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Consider the following algorithmic procedure: Input :Non-negative integers x, y Output :Non-negative integer r:= x; s:= y; a := 0; while s #0 do if 3 |s then

Consider the following algorithmic procedure: Input :Non-negative integers x, y Output :Non-negative integer r:= x; s:= y; a := 0; while s #0 do if 3 |s then

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Consider the following algorithmic procedure: Input :Non-negative integers x, y Output :Non-negative integer a r:= x; s:= y; a := 0; while s + 0 do if 3 | s then r:=r+r+r; s:= s/3; end else if 3| (s - 1) then a := a+r; r:=r+r+r; s:= (s – 1)/3; end else a := a+r+r; r:=r+r+r; s:= (s – 2)/3; end end return a Algorithm 1: Multiply We can model Algorithm 1 as a state machine whose states are triples of non-negative integers (r, s,a). The initial state is (x, y, 0). The transitions are given by the rule & that for s > 0: (3r, s/3,a) 8(r, s, a) = {(3r, (s – 1)/3,a+r) if 3 | s if 3 | (s – 1) (3r, (s – 2)/3,a+ 2r) otherwise. 1. List the sequence of steps that appears in the execution of the algorithm for inputs x = 5 and y = 10. In other words, walk us through the algorithm starting from the initial state and ending with the final state. 2. Verify the partial correctness of Algorithm 1. (a) Define a predicate P on the states that you believe is a preserved invariant. (Suggestion: Think about why the loop condition is s # 0 and look at your sequence of steps from Part 1.) (b) Prove that P is a preserved invariant. (c) Apply the Invariant Principle to prove that if s = 0, then a = xy. 3. Verify that Algorithm 1 terminates. (a) Define a strictly decreasing derived variable on the states. (b) Prove that the algorithm terminates after at most 1+ log, y executions of the body of the do statement.
Transcribed Image Text:Consider the following algorithmic procedure: Input :Non-negative integers x, y Output :Non-negative integer a r:= x; s:= y; a := 0; while s + 0 do if 3 | s then r:=r+r+r; s:= s/3; end else if 3| (s - 1) then a := a+r; r:=r+r+r; s:= (s – 1)/3; end else a := a+r+r; r:=r+r+r; s:= (s – 2)/3; end end return a Algorithm 1: Multiply We can model Algorithm 1 as a state machine whose states are triples of non-negative integers (r, s,a). The initial state is (x, y, 0). The transitions are given by the rule & that for s > 0: (3r, s/3,a) 8(r, s, a) = {(3r, (s – 1)/3,a+r) if 3 | s if 3 | (s – 1) (3r, (s – 2)/3,a+ 2r) otherwise. 1. List the sequence of steps that appears in the execution of the algorithm for inputs x = 5 and y = 10. In other words, walk us through the algorithm starting from the initial state and ending with the final state. 2. Verify the partial correctness of Algorithm 1. (a) Define a predicate P on the states that you believe is a preserved invariant. (Suggestion: Think about why the loop condition is s # 0 and look at your sequence of steps from Part 1.) (b) Prove that P is a preserved invariant. (c) Apply the Invariant Principle to prove that if s = 0, then a = xy. 3. Verify that Algorithm 1 terminates. (a) Define a strictly decreasing derived variable on the states. (b) Prove that the algorithm terminates after at most 1+ log, y executions of the body of the do statement.
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