Prolog

1
Prolog

• Prolog Programming in Logic
• Prolog programs consist of clauses and rules.
• Declarative programming
• Emphasis is on data and relations between
  objects.
• No familiar procedural controls.
• Built-in inference engine.

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Where is declarative programming useful?

• Data-driven programming tasks
• Verification
• Natural Language
• Expert Systems
• Relational Databases
• Diagnostic Systems

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Logical Operators
AND Conjunction
OR Disjunction
NOT Inversion
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Logical Operators
Implication
Equivalence
A-gtB is equivalent to !AvB
Alt-gtB is equivalent to A-gtB B-gtA (A B) v
(!A !B)
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Propositional Calculus

• Represent facts as symbols.
• Homer_likes_beer.
• Lisa_likes_tofu.
• !(Bart_likes_school)
• Bart_likes_school -gt Bart_does_homework.
• Lisa_likes_school Lisa_likes_music
• Easy, but overly verbose.
• No reuse.

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Predicate Calculus

• A predicate is a function that maps from zero or
  more objects to T,F.
• Likes(Homer, beer)
• !Likes(Bart, school)
• Likes(Bart, school) -gt Does(Bart, homework).
• Happy(Lisa).
• Gives(Moe, Homer, beer) -gt Happy(Homer).
• Predicates provides a syntactic shorthand for
  propositions.
• No extra representational power.

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Inference Rules

• AND-elimination
• If AB is true, then A is true.
• likes(Homer,beer) likes(Homer,
  food)-gtlikes(Homer,beer)
• OR-introduction
• If A is true, then AvB is true.
• likes(Homer,food)-gtlikes(Homer,food)v
  likes(Homer,work).
• Modus Ponens
• If A-gtB is true, and A is true, then B is true.
• likes(Lisa,school)-gtdoes(Lisa,homework)
  likes(Lisa,school)
• -gt Does(Lisa,homework)

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Quantification

• We would like to be able to express facts such
  as
• Everyone who likes school does their homework.
• No one who likes beef also likes tofu.
• Someone likes beef.
• This requires us to be able to quantify over
  variables in the domain.

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First-Order Logic

• First order logic allows quantification over
  objects.
• Universal quantification For all x
• ?x likes(x,school) -gt does(x, homework).
• Everyone who likes school does their homework.
• Existential quantification There exists an x
• ?x likes(x,tofu).
• There exists someone who likes tofu.

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Computational Issues

• Full FOL is too computationally difficult to work
  with.
• Nested quantifiers are semi-decidable.
• ?x?y?z likes(x,y) likes(y,z).
• Everyone likes a person who likes everyone.
• Implications with an AND in the consequent are
  exponentially hard.
• ?x likes(x, school) -gt does(x,homework)
  attends(x,class)
• Everyone who likes school does their homework
  and attends class.

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Horn clauses

• Horn clauses are implications of the form
• x y z -gt a
• Only a single term in the consequent.
• Horn clause inference is computationally
  tractable.
• Prolog uses Horn clauses.

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Knowledge Representation in Prolog

• Facts (unit clauses)
• likes(homer, beer).
• Constants are lower case.
• Names of relations are lower case.
• Conjunctions
• likes(homer,beer).
• likes(homer,food).
• Statement is represented as separate Prolog
  clauses.

lt- ends with a period
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Knowledge Representation in Prolog

• Rules
• does(lisa,homework) - likes(lisa,school).
• Equivalent to
• likes(lisa,school) -gtdoes(lisa,homework)
• Read as Lisa does homework if she likes
  school. or
• To prove that Lisa does homework, prove that she
  likes school.

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Variables

• Variables are all upper-case.
• Universal quantification is achieved by using
  variables in rules.
• does(X,homework) - likes(X,school).
• Everyone who likes school does their homework.
• To prove that someone does their homework, prove
  that they like school.