Predicates and quantifiers

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Predicates and quantifiers

• Chapter 8
• Formal Specification using Z

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Predicates

• A predicate is a logical statement that depends
  on a value or values. When a predicate is applied
  to a particular value it becomes a proposition.
  For example
• prime(x)
• depends on some numeric value of x
• prime(7)
• is a true proposition the only devisors of 7 are
  1 and 7.
• Prime(6) is a false proposition 6 can be divided
  by 2,3.
• A predicate can be viewed as a propositional
  function, that for some set of arguments returns
  to either true or false

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Quantifiers

• Consider the following statements
• All cats have tails.
• Some people like cake.
• Everyone gets a break once in a while.
• All these statements indicate how frequently
  certain things are true. In predicate calculus we
  use quantifiers in this context.

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Universal Quantifier

• The Universal Quantifier is written
• A
• and is pronounced for all. It is often used in
  the form
• A declaration constraint predicate
• Which states the for the declarations given,
  restricted to certain values (the constraint),
  the predicate holds.

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Universal Quantifier

• The constraint may be omitted.
• The declaration introduces a typical element that
  is then optionally constrained or restricted in
  some some way For example
• A i N i lt 10 i2 lt 100
• Which states that for all natural numbers less
  than 10 their square is less that 100 .

Constraint
Predicate
Separators
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Universal Quantifier

• A universal quantifier can be viewed as a chain
  of conjunctions. So
• A i N i lt 10 i2 lt 100
• is equivalent to
• 02 lt 100 L 12 lt 100 L . . L 92 lt 100
• If the set of values over which the variable is
  universally quantified is empty, then the
  quantification is defined as true.
• A i N 0 i lt 0 i2 lt 100
• is true.

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Existential Quantifier

• The Existential Quantifier is written
• E
• and is pronounced there exists. It is often
  used in the form
• E declaration constraint predicate
• The declaration introduces a typical element that
  is then optionally constrained or restricted in
  some way For example
• E i N i lt 10 i2 lt 100

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Existential Quantifier

• There may be more than one value of i for which
  this is true, in the previous example there are
  ten values, 0 to 9, that satisfy the predicate.
  Another example
• Even(x) ? E k Z k 2 x
• The existential quantifier can be considered as a
  chain of disjunctions (ors). If the set of values
  over which the variable is existentially
  quantified is empty, then the quantification is
  defined as false.
• E i N 0 i lt 0 i2 lt 100

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Unique Quantifier

• The unique quantifier is similar to the
  existential quantifier except that it states that
  there exists only one value for which the
  predicate is true. It is written as
• E1
• An example
• E1i N i lt 1 0 i2 lt 100 ? i2 gt 80

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Unique Quantifier

• The previous unique quantifier example is
  equivalent to saying that the predicate holds for
  i, but there is no value j for which it holds
  This is written as
• E1i N i lt 1 0 i2 lt 100 ? i2 gt 80
• ?
• E1i N i lt 1 0 i2 lt 100 ? i2 gt 80
• ? (Ej N j lt 1 0 i j j2 lt 100 ? j2 gt
  80)

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Counting Quantifier

• Some notations use a counting quantifier that
  counts for how many values of a variable the
  predicate holds. In Z this is not needed, we use
  set comprehension instead.

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Set Comprehension

• Set Comprehension
• It is possible to give the value of a set by
  giving a condition (a predicate) which must hold
  true for every member of the set. The general
  form is
• declaration constraint expression
• SetA xZ Even(x) xx

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Set Comprehension

• declaration constraint expression
• The declaration is for a typical element and it
  gives the elements name and type.
• The constraint restricts the possible values of
  the typical element. It is a logical expression
  which must be true for that value of the typical
  element to be included.
• The expression indicates the value to be held in
  the set

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Set Comprehension

• SetA xZ Even(x) xx
• The above defines a set of type integer. The
  value of x is constrained to be even. Each
  element will be the square of an even integer.
• SetB xZ Even(x) x
• The above defines a set of type integer. The
  value of x is constrained to be even. Each
  element will be an even integer.

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Ranges of numbers

• We have already introduced m..n. This is
  shorthand for
• iZ m i ? n i i

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Relationship between Logic and Set Theory

• There is a direct relationship between some of
  the operations of logic and operations on sets
• X any set S,T PX
• S ? T xX x? S ? x ? T x
• S ? T xX x? S ? x ? T x
• S \ T xX x? S ? x ? T x

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Summary of Quantifier

• A xT P
• for all x of type T, P is true.
• E xT P
• there exists an x of type T, such that P is
  true.
• E1 x T P
• there exists a unique x of type T, such that P
  is true.

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Summary of Quantifier

• D P t
• The set of ts declared by D where P is true.

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Mixing Quantifiers

• What is the precedence or associativity used. Is
  forAll(x),forAll(y) equivalent to
  forAll(y),forAll(x)?
• Assuming standard notation, they are read left to
  right two universal quantifiers following one
  another commute (just like multiplication
  abba).

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Mixing Quantifiers

• Likewise, if you have two existential
  quantifiers, they commute, because
• Exists(x),Exists(y), p(x,y)
• is true if and only if
• Exists(y),Exists(x), p(x,y)
• is true.

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Mixing Quantifiers

• But if you have mixed universal and existential
  quantifiers, you have to be careful. If you write
• A(x), E(y), p(x,y)
• that means that for each x, there exists a y,
  which may depend on x, for which p(x,y) is true.
• But if you write
• E(y), A(x), p(x,y)
• you are saying that there is a y which works for
  any given x (including yx), that make p(x,y)
  true.

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Mixing Quantifiers
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Mixing Quantifiers
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Exercises

• Let O(x,y) be the predicate (or propositional
  function) x is older than y. The domain of
  discourse consists of three people Garry, who is
  30 Ellen who is 20 and Martin who is 35. Write
  each of the following propositions in words and
  state whether they are true or false.

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Exercises
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Exercises
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Exercise
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Exercise