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Applications Computational LogicLecture 11 Michael Genesereth Spring 2004

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2 Plan First Lecture: Pattern Matching Unification Second Lecture: Relational Clausal Form Resolution Principle Resolution Theorem Proving Third Lecture: True or False Questions Fill in the Blank Questions Residue Fourth Lecture: Strategies to enhance efficiency

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3 Determining Logical Entailment To determine whether a set of sentences logically entails a closed sentence, rewrite { } in clausal form and try to derive the empty clause.

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4 Example Show that (p(x) q(x)) and p(a) logically entail z.q(z).

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5 Alternate Method Basic Method: To determine whether a set of sentences logically entails a closed sentence, rewrite { } in clausal form and try to derive the empty clause. Alternate Method: To determine whether a set of sentences logically entails a closed sentence, rewrite { goal} in clausal form and try to derive goal. Intuition: The sentence ( goal) is equivalent to the sentence ( goal).

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6 Example Show that (p(x) q(x)) and p(a) logically entail z.q(z).

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7 Answer Extraction Method Alternate Method for Determining Logical Entailment: To determine whether a set of sentences logically entails a closed sentence, rewrite { goal} in clausal form and try to derive goal. Method for Answer Extraction: To get values for free variables 1,…, n in for which logically entails, rewrite { goal( 1,…, n )} in clausal form and try to derive goal( 1,…, n ). Intuition: The sentence (q(z) goal(z)) says that, whenever, z satisfies q, it satisfies the goal.

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8 Example Given (p(x) q(x)) and p(a), find a term such that q( ) is true.

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9 Example Given (p(x) q(x)) and p(a) and p(b), find a term such that q( ) is true.

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10 Example Given (p(x) q(x)) and (p(a) p(b)), find a term such that q( ) is true.

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11 Kinship Art is the parent of Bob and Bud. Bob is the parent of Cal and Coe. A grandparent is a parent of a parent.

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12 Is Art the Grandparent of Coe?

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13 Who is the Grandparent of Coe?

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14 Who Are the Grandchildren of Art?

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15 People and their Grandchildren?

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16 Map Coloring Problem Color the map with the 4 colors red, green, blue, and purple such that no two regions have the same color.

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17 Acceptable Neighbors

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18 Question

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19 Residue Problem: Given the database shown below, find conditions under which s(x) is true expressed in terms of m, n, and o. p(x) q(x) s(x) m(x) p(x) n(x) q(x) o(x) r(x) Residue: m(x) n(x) In other words: m(x) n(x) s(x)

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20 Method To obtain the residue for given in terms of 1,..., n, rewrite { } in clausal form and try to derive a clause with constants restricted to 1,..., n. The residue is the negation of any clause so derived.

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21 Example

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22 Multiple Residues

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23 Databases

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24 Databases as Ground Atomic Sentences

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25 Database Views Father: p(x,y) male(x) f(x,y) Mother: p(x,y) female(x) m(x,y) Grandfather: f(x,y) p(y,z) gf(x,z) Grandmother: m(x,y) p(y,z) gm(x,z)

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26 Database Query Planning Query: gf(u,v) Tables: male, female, p. Reformulated Query: p(u,y) male(u) p(y,v)

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27 Optimizations Non-Optimal Query:Better Query: p(x,y) p(y,z) p(y,x) False f(x,y) p(y,z) p(x,y) f(x,y) p(y,z) f(x,y) m(x,y) p(x,y) p(x,y)

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28 Constraints Parenthood is antisymmetric: p(x,y) p(y,x) Fathers are parents: f(x,y) p(x,y) Mothers are parents: m(x,y) p(x,y)

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29 Conjunction Elimination Example: p(x,y) p(y,z) p(y,x) False Constraint: p(x,y) p(y,x){ p(x,y), p(y,x)} Technique: Check conjunction for contradiction. If found, delete the entire conjunction. Clausal Form of x. y. z.(p(x,y) p(y,z) p(y,x)): {p(a,b)} {p(b,c)} {p(b,a)}

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30 Conjunctive Minimization Example: f(x,y) p(y,z) p(x,y) f(x,y) p(y,z) Constraint: f(x,y) p(x,y) { f(x,y),p(x,y)} Technique: Check if conjunct is implied by others. If so, delete. f(x,y) p(y,z) p(x,y) f(x,y) p(x,y) p(y,z) p(y,z) p(x,y) f(x,y) Clausal Form of x. y. z.(f(x,y) p(y,z) p(x,y)): {f(a,b)} {p(b,c)} { p(a,b)}

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31 Disjunctive Minimization Example: f(x,y) m(x,y) p(x,y) p(x,y) Constraints: f(x,y) p(x,y) { f(x,y),p(x,y)} Technique: Check if disjunct logically entails others. If so, delete the disjunct. f(x,y) m(x,y) p(x,y) m(x,y) f(x,y) p(x,y) p(x,y) f(x,y) m(x,y) Clausal Form of x. y. z.(f(x,y) m(x,y) p(x,y)): {f(a,b)} { m(a,b)} { p(a,b)}

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32 Recursion Suppose we are given three tables p, male, and female. The ancestor relation a is the transitive closure of the parent relation p. How do we rewrite a(u,v) in terms of p?

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33 Recursion

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34 Recursion Infinitely many residues! Solution: Include recursive relations among the residue relations. Let database handle recursive queries.

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