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Disciplined Software Engineering Lecture #13

This lecture covers trace tables, symbolic execution, case checking, and program proofs for design verification.

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Disciplined Software Engineering Lecture #13

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  1. Disciplined Software Engineering Lecture #13 • Software Engineering Institute • Carnegie Mellon University • Pittsburgh, PA 15213 • Sponsored by the U.S. Department of Defense

  2. Design Verification, Part 2 • This is the second lecture on design verification. • It covers • trace tables • symbolic execution • case checking • mathematical program verification • program proofs

  3. Trace Tables • Trace tables are similar to execution tables, but more general. • Trace tables examine general program behavior rather than verifying individual cases. • Trace tables use • symbolic execution • case checking

  4. Symbolic Execution - 1 • In symbolic execution • the program is represented symbolically • program behavior is examined analytically • Symbolic execution can produce quite general verifications. • Symbolic execution is not always practical.

  5. Symbolic Execution - 2 • In symbolic execution, the approach is to • assign algebraic symbols to the program variables • restate the program as one or more equations in these symbols • analyze the behavior of these equations

  6. Symbolic Execution - 3 • Some questions to ask are • does the program converge on a result? • how does the program behave for both normal and abnormal input values? • does the program always produce the desired results?

  7. Symbolic Execution Example • begin • V1 = V2 • V2 = V4 • V3 = V1 • V4 = V3 • end • Given the following simple program segment • What would be the result for various values of V1, V2, V3, and V4?

  8. The trace table for this example would look like this: • Thus the result, for inputs A, B, C, and D is B, D, B, and B. V1 # Instruction V3 V4 V2 A C D B Initial values 1 V1 = V2 B 2 V2 = V4 D 3 B V3 = V1 4 B V4 = V3 Final values B B D B

  9. Symbolic Execution Exercise • Determine the values for which the following program will terminate. • Calculate(x: real) • if x>0 • x=sqrt(x)-1 • else • x=-x • if x<>0 • Calculate(x) • end.

  10. Exercise Result - 1 • The variables can be represented by x1 and x2. • The equation for the result of each cycle is • if x1>0: x2 = sqrt(x1) -1 • if x1 <= 0: x2 = -x1

  11. Exercise Result - 2 • For x1 = 0, the program terminates on the first cycle. • For x1 = 1, the program terminates on the first cycle. • For x1 = -1, the program terminates on the second cycle.

  12. Exercise Result - 3 • Working backwards • for x2 to be 1, x1 would have to be 4 • for x2 to be 4, x1 would have to be 25 • for x2 to be 25, x1 would have to be 676 • for x2 to be 676, x1 would have to be 458329 • and so forth • Thus, the program terminates for input values of + or - 1, 4, 25, 676, 458329, etc.

  13. Exercise Result - 4 • For other large positive or negative values of x1, the value of x2 will converge to a non-zero number between -1 and +1. • What happens for non-zero values between -1 and +1?

  14. Exercise Result - 5 • Pick x1=a, a very small number • x2=sqrt(a)-1, a negative number • next, x2 is made positive • in 2 more cycles, x2 is approximately sqrt(a)/2, or slightly larger than a • For small x1, the value of x2 thus gradually grows larger than x1. • The value of x2 will thus not reach 0 but will converge on some number less than 1.

  15. Exercise Result - 6 • When the value converges, x1 = x2 = x and the result would be given by the equation • sqrt(x) - 1 = -x • the solution to this equation is • x = [3 - sqrt(5)]/2 = 0.381966 • Thus, the program will end up cycling between the values of + and - 0.381966.

  16. Symbolic Execution Advantages • Symbolic proofs can be general. • Symbolic proofs typically involve less work than other methods. • Symbolic proofs are less error prone than executing test cases.

  17. Symbolic Execution Disadvantages • Symbolic execution is hard to use except for algorithmic and substitution problems. • Symbolic execution proofs are manual and thus error prone. • Symbolic execution is difficult to use with complex logic.

  18. Case Checking - 1 • Case checking involves the following steps • 1. Examine the program and determine the • various categories of program behavior. • 2. For these categories, identify the cases to be • checked. • 3. Check the cases to ensure that they cover all • possible situations.

  19. Case Checking - 2 • 4. Using trace tables, either alone or with • symbolic execution or proof by induction, • examine program behavior for each case. • 5. Once every case has been verified, program • behavior has been verified.

  20. Trace Table Example - 1 • The ClearSpaces program requirements • clears the leading and trailing spaces from a character string • leaves the embedded spaces in place • produces a state value 0 for an empty string • produces a state value 1 for a 1 space string • produces a state value 2 for a 2 or more space string • produces a state value 3 for a string containing non-space characters.

  21. ClearSpaces(char Input[ ], int State) {Length = strlen(Input); if (Length > 0 ) while(State!=3 && Length!=0) if (Input[Length-1] = ‘ ‘) {Length = Length - 1; if (State < 2) State=State+1; } else State = 3; Trace Table Example - 2 • The portion of ClearSpaces for removing trailing spaces is

  22. Trace Table Example - 3 • To define the cases for trailing spaces, establish the following conventions • 1. A message string is a string starting and • ending with a non-space character. • 2. An input string will consist of m message • characters and t trailing spaces • 3. m is either >0 or 0 • 4. t is either >0 or 0

  23. Trace Table Example - 4 • There are thus a total of 4 cases • 1. 0-0: m = 0, t = 0 • 2. 0-t: m = 0, t > 0 • 3. m-0: m > 0, t = 0 • 4. m-t: m > 0, t > 0

  24. Trace Table Example - 5 • The strategy is to verify program behavior with each case, using whatever method is most appropriate. • 1. Check the 0-0 case by itself, using an • execution table. • 2. Check the other cases, using proof by • induction. • 3. Do each proof with a trace table, using • the execution table method described in • lecture 12.

  25. Verifying Program Correctness • These methods are based on treating programs as mathematical theorems. • They are particularly useful during design and design review. • Because program verification often involves sophisticated reasoning, it is important to check your work carefully.

  26. Correctness Verification • The correctness verification approach is to • identify all the program cases. • use trace tables to evaluate all the conditions the program must satisfy for every case • use proof by induction where appropriate • The program is verified if all the conditions are satisfied for every case. • While this verification approach works with most design constructs, only loops are described in this lecture.

  27. ForLoop Verification - 1 • The ForLoop is assumed to have the following form • ForLoop • for n = First to Last • begin • nPart • end

  28. ForLoop Verification - 2 • The for-loop conditions require that the answer to the following question be true • Does ForLoop = FirstPart+SecondPart+...+LastPart

  29. ForLoop Example - 1 • Verify that the following program produces the factorial of n. • long factorial(int n) • { • x=1; • for (i=1; i <=n, i++) • x = x*I; • return x; • }

  30. ForLoop Example - 2 • First, identify the cases. • The cases are for n = 1 and for n > 1. • For n = 1 • ForLoop = 1! = 1 • FirstPart+SecondPart+...=FirstPart = 1*1=1 • The case for n = 1 is thus valid.

  31. ForLoop Example - 3 • For n > 1, say 3 • ForLoop = 3! = 6 • FirstPart+SecondPart+ThirdPart = 1*1*2*3 = 6 • The case for n = 3 is thus true.

  32. ForLoop Example - 4 • Next, use proof by induction to verify performance at all higher values of n. • Assume that operation is verified for n = k. • ForLoop = k! and • FirstPart+...LastPart = 1*1*2*3*4*...*k = k!

  33. ForLoop Example - 5 • Now, for k+1 • ForLoop = k!*(k+1) = (k+1)! • FirstPart+...LastPart = 1*1*2*3*4*...*k*(k+1) • = (k+1)! • The program is thus valid for k+1 so, by induction, it is valid for all values of n => 1.

  34. WhileLoop Verification - 1 • The WhileLoop is assumed to have the following form • WhileLoop • while WhileTest • begin • LoopPart • end

  35. WhileLoop Verification - 2 • Question 1. Is loop termination guaranteed for any argument of WhileTest? • Question 2. When WhileTest is true does • WhileLoop = LoopPart followed by WhileLoop? • Question 3. When WhileTest is false does • WhileLoop = identity?

  36. WhileLoop Example - 1 • Use the first part of a program to reduce a floating point number to standard form. • Exp is a positive integer • Mant is to be made a positive non- zero digit before the decimal point. • void NormalizeReal(double& Mant, int& Exp, int NegEx) • while (Mant >= 10) • { Mant = Mant/10; • if (NegEx) Exp=Exp-1; else Exp = Exp + 1; }

  37. WhileLoop Example - 2 • Since the value of Exp is not material, there are 3 cases of interest. • Each of these cases is examined for NegEx true and NegEx false. • 1. Mant <10 • 2. Mant = 10 • 3. Mant > 10

  38. WhileLoop Example - 3 • Question 1. Is loop termination guaranteed for any argument of WhileTest? • Case 1. Mant < 10, the while test fails immediately, so the loop terminates. • Case 2. Mant = 10, the while test will fail after one cycle, thus terminating the loop. • In both cases, this happens regardless of the value of NegEx.

  39. WhileLoop Example - 4 • Question 1, continued • Case 3. Mant > 10, can be shown by induction • Mant = k*10**n, where 1 <= k < 10 • when n = 1, the test fails after 1 cycle • if the test is true for some value n=m then after m cycles, Mant = k • thus for n=m+1, after m cycles, Mant = k*10 and the WhileTest will fail in one more cycle • Since the value of NegEx does not affect the result, question 1 is satisfied forall n => 1.

  40. A WhileLoop Example - 5 • Question 2. When WhileTest is true does • WhileLoop = LoopPart followed by WhileLoop? • Case 1. Mant < 10, is not pertinent since WhileTest is never true.

  41. A WhileLoop Example - 6 • To examine the other cases, check that this program gives the same result as WhileLoop (the check is described in the next foil). • Mant = Mant/10; • if (NegEx) Exp=Exp-1; • else Exp = Exp + 1; • while (Mant >= 10) • { Mant = Mant/10; • if (NegEx) Exp=Exp-1; • else Exp = Exp + 1;}

  42. A WhileLoop Example - 7 • Question 2. When WhileTest is true does • WhileLoop = LoopPart followed by WhileLoop? • For case 2. Mant = 10, the value of NegEx is immaterial • WhileLoop has one cycle, leaving Mant = 1 • LoopPart followed by WhileLoop starts with Mant = 10, LoopPart leaves value 1, and WhileLoop leaves the same value 1 • Since both give the same result, the answer is true.

  43. WhileLoop Example - 8 • Question 2 - Continued • Case 3. Mant > 10, can be shown by induction. • Mant = k*10**n, where 1<=k<10 • when n = 1, WhileLoop would take one cycle and the WhileLoop following the LoopPart would take none • when n > 1 the While Loop following LoopPart would take one less cycle with the same result • Question 2 is thus satisfied for all cases.

  44. WhileLoop Example - 9 • Question 3. When WhileTest is false does WhileLoop = identity? • Case 1. Mant < 10 is the only case where WhileTest is false. • Since, in this case the program is not executed, the conditions are left undisturbed which is the definition of identity. • Question 3 is thus satisfied.

  45. RepeatUntil Verification - 1 • RepeatUntil has the following form • RepeatUntil • begin • LoopPart • UntilTest • end

  46. RepeatUntil Verification - 2 • Repeat-until verification is satisfied if the answers to the following 3 questions are true. • Question 1. Is loop termination guaranteed for any argument of UntilTest? • Question 2. When the UntilTest after LoopPart is false, does • RepeatUntil = • LoopPart followed by RepeatUntil?

  47. RepeatUntil Verification - 3 • Question 3. When UntilTest after LoopPart is true, does • RepeatUntil = LoopPart? • The repeat-until verification follows the same logical format as WhileLoop verification so no example is given.

  48. Comments on Program Verification - 1 • If done correctly, these verification methods are quite general and can guarantee program correctness. • The verification methods are tricky and require skill. • While you should attempt to use these methods, when in doubt, check with a trace table or an execution table.

  49. Comments on Program Verification - 2 • With practice, you can do program verification quite rapidly. • The careful examination of all possible cases will quickly reveal most defects. • The loop termination test is critical because endless loops are hard to identify with other methods.

  50. Lecture #13 Assignment • Start work on the final report. • Update the process used to develop the mid-term report and use it to develop the final report. • Turn in the updated process, the plan for the work, the actual data for the work, and the final report. • Consult the report specifications in Appendix D.

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