Assertions and Propositional Calculus

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Assertions andPropositional Calculus

• COMP 114
• Tuesday March 5

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Announcements

• No class Thursday, March 21
• Test 2 is on April 9
• Homework 3 due March 21
• All electronic, Word file on web page
• Submit via Blackboard system
• No office hours Th 3/7 and Mon 3/18
• Email if youd like to meet well set time

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Read

• Chapter 3 of Stanat and Weiss
• http//www.cs.unc.edu/weiss/COMP114/BOOK/03RulesO
  fPgm.doc

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Topics

• Last Time
• Complexity

• This Class
• Program State
• Assertions
• Pre- and Post- Conditions
• Propositional Calculus

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Program Documentation

• Programs very hard to understand!
• Typical documentation in English

• Typical documentation
• // Sets i to 5
• i 5

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Formal Documentation

• Ways to make documentation precise.
• More like programs.
• Why? To make you think more formally about what
  program is doing.

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Program State

• All of the information that would be necessary to
  restart at same place
• Program counter
• Values of variables
• Also something about I/O
• (we will typically ignore the state of I/O)

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Assertions

• An assertion is a statement that something is
  true.
• Precondition an assertion about state before a
  program (or chunk of code).
• Postcondition assertion after code

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Silly Example

• // Precondition x 6
• x
• // Postcondition x 7
• The behavior of the program is described.
• If x 6 at the beginning,
• then x 7 at the end.

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Pre and Post conditions

• Says nothing about implementation!
• Could be
• // Precondition x 6
• x 7
• // Postcondition x 7
• This description is called the specification
• Also says nothing about termination.

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Definition

• A program C is correct with respect to
  precondition P and a postcondition Q if,
• whenever condition P holds prior to execution of
  program C,
• and C terminates,
• then condition Q will (always!) hold after C has
  finished execution.

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What if Precondition is False?

• Then the assertions mean nothing
• The statements are only based on the state being
  what you expect on entry to the block.

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Propositional Calculus

• A language of Boolean expressions
• Values are true or false
• Weve seen this before
• if( x gt 10 y lt 5 )

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Propositions

• Can be constructed using Boolean operators
• , , , !, xor ( in Java)
• Example
• (p q) !q

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Boolean Operators
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One and Zero as True and False
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Implication

• Less common Boolean operator gt
• Read as p implies q
• q is true whenever p is true

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Combination of Arithmetic and Boolean Expressions

• Just like Java
• ( (x y) lt 7 ) ( y gt 9 )

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Weak vs. Strong Assertions

• A gt B
• A is said to be stronger than B (or B weaker than
  A)
• Example
• ((x gt 3) (x lt 7)) gt
• ((x gt 0) (x lt 10))

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Other Examples

• (p r ) gt p

If p true, true gt true false gt true If p
false, false gt false

• p gt (p r)

If p true, true gt true If p false,
false gt false false gt true
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Other Examples

• Is this true?
• p gt true

• How about this?
• false gt q

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No Implication Operator in Java?

• No problem
• You can make one
• How?

!p q
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Simple Assert Method

• public static boolean assert(boolean b,
• String error)
• if (!b)
• System.out.println(
• "Assertion failure error)
• System.exit(0)
• return true

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Assertion Exception Class

• public class AssertionEx extends
  RuntimeException
• AssertionEx()
• super("Assertion failed")
• AssertionEx(String s)
• super("Assertion failed "s)

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New Assert Class

• public class Assert
• public static void assert(
• boolean b,String s)
• if (!b)
• throw new AssertionEx(s)

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Now We Can Code Assertions

• Our silly example
• Assert.assert( x 6, pre x ! 6)
• x
• Assert. assert( x 7, post x ! 7)
• We can also save values of variables from
  assertion to assertion.

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Quantifiers

• Still missing ability to say something about sets
  of variables
• How could we test that list is sorted?
• Quantifiers allow you to say
• All of the entries of array B are gt 0
• There is an element of B that is zero

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Universal Quantifier

• For all integers x, x gt 3
• We can write this as
• "x x gt 3
• Or, if you dont have cool symbols
• Ax x gt 3
• Read as
• For all x, x is greater than 3.
• Generally false, of course.

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Existential Quantifier

• There exists an integer x, such that x gt 3
• x x gt 3
• or
• Ex x gt 3

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Quantified Assertions

• In the Stanat Weiss text, they write assertions
  as
• (QxD(x)P(x))
• where
• Qx is the quantifier Ai or Ei
• D(x) is a domain predicate 4 lt i gt 10
• P(x) is the assertion Bi gt 0
• (Ai 4 lt i lt 10 Bi gt 0)

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Examples

• Informal
• The value 4 occurs in the array B
• Formal
• There exists a value of i between 0 and n-1,
  inclusive, such that Bi 4.
• Notation
• (Ei 0 lt i lt n Bi 4)

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Examples

• Informal The first element of the array B is the
  largest.
• Could mean
• The value of B0 is at least as large as every
  entry of B.
• (Ai 0 lt i lt n Bi lt B0)
•  or
• The value of B0 is strictly larger than every
  other entry of B.
• (Ai 0 lt i lt n Bi lt B0)

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Examples

• Informal The array B is sorted in non-decreasing
  order.
• If i is less than j, then Bi is less than or
  equal to Bj
• (Ai Aj 0 lt i lt j lt n Bi lt Bj)

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Example

• Code segment that initializes all entries of an
  array Bn to 0.
• assert (true) // Precondition
• // (Ai 0 lt i lt n Bi 0) Post

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Summary

• Assertions as pre- and post- conditions formally
  specify program behavior
• If the precondition is true, then
• if the postcondition is true, the program behaves
  as specified.
• Some assertions can be coded in Java and checked
  at run time

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Next Time

• Loop invariants
• Way to say something about whats happening in a
  loop