CISC/CMPE 223 - Assignment 5 (Winter 2022) Due: Thursday April 7, 2:00 PM Regulations on assignments · The assignments are graded according to the correctness, preciseness and legibility of the solutions. All handwritten parts, including figures, should be clear and legible. This assignment is marked out of 20 possible marks. . Please submit your solution in onQ before the due time. The submission must be in one of formats: . PDF, .JPG, .PNG, .DOCX. · The assignment must be based on individual work. Copying solutions from other students is a violation of academic integrity. See the course onQ site for more information. 1. Use the array-component assignment axiom (two times in case (b)) to find the weakest sufficient pre-condition P for the following code fragments: (a) (1 mark) ASSERT( P ) /* determine what is P */ A[j] = A[i]; ASSERT ( A[k] >= A[j] ) (b) (3 marks) ASSERT( P ) /* determine what is P */ A[j] = x; A[i] = A[k] ; ASSERT( A[j] == 2 ) Above x is an integer variable, A is an array of integers and we assume that all the subscripts are within the range of subscripts for A. In both cases, write the assertion P first using the notation from the array-component assignment axiom, and then rewrite P in a logically equivalent and simplified form that does not contain any notation (A | I +> E) (as in examples on pp. 85-86 and in examples covered in class).
2. (6 marks) Assume a declarative interface where n and max are constant integers. Also A is an array of integers and we know that the entries in the segment A [0 : max] are defined. Consider the following (partial) correctness statement: ASSERT ( 1 <= n < max ) { int i; i = 1; A [0] = 2; while( i < n ) { A[i] = A[i-1] + 2*i; i = i+1; } //end while } ASSERT ( ForAll (k = 0; k < n) A [k] == k*k + k + 2 ) Choose a loop invariant and give a complete proof tableau by adding all the intermediate assertions. Clearly state any mathematical facts used. Also make an argument for termination by including a suitable assertion in the loop invariant. 3. (2 marks) Show that the following inference rule for correctness statements is not generally valid: PA{C}Q1 P2{C}Q2 P1 | P2{C}Q1 & & Q2 Hint. In order to show that the inference rule is not valid (i.e., it is unsound), you need to give an example of valid premises such that the conclusion is invalid. 4. (3 marks) For an integer z, power(2, z) denotes "two to the power z", (that is, 22). Is the following specification implementable? Justify your answer! bool SizeOfUniverse (int z) ; /* Returns true if power(2, z) is greater than the maximum number of atoms in the universe any time in the past or future; otherwise returns false. */ Note: If the number of atoms