We define $$n : T → \mathbb{N}$$ by the rules $$n(nil) := 0$$ and $$n(node(a,t_1,t_2)) := 1 + n(t_1) + n(t_2)$$.

Claim: for all $$t \in T$$ and for all $$k \in \mathbb{N}$$, If $$t$$ is balanced of height $$k$$ then $$n(t) = 2^{k}-1$$.

Proof: By structural induction on $$t$$. Let $$P(t)$$ be the statement "for all $$k \in \mathbb{N}$$, if $$t$$ is balanced of height $$k$$, then $$n(t) = 2^{k}-1$$." We must show $$P(nil)$$ and $$P(node(a,t_1,t_2))$$.

We start by proving $$P(nil)$$, i.e. that for all $$k$$, if $$nil$$ is balanced of height $$k$$ then $$n(nil) = 2^k-1$$. Well, the only way for $$nil$$ to be balanced of height $$k$$ is if $$k = 0$$. Therefore $$2^k - 1 = 2^0 - 1 = 0$$. The definition of $$n$$ shows that $$n(nil)$$ is also 0, so $$n(nil) = 2^k-1$$ in this case.

For the $$node$$ case, we must show that if $$node(a,t_1,t_2)$$ is balanced of height $$k$$ for some $$k$$, then $$n(node(a,t_1,t_2)) = 2^k-1$$. We get to assume the inductive hypotheses: $$P(t_1)$$ says that if $$t_1$$ is balanced of height $$k'$$ for some $$k'$$ then $$n(t_1) = 2^{k'}-1$$, and similarly for $$t_2$$.

Since $$node(a,t_1,t_2)$$ is balanced of height $$k$$, we know that $$t_1$$ and $$t_2$$ must both be balanced of height $$k-1$$ (this is the definition of balanced of height $$k$$). Therefore, by $$P(t_1)$$ we see that $$n(t_1) = 2^{k-1}-1$$, and $$n(t_2) = 2^{k-1}-1$$. Therefore, by definition of $$n$$, we see

$n(node(a,t_1,t_2)) = 1 + n(t_1) + n(t_2) = 1 + (2^{k-1}-1) + (2^{k-1}-1) = 2^k$

as required.