5.3 Recursive definitions and structural induction
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1 Recall factorial function: n! = n (n-1) 1, for n 0 and 0! = 1 Then 4! = = 24 5! = = 120 1
2 Recall factorial function: f(n) = n! = n (n-1) 1, for n 0 and f(0) =0! = 1 Then f(4) = 4! = = 24 f(5) = 5! = = 120 Recursive definition of factorial function: f(0) = 1 f(n) = n f(n-1), for n 1 2
3 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 3
4 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) 4
5 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) 5
6 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) n=2 2* f(1) 6
7 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) n=2 2* f(1) n=1 1 * f(0) 7
8 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) n=2 2* f(1) n=1 1 * f(0) n=0 1 8
9 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) n=2 2* f(1) n=1 1 * f(0) n=0 1*1=
10 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) n=2 2* 2*1=2 f(1) 1 n=1 1 * f(0) n=0 1*1=
11 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 3* f(2) n=2 3*2=6 2 2* 2*1=2 f(1) 1 n=1 1 * f(0) n=0 1*1=
12 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 4* f(3) n=3 4*6=24 6 3* f(2) n=2 3*2=6 2 2* 2*1=2 f(1) 1 n=1 1 * f(0) n=0 1*1=
13 Recursive definition of factorial function: Inductive definition f(0) = 1 f(n) = n f(n-1), for n 1 f(4) n=4 24 4* f(3) 4*6=24 6 n=3 3* f(2) n=2 3*2=6 2 2* 2*1=2 f(1) 1 n=1 1 * f(0) n=0 1*1=
14 Structural recursion examples: 7-level pyramid 14
15 Structural recursion examples: 6-level pyramid + 7-level pyramid bottom level 15
16 Structural recursion examples: a 6-rings bullseye 16
17 Structural recursion examples: + a 6-rings bullseye 1 ring a 5-rings bullseye 17
18 Recursively(Inductively) defined functions domain: non-negative integers Basis step: specify the value of the function at zero Recursive step: give a rule for finding its value at an integer from its values at smaller integers 18
19 Recursively(Inductively) defined functions domain: non-negative integers Basis step: specify the value of the function at zero Recursive step: give a rule for finding its value at an integer from its values at smaller integers Recursively defined functions are well defined, i.e. for every positive integer the value of a function at this integer is determined in an unambiguous way. 19
20 Recursively(Inductively) defined functions Examples: 1) Give a recursive definition of x n, where x,n Z and n 0, x 0 20
21 Recursively(Inductively) defined functions Examples: 1) Give a recursive definition of x n, where x,n Z and n 0, x 0 Answer: x 0 = 1 x n+1 = x n x, for n = 0, 1, 2, 21
22 Recursively(Inductively) defined functions Examples: 2) Find the recursive definition that matches the nonrecursive definition of the function f (n)= i=0 n i 2 22
23 Recursively(Inductively) defined functions Examples: 2) Find the recursive definition that matches the nonrecursive definition of the function Solution: n f (n)= i=0 f (n)= i=0 i 2 = n 2 n i 2 23
24 Recursively(Inductively) defined functions Examples: 2) Find the recursive definition that matches the nonrecursive definition of the function Solution: n f (n)= i=0 f (n)= i=0 i 2 = n 2 when n = 0, n i 2 0 f (0)= i=0 i 2 =0 2 =0 24
25 Recursively(Inductively) defined functions Examples: 2) Find the recursive definition that matches the nonrecursive definition of the function Solution: for n+1: n f (n)= i=0 n+1 f (n+1)= i=0 f (n)= i=0 i 2 = n 2 i 2 when n = 0, n i 2 0 f (0)= i=0 i 2 =0 2 =0 25
26 Recursively(Inductively) defined functions Examples: 2) Find the recursive definition that matches the nonrecursive definition of the function Solution: for n+1: n f (n)= i=0 n+1 f (n+1)= i=0 f (n)= i=0 i 2 = n 2 when n = 0, n i 2 = i=0 n i 2 0 f (0)= i 2 =0 2 =0 i=0 i 2 +(n+1) 2 =f (n)+(n+1) 2 26
27 Recursively(Inductively) defined functions Examples: 2) Find the recursive definition that matches the nonrecursive definition of the function Solution: for n+1: n f (n)= i=0 n+1 f (n+1)= i=0 f (n)= i=0 i 2 = n 2 when n = 0, Answer: f(0)=0 f(n+1) = f(n) + (n+1) 2, for n = 0, 1, 2, n i 2 = i=0 n i 2 0 f (0)= i 2 =0 2 =0 i=0 i 2 +(n+1) 2 =f (n)+(n+1) 2 27
28 Recursively defined sets and structures Certain kinds of sets are most naturally specified with recursive definitions. A recursive definition of a set shows how to construct elements in the set by putting together smaller elements. 28
29 Recursively defined sets and structures Certain kinds of sets are most naturally specified with recursive definitions. A recursive definition of a set shows how to construct elements in the set by putting together smaller elements. Example: consider a set of all positive odd integers S = {1, 3, 5, 7,.} 29
30 Recursively defined sets and structures Certain kinds of sets are most naturally specified with recursive definitions. A recursive definition of a set shows how to construct elements in the set by putting together smaller elements. Example: consider a set of all positive odd integers S = {1, 3, 5, 7,.} Recursive definition: 1 S if x S then x+2 S and x-2 S 30
31 Recursively defined sets and structures Certain kinds of sets are most naturally specified with recursive definitions. A recursive definition of a set shows how to construct elements in the set by putting together smaller elements. Example: consider a set of all positive odd integers S = {1, 3, 5, 7,.} Recursive definition: 1 S if x S then x+2 S and x-2 S 31
32 Components of a recursive definition of a set: a basis explicitly states that one or more specific elements are in the set a recursive rule shows how to construct larger elements in the set from elements already known to be in the set. (can be more than one recursive rule) an exclusion statement states that an element is in the set only if it is given in the basis or can be constructed by applying the recursive rules repeatedly to elements given in the basis 32
33 Recursively defined sets and structures Example: consider a set of all positive odd integers S = {1, 3, 5, 7,.} Recursive definition: 1 S if x S then x+2 S Exclusion statement: k S only if it is given in the basis or can be calculated by applying the recursive rule to elements of set S. 33
34 Recursively defined sets and structures Recursive definitions play an important role in the study of strings. 34
35 Recursively defined sets and structures Recursive definitions play an important role in the study of strings. Example: assume that is a finite/infinite sequence of symbols, called alphabet a string over an alphabet is a finite sequence of symbols from 35
36 Recursively defined sets and structures Recursive definitions play an important role in the study of strings. Example: assume that is a finite/infinite sequence of symbols, called alphabet a string over an alphabet is a finite sequence of symbols from The set * of strings over the alphabet is defined recursively by: 36
37 Recursively defined sets and structures Recursive definitions play an important role in the study of strings. Example: assume that is a finite/infinite sequence of symbols, called alphabet a string over an alphabet is a finite sequence of symbols from The set * of strings over the alphabet is defined recursively by: lambda Basis step: * (where is the empty string, i.e.no symbols) Recursive step: if w * and x *, then wx * 37
38 Recursively defined sets and structures Example: let alphabet = {0,1} Basis step: * Recursive step: if x * then x0 * and x1 * Bit-strings 38
39 Recursively defined sets and structures Example: let alphabet = {0,1} Basis step: * Recursive step: if x * then x0 * and x1 * * 101 * 1 * Bit-strings 39
40 Recursively defined sets and structures Example: let alphabet = {0,1} Basis step: * Recursive step: if x * then x0 * and x1 * * 101 * 1 * How about 001? Does 001 *? Bit-strings 40
41 Recursively defined sets and structures Example: properly nested parentheses Basis: The sequence () is properly nested. Recursive rules: If u and v are properly-nested sequences of parentheses then: (u) is properly nested. uv is properly nested. Exclusion statement: a string is properly nested only if it is given in the basis or can be constructed by applying the recursive rules to strings in the basis. 41
42 Recursively defined sets and structures Example: properly nested parentheses Basis: The sequence () is properly nested. Recursive rules: If u and v are properly-nested sequences of parentheses then: (u) is properly nested. uv is properly nested. Exclusion statement: a string is properly nested only if it is given in the basis or can be constructed by applying the recursive rules to strings in the basis. (()()) 42
43 Recursively defined sets and structures Example: properly nested parentheses Basis: The sequence () is properly nested. Recursive rules: If u and v are properly-nested sequences of parentheses then: (u) is properly nested. uv is properly nested. Exclusion statement: a string is properly nested only if it is given in the basis or can be constructed by applying the recursive rules to strings in the basis. (()()) (((()())())()()) 43
44 Recursively defined sets and structures Example: strings concatenation abc bbc = abcbbc Let's define concatenation of strings recursively: 44
45 Recursively defined sets and structures Example: strings concatenation abc bbc = abcbbc Let's define concatenation of strings recursively: Basis step: if w *, then w = w, where is empty string Inductive step: if w 1 *, w 2 *, and x *, then w 1 (w 2 x) = (w 1 w 2 )x 45
46 Recursively defined sets and structures Example: strings concatenation abc bbc = abcbbc Let's define concatenation of strings recursively: Basis step: if w *, then w = w, where is empty string Inductive step: if w 1 *, w 2 *, and x *, then w 1 (w 2 x) = (w 1 w 2 )x Very often instead of w 1 w 2 we simply write w 1 w 2 Example: If w 1 =lala and w 2 =land then w 1 w 2 = lalaland 46
47 Recursively defined sets and structures Example: well-formed formulae in Propositional Logic 47
48 Recursively defined sets and structures Example: well-formed formulae in Propositional Logic Basic step: T, F, and s (propositional variable) are wellformed formulae Recursive step: if E and F are well-formed formulae, then ( E),(E F), (E F), (E F) and (E F) are well-formed formulae. 48
49 Recursively defined sets and structures Example: well-formed formulae in Propositional Logic Basic step: T, F, and s (propositional variable) are wellformed formulae Recursive step: if E and F are well-formed formulae, then ( E),(E F), (E F), (E F) and (E F) are well-formed formulae. Example: p, q (p q) (p q) ( (p q)) q) 49
50 Recursively defined sets and structures Trees (rooted trees) A trees consist of a root, vertices and edges vertices / nodes vertex node edges root 50
51 Recursively defined sets and structures Trees (rooted trees) [Def] A rooted tree consist of a set of vertices with a distinguished vertex called the root, and edges connecting there vertices, and can be defined recursively: Basis step: a single vertex r is a rooted tree Inductive step: assume T 1,T 2,..., T n are disjoint rooted trees with roots r 1, r 2,, r n, respectively. Then the graph formed by adding an edge from each tree to a vertex r (which is not in any of the trees T 1,..., T n ). Vertex r is the root of the newly created rooted tree. 51
52 Recursively defined sets and structures Trees (rooted trees) Basis step: Step 1:... Step 2:
53 Recursively defined sets and structures Binary Trees 53
54 Structural Induction - is a type of induction used to prove theorems about recursively defined sets - follows the structure of the recursive definition. 54
55 Structural Induction Example: Show that the set S defined below is a set of positive integers. Basis step: 1 S, (1) Recursive step: s+t S, if s,t S (2) 55
56 Structural Induction Example: Show that the set S defined below is a set of positive integers. Basis step: 1 S, (1) Recursive step: s+t S, if s,t S (2) Proof (by structural induction): Base case: the element 1 from the basis step is a positive integer - show that the result holds for all elements specified in the basis step of the rec. definition to be in the set 56
57 Structural Induction Example: Show that the set S defined below is a set of positive integers. Basis step: 1 S, (1) Recursive step: s+t S, if s,t S (2) Proof (by structural induction): Base case: the element 1 from the basis step is a positive integer Inductive/recursive step: assume that two positive integers k,t S were constructed by applying a sequence of recursive steps starting with number 1. Obviously a sum of two positive integers (k+t) is a positive 57 integer, in addition by (2) (k+t) S q.e.d.
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