Intelligent Systems, part III Intelligent Agents and Knowledge Representation

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Intelligent Systems, part IIIIntelligent Agents
and Knowledge Representation

• Lecture 1 Logic and Proofs
• Leon van der Torre

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Slides

• Slides can be found on the internet
• These slides are based on MIT open courseware,
  by S. Devadas and E. Lehmann

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What do these sentence mean?

• You may have cake or you may have ice cream.
• If pigs can fly, then you can understand the
  Chernoff bound.
• If you can solve any problem we come up with,
  then you get a 20 for the course.
• Every American has a dream.

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Propositions and Boolean variables

• Proposition is a statement that is true or false.
  
• NotWhats a surjection, again? or Learn
  logarithms!
• All Greeks are human, All humans are mortal.
• 2  3  5
• a4 b4  c4  d4 has no solution where a, b, c
  are positive integers. (Euler)
• Every even integer greater than 2 is the sum of
  two primes. (Goldbach)

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Combining propositions NOT

• If all humans are mortal and all Greeks are
  human, then all Greeks are mortal.
• Truth tables if P denotes an arbitrary
  proposition, then the validity of the proposition
  not P is defined by
• First row when P is true (T), not P is false
  (F).
• Second line when P is false, not P is true.

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Combining propositions AND, OR

• True / false for each possible setting of
  variables.
• If a mathematician says You may have cake or
  your may have ice cream, then you can have both.

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Combining propositions Implies

• If Goldbachs Conjecture is true, then x 0 for
  every real number x.
• If pigs fly, then you can understand the
  Chernoff bound.
• If the moon is white, then the moon is made of
  white cheddar.
• An implication is true when the if-part is false
  or the then-part is true.

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Combining propositions If and only if

• P if and only if Q P and Q logically
  equivalent
• That is, either both are true or both are false.
• x2 - 4  0 if and only if x 2
• Doesnt arise in ordinary speech, abbreviated
  iff.

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Propositional Logic in Computer Programs

• if ( x gt 0 (x lt 0 y gt 100) )
• (further instructions)
  or, and.
• A or ((not A) and B)  for A
  x gt0, B y gt 100.
• if ( x gt 0 y gt 100 )
• (further instructions)
• Chip designers minimize physical devices on a
  chip

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Two Cryptic Notations

• If P and not Q, then R (P ? ?Q) ? R

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Are these sentences saying the same?

• If I am hungry (P) , then I am grumpy (Q).
• If I am not grumpy, then I am not hungry.

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Are these sentences saying the same?

• If I am hungry (P) , then I am grumpy (Q).
• If I am grumpy, then I am hungry.
• An implication is logically equivalent to its
  contrapositive, but not to its converse.

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Are these sentences saying the same?

• These two statements together
• If I am grumpy, then I am hungry.
• If I am hungry, then I am grumpy.
• I am grumpy if and only if I am hungry.

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Logic exercise

• In computer chip design, we often use nand
• Using nand and not, find an equivalent expression
  for A and B, A or B, and A implies B

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Tautologies

• P P is a tautology P is always true
• (P ? Q) ? (Q ? P)
• P ? (Q ? R) ? (P ? Q) ? R
• P ? (Q ? R) ? (P ? Q) ? (P ? R)
• (P ? P) ? P
• ??P ? P
• ? (P ? Q) ? ?P ? ?Q

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SAT

• Proposition is satisfiable if some setting of
  variables makes it true.
• P ? Q is satisfiable because it is true when
  P is true and Q is false.
• P ? P is not satisfiable because it is false for
  both settings of P.
• How about (P?Q?R)?(?P??Q)?(?P??R)?(?R??Q) ?
• Deciding whether a proposition is satisfiable is
  called SAT.
• Construct a truth table and check whether or not
  a T ever appears.
• Not very efficient a proposition with
  n variables has a truth table with 2n lines.
  With just 30 variables, thats already over a
  billion!
• Is there an efficient solution to SAT? No one
  knows
• An efficient solution to SAT would immediately
  imply efficient solutions to many, many other
  important problems involving packing, scheduling,
  routing, and circuit verification, and worldwide
  chaos
• Decrypting coded messages would also become an
  easy task (for most codes). Online financial
  transactions would be insecure and secret
  communications could be read by everyone.

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Predicates

• A predicate is a proposition whose truth depends
  on the value of one or more variables. For
  example,
• n is a perfect square
• The predicate is true for n 4 since 4 is a
  perfect square, but false for n 5 since 5 is not
  a perfect square.
• Function-like notation.
• P(n) n is a perfect square
• P(4) is true, and P(5) is false.
• Note If P is a predicate, then P(n) is either
  true or false, depending on the value of n. If
  P is an ordinary function, like n2  1, then P(n)
  is a numerical quantity.

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Quantifying a Predicate

• Two ways to use variables
• x2  0 is always true when x is a real number.
• x2 - 9 0 is only sometimes true (when x  
  3).
• Always True a universal quantification
• For all x, x2  0. or x2  0 for every x.
• Sometimes True an existential quantification
• There exists an x such that x2 - 9 0.
• x2 - 9 0  for some x.
• x2 - 9 0  for at least one x.

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Natural language may be confusing

• If you can solve any problem we come up with,
  then you get a 20 for the course.
• Can be interpreted as universal or existential
  quantification
• you can solve every problem we come up with
• you can solve at least one problem we come up
  with
• Note that quantified statements are themselves
  propositions and can be combined with and, or,
  implies, etc. just like any other proposition.

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More Cryptic Notation

• Predicate P(n) is true for all values of n in
  some set S
• ?n ?S P(n)  for all n in S, P(n) is true.
• Predicate P(n) is true for at least one value of
  n in S
• ?p?S P(n) There exists an n in S such that P(n)
  is true.
• ? and ? are always followed by a variable and
  predicate.
• Let P be the set of problems we come up with,
  S(x) be the predicate You can solve problem x,
  and A be the proposition, You get an 20 for the
  course.
• (?x ?P S(x)) ? A you can solve every problem we
  come up with
• (?x ?P S(x)) ? A you can solve at least one
  problem we come up with

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Mixing of Quantifiers

• Every even integer greater than 2 is the sum of
  two primes. (Goldbach)
• For every even integer n greater than 2, there
  exist primes p and q such that npq.
• Let E be the set of even integers greater than 2,
  and let P be the set of primes
• ?n ?E ? p?P ? q?P npq

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Order of Quantifiers

• Every American has a dream.
• Let A be the set of Americans, D be the set of
  dreams, and predicate H(a,d) be American a has
  dream d..
• There is a single dream every American shares
• ?d ? D ?a ? A H(a,d)
• Every American has an individual dream
• ?a ? A ?d ? D H(a,d)
• Swapping quantifiers in Goldbachs Conjecture
• ?p ? P ?q ? P ?n ? E npq

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Negating Quantifiers

• The following two sentences mean the same thing
• It is not the case that everyone likes to
  snowboard.
• There exists someone who does not like to
  snowboard.
• Similarly, these sentences mean the same thing
• There does not exist anyone who likes skiing
  over magma.
• Everyone dislikes skiing over magma.
• Moving a not across a quantifier changes
  quantifier.
• ??x P(x) ? ?x ?P(x) ??x P(x) ? ?x
  ?P(x)

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Logic test

• Prove using truth tables that ? (P ? Q) ? ?P
  ? ?Q
• Let Cn(S) p S p Give ten distinct
  elements of Cn(p/\q)
• Formalize in first order logic
• "All red cars are owned by students
• "Everybody, who owns a red car, is rich."
• "At least two students own a red car."
• "Fred owns a blue car."
• "Every student living in L. owns at least one
  blue car
• "Some students are rich."