CSC5125 Program Verification

1
CSC5125 Program Verification

• Developing an approach towards reasoning about
  programs - 3

2
Extending the Propositional Calculus and
introducing a functional view of arrays

• Material from Chapters 4 and 5
• Quick review of quantification
• General forms
• (E i R E) and (A i R E)
• Note that R refers to some range of values (not
  necessarily numeric), and E is some predicate
• Example of non-numeric range
• (A i person(i) Mortal(i))

3
Another couple notes on quantification

• We assume the type of the value being quantified
  over (normally this is reasonable, but we will
  have to be careful later)
• Any tautology E is equivalent to the same
  predicate, but with all of its identifiers
  universally quantified
• This fact will help us in describing initial
  values in a program

4
Revisiting substitution

• This is actually hard. We will be glossing over
  most of the particulars until we get to
  denotational semantics
• Notation that will be used in the text

This notation refers to substituting all free
occurrences of x in predicate E with e
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Textual substitution

• Examples
• Text error (4.4.5) should read

6
Substitution continued
7
Substitution and states

• Remember that s(e) represents the value of
  expression e in state s.
• s (s xv) says that state s is the same as
  state s except that identifier x has value v.

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Substitution lemmas

• 1) Substituting an expression e for x in E and
  then evaluating E in a state s yields the same
  result as substituting the value of e in state s
  for x and then evaluating
• In other words
• Proof is by structural induction

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Substitution lemmas
10
Arrays

• We talk about arrays as a special topic due to
  the fact that the way we will reason about them
  is slightly different that for Booleans and ints.
• We will view arrays functionally namely as a
  map from a subset of the natural numbers to some
  type (the type of the array)

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Arrays - notation

• We will generally begin arrays at position 0,
  though there is no particular reason for doing so
  (except that Java does as well).
• (b ie) (example (b 0T)) is just like the
  array b except that position i has value e.
• Let b0..3 1,3,4,2. Then what is the
  following value of the array? (b 08 19 07)
  
• perm((c 0x), C) is a permutation of array C but
  with x at position 0.

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Array notation

• Given array b
• (A i 0lt i lt n bi lt x) we write b lt x
• (A i 0lt i lt n bi 0) we write b 0
• b37 x
• b y (but note that b y is not the same as
  !(b!y)
• (E i 0 lt i lt n bi x) we write x e b
• Can also use picture notation

13
Arrays - definition
Note the similarity of the definition of how
arrays work to how states work!