Formal Methods in Computer Science CS1502 Intro to Quantification

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Formal Methods in Computer ScienceCS1502Intro
to Quantification

• Patchrawat Uthaisombut
• University of Pittsburgh

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Exam result

• No students can do any problem.

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Exam result

• All students can do all problems.

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Exam result

• All students can do at least one problem on the
  exam.
• There is some problem that all students can do.
• There is some problem that not all students can
  do.
• There is some student that can do all the
  problems.
• There are no problems that no students can do.

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Goals

• To understand the limit of propositional logic
  (with /\, \/, ) and the need for first-order
  logic (with ?, ?).
• To learn the definition of well-formed formulas
  (wffs) and their use in constructing quantified
  sentences.
• To learn the notion of satisfaction of wffs and
  its use to systematically determine the truth
  value of quantified sentences.

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Concepts

• domain of discourse
• variables
• atomic well-formed formulas
• well-formed formulas
• quantifiers
• sentence
• translation between English and logic
• satisfaction
• truth value of a quantified sentence

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Sentences expressible in first-order logic

• Algorithm A1 correctly solves the sorting
  problem.
• Algorithm A2 has a worst-case running time of
  O(n2).
• Algorithm A3 is a 2-approximation algorithm.
• No algorithm can sort in time better than ?(n).
• No algorithm can correctly solve the Halting
  problem.
• Function f1 is continuous.
• Function f2 is one-to-one but not on-to.

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Domain of discourse

• Domain of discourse (or just domain)
• The set of objects of interest.
• If we have a specific world in mind, the domain
  often (but not necessarily) refers to the set of
  all objects in the world.

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Whats wrong with propositional logic

• Suppose we want to say All objects are cubes.
• What if the domain of discourse is infinite?
• What we can claim with propositional logic.
• 1 atomic sentence
• 1 property of 1 object Cube(b)
• 1 relationship of a fixed number of objects
• Between(a,b,c)

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• Connectives
• Finite number of relationships (properties), each
  with a fixed number of objects
• Conjunction, disjunction, conditional symbol,
• Cube(b) /\ (Tet(a) \/ (Dodec(c) ?LeftOf(a,b))

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• Objects must be named.
• Finite claims
• Can we use propositional logic to claim something
  like the following?
• All objects are small.
• No objects are tetrahedron.
• Some object is large.

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• Propositional logic is still possible for finite
  domain.
• If the domain of discourse is infinite, we cannot
  express claims like all objects are cubes. in
  propositional logic.

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Sentences expressible in first-order logic

• We will learn how to use them to make claims such
  as
• Algorithm A1 correctly solves the sorting
  problem.
• Algorithm A2 has a worst-case running time of
  O(n2).
• Algorithm A3 is a 2-approximation algorithm.
• No algorithm can sort in time better than ?(n)
• No algorithm can correctly solve the Halting
  problem.
• Function f1 is continuous.
• Function f2 is one-to-one but not on-to.

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Creating quantified sentences

• Redefining terms
• Defining well-formed formulas
• Redefining sentences

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Definition of terms (so far)

• Names are terms.
• also called basic terms.
• Applications of function symbols on terms are
  terms.
• Examples
• john
• father(john)
• mother(father(john))

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atomic sentences (so far)

• formed by a single predicatefollowed by one or
  more terms
• Maxs father is tall Tall(father(max))
• e is larger than bc Larger(e,bc)
• e is identical to a e a
• A sentence expresses a claim that is either true
  or false

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Definition of terms (modified)

• Names and variables are terms.
• also called basic terms.
• Applications of function symbols on terms are
  terms.
• Examples
• x
• father(x)
• mother(father(x))

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atomic well-formed formulas

• formed by a single predicate followed by one or
  more terms
• xs father is tall Tall(father(x))
• x is larger than bc Larger(x,bc)
• e is identical to x e x
• An atomic wff may not be a sentence and thus may
  not have truth value.
• Any variable that appears in an atomic wff is
  free or unbound.

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Complex Sentences (so far)

• Suppose P and Q are sentences of FOL (atomic or
  complex)
• Then the followings are also sentences of FOL
• ?P
• P?Q
• P?Q
• P ? Q
• P ? Q
• Examples
• ? Cube(a)
• LeftOf(a,b) ? LeftOf(b,c)
• ?(Large(a) ? Medium(a)) ? Tet(c)

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Well-formed formulas

• Suppose P and Q are wffs (atomic or complex), and
  x is a variable.
• Then the followings are also wffs of FOL
• ?P P?Q P?Q P ? Q P ? Q
• ?x P ?x P
• Any free occurrence of x in P is bound in ?x P.
  Same for ?x P.
• Examples
• ?x ?Cube(x)
• ?x LeftOf(x,b) ? ?y LeftOf(b,y)
• ?y (?x ?(Large(x) ? Medium(x)) ? Tet(y))

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Sentences (redefined)

• A sentence is a wff with no free variables.

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Examples wff or sentence

• Gave(x,y,z)
• ?y Gave(claire, y, max)
• ?y (Gave(claire, y, max) /\ (y scruffy))
• ?x Between(a,x,b)
• ?a Between(a,x,b)

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Examples

• ?y Cube(y) /\ Large(y)
• ?x (?y Cube(y) /\ FrontOf(y,x))
• ?x (?y ?z (LeftOf(y,z) \/ FrontOf(x,y)) /\ ?x
  Cube(x)
• ?x (?x (?x Cube(x) /\ Tet(x)) /\ Dodec(x))

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Translation

• English vs. FOL

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Universal quantifier ?

• everything, each thing, all things,
  anything
• ?x reads for every object x.
• ?x Cube(x)
• Everything is a cube.
• For every object x, x is a cube.
• ?x (Major(x,cs) ? Smart(x))
• Every CS major is smart.
• For every object x, if x majors in CS, x is
  smart.

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Existential quantifier ?

• some thing, at least one thing, a, an.
• ?x reads for some object x
• ?x Cube(x)
• Something is a cube.
• For some object x, x is a cube.
• ?x (Major(x,cs) /\ Smart(x))
• Some CS major is smart.
• For every object x, x majors in CS and x is
  smart.

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Nothing

• How to translate these?
• Nothing is both a cube and a tet.
• Nothing is P
• Not one thing is P Ex P(x)
• Everything is not P ?x P(x)

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Determining truth value

• Satisfaction of wff
• Truth values of quantified sentences
• How to do it systematically

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Truth value

• We know how to determine truth value of sentences
  in propositional logic.
• Complex sentences are built from smaller
  sentences.
• How to determine truth value of sentences in
  first-order logic?
• Complex sentences are built from smaller
  sentences.

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Domain of discourse

• Is the sentence ?x Cube(x) true or false?
• In general, a quantified sentence is true or
  false relative to some domain of discourse or
  domain of quantification.
• This is the set of objects that we are interested
  in.
• The domain of discourse could be
• the set of all possible objects.
• the set of objects in the world of interest.
• etc.

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Examples

• ?x Cube(x) is true in this world.
• ?x (Integer(x) ? Integer(x1)) is true when the
  domain of discourse contains everything.

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S(x)

• In propositional logic,
• we use single capital letters to represents
  sentences.
• A Tet(a) /\ Cube(b)
• P Tall(max) \/ Home(claire)
• S (Large(b) ? Small(c))
• In first-order logic,
• we use single capital letters followed by one or
  more variables to represents wffs with free
  variables.
• S(x) is a wff with x as the only free variable.
• A(x) Tet(x) /\ Cube(a)
• P(x,y) ?z Tet(z) \/ Cube(x) \/ Tet(y)

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Replacing variables with names

• Suppose S(x) is a wff
• Cube(x)
• Small(x) /\ Cube(x) /\ ?x Tet(x)
• Consider an object c in the domain of discourse.
• Replace all (free) occurrences of x by c. S(c)
  becomes a sentence.
• Cube(c)
• Small(c) /\ Cube(c) /\ ?x Tet(x)

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Satisfying a wff

• Object c satisfies S(x) if and only ifS(c) is
  true.

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Discussion satisfying a wff

• Object c satisfies S(x) if and only if S(c) is
  true.
• A sentence has a truth value.
• Everything (names, predicates) is determinate.
• A name refer to a specific object.
• A wff with free variables does not have a truth
  value
• A variable does not refer to any specific objec
• If we replace variables with some names,
  everything becomes determinate. Wff becomes a
  sentence, and we can determine truth value.

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Truth value of quantified sentences

• Suppose S(x) is a wff with x as the only free
  variable.
• Cube(x)
• Small(x) /\ Cube(x) /\ ?x Tet(x)
• ?x S(x) is a sentence (no free variable).
• ?x S(x) is a sentence (no free variable).

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Truth value of quantified sentences

• Truth value of a quantified sentence ?x S(x) is
  defined in term of satisfaction of the
  constituent wff S(x).
• ?x S(x) is true if and only if at least one
  object in the domain satisfies S(x).
• ?x S(x) is true if and only if every object in
  the domain satisfies S(x).

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Examples / Exercises

• ?x ?y SameRow(x,y)
• ?x ?y SameRow(x,y)
• ?x ?y SameRow(x,y)
• ?x ?y SameRow(x,y)