CS 1110: Introduction to Computing Using Python Loop Invariants
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1 CS 1110: Introduction to Computing Using Python Lecture 21 Loop Invariants [Andersen, Gries, Lee, Marschner, Van Loan, White]
2 Announcements Prelim 2 conflicts due by midnight tonight Lab 11 is out Due in 2 weeks because of Prelim 2 Review Prelim 2 announcements from previous lecture A4 is due Thursday at midnight There will only be 5 assignments. Can look at webpage for redistributed weights
3 Loop Invariants: Eat your Vegetables! source: Wikipedia 4/18/17 Loop Invariants 3
4 Recall: The while-loop while <condition>: statement 1 statement n true condition repetend repetend or body Relationship to for-loop Must explicitly ensure condition becomes false You explicitly manage what changes per iteration false 4/18/17 Loop Invariants 4
5 Example: Sorting 0 n pre: b? post: b 0 n sorted i = 0 while i < n: # Find minimum val in b[i..] # Swap min val with val at i i = i+1 i i i n n n 4/18/17 Loop Invariants 5
6 Recall: Important Terminology assertion: true-false statement placed in a program to assert that it is true at that point Can either be a comment, or an assert command invariant: assertion supposed to "always" be true If temporarily invalidated, must make it true again Example: class invariants and class methods loop invariant: assertion supposed to be true before and after each iteration of the loop iteration of a loop: one execution of its body 4/18/17 Loop Invariants 6
7 Preconditions & Postconditions n precondition # x = sum of 1..n-1 x = x + n n = n + 1 # x = sum of 1..n-1 postcondition Precondition: assertion placed before a segment Postcondition: assertion placed after a segment x contains the sum of these (6) n x contains the sum of these (10) Relationship Between Two If precondition is true, then postcondition will be true 4/18/17 Loop Invariants 7
8 Solving a Problem precondition # x = sum of 1..n n = n + 1 # x = sum of 1..n What statement do you put here to make the postcondition true? A: x = x + 1 B: x = x + n C: x = x + n+1 D: None of the above E: I don t know postcondition 4/18/17 Loop Invariants 8
9 Solving a Problem precondition # x = sum of 1..n n = n + 1 # x = sum of 1..n What statement do you put here to make the postcondition true? A: x = x + 1 B: x = x + n C: x = x + n+1 D: None of the above E: I don t know postcondition Remember the new value of n 4/18/17 Loop Invariants 9
10 # x = sum of 1..n Solving a Problem precondition n n+1 x contains the sum of these (10) n = n + 1 # x = sum of 1..n A: x = x + 1 B: x = x + n C: x = x + n+1 D: None of the above E: I don t know postcondition n x contains the sum of these (15) Remember the new value of n 4/18/17 Loop Invariants 10
11 Invariants: Assertions That Do Not Change Loop Invariant: an assertion that is true before and after each iteration (execution of repetend) x = 0; i = 2 while i <= 5: x = x + i*i i = i +1 # x = sum of squares of 2..5 Invariant: x = sum of squares of 2..i-1 in terms of the range of integers that have been processed so far i = 2 # invariant i <= 5 true x = x + i*i false i = i +1 The loop processes the range /18/17 Loop Invariants 11
12 Invariants: Assertions That Do Not Change Loop Invariant: an assertion that is true before and after each iteration (execution of repetend) Should help you understand the loop There are good invariants and bad invariants Bad: 2!= 1 Good: s[0 k] is sorted True, but doesn t help you understand the loop Seems useful in order to conclude that s is sorted. 4/18/17 Loop Invariants 12
13 Key Difference x = 0; i = 2 # Inv: x = sum of squares of 2..i-1 Invariant: while i <= 5: True when loop terminates x = x + i*i Loop termination condition: i = i +1 False when loop terminates # Post: x = sum of squares of /18/17 Loop Invariants 13
14 Invariants: Assertions That Do Not Change x = 0; i = 2 # Inv: x = sum of squares of 2..i-1 while i <= 5: x = x + i*i i = i +1 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: x 0 i? i = 2 # invariant true i <= 5 false x = x + i*i i = i +1 The loop processes the range /18/17 Loop Invariants 14
15 Invariants: Assertions That Do Not Change x = 0; i = 2 # Inv: x = sum of squares of 2..i-1 while i <= 5: x = x + i*i i = i +1 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: 2..1 (empty) x 0 i? 2 i = 2 # invariant true i <= 5 false x = x + i*i i = i +1 The loop processes the range /18/17 Loop Invariants 15
16 Invariants: Assertions That Do Not Change x = 0; i = 2 # Inv: x = sum of squares of 2..i-1 while i <= 5: x = x + i*i i = i +1 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: (empty) x 0 i? 2 i = # invariant true i <= 5 false x = x + i*i i = i +1 The loop processes the range /18/17 Loop Invariants 16
17 Invariants: Assertions That Do Not Change x = 0; i = 2 # Inv: x = sum of squares of 2..i-1 while i <= 5: x = x + i*i i = i +1 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: 2, (empty) x 0 i? 2 i = # invariant true i <= 5 false 4 x = x + i*i i = i +1 The loop processes the range /18/17 Loop Invariants 17
18 Invariants: Assertions That Do Not Change x = 0; i = 2 # Inv: x = sum of squares of 2..i-1 while i <= 5: x = x + i*i i = i +1 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: 2, 3, (empty) x 0 i? 2 i = # invariant true i <= 5 false x = x + i*i i = i +1 The loop processes the range /18/17 Loop Invariants 18
19 Invariants: Assertions That Do Not Change x = 0; i = 2 x # Inv: x = sum of squares of 2..i-1 while i <= 5: i? x = x + i*i i = i +1 i = 2 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: 2, 3, 4, (empty) # invariant true i <= 5 x = x + i*i false i = i +1 The loop processes the range /18/17 Loop Invariants 19
20 Invariants: Assertions That Do Not Change x = 0; i = 2 x # Inv: x = sum of squares of 2..i-1 while i <= 5: i? x = x + i*i i = i +1 i = 2 # Post: x = sum of squares of 2..5 Integers that have been processed: Range 2..i-1: 2, 3, 4, (empty) # invariant true i <= 5 x = x + i*i Invariant was always true just before test of loop condition. So it s true when loop terminates false 4/18/17 Loop Invariants i = i +1 The loop processes the range
21 Designing Integer while-loops # Process integers in a..b # inv: integers in a..k-1 have been processed k = a while k <= b: process integer k k = k + 1 # post: integers in a..b have been processed Command to do something Equivalent postcondition init invariant cond true Process k invariant false k= k +1; 4/18/17 Loop Invariants 21
22 Designing Integer while-loops 1. Recognize that a range of integers b..c has to be processed 2. Write the command and equivalent postcondition 3. Write the basic part of the while-loop 4. Write loop invariant 5. Figure out any initialization 6. Implement the repetend (process k) 4/18/17 Loop Invariants 22
23 Designing Integer while-loops 1. Recognize that a range of integers b..c has to be processed 2. Write the command and equivalent postcondition 3. Write the basic part of the while-loop 4. Write loop invariant 5. Figure out any initialization 6. Implement the repetend (process k) # Process b..c # Postcondition: range b..c has been processed 4/18/17 Loop Invariants 23
24 Designing Integer while-loops 1. Recognize that a range of integers b..c has to be processed 2. Write the command and equivalent postcondition 3. Write the basic part of the while-loop 4. Write loop invariant 5. Figure out any initialization 6. Implement the repetend (process k) # Process b..c while k <= c: k = k + 1 # Postcondition: range b..c has been processed 4/18/17 Loop Invariants 24
25 Designing Integer while-loops 1. Recognize that a range of integers b..c has to be processed 2. Write the command and equivalent postcondition 3. Write the basic part of the while-loop 4. Write loop invariant 5. Figure out any initialization 6. Implement the repetend (process k) # Process b..c # Invariant: range b..k-1 has been processed while k <= c: k = k + 1 # Postcondition: range b..c has been processed 4/18/17 Loop Invariants 25
26 Designing Integer while-loops 1. Recognize that a range of integers b..c has to be processed 2. Write the command and equivalent postcondition 3. Write the basic part of the while-loop 4. Write loop invariant 5. Figure out any initialization 6. Implement the repetend (process k) # Process b..c Initialize variables (if necessary) to make invariant true # Invariant: range b..k-1 has been processed while k <= c: # Process k k = k + 1 # Postcondition: range b..c has been processed 4/18/17 Loop Invariants 26
27 Finding an Invariant # Make b True if n is prime, False otherwise Command to do something # b is True if no int in 2..n-1 divides n, False otherwise Equivalent postcondition What is the invariant? 4/18/17 Loop Invariants 27
28 Finding an Invariant # Make b True if n is prime, False otherwise Command to do something while k < n: # Process k; k = k +1 # b is True if no int in 2..n-1 divides n, False otherwise What is the invariant? Equivalent postcondition 4/18/17 Loop Invariants 28
29 Finding an Invariant Command to do something # Make b True if n is prime, False otherwise # invariant: b is True if no int in 2..k-1 divides n, False otherwise while k < n: # Process k; k = k +1 # b is True if no int in 2..n-1 divides n, False otherwise What is the invariant? Equivalent postcondition k-1 k k+1 n 4/18/17 Loop Invariants 29
30 Finding an Invariant Command to do something # Make b True if n is prime, False otherwise b = True k = 2 # invariant: b is True if no int in 2..k-1 divides n, False otherwise while k < n: # Process k; k = k +1 # b is True if no int in 2..n-1 divides n, False otherwise What is the invariant? Equivalent postcondition k-1 k k+1 n 4/18/17 Loop Invariants 30
31 Finding an Invariant Command to do something # Make b True if n is prime, False otherwise b = True k = 2 # invariant: b is True if no int in 2..k-1 divides n, False otherwise while k < n: # Process k; if n % k == 0: b = False k = k +1 # b is True if no int in 2..n-1 divides n, False otherwise What is the invariant? Equivalent postcondition k-1 k k+1 n 4/18/17 Loop Invariants 31
32 Finding an Invariant # set x to # adjacent equal pairs in s Command to do something for s = 'ebeee', x = 2 while k < len(s): # Process k k = k + 1 # x = # adjacent equal pairs in s[0..len(s)-1] Equivalent postcondition k: next integer to process. Which have been processed? A: 0..k B: 1..k C: 0..k 1 D: 1..k 1 E: I don t know
33 Finding an Invariant # set x to # adjacent equal pairs in s Command to do something for s = 'ebeee', x = 2 while k < len(s): # Process k k = k + 1 # x = # adjacent equal pairs in s[0..len(s)-1] Equivalent postcondition k: next integer to process. Which have been processed? A: 0..k B: 1..k C: 0..k 1 D: 1..k 1 E: I don t know What is the invariant? A: x = no. adj. equal pairs in s[1..k] B: x = no. adj. equal pairs in s[0..k] C: x = no. adj. equal pairs in s[1..k 1] D: x = no. adj. equal pairs in s[0..k 1] E: I don t know
34 Finding an Invariant # set x to # adjacent equal pairs in s Command to do something # inv: x = # adjacent equal pairs in s[0..k-1] while k < len(s): # Process k k = k + 1 # x = # adjacent equal pairs in s[0..len(s)-1] for s = 'ebeee', x = 2 Equivalent postcondition k: next integer to process. What indices have been considered? What is the invariant? A: 0..k B: 1..k C: 0..k 1 D: 1..k 1 E: I don t know A: x = no. adj. equal pairs in s[1..k] B: x = no. adj. equal pairs in s[0..k] C: x = no. adj. equal pairs in s[1..k 1] D: x = no. adj. equal pairs in s[0..k 1] E: I don t know
35 Finding an Invariant # set x to # adjacent equal pairs in s x = 0 # inv: x = # adjacent equal pairs in s[0..k-1] while k < len(s): # Process k k = k + 1 # x = # adjacent equal pairs in s[0..len(s)-1] Command to do something for s = 'ebeee', x = 2 Equivalent postcondition k: next integer to process. What is initialization for k? A: k = 0 B: k = 1 C: k = 1 D: I don t know
36 Finding an Invariant # set x to # adjacent equal pairs in s x = 0 k = 1 # inv: x = # adjacent equal pairs in s[0..k-1] while k < len(s): # Process k k = k + 1 # x = # adjacent equal pairs in s[0..len(s)-1] Command to do something for s = 'ebeee', x = 2 Equivalent postcondition k: next integer to process. What is initialization for k? A: k = 0 B: k = 1 C: k = 1 D: I don t know Which do we compare to process k? A: s[k] and s[k+1] B: s[k-1] and s[k] C: s[k-1] and s[k+1] D: s[k] and s[n] E: I don t know
37 Finding an Invariant # set x to # adjacent equal pairs in s x = 0 k = 1 # inv: x = # adjacent equal pairs in s[0..k-1] while k < len(s): # Process k x = x + 1 if (s[k-1] == s[k]) else 0 k = k + 1 # x = # adjacent equal pairs in s[0..len(s)-1] Command to do something for s = 'ebeee', x = 2 Equivalent postcondition k: next integer to process. What is initialization for k? A: k = 0 B: k = 1 C: k = 1 D: I don t know Which do we compare to process k? A: s[k] and s[k+1] B: s[k-1] and s[k] C: s[k-1] and s[k+1] D: s[k] and s[n] E: I don t know
38 Reason carefully about initialization # s is a list of ints; len(s) >= 1 # Set c to largest element in s c =?? Command to do something k =?? # inv: while k < len(s): # Process k k = k+1 # c = largest int in s[0..len(s) 1] 1. What is the invariant? Equivalent postcondition 4/18/17 Loop Invariants 38
39 Reason carefully about initialization # s is a list of ints; len(s) >= 1 # Set c to largest element in s c =?? Command to do something k =?? # inv: c is largest element in s[0..k 1] while k < len(s): # Process k k = k+1 # c = largest int in s[0..len(s) 1] 1. What is the invariant? Equivalent postcondition 4/18/17 Loop Invariants 39
40 Reason carefully about initialization # s is a list of ints; len(s) >= 1 # Set c to largest element in s c =?? Command to do something k =?? # inv: c is largest element in s[0..k 1] while k < len(s): # Process k k = k+1 # c = largest int in s[0..len(s) 1] Equivalent postcondition 1. What is the invariant? 2. How do we initialize c and k? A: k = 0; c = s[0] B: k = 1; c = s[0] C: k = 1; c = s[1] D: k = 0; c = s[1] E: None of the above 4/18/17 Loop Invariants 40
41 Reason carefully about initialization # s is a list of ints; len(s) >= 1 # Set c to largest element in s c =?? Command to do something k =?? # inv: c is largest element in s[0..k 1] while k < len(s): # Process k k = k+1 # c = largest int in s[0..len(s) 1] Equivalent postcondition 1. What is the invariant? 2. How do we initialize c and k? A: k = 0; c = s[0] B: k = 1; c = s[0] C: k = 1; c = s[1] D: k = 0; c = s[1] E: None of the above An empty set of characters or integers has no maximum. Therefore, be sure that 0..k 1 is not empty. You must start with k = 1. 4/18/17 Loop Invariants 41
42 What is the Invariant? 0 n pre: b? post: b 0 n sorted Insertion Sort: 0 i n inv: b sorted, sorted b[i..] b[0..i-1]? i = 0 while i < n: # Find minimum val in b[i..] # Swap min val with val at i i = i+1 First segment always contains smaller values i i i n n n 4/18/17 Loop Invariants 42
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