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Recursive Definitions: functions, structures, languages

Recursive Definitions: functions, structures, languages. Sections 5.3 & 5.4 Tuesday, June 10. 1. Recursively Defined Sets. A set S can be defined recursively as follows: Specify one or more basis elements .

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Recursive Definitions: functions, structures, languages

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  1. Recursive Definitions: functions, structures, languages Sections 5.3 & 5.4 Tuesday, June 10 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS 1

  2. Recursively Defined Sets • A set S can be defined recursively as follows: • Specify one or more basis elements. • Specify one or more recursive formation rules indicating how to create an element of S from other elements in S. • An element x belongs to S iff either • x is a basis element or • x is obtained by applying a formation rule to some elements, x1, x2, …, xk , all of which belong to S. 2 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  3. Example • Define set S recursively, as follows: • Basis elements: 2 S • Recursive formation rules: R1: If x S and y S then x + y SR2: If x S and y S then x − y S • What set is this? 3 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  4. Example • Define set A recursively, as follows: • Basis element: 2 A • Recursive Rules: R1: If x A and n N then xn A • What set is this? 4 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  5. Exercise • Define set A recursively, as follows: • Basis element: 2 A and 3 A • Recursive Rules: R1: If x, y A then x− y A • What set is this? 5 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  6. Numeric expressions basis The set of numeric expressions is defined recursively: • An integer is a numeric expression • If e is a numeric expression, then the following are both numeric expressions: – e (e) • if e1 and e2 are numeric expressions, then the following are all numeric expressions: e1 + e2 e1 – e2 e1 * e2 e1 dive2 e1 mode2  e1 / e2   e1 / e2 recursive formation rules RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  7. Exercise • According to this definition which of the following are numeric expressions? 36 – (6 div 2) 1 = 16 (mod 5) 3.15 * 100 + 9 4 / 0div 3 4 / 0 div 3 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  8. Strings • A stringover an alphabet Σis a finite sequence of symbols from Σ, e.g. • Bit strings are strings over {0, 1}, e.g., 0 1 00 01 10 11… • Strings over {a, b, … z}: helloworld • The set of stringsΣ*over the alphabet Σ is defined recursively by: • Basis element: λΣ*, whereλ, called the empty string, denotes the sequence containing no elements. • Formation rule: If wΣ* and xΣ then wxΣ*(By convention, when w =λ, we also write x for λx.) 8 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  9. Function on Recursively Defined Set • To define a function f on a recursively defined set S: • For each basis element x:Define f (x) • For each formation rule specifying how x is created from elements, x1, x2, …, xk , of S: Give a formation rule specifying how f (x) is created from f (x1), f (x2), …, f (xk). RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  10. Length of a string • The length function, l : Σ* N, computes the length of a string and is defined recursively as follows: • l(λ) = 0, and • l(wx) = l(w) + 1, if wΣ* and xΣ. • Example: • l(0) = l(λ0) = l(λ) + 1 = 0 + 1 = 1 • l(00) = l(0) + 1 = 1 + 1 = 2 • l(001011) = l(00101) + 1 = l(0010) + 2 = … = 6 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  11. Example: Palindromes • The empty string is a palindrome. • A string of length 1 is a palindrome. • If w is a palindrome and a is a character, then the string awa is a palindrome. • If w is a palindrome and b is the space character, then the strings bw and wb are both palindromes • (Nothing else is a palindrome.) RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  12. Example • “kayak” is a palindrome since • “y” is a palindrome by rule 2 • “aya” is a palidrome by rule 3 and previous • “kayak” is a palindrome by rule 3 and previous • Show: “race car” is a palindrome • Exercise: Sketch a procedure is_palindrome(w), which determines if the input string w is a palindrome. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  13. Concatenation of Strings • Defn: Given an alphabet Σ, the concatenation of two strings fromΣ*is defined recursively by: • Basis: If wΣ*, then w∙λ = w, where λ is the empty string. • Recursive rule: If w1Σ* and w2Σ* andxΣ, then w1∙(w2 x) = (w1∙w2 ) x • Example: Consider Σ = {0, 1}. • 00∙1 = 00∙(λ1) = (00∙λ) 1 = 001 • 00∙11 = (00∙1)1 = 0011 • 10100∙1100 = … = 101001100 • In general, if w1Σ* and w2Σ*, then w1∙w2 consists of the symbols in w1 followed by the symbols in w2 13 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  14. Structural Induction • To prove P(s), where s ranges over a recursively defined set S, use structural induction. • Basis step: For each basis element b, show P(b) is true. • Induction step: For each formation rule that says how to construct an element s in S from other elements, s1, …, sk, in S: • Assume P(s1), …, P(sk), are true. (IH) • Show that, under this assumption, then P(s) is also true. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  15. Exercise: Structural Induction Worksheet Defn [Bit Strings]. The set of bit strings B*, where B = { 0, 1 }, is defined inductively, as follows Basis: The empty string, denoted λ , is in B* — that is, λ  B*. Recursive rule: If w  B* and x  B then wx  B*. Nothing else is a bit string. Additionally, the bit string λx is abbreviated as just x. Defn [Concatenation]. The concatenation of two bit strings is defined recursively as follows: Basis: If w  B*, then w∙λ = w, where λ is the empty string. Recursive rule: If w1 B* and w2 B*and x  B, then w1∙(w2 x) = (w1∙w2)x Defn [Length]. The length l of a bit string is defined recursively as follows: Basis: l(λ) = 0. Recursive rule: If w  B* and x  B, then l(wx) = l(w) + 1. Prove the following claim using structural induction: Claim: If w1 B* and w2 B*, then l(w1∙w2) = l(w1) + l(w2). RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  16. Exercise: Structural Induction Worksheet • State what you need to prove for the basis step: l(w1∙ λ) = l(w1) + l(λ). • Proof: l(w1∙ λ) = l(w1) by defn of concatenation (basis) = l(w1) + 0 by arithmetic = l(w1) + l(λ) by defn of length function (basis) • P(w2) : l(w1∙w2) = l(w1) + l(w2) RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  17. Exercise: Structural Induction Worksheet • P(w2) : l(w1∙w2) = l(w1) + l(w2) • Induction step : Under the assumption that wB*and xB, and that l(w1∙w) = l(w1) + l(w), we need to prove that l(w1∙(wx)) = l(w1) + l(wx). • Proof: l(w1∙(wx)) = l((w1∙w)x) by defn of concatenation (rec rule) = l(w1∙w) + 1 by defn of l (rec rule) = (l(w1) + l(w)) + 1 induction hypothesis = l(w1) + (l(w) + 1) by arithmetic (assoc. law) = l(w1) + l(wx) by defn of l (rec rule) Induction Hypothesis RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  18. Example: Recursive definition of a list A list l is either • The empty list: [ ] • A nonempty list: A first element, l[0], called the head of the list.followed by a list,l[1:len(l)],called the tail of the list. basis element recursive formation rule l = [5, -3, 2 ] 5 -3 2 [ ] l[0] = 5 l[1:3] = [-3, 2 ] l[1:3](tail of l ) l[0](head of l ) RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  19. Function on Recursively Defined Set • To define a function f on a recursively defined set S: • For each basis element x:Define f (x) • For each formation rule specifying how x is created from elements, x1, x2, …, xk , of S: Give a formation rule specifying how f (x) is created from f (x1), f (x2), …, f (xk). • Example: • Definition of the sum function s on a list l:s(l) = 0, if l = []; and s(l) = s[0] + s(l[1:len(l)]), if l ≠ []. • Definition of the member function:m(l, x) = false, if l = []; m(l, x) = true, if l ≠ [] and x = l[0];m(l, x) = m(l[1:len(l)], x), if l ≠ [] and x ≠ l[0]. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  20. Recursive summing of elements in a list def sumlist(l): if l == [ ]: return 0 else: return l [0] + sumlist(l [1:len(l)]) RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  21. Recursive linear search of list L for element E def member(L, X): if L == [ ]: return False elif L[0] == X: return True else: return member(L[1:len(L)], X) RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  22. Prove the Python member function is correct: Let P(L) be the statement: member(L, X) = True iff X is in L. • Base case: By the Python def, member([],X) = False. As X is not an element of [], this proves the base case (i.e, P([]). • Inductive step: Assume that L is non-empty and that member(L[1:len(L)], X) = True iff X is in L[1:len(L)] (IH). We need to prove: member(L, X) = True iff X is in L (*). Consider the two possible outcomes of the test “L[0] == X”: If L[0] = X, then member(L, X) = True and X is (the first element) in L, so (*) holds. Otherwise, L[0] ≠ X, and so X is in L iff X is in L[1:len(L)] andmember(L,X) = member(L[1:len(L)], X), by the Python def. Thus: X is in L iff X is in L[1:len(L)] iff member(L[1:len(L)], X) = True (by IH) iff member(L, X) = True. which proves (*) in this case, also. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  23. Rooted Trees • A rooted tree is a special kind of graph • one vertex is distinguished as the root, and • a (possibly empty) set of edges connects the root to a set of child vertices, which are themselves roots of disjoint trees. • a vertex with no children is called a leaf. • The set of rooted trees is defined recursively by: Basis elements: A single vertex r is a rooted tree with root r. Formation rule: If T1, T2, …,Tn are disjoint rooted trees with roots r1, r2,…,rn, respectively. Then the graph formed by starting with a new root r, not in any of T1, T2, …,Tn, and adding an edge from r to each of the vertices r1, r2,…,rn, is also a rooted tree. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  24. Building Up Rooted Trees RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  25. Full Binary Trees • A full binary tree is a rooted tree in which each vertex either is a leaf or has two child vertices, which are roots of a left subtree and a right subtree. • The set of full binary trees can be defined recursively: • Basis element: A single vertex r is a full binary tree. • Formation rule: If T1 and T2 are disjoint full binary trees, then T1∙T2 denotes a full binary tree consisting of a new root r together with two new edges: an edge connecting r to the root of the left subtree T1 and an edge connecting r tothe right subtree T2. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  26. Example: Construction of a FBT T9 = T5∙T8 T8 = T6∙T7 T5 T9 T2 T4 T7 T5 = T3∙T4 T3 T1 T6 T8 T3 = T1∙T2 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  27. Building Up Full Binary Trees RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  28. Functions on full binary trees • Define the number of verticesn(T) of a full binary tree T recursively as follows: • The number of vertices of a full binary tree T consisting of only a root r is n(T) = 1. • If T1 and T2are full binary trees, then n(T1∙T2) = 1 + n(T1) + n(T2). RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  29. Exercise: Use this definition to find n(T9) n(T9)= n(T5)+ n(T8)+ 1 = n(T5)+ n(T6)+ n(T7)+ 2 = n(T5)+ 4 = n(T3)+ n(T4)+ 5 = n(T3)+ 6 = n(T1)+ n(T2)+ 7 = 9 T5 T9 T2 T4 T7 T3 T1 T6 T8 The number of vertices of a full binary tree T consisting of only a root r is n(T) = 1. If T1 and T2are full binary trees, then n(T1∙T2) = 1 + n(T1) + n(T2). RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  30. Functions on full binary trees • Define the heighth(T)of a full binary tree T recursively as follows: • The height of a full binary tree T consisting of only a root r is h(T) = 0. • If T1 and T2are full binary trees, then h(T1∙T2) = 1 + max(h(T1),h(T2)). RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  31. Exercise: Use this definition to find h(T9) h(T1)= h(T2)= h(T4) = h(T6)= h(T7)= 0 h(T3) = max(h(T1),h(T2))+ 1 = 1 h(T5) = max(h(T3),h(T4))+ 1 = 1 + 1 = 2 . . . h(T9) = max(h(T5),h(T8))+ 1 = 2 + 1 = 3 T5 T9 T2 T4 T7 T3 T1 T6 T8 The height of a full binary tree T consisting of only a root r is h(T) = 0. If T1 and T2are full binary trees, then h(T1∙T2) = 1 + max(h(T1),h(T2)) RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  32. Proving results about full binary trees using structural induction • To prove P(T) where T ranges over the set of full binary trees • Basis Step: Prove P(T) is true when T is a single vertex. • Induction Step: Assume T1 and T2 are full binary trees and P(T1) and P(T2) are true. (IH) Prove that, under these assumptions, P(T1∙T2) is also true. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  33. Example: structural induction on FBT’s Theorem: If T is a full binary tree, then n(T) ≤ 2h(T)+1– 1. Proof: Use structural induction. • BASIS STEP: If T is a full binary tree consisting only of a root, then n(T) = 1 and h(T) = 0. Hence, n(T) = 1 ≤ 20+1– 1 = 2h(T)+1– 1. • RECURSIVE STEP: Assume n(T1) ≤ 2h(T1)+1– 1 and n(T2) ≤ 2h(T2)+1 – 1 where T1 and T2are full binary trees. Then: n(T1∙T2) = 1 + n(T1) + n(T2) (by recursive definition of n(T)) ≤ 1 + (2h(T1)+1– 1) + (2h(T2)+1 – 1) (by induction hypothesis) = 2h(T1)+1 + 2h(T2)+1– 1 ≤ 2∙max(2h(T1)+1, 2h(T2)+1 )– 1 (since x + y ≤2∙max(x, y)) = 2∙2max(h(T1)+1,h(T2)+1) – 1 (since max(2x , 2y)= 2max(x,y) ) = 2∙2max(h(T1),h(T2))+1– 1 (since max(x+a, y+a)= max(x,y)+a ) = 2∙2h(T)– 1 (by recursive definition of h(T)) = 2h(T)+1– 1 RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  34. Example: Generalized Recursion • C(n, k): the number of subsets of cardinality k that can be formed using elements from a set of cardinality n. • Example: C(4, k) the number of k-element subsets that can be formed using elements of {1, 2, 3, 4}. k = 0 {1, 2, 3, 4} { } C(4,0) = 1 {1, 2, 3, 4} {1} {2} {3} {4} k = 1 C(4,1) = 4 {1, 2, 3, 4} {1, 2} {1, 3} {2, 3} {1, 4} {2, 4} {3, 4} C(4,2) = 6 k = 2 {1, 2, 3, 4} {1, 2, 3} {1, 2, 4} {1, 2, 4} {2, 3, 4} C(4,3) = 4 k = 3 k = 4 {1, 2, 3, 4} C(4,4) = 1 {1, 2, 3, 4} RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  35. C(n, k): Combinations of n objects k at a time • For n ≥ 0 and n ≥ k ≥ 0, we can define C(n, k) recursively as follows: • C(n, 0) = 1 • C(n, n) = 1 • C(n, k) = C(n −1, k) + C(n −1, k −1), if n > k > 0 Number of subsets containing n Number of subsets not containing n RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  36. This is the structure of the Pascal Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Notes: element k in row n is C(n, k). It is the sum of the 2 closest elements in the previous row. The elements in row n account for all the subsets of sizes {0, 1, 2, 3, … , n }. Thus, the sum of elements in row n is 2n RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  37. Remaining slides are optional. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  38. A recursive fill or coloring algorithm Common in computer graphics and image processing. We will prove that the algorithm is correct. RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  39. paint/fill algorithm • Object region must be bounded by color C • Start at any pixel [r,c] inside boundary of C • Recursively color neighbors RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  40. Note: what is a binary image? It has Rows = {1, 2, 3, …, R} It has Columns={1,2,3, …, C} A binary image I is a function that maps Rows  Columns into {0, 1}. In defining a set of “ON” pixels, it is also a “characteristic function” RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  41. A recursive fill algorithm Fills a bounded region in a 2D image with a given color C; Must have starting pixel location P=[r,c]; Color P with color C, then color all neighbors RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  42. Why a good example? • important image operation • recursive example • can prove correctness • very general base algorithm with extensions to determine connectedness, solve a maze, find objects, etc RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  43. Paint/fill algorithm overview • if image pixel I[r, c] already colored, return • color pixel: I[r, c] = C; • recurse on each neighbor of I[r, c] Note: In C++, the image reference will be I[r][c] RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  44. Pixel neighborhoods 4-neighbors of PP 8-neighbors of PP N4 N3 N2 N2 r N3 PP N1 N5 PP N1 N6 N7 N8 N4 c RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  45. Pre and Post conditions • Pre: region boundary marked with C is closed (so N-neighbors cannot leak out) • Pre: rest of image has color not C • Pre: pixel [r, c] is inside boundary • Post: every pixel inside boundary has color C • We will prove that the algorithm does yield the post condition RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  46. Fill algorithm: next level of detail void fill ( image& I, const int r, c, const color C ) { if I[r, c] == C then return; I[r, c] = C; // set the pixel to the fill color for all neighbors I[r + delta_row, c + delta_col]: fill ( image& I, r+delta_row, c+delta_col, C ) } e.g., take (delta_row, delta_col) to be, in order: (1,0), (0,1), (-1,0), (0,-1); “right, up, left, down” RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  47. Visiting pixels using a stack: optional void paint ( Image& I, int r, int c, int color ) { push r, c onto STACK; // replaces call stack of recursive fill while ( STACK not empty ) // are there more region pixels to visit? { pop c, r off STACK; if ( I[r][c] != background 0 or the current color ) { I[r][c] = color; // paint the current pixel for all neighbors of this pixel, push r+deltar, c+deltac onto STACK } } // when STACK is empty, all connected pixels have been visited } RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  48. Worksheet for practice • consider given examples • trace algorithm and label pixels with color C and order k = 1, 2, 3, 4, … • consider how to compute area and bounding box • consider how to compute centroid RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  49. Example 0 1 2 3 4 5 c 0 Start at pixel [ 3, 2 ] Use neighbor order right, up, left, down as (1, 2, 3, 4) Number each pixel that will be colored C with the order number in which it is reached and colored using the above recursive fill algorithm 1 C C C 2 C C r 3 C C 4 C C C 5 C RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

  50. Correctness proof by contradiction • suppose the pre conditions hold • prove the post condition follows • by contradiction: show that negative of post condition leads to a contradiction P RECURSIVE DEFINITIONS, STRUCTURES & ALGORITHMS

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