Logic Programming

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Logic Programming
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Announcements

• Test No. 5 This Friday
• Since Last Exam 50
• Event-driven programming
• Logic programming
• Previous lessons 50
• Emphasis on concepts that should carry over
  beyond the course
• Hint Review missed questions in previous tests
• Multiple Choice
• Closed book closed notes
• Homework 8 is also due

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Logic Programming

• Algorithm logic control
• Lot of effort in programming is to manage control
• Logic programming separates them
• Use facts and rules to represent information
• Use deduction to answer queries

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Logic Programming

• (define overlap?
• (lambda (set1 set2)
• (cond
• ((null? set1) f)
• (else (or
• (member? (car set1) set2)
• (overlap? (cdr set1) set2)))))))

overlap(X,Y) - member(M,X), member(M,Y).
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Declarative Programming

• Programmer declares the goals of the computation
• Not the detailed algorithm by which the goals can
  be reached
• Two major domains
• Databases, e.g. SQL
• Artificial intelligence (Prolog)
• Logic Programming
• Express the computation using mathematical logic
• Motivated by natural language processing and
  automatic theorem proving

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Logic Programming

• Prolog is the principal language
• Based on two powerful principles
• Resolution
• Unification
• Two interesting features
• Nondeterminism find multiple solutions
• Backtracking part of the language

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Propositional Logic

• Provides the formal foundation for Boolean
  expressions in languages
• Basic Rules

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Propositional Logic

• Precedence of operators
• Negation, conjunction, disjunction, implication,
  equivalence
• Propositions provide symbolic representation for
  logic expressions
• Can be interpreted as true or false.

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Propositional LogicExample

• p Mary speaks Russian
• q Bob speaks Russian
• p?q Either Mary or Bob speak Russian
• p?q Both Mary and Bob speak Russian
• r Bob and Mary can communicate with each other
• p?q?r true

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Predicate Logic

• Includes all of Propositional logic
• Includes additional entities
• Variables in various domains
• Boolean valued functions
• Quantifiers

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Predicate LogicExamples
True if x is between 0 and 1 false otherwise
True if both speak Russian means x talks with y
True if everybody speaks Russian
True if for every person can speak a language
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Properties of Predicate Logic
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Predicate Logic

• Tautology Propositions that are true for all
  possible values of variables
• p?p
• Satisfiability Predicates that are true for some
  particular assignment of variables
• speaks(x,Russian) (not valid)
• Validity Predicates that are true for all
  possible assignments of values to the variables
• even(x)?odd(x)

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Horn Clauses

• Consists of two parts
• Head (predicate)
• Body (a set of predicates)

• h is true if the predicates in the body are all
  true simultaneously
• i.e. logical conjunction

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Horn Clauses

• Example
• snowing(city)?precipitation(city),freezing(city)
• Correspondence between Horn clauses and
  predicates
• precipitation(city),freezing(city)? snowing(city)
• Equivalent to
• ?(precipitation(city)?freezing(city))?
  snowing(city)
• ?precipitation(city) ?? freezing(city) ?
  snowing(city)

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Horn Clauses

• Underlying representation for Prolog
• Any Horn clause can be written as a predicate
• Not all predicates can be written as Horn clauses
• Systematic procedure exists to convert a
  predicate into a Horn clause
• Can be automated

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Horn Clauses

• Eliminate implication using the following rule
• p?q is equivalent to ?p?q
• Move negation inward in p using deMorgan and
  quantification properties so that only individual
  terms are negated
• ?(p?q) is equivalent to ?p??q

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Horn Clauses

• Eliminate existential quantifiers
• Replace the existentially quantified variable by
  a unique constant
• ?x p(x) is replaced by p(c), c is a constant
• Called skolemization
• Move all universal quantifiers to the beginning
  of p and then drop them
• Avoid naming conflicts
• Does not change the meaning of p

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Horn Clauses

• Use Distributive, Associative and Commutative
  properties to convert p to a conjunctive normal
  form
• Conjunction of disjunctions
• Convert embedded disjunctions to implications in
  the Horn clause format

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Horn ClausesExample
literate(x)?writes(x) literate(x)
?reads(x,y),book(y)
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Horn ClausesExample

• Conversion of a predicate to CNF does not
  guarantee a set of Horn clauses
• Example
• ?x(literate(x)?reads(x)?writes(x))
• ?literate(x)?reads(x)?writes(x)
• literate(x)?reads(x)?writes(x)
• Not a single term (head) in the right