1. CSI 3125, Axiomatic Semantics, page 1 Axiomatic semantics The assignment statement Statement composition The "if-then-else" statement The "while" statement Narrowing and widening Termination Two diversions The greatest common divisor The "if-then" statement Points to discuss:

2. CSI 3125, Axiomatic Semantics, page 2 Program verification Program verification includes two steps. 1.Associate a formula with every meaningful step of the computation. 2.Show that the final formula logically follows from the initial one through all intermediate steps and formulae.

3. CSI 3125, Axiomatic Semantics, page 3 What is axiomatic semantics? Axiomatic semantics of assignments, compound statements, conditional statements, and iterative statements has been developed by Professor C. A. R. Hoare. The elementary building blocks are the formulae for assignments and conditions. The effects of other statements are described by inference rules that combine formulae for assignments (just as statements themselves are combinations of assignments and conditions).

4. CSI 3125, Axiomatic Semantics, page 4 The assignment statement Let  be a logical formula that contains variable v.  v  e is a formula which we get from  when we replace all occurrences of variable v with expression e.

5. CSI 3125, Axiomatic Semantics, page 5 Replacement, an example Before replacement:   h >= 0 & h 0  h  0  0 >= 0 & 0 0 after replacement

6. CSI 3125, Axiomatic Semantics, page 6 Another example  m == min( 1 <= i & i <= k–1: a i ) & k–1 <= N  k  k+1  m == min( 1 <= i & i <= (k+1) – 1: a i ) & (k+1)–1 <= N  m == min( 1 <= i & i <= k: a i ) & k <= N

7. CSI 3125, Axiomatic Semantics, page 7 The axiom for the assignment statement {  v  e } v = e {  } Example: { 0 >= 0 & 0 0 } x = 0; { x >= 0 & x 0 }

8. CSI 3125, Axiomatic Semantics, page 8 Two small puzzles { ??? } z = z + 1; { z <= N } { a > b } a = a – b; { ??? }

9. CSI 3125, Axiomatic Semantics, page 9 Statement composition ASSUME THAT {  ´ } S ´ {  ´´ } and {  ´´ } S ´´ {  ´´´ } CONCLUDE THAT {  ´ } S ´ S ´´ {  ´´´ } In other words: {  ´ } S ´ {  ´´ } S ´´ {  ´´´ }

10. CSI 3125, Axiomatic Semantics, page 10 A more complicated example x = 0; f = 1; while (x != n) { x = x + 1; f = f * x; } We want to prove that { f == x! } x = x + 1; f = f * x; { f == x! }

11. CSI 3125, Axiomatic Semantics, page 11 Let's apply the inference rule for composition.  ´ isf == x!  ´´´ isf == x! S ´ isx = x + 1; S ´´ isf = f * x; The factorial

12. CSI 3125, Axiomatic Semantics, page 12 We need to find a  ´´ for which we can prove: { f == x! } x = x + 1; {  ´´ } f = f * x; { f == x! } Observe thatf == x!  f == ((x + 1) – 1)! and thereforef == (x – 1)! x  x + 1  f == x! That is: { f == x! } x = x + 1; {f == (x – 1)! } ´´  ´´ S ´S ´ The factorial (2)

13. CSI 3125, Axiomatic Semantics, page 13 Now, let us observe that f == (x – 1)!  f * x == (x – 1)! * x == x! So, we have f == x! f  f * x  f == (x – 1)! That is, {f == (x – 1)! } f = f * x; {f == x! }  ´´  ´´´ S ´´ The factorial (3) QED

14. CSI 3125, Axiomatic Semantics, page 14 The "if-else" statement ASSUME THAT {  &  } S´ {  } and {  &   } S´´ {  } CONCLUDE THAT {  } if (  ) S ´ else S ´´ {  } Both paths through the if-else statement establish the same fact . That is why the whole conditional statement establishes this fact.

15. CSI 3125, Axiomatic Semantics, page 15 The statement if ( a < 0 ) b = -a; else b = a; makes the formula b == abs(a) true. Specifically, the following fact holds: {true} if ( a < 0 ) b = -a; else b = a; { b == abs(a) } Here:  istrue  isb == abs(a)  isa < 0 Also:S´ isb = -a; S´´ isb = a; "if-else", an example

16. CSI 3125, Axiomatic Semantics, page 16 We will consider cases. First, we assume that  is true: true & a < 0  a < 0  – a == abs(a) Therefore, by the assignment axiom: {– a == abs(a)} b = -a; {b == abs(a)} Similarly, when we assume  , we get this: true &  a < 0  a  0  a == abs(a) Therefore: {a == abs(a)} b = a; {b == abs(a)} "if-else", an example (2)

17. CSI 3125, Axiomatic Semantics, page 17 This shows that both S´ and S´´ establish the same condition: b == abs(a) Our fact has been proven: {true} if ( a < 0 ) b = -a; else b = a; { b == abs(a) } In other words, our conditional statement computes abs(a). It does so without any preconditions: "true" means that there are no restrictions on the initial values of a and b. "if-else", an example (3)

18. CSI 3125, Axiomatic Semantics, page 18 The "while" statement A loop invariant is a condition that is true immediately before entering the loop, stays true during its execution, and is still true after the loop has terminated. ASSUME THAT {  &  } S {  } [That is, S preserves .] CONCLUDE THAT {  } while (  ) S {  &   } provided that the loop terminates.

19. CSI 3125, Axiomatic Semantics, page 19 The factorial again... x = 0; f = 1; while ( x != n ) { x = x + 1; f = f * x; } Assume for now that n ≥ 0. After computing x = 0; f = 1; we have f == x! because it is true that 1 == 0! We showed earlier that { f == x! } x = x + 1; f = f * x; { f == x! }

20. CSI 3125, Axiomatic Semantics, page 20 Now,  isf == x!  isx != n   isx == n Using the inference rule for "while" loops: { f == x! } while ( x != n ) { x = x + 1; f = f * x; } { f == x! & x == n} The factorial again... (2)

21. CSI 3125, Axiomatic Semantics, page 21 Notice that f == x! & x == n  f == n! This means two things: { true }x = 0; f = 1; { f == x! } AND { f == x! }while ( x != n ) { x = x + 1; f = f * x; } { f == n!} The factorial again... (3)

22. CSI 3125, Axiomatic Semantics, page 22 In other words, the program establishes f == n! without any preconditions on the initial values of f and n, assuming that we only deal with n ≥ 0. The axiom for statement composition gives us: { true }x = 0; f = 1; while ( x != n ) { x = x + 1; f = f * x; } { f == n!} So: this program does compute the factorial of n. The factorial again... (4)

23. CSI 3125, Axiomatic Semantics, page 23 Our reasoning agrees with the intuition of loop invariants: we adjust some variables and make the invariant temporarily false, but we re-establish it by adjusting some other variables. { f == x! } x = x + 1; {f == (x – 1)! } the invariant is "almost true" {f == (x – 1)! } f = f * x; {f == x! } the invariant is back to normal This reasoning is not valid for infinite loops: the terminating condition  &   is never reached, and we know nothing of the situation following the loop. The factorial again... (5)

24. CSI 3125, Axiomatic Semantics, page 24 Narrowing and widening ASSUME THAT  ´   and {  } S {  } CONCLUDE THAT {  ´ } S {  } ASSUME THAT {  } S {  } and   ´  ´ CONCLUDE THAT {  } S {  ´ } These rules can be used to narrow a precondition, or to widen a postcondition.

25. CSI 3125, Axiomatic Semantics, page 25 n! is computed, for any nonnegative n, with true as the precondition (it is always computed successfully); So, n! will in particular must be computed successfully if initially n == 5. Narrowing and widening, a small example

26. CSI 3125, Axiomatic Semantics, page 26 A larger example (in a more concise notation) { N >= 1 }  { N >= 1 & 1 == 1 & a 1 == a 1 } i = 1; s = a 1 ; { N >= 1 & i == 1 & s == a 1 }  { N >= 1 & s == a 1 + … + a i }  INVARIANT while ( i != N ) { { N >= 1 & s == a 1 + … + a i & i != N } i = i + 1; { N >= 1 & s == a 1 + … + a i–1 & i – 1 != N } s = s + a i ; { N >= 1 & s == a 1 + … + a i } } { N >= 1 & s == a 1 + … + a i & i == N }  { N >= 1 & s == a 1 + … + a N }

27. CSI 3125, Axiomatic Semantics, page 27 We have shown that this program computes the sum of a 1,..., a N. The precondition N >= 1 is only necessary to prove termination. A larger example (2)

28. CSI 3125, Axiomatic Semantics, page 28 Termination Proofs like these show only partial correctness. –Everything is fine if the loop stops. –Otherwise we don't know (but the program may be correct for most kinds of data). A reliable proof must show that all loops in the program are finite. We can prove termination by showing how each step brings us closer to the final condition.

29. CSI 3125, Axiomatic Semantics, page 29 Once again, the factorial… Initially, x == 0. Every step increases x by 1, so we go through the numbers 0, 1, 2,... n >= 0 must be found among these numbers. Notice that this reasoning will not work for n < 0: the program loops.

30. CSI 3125, Axiomatic Semantics, page 30 A decreasing function A loop terminates when the value of some function of program variables goes down to 0 during the execution of the loop. For the factorial program, such a function could be n – x. Its value starts at n and decreases by 1 at every step. For summation, we can take N – i.

31. CSI 3125, Axiomatic Semantics, page 31 Multiplication by successive additions { B >= 0 & B == B & 0 == 0}  FOR TERMINATION b = B; p = 0; { b == B & p == 0 }  { p == A * (B – b) }  INVARIANT while ( b != 0 ) { p = p + A; { p == A * (B – (b – 1)) } b = b - 1; { p == A * (B – b) } } { p == A * (B – b) & b == 0}  { p == A * B } The loop terminates, because the value of the variable b goes down to 0.

32. CSI 3125, Axiomatic Semantics, page 32 Two diversions Prove that the sequence p = a; a = b; b = p; exchanges the values of a and b : { a == A & b == B } p = a; a = b; b = p; { b == A & a == B } The highlights of a proof: { a == A & b == B }p = a; { p == A & b == B }a = b; { p == A & a == B }b = p; { b == A & a == B }

33. CSI 3125, Axiomatic Semantics, page 33 x = x + y; y = x - y; x = x - y; Two diversions (2) Discover and PROVE the behaviour of the following sequence of statements for integer variables x, y:

34. CSI 3125, Axiomatic Semantics, page 34 {x == X & y == Y }  {x + y == X + Y & y == Y }x = x + y; {x == X + Y & y == Y }  {x == X + Y & x - y == X }y = x - y; {x == X + Y & y == X }  { x - y == Y & y == X }x = x - y; { x == Y & y == X } Two diversions (3)

35. CSI 3125, Axiomatic Semantics, page 35 The greatest common divisor { X > 0 & Y > 0 } a = X; b = Y; {  }  what should the invariant be? while ( a != b ) {  & a != b } { if ( a > b ) {  & a != b & a > b } a = a - b; else {  & a != b &  (a > b) } b = b - a; } {  &  (a != b) } { GCD( X, Y ) == a }

36. CSI 3125, Axiomatic Semantics, page 36 We will need only a few properties of greatest common divisors: GCD( n + m, m ) == GCD( n, m ) GCD( n, m + n ) == GCD( n, m ) The first step (very formally): { X > 0 & Y > 0 }  { X > 0 & Y > 0 & X == X & Y == Y } a = X; b = Y; { a > 0 & b > 0 & a == X & b == Y } GCD (2)

37. CSI 3125, Axiomatic Semantics, page 37 When the loop stops, we get a == b & GCD( a, b ) == a We may want this condition in the invariant: a == b & GCD( X, Y ) == GCD( a, b ) At the beginning of the loop, we have: { a > 0 & b > 0 & a == X & b == Y }  {a > 0 & b > 0 & GCD( X, Y ) == GCD( a, b ) } So, the invariant could be this: a > 0 & b > 0 & GCD( X, Y ) == GCD( a, b ) GCD (3)

38. CSI 3125, Axiomatic Semantics, page 38 We should be able to prove that {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b) & a != b} while...... {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b)} The final condition will be a > 0 & b > 0 & GCD(X, Y) == GCD(a, b) & a == b and this will imply GCD( X, Y ) == a GCD (4)

39. CSI 3125, Axiomatic Semantics, page 39 The loop consists of one conditional statement. Our proof will be complete if we show this: {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b) & a != b} if ( a > b ) a = a - b; else b = b - a; {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b)} GCD (5)

40. CSI 3125, Axiomatic Semantics, page 40 Consider first the case of a > b. {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b) & a != b & a > b }  {a – b > 0 & b > 0 & GCD(X, Y) == GCD(a – b, b)} a = a - b; {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b)} GCD (6)

41. CSI 3125, Axiomatic Semantics, page 41 Now, the case of  a > b. {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b) & a != b &  (a > b) }  {a > 0 & b – a > 0 & GCD(X, Y) == GCD(a, b – a)} b = b - a; {a > 0 & b > 0 & GCD(X, Y) == GCD(a, b)} GCD (7)

42. CSI 3125, Axiomatic Semantics, page 42 Both branches of if-else give the same final condition. We will complete the correctness proof when we show that the loop terminates. We show how the value of max( a, b ) decreases at each turn of the loop. Let a == A, b == B at the beginning of a step. Assume first that a > b: max( a, b ) == A, so a – b < A, b < A, therefore max( a – b, b ) < A. GCD (8)

43. CSI 3125, Axiomatic Semantics, page 43 Now assume that a < b: max( a, b ) == B, b – a < B, a < B, therefore max( a, b – a ) < B. Since a > 0 and b > 0, max( a, b ) > 0. This means that decreasing the values of a, b cannot go forever. QED GCD (9)

44. CSI 3125, Axiomatic Semantics, page 44 The "if" statement ASSUME THAT {  &  } S {  } and  &     CONCLUDE THAT {  } if (  ) S {  }

45. CSI 3125, Axiomatic Semantics, page 45 We will show the following: { N > 0 } k = 1; m = a 1 ; while ( k != N ) { k = k + 1; if ( a k < m ) m = a k ; } { m == min( 1 <= i & i <= N: a i ) } An example with "if"

46. CSI 3125, Axiomatic Semantics, page 46 Loop termination is obvious: the value of N – k goes down to zero. Here is a good invariant: at the k th turn of the loop, when we have already looked at a 1,..., a k, we know that m == min( 1 <= i & i <= k : a i ). Initially, we have this: { N > 0 } k = 1; m = a 1 ; { k == 1 & m == a 1 }  { k == 1 & m == min( 1 <= i & i <= k : a i ) } Minimum

47. CSI 3125, Axiomatic Semantics, page 47 We must prove the following: { m == min( 1 <= i & i <= k : a i ) & k != N } k = k + 1; if ( a k < m ) m = a k ; { m == min( 1 <= i & i <= k : a i ) } Minimum

48. CSI 3125, Axiomatic Semantics, page 48 { m == min( 1 <= i & i <= k : a i ) & k != N }  { m == min( 1 <= i & i <= (k + 1) – 1: a i ) & (k + 1) – 1 != N } k = k + 1; { m == min( 1 <= i & i <= k – 1: a i ) & k – 1 != N } Note that k – 1 != N ensures the existence of a k. Minimum (2)

49. CSI 3125, Axiomatic Semantics, page 49 This remains to be shown: { m == min( 1 <= i & i <= k – 1: a i ) & k – 1 != N } if ( a k < m ) m = a k ; { m == min( 1 <= i & i <= k: a i ) } The fact we will use is this: min( 1 <= i & i <= k: a i ) == min2( min( 1 <= i & i <= k – 1: a i ), a k ) Minimum (3)

50. CSI 3125, Axiomatic Semantics, page 50 We will consider two cases of the conditional statement. First,  (a k < m). {m == min(1 <= i & i <= k – 1: a i ) & k – 1 != N &  (a k < m)}  {m == min2(min( 1 <= i & i <= k – 1: a i ), a k )}  {m == min(1 <= i & i <= k: a i )} Minimum (4)

51. CSI 3125, Axiomatic Semantics, page 51 Now, a k < m. {m == min(1 <= i & i <= k – 1: a i ) & k – 1 != N & a k < m}  {a k == min2( min( 1 <= i & i <= k – 1: a i ), a k )}  {a k == min(1 <= i & i <= k: a i )} m = a k ; {m == min(1 <= i & i <= k: a i )} So, the body of the loop preserves the condition m == min( 1 <= i & i <= k: a i ) Minimum (5)

52. CSI 3125, Axiomatic Semantics, page 52 Now, the whole loop works as follows: { m == min( 1 <= i & i <= k: a i ) } while ( k != N ) } k = k + 1; if ( a k < m ) a k = m; } { m == min( 1 <= i & i <= k: a i ) & k == N }  { m == min( 1 <= i & i <= N: a i ) } All in all, we have shown that our program finds the minimum of N numbers, if only N > 0. QED Minimum (6)

53. CSI 3125, Axiomatic Semantics, page 53 Yet another "while" loop { B > 0 }  FOR TERMINATION b = 1; p = A; while ( b != B ) { b = b + 1; p = p * A; } { ??? } Examples

54. CSI 3125, Axiomatic Semantics, page 54 Yet another "while" loop (2) { B > 0 & 1 == 1 & A == A}  FOR TERMINATION b = 1; p = A; { b == 1 & p == A }  { p == A ** b }  INVARIANT while ( b != B ) { b = b + 1; { p == A ** (b - 1) } p = p * A; { p == A ** b } } { p == A ** b & b == B}  { p == A ** B } The loop terminates: the value B - b goes down to 0. Examples

55. CSI 3125, Axiomatic Semantics, page 55 { N > 0 }  FOR TERMINATION k = 1; while ( k != N ) { if ( A k > A k+1 ) { p = A k ; A k = A k+1 ; A k+1 = p; } k = k + 1; } { ??? } Another example with "if" Examples

56. CSI 3125, Axiomatic Semantics, page 56 { N > 0 }  FOR TERMINATION k = 1; { A k == max( 1 <= i & i <= k: A i ) }  INVARIANT while ( k != N ) { { A k == max( 1 <= i & i <= k: A i ) & k != N } if ( A k > A k+1 ) { p = A k ; A k = A k+1 ; A k+1 = p; } { A k+1 == max( 1 <= i & i <= k+1: A i ) } k = k + 1; { A k == max( 1 <= i & i <= k: A i ) } } {A k == max( 1 <= i & i <= k: A i ) & k == N }  {A N == max( 1 <= i & i <= N: A i ) } Another example with "if" (2) Examples

57. CSI 3125, Axiomatic Semantics, page 57 {A k == max( 1 <= i & i <= k: A i ) & k != N } case 1: A k > A k+1 { A k == max( 1 A k+1 } p = A k ; { p > A k+1 } A k = A k+1 ; { p > A k } A k+1 = p; { A k+1 > A k } { A k+1 == max( 1 <= i & i <= k+1: A i ) } case 2: A k <= A k+1 { A k == max( 1 <= i & i <= k: A i ) & k != N & A k <= A k+1 }  { A k+1 == max( 1 <= i & i <= k+1: A i ) } Another example with "if" (3) Examples