README - Syntax. Skolemization. Skolemization - Example 1. Skolemization - Example 1. Skolemization - Example 1. Skolemization - Example 2
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1 README - Syntax Skolemization Logical implication is commonly included in the syntax of first-order and propositional logical languages. The symbol used to denote logical implication differs from language to language. As such, in any particular languae you may see one of the following: In these notes we use. Conversion of sentences FOL to CNF requires skolemization. Skolemization: remove existential quantifiers by introducing new function symbols. How: For each existentially quantified variable introduce a n-place function where n is the number of previously appearing universal quantifiers. Special case: introducing constants (trivial functions: no previous universal quantifier). Skolemization - Example 1 Skolemization - Example 1 Every philosopher writes at least one book. x[philo(x) y[book(y) Write(x, y)]] Every philosopher writes at least one book. x[philo(x) y[book(y) Write(x, y)]] Eliminate Implication: x[ Philo(x) y[book(y) Write(x, y)]] Skolemization - Example 1 Skolemization - Example 2 Every philosopher writes at least one book. x[philo(x) y[book(y) Write(x, y)]] Eliminate Implication: x[ Philo(x) y[book(y) Write(x, y)]] Skolemize: substitute y by g(x) x[ Philo(x) [Book(g(x)) Write(x, g(x))]] All students of a philosopher read one of their teacher s books. x y[philo(x) StudentOf (y, x) z[book(z) Write(x, z) Read(y, z)]]
2 Skolemization - Example 2 Skolemization - Example 2 All students of a philosopher read one of their teacher s books. x y[philo(x) StudentOf (y, x) z[book(z) Write(x, z) Read(y, z)]] Eliminate Implication: x y[ Philo(x) StudentOf (y, x) z[book(z) Write(x, z) Read(y, z)]] All students of a philosopher read one of their teacher s books. x y[philo(x) StudentOf (y, x) z[book(z) Write(x, z) Read(y, z)]] Eliminate Implication: x y[ Philo(x) StudentOf (y, x) z[book(z) Write(x, z) Read(y, z)]] Skolemize: substitute z by h(x, y) x y[ Philo(x) StudentOf (y, x) [Book(h(x, y)) Write(x, h(x, y)) Read(y, h(x, y))]] Skolemization - Example 3 Skolemization - Example 3 There exists a philosopher with students. x y[philo(x) StudentOf (y, x)] There exists a philosopher with students. x y[philo(x) StudentOf (y, x)] Skolemize: substitute x by a and y by b Philo(a) StudentOf (b, a) Most General Unifier MGU - Example 1 Least specialized unification of two clauses. We can compute the MGU using the disagreement set D k = {e 1, e 2 }: the pair of expressions where two clauses first disagree. REPEAT UNTIL no more disagreement found MGU. IF either e 1 = V, a variable, and e 2 = t, a term not containing V (or vice versa) then: 1. σ k+1 = σ k {V = t} Compose the additional substitution. 2. S k+1 = S k {V = t} Apply the additional substitution 3. Find disagreement set D k+1. ELSE unification is not possible.
3 MGU - Example 1 MGU - Example 1 D 0 = {f (a), Y } σ = {Y = f (a)} D 0 = {f (a), Y } σ = {Y = f (a)} S 1 = {p(f (a), g(x )) ; p(f (a), f (a))} MGU - Example 1 MGU - Example 1 D 0 = {f (a), Y } σ = {Y = f (a)} S 1 = {p(f (a), g(x )) ; p(f (a), f (a))} D 1 = {g(x ), f (a)} D 0 = {f (a), Y } σ = {Y = f (a)} S 1 = {p(f (a), g(x )) ; p(f (a), f (a))} D 1 = {g(x ), f (a)} no unification possible!
4 D 2 = {g(a), Y }
5 D 2 = {g(a), Y } σ = {Z = a, X = h(g(a)), Y = g(a)} D 2 = {g(a), Y } σ = {Z = a, X = h(g(a)), Y = g(a)} S 3 = {p(a, h(g(a)), h(g(a))) ; p(a, h(g(a)), h(g(a)))} D 2 = {g(a), Y } σ = {Z = a, X = h(g(a)), Y = g(a)} S 3 = {p(a, h(g(a)), h(g(a))) ; p(a, h(g(a)), h(g(a)))} No disagreement σ = {Z = a, X = h(y ), Y = g(a)} is MGU σ = {X = Y }
6 σ = {X = Y } S 1 = {p(y, Y ) ; p(y, f (Y ))} σ = {X = Y } S 1 = {p(y, Y ) ; p(y, f (Y ))} D 1 = {Y, f (Y )} Full example problem σ = {X = Y } S 1 = {p(y, Y ) ; p(y, f (Y ))} D 1 = {Y, f (Y )} no unification possible! Given the following sentences, answer the question What is connected to the Galbraith building? using resolution with answer extraction: Connected is a binary symmetric relation. An object X is part of another object Y iff everything X is connected to, Y is also connected to. Room GB221 is part of Galbraith building. Room GB221 is connected to itself. Full example problem - Representation in FOL Full example problem - Representation in FOL (a) Represent these sentences in first order logic. Connected is a binary symmetric relation. An object X is part of another object Y iff everything X is connected to, Y is also connected to. Room GB221 is part of Galbraith building. Room GB221 is connected to itself. Connected is a symmetric relation. ( X, Y ) connected(x, Y ) connected(y, X )
7 Full example problem - Representation in FOL Full example problem - Representation in FOL An object X is part of another object Y iff everything X is connected to, Y is also connected to. ( X, Y ) (part(x, Y ) (( Z ) connected(z, X ) connected(z, Y ))) Room GB221 is part of Galbraith building. part(gb221, galbraith) Full example problem - Representation in FOL Full example problem - CNF Conversion Room GB221 is connected to itself. connected(gb221, gb221) (b) Convert the formulas to clausal form. Indicate any Skolem functions or constants used. ( X, Y ) connected(x, Y ) connected(y, X ) ( X, Y ) part(x, Y ) ( Z ) connected(z, X ) connected(z, Y ) part(gb221, galbraith) connected(gb221, gb221) Full example problem - CNF Conversion Full example problem - CNF Conversion ( X, Y ) connected(x, Y ) connected(y, X ) [ connected(x, Y ), connected(y, X )] ( X, Y ) part(x, Y ) ( Z ) connected(z, X ) connected(z, Y ) : [ part(x, Y ), connected(z, X ), connected(z, Y )] : [part(x, Y ), connected(f (X, Y ), X )] [part(x, Y ), connected(g(x, Y ), Y )]
8 Full example problem - CNF Conversion Full example problem - CNF Conversion part(gb221, galbraith) [part(gb221, galbraith)] connected(gb221, gb221) [connected(gb221, gb221)] Full example problem - Goal Full example problem - Goal (c) Convert the negation of the statement What is connected to the Galbraith building? to clause form (using an answer literal). FOL: ( X ) connected(galbraith, X ) (c) Convert the negation of the statement What is connected to the Galbraith building? to clause form (using an answer literal). FOL: ( X ) connected(galbraith, X ) negate goal!! ( X ) connected(galbraith, X ) Full example problem - Goal (c) Convert the negation of the statement What is connected to the Galbraith building? to clause form (using an answer literal). FOL: ( X ) connected(galbraith, X ) negate goal!! ( X ) connected(galbraith, X ) CNF with answer literal: ( X ) connected(galbraith, X ) [ connected(galbraith, X ), ans(x )] (d) Answer the question using resolution and answer extraction. Use the notation developed in class: every new clause must be labeled by the resolution step that was used to generate it. For example, a clause labeled R[4c, 1d]x = a, y = f(b) means that it was generated by resolving literal c of clause 4 against literal d of clause 1, using the MGU x = a, y = f(b). Our clauses:
9 8. R[7a, 1b] {S = galbraith, R = U} [ connected(a, galbraith), ans(a)] 8. R[7a, 1b] {S = galbraith, R = U} [ connected(a, galbraith), ans(a)] 9. R[8a, 2c] {V = A, U = galbraith} [ part(t, galbraith), connected(a, T ), ans(a)] 8. R[7a, 1b] {S = galbraith, R = U} [ connected(a, galbraith), ans(a)] 9. R[8a, 2c] {V = A, U = galbraith} [ part(t, galbraith), connected(a, T ), ans(a)] 10. R[9a, 5] {T = gb221} [ connected(a, gb221), ans(a)] 8. R[7a, 1b] {S = galbraith, R = U} [ connected(a, galbraith), ans(a)] 9. R[8a, 2c] {V = A, U = galbraith} [ part(t, galbraith), connected(a, T ), ans(a)] 10. R[9a, 5] {T = gb221} [ connected(a, gb221), ans(a)] 11. R[10a, 6] {A = gb221} [ans(gb221)]
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