1. Mathematical Structures for Computer Science Chapter 1
Formal Logic
Mathematical Structures for Computer Science
Chapter 1
Copyright © 2006 W.H. Freeman & Co. MSCS Slides Formal Logic

2. Proof of Correctness
Program verification attempts to ensure that a computer program is correct.
A program is correct if it behaves in accordance with its specifications.
This does not necessarily mean that the program solves the problem that it was intended to solve; the program’s specifications may be at odds with or not address all aspects of a client’s requirements.
Program validation attempts to ensure that the program indeed meets the client’s original requirements.
Program testing seeks to show that particular input values produce acceptable output values.
Proof of correctness uses the techniques of a formal logic system to prove that if the input variables satisfy certain specified predicates or properties, the output variables produced by executing the program satisfy other specified properties.
Section 1.6
Proof of Correctness

3. Assertions
Let us denote by X an arbitrary collection of input values to some program or program segment P.
The actions of P transform X into a corresponding group of output values Y.
The notation Y = P(X) suggests that the Y values depend on the X values through the actions of program P.
A predicate Q(X) describes conditions that the input values are supposed to satisfy. Q is the precondition.
A predicate R describes conditions that the output values are supposed to satisfy. These conditions will often involve the input values, so R has the form R(X, Y) or R[X, P(X)]. R is the postcondition.
Section 1.6
Proof of Correctness

4. Assertions
For example, if a program is supposed to find the square root of a positive number, then X consists of one input value, x, and Q(x) might be “x > 0.”
If y is the single output value, then y is supposed to be the square root of x, so R(x, y) would be “y2 = x.”
Program P is correct if the following implication is valid:
("X)(Q(X)  R[X, P(X)])
For the square root case, it is:
("x)(x > 0  [P(x)]2 = x )
The traditional program correctness notation (called a Hoare triple) is:
{Q}P{R}
Section 1.6
Proof of Correctness

5. Assertions
A program or program segment is broken down into individual statements si, with predicates inserted between statements as well as at the beginning and end.
These predicates are also called assertions because they assert what is supposed to be true about the program variables at that point in the program.
{Q} s0
{R1} s1
{R2}
sn1 .
{R}
Where Q, R1, R2, ... , Rn = R are assertions. The intermediate assertions are often obtained by working backward from the output assertion R.
Section 1.6
Proof of Correctness

6. Assertions
P is provably correct if each of the following implications holds:
{Q}s0{Rl}
{Rl}sl{R2}
{R2}s2{R3} .
{Rn1}sn1{R}
A proof of correctness for P consists of producing this sequence of valid implications.
Some new rules of inference can be used, based on the nature of the program statement si.
Section 1.6
Proof of Correctness

7. Assignment Rule
Suppose that statement si is an assignment statement of the form x = e for some expression e.
The Hoare triple to prove correctness of this one statement has the form: {Ri} x = e {Ri + l}.
For this triple to be valid, the assertions Ri and Ri +1 must be related in a particular way.
The appropriate rule of inference for assignment statements is the assignment rule.
It says that if the precondition and postcondition are appropriately related, the Hoare triple can be inserted at any time in a proof sequence without having to be inferred from something earlier in the proof sequence. It has the following conditions:
si has the form x = e.
Ri is Ri + 1 with e substituted everywhere for x.
Section 1.6
Proof of Correctness

8. Assignment Rule For example, the Hoare triple:
{x  1 > 0} x = x  1 {x > 0}
is valid by the assignment rule.
The postcondition is:
x > 0
Substituting x  1 for x throughout the postcondition results in:
x – 1 > 0 or x > 1
which is the precondition.
Section 1.6
Proof of Correctness

9. Conditional Rule
A conditional statement is a program statement of the form:
if condition B then P1
else P2
end if
A conditional rule of inference determines when a Hoare triple can be inserted in a proof sequence if si is a conditional statement.
The Hoare triple is inferred from two other Hoare triples:
{Q /\ B} P1 {R} if B is true
{Q /\ B} P2 {R} if B is false
This simply says that each branch of the conditional statement must be proved correct.
Section 1.6
Proof of Correctness

10. Conditional Rule For example: We must prove:
{n = 5} if n >= 10 then y = 100 else y = n + 1 end if {y = 6}
We must prove:
{n = 5 and n >= 10} y = 100 {y = 6}
{n = 5 and n < 10} y = n + 1 {y = 6}
Using the assignment rule we get:
{n = 5} y = n + 1 {y = 6}
{n = 5 and n < 10} y = n + 1 {y = 6}
The conditional rule allows us to conclude that the program segment is correct.
Section 1.6
Proof of Correctness