ApplicationsPlanDetermining Logical EntailmentExampleAlternate MethodSlide 6Answer Extraction MethodSlide 8Slide 9Slide 10KinshipIs Art the Grandparent of Coe?Who is the Grandparent of Coe?Who Are the Grandchildren of Art?People and their Grandchildren?Variable Length ListsList MembershipList ConcatenationList ReversalNatural Language ProcessingSentence Generation as Answer ExtractionMap Coloring ProblemMap Coloring Problem Approach #1Map Coloring Problem Approach #2ResidueMethodSlide 27Multiple ResiduesDatabasesDatabases as Ground Atomic SentencesDatabase ViewsDatabase Query PlanningOptimizationsConstraintsConjunction EliminationConjunctive MinimizationDisjunctive MinimizationRecursionSlide 39Slide 40ApplicationsComputational Logic Lecture 12Michael Genesereth Autumn 20102PlanFirst Lecture - Resolution PreliminariesUnificationRelational Clausal FormSecond Lecture - Resolution PrincipleResolution Principle and FactoringResolution Theorem ProvingThird Lecture - Resolution Applications Theorem ProvingAnswer ExtractionResidue3Determining Logical EntailmentTo determine whether a set of sentences logically entails a closed sentence , rewrite {} in clausal form and try to derive the empty clause.4ExampleShow that (p(x) q(x)) and p(a) logically entail z.q(z).1. {¬p(x),q(x)} Premise2. {p(a)} Premise3. {¬q(z)} Goal4. {¬p(z)} 1,35. {} 2,45Alternate MethodBasic 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).6ExampleShow that (p(x) q(x)) and p(a) logically entail z.q(z).1. {¬p(x),q(x)} p(x)⇒ q(x)2. {p(a)} p(a)3. {¬q(z),goal} ∃z.q(z)⇒ goal4. {¬p(z),g oal} 1,35. {goal} 2,47Answer Extraction MethodAlternate Method for 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”.8ExampleGiven (p(x) q(x)) and p(a), find a term such that q() is true.1. {¬p(x),q(x)} p(x) ⇒ q(x)2. {p(a)} p(a)3. {¬q(z),goal(z)} q(z) ⇒ goal(z)4. {¬p(z),goal(z)} 1,35. {goal(a)} 2,49ExampleGiven (p(x) q(x)) and p(a) and p(b), find a term such that q() is true.€ 1. {¬ p(x),q(x)} p(x) ⇒ q(x)2. {p(a)} p(a)3. {p(b)} p(b)4. {¬ q(z), goal(z)} q(z) ⇒ goal(z)5. {¬ p(z), goal(z)} 1, 46. {goal(a)} 2,57. {goal(b)} 3,510ExampleGiven (p(x) q(x)) and (p(a) p(b)), find a term such that q() is true.1. {¬p(x),q(x)} p(x)⇒ q(x)2. {p(a),p(b)} p(a)∨ p(b)3. {¬q(z),goal(z)} q(z) ⇒ goal(z)4. {¬p(z),goal(z)} 1,35. {p(b),goal(a)} 2,46. {goal(a),g oal(b)} 4,511KinshipArt is the parent of Bob and Bud.Bob is the parent of Cal and Coe.A grandparent is a parent of a parent.p(art,bob)p(art,bud)p(bob,cal)p(bob,coe)p(x,y)∧ p(y,z) ⇒ g(x,z)12Is Art the Grandparent of Coe?1. {p(art,bob)} p(art,bob)2. {p(art,bud)} p(art,bud)3. {p(bob,cal)} p(bob,cal)4. {p(bob,coe)} p(bob,coe)5. {¬p(x,y),¬p(y,z),g(x,z)} p(x,y)∧ p(y,z)⇒ g(x,z)6. {¬g(art,coe),g oal} g(art,coe)⇒ goal7. {¬p(art,y),¬p(y,coe),goal} 5,68. {¬p(bob,coe),goal} 1,79. {goal} 4,813Who is the Grandparent of Coe?1. {p(art,bob)} p(art,bob)2. {p(art,bud)} p(art,bud)3. {p(bob,cal)} p(bob,cal)4. {p(bob,coe)} p(bob,coe)5. {¬p(x,y),¬p(y,z),g(x,z)} p(x,y) ∧ p(y,z)⇒ g(x,z)6. {¬g(x,coe),goal(x)} g(x,coe)⇒ goal(x)7. {¬p(x,y),¬p(y,coe),goal(x)} 5,68. {¬p(bob,coe),goal(art)} 1,79. {goal(art)} 4,814Who Are the Grandchildren of Art?1. {p(art,bob)} p(art,bob)2. {p(art,bud)} p(art,bud)3. {p(bob,cal)} p(bob,cal)4. {p(bob,coe)} p(bob,coe)5. {¬p(x,y),¬p(y,z),g(x,z)} p(x,y)∧ p(y,z)⇒ g(x,z)6. {¬g(art,z),goal(z)} g(art,z)⇒ g oal(z)7. {¬p(art,y),¬p(y,z),goal(z)} 5,68. {¬p(bob,z),goal(z)} 1,79. {¬p(bud,z),goal(z)} 2,710. {goal(cal)} 3,811. {goal(coe)} 4,815People and their Grandchildren?1. {p(art,bob)} p(art,b ob)2. {p(art,bud)} p(art,b ud)3. {p(bob,cal)} p(bob,cal)4. {p(bob,coe)} p(bob,coe)5. {¬p(x,y),¬p(y,z),g(x,z)} p(x,y)∧ p(y,z)⇒ g(x,z)6. {¬g(x,z),goal(x,z)} g(x,z)⇒ goal(x,z)7. {¬p(x,y),¬p(y,z),g oal(x,z)} 5,68. {¬p(bob,z),goal(art,z)} 1,79. {¬p(bud,z),goal(art,z)} 2,710. {goal(art,cal)} 3,811. {goal(art,coe)} 4,816Variable Length ListsExample[a,b,c,d]Representation as Termcons(a,cons(b,cons(c,cons(d,nil))))Shorthand(a . (b . (c . (d . nil))))Shorthand[a,b,c,d]a b c d17List MembershipMembership axioms:member(u, u . x)member(u, v . y) member(u, y)Membership Clauses:{member(u, u . x)}{member(u, v . y), member(u, y)}Answer Extraction for member(w, [a, b, c]){member(w, a.b.c. nil),goal(w)}{goal(a)}{member(w, b.c. nil)}{goal(b)}18List ConcatenationConcatenation Axioms:append(nil,y,y)append(w. x, y, w.z) append(x, y, z)Concatenation Clauses:{append(nil,y,y)}{append(w. x, y, w.z), append(x, y, z)}Answer Extraction for append([a,b],[c,d],z):{append(a.b.nil, c.d.nil, z), goal(z)}{append(b.nil, c.d.nil, z1), goal(a.z1)}{append(nil, c.d.nil, z2), goal(a.b.z2)}{goal(a.b.c.d.nil)}19List ReversalReversal Example:reverse([a,b,c,d], [d,c,b,a])Reversal Axioms:reverse(x, y) reverse2(x, nil, y)reverse2(nil, y, y)reverse2(w.x, y, z) reverse2(x, w.y, z)Answer Extraction for reverse([a,b,c,d],z):{reverse(a.b.c.d.nil, z), goal(z)} …{goal(d.c.b.a.nil)}20Natural Language ProcessingGrammar:S NP VPNP NounNP Noun and NounVP Verb NPNoun Harry | Ralph | MaryVerb hate | hatesLogical Form: S(z) NP(x) VP(y) append(x,y,z) NP(z) Noun(z) NP(z) NP(x) NP(y) append(x,and,x1) append(x1,y,z) VP(z) Verb(x) NP(y) append(x,and,x1) append(x1,y,z) Noun(Harry) Noun(Ralph) Noun(Mary) Verb(hate) Verb(hates)21Sentence Generation as Answer
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