The need for formal methods and other introductory comments

1
The need for formal methodsand other
introductory comments

• Covering
• Ch 1 and 2 of Potter Sinclair and Till

2
The Importance of Specification

• Specifications can be used as contractual
  agreements of what the software is supposed to
  do.
• Must say what without saying how.
• Need to be precise and detailed but without
  resorting to implementations.

3
Need for Abstraction

• Because specification has to avoid implementation
  details, relevant aspects of the design need to
  be abstracted.
• Hence need to select a notation that provides
  enough details but is not too cumbersome.

4
Notation

• Select the Z specification language based on
• Set theory (is again based on logic)
• Logic
• Plan
• Introduce them informally (as stated in Ch 2 of
  PST) and,
• Discuss them more formally(as in Ch 1 and 2 of HR)

5
Set Theory

• Set collection of items
• Ex 1,2,3, SWE-620, INFS-612
• set of all prime numbers PRIME (say)
• Set-membership relation
• 3, 17 e PRIME and
• SWE-623 \e SWE-620, INFS-612

6
Set Operators

• Intersection A/\B
• Elements in both A and B
• Union AUB
• Elements in A or B
• Difference A\B
• Elements in A but not B
• Comprehension xeAf(x)
• Elements of A satisfying f
• Power-set P(A)
• The set of all subsets of A

7
Some Examples

• EU Belgium, France, Denmark
• NATO Belgium,France,Canada
• EU/\NATO Belgium,France
• EU U NATO Belgium,France,Denmark,Canada
• EU \ NATO Denmark
• P(EU) Belgium,France,Denmark,
  Belgium,France,Belgium,Denmark, ..

8
Set Comparisons

• A is a subset of B A ( B
• Meaning every element of A is an element of B.
• A B if and only if A ( B and B ( A
• That means A and B have same elements

9
Empty/Universal Sets and Complements

• Universal set U
• Consists of all elements under discourse
• Empty set f
• Subset of U with no elements
• Complement Ac
• U \ A elements not in A.

10
Some Laws of Set Theory

• A/\(B u C) (A/\B) u (A /\ C)
• A u (B/\C) (A u B) /\ (A u C)
• (A u B)c Ac /\ Bc
• A /\ f A
• A /\ U U
• Can be shown with a simple logical arguments

11
Sample Proof of A/\ f f

• Need to show, that for every element x e U
• If x e A /\ f then x e f
• If x e f then x e A /\ f

• Suppose x e f.
• Then x e A /\ f
• As x e f and
• false gt any statement

Suppose x e A /\ f. Then x e A and x e f.
Thus x e f.
12
The Need for Logical Reasoning

• As we saw simple proofs use logic.
• Hence we need to
• know what proofs are and
• How to produce proofs.
• So we study Logic
• Prepositional
• To reason about ands ors and nots.
• Predicate
• To reason about properties of individuals and
  classes.