SLIDE 1
CSE 311: Foundations of Computing
Fall 2013 Lecture 17: Recursive definitions and structural induction
announcements
Reading assignment Induction
5.3, 7th edition 4.3, 6th edition
Homework 5 solutions out today Midterm
Monday, November 4th, IN CLASS Topics: Everything up to ordinary induction and recursive definition of functions Sample questions from old midterms now posted. Solutions posted later today. Review sessions: Today 3:30, Sunday 4:00 in EEB 125
review: recursive definition of sets
Recursive definition – Basis step: 0 ∈ – Recursive step: if ∈ , then + 2 ∈ – Exclusion rule: Every element in follows from basis steps and a finite number of recursive steps
review: recursive definitions of sets
Basis: 6 ∈ ; 15 ∈ ; Recursive: if , ∈ , then + ∈ ; Basis: 1, 1, 0 ∈ , 0, 1, 1 ∈; Recursive: if , , ∈ , ∈ ℝ, then α, , ∈ if [1, 1, 1], [2, 2, 2] ∈ then [1 + 2, 1 + 2, 1 + 2] ∈ Powers of 3: 1 ∈ and ∈ ⇒ 3 ∈
SLIDE 2 recursive definitions of sets: general form
Recursive definition – Basis step: Some specific elements are in – Recursive step: Given some existing named elements in some new objects constructed from these named elements are also in . – Exclusion rule: Every element in follows from basis steps and a finite number of recursive steps
review: strings
Σ Σ Σ is any finite set of characters
Σ Σ Σ* * * * of strings over the alphabet Σ Σ Σ Σ is defined by – Basis: Basis: Basis: Basis: λ ∈ Σ* (λ is the empty string) – Recursive: Recursive: Recursive: Recursive: if ∈ Σ*, ∈ Σ, then ∈ Σ*
review: palindromes
Palindromes are strings that are the same backwards and forwards Basis: Basis: Basis: Basis: λ is a palindrome and any ∈ Σ is a palindrome Recursive step: Recursive step: Recursive step: Recursive step: If ∈ Σ∗ is a palindrome then is a palindrome for every ∈ Σ
review: all binary strings with no 1’s before 0’s
SLIDE 3 function definitions on recursively defined sets
len (λ) = 0; len () = 1 + len(); for ∈ Σ∗, ∈ Σ Reversal: = = for ∈ Σ*, ∈ Σ Concatenation:
- λ = for ∈ Σ*
- = ( • ) for , ∈ Σ*, ∈ Σ
rooted binary trees
Basis: Basis: Basis: • is a rooted binary tree
Recursive step: Recursive step: Recursive step: If and are rooted binary trees then so is: T1 T2 T1 T2 functions defined on rooted binary trees
- size(•)=1
- size( ) = 1+size(T1)+size(T2)
- height(•)=0
- height( )=1+max{height(T1),height(T2)}
T1 T2 T1 T2
structural induction
How to prove ∀ ∈ , () is true:
– Base Case: Show that is true for all specific elements of mentioned in the Basis step – Inductive Hypothesis: Assume that is true for some arbitrary values of each of the existing named elements mentioned in the Recursive step – Inductive Step: Prove that holds for each of the new elements constructed in the Recursive step using the named elements mentioned in the Inductive Hypothesis – Conclude that ∀ ∈ , ()
SLIDE 4 structural induction vs. ordinary induction
Ordinary induction is a special case of structural induction: Recursive definition of ℕ Basis: 0 ∈ ! Recursive Step: If k ∈ ! then k+1 ∈ ! Structural induction follows from ordinary induction: Let "(#) be true iff for all ∈ that take # recursive steps to be constructed, () is true.
using structural induction
– Basis: Basis: Basis: Basis: 6 ∈ ; 15 ∈ ; – Recursive: Recursive: Recursive: Recursive: if , ∈ , then + ∈ .
Claim: Claim: Claim: Every element of is divisible by 3. Claim: Claim: Claim: Claim: Every element of is divisible by 3.
structural induction for strings
- Let be a set of strings over {, %} defined as
follows – Basis: ∈ – Recursive:
If ∈ then ∈ and % ∈ If ' ∈ and ( ∈ then '( ∈
- Claim: If ∈ then has more ’s than %’s
SLIDE 5 Claim: If ∈ then has more ’s than %’s
Basis: ∈ Recursive: If ∈ then ∈ and % ∈ If ' ∈ and ( ∈ then '( ∈
function definitions on recursively defined sets
len (λ) = 0; len () = 1 + len(); for ∈ Σ∗, ∈ Σ Reversal: = = for ∈ Σ*, ∈ Σ Concatenation:
- λ = for ∈ Σ*
- = ( • ) for , ∈ Σ*, ∈ Σ
len( • ) = len() + len() for all strings and
Let () be “len( • ) = len() + len() for all strings ”
len( • ) = len() + len() for all strings and
Let () be “len( • ) = len() + len() for all strings ”
SLIDE 6 functions defined on rooted binary trees
- size(•)=1
- size( ) = 1+size(T1)+size(T2)
- height(•)=0
- height( )=1+max{height(T1),height(T2)}
T1 T2 T1 T2
For every rooted binary tree ,, -./(,) ≤ 2123415(6)78 − 1 For every rooted binary tree ,, -./(,) ≤ 2123415(6)78 − 1
