Inference First Order Logic
Chastine Fatichah
Teknik Informatika
Institut Teknologi Sepuluh Nopember
November 2012
12/10/2012 1 Inference First Order Logic @ Kecerdasan Buatan
(KI092301)
Kecerdasan Buatan (KI092301)
Pokok Bahasan
• Mengubah First Order Logic ke Propotional
Logic
• Unification
• Generalized Modus Ponens
• Forward chaining
• Backward chaining
• Resolution
• Ringkasan
12/10/2012 2 Inference First Order Logic @
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Sejarah Reasoning
• 450b.c. Stoics propositional logic, inference
• 322b.c. Aristotle “syllogisms" (inference rules), quantifiers
• 1565 Cardano probability theory (propositional logic +
uncertainty)
• 1847 Boole propositional logic (again)
• 1879 Frege logic
• 1922 Wittgenstein proof by truth tables
• 1930 Gӧdel complete algorithm for FOL
• 1930 Herbrand complete algorithm for FOL (reduce to
propositional)
• 1931 Gӧdel complete algorithm for arithmetic
• 1960 Davis/Putnam “practical" algorithm for propositional logic
• 1965 Robinson “practical" algorithm for FOL resolution
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Pokok Bahasan
• Mengubah First Order Logic ke Propotional
Logic
• Unification
• Generalized Modus Ponens
• Forward chaining
• Backward chaining
• Resolution
• Ringkasan
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Universal instantiation (UI)
Semua kalimat dengan universal quantifier () meng-entail
semua instantiation-nya:
v α
Subst({v/g}, α)
untuk sembarang variabel v and ground term g
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x King(x) Greedy(x) Evil(x) meng-entail:
King(John) Greedy(John) Evil(John)
King(Richard) Greedy(Richard) Evil(Richard)
King(Father(John)) Greedy(Father(John)) Evil(Father(John))
Contoh
Existential instantiation (EI)
Untuk sembarang kalimat α, variabel v, and konstanta k
yang tidak muncul di knowledge-base:
v α
Subst({v/k}, α)
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x Crown(x) OnHead(x,John) meng-entail:
Crown(C1) OnHead(C1,John) Dengan syarat C1 adalah konstanta baru, disebut konstanta Skolem
Contoh
Existential instantiation (EI)
• Universal instantiation bisa digunakan berkali-
kali untuk menambahkan kalimat baru.
• KB yang baru logically equivalent dengan yang
lama.
• Existential instantiation cukup digunakan sekali
untuk menggantikan kalimat existential.
• KB yang baru tidak logically equivalent dengan
yang lama, tetapi satisfiable iff KB yang lama
juga satisfiable inferentially equivalent
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Mengubah ke Propositional Inference Misalkan KB berisi kalimat-kalimat berikut:
Dan jika kita mengambil semua kemungkinan instantiation dari kalimat universal, maka kita dapatkan KB sbb:
KB yang baru disebut propositionalized: proposition symbolnya adalah:
King(John), Greedy(John), Evil(John), King(Richard), dll.
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x King(x) Greedy(x) Evil(x) King(John) Greedy(John) Brother(Richard,John)
King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(John) Greedy(John) Brother(Richard,John)
Mengubah ke Propositional
Inference • Klaim : Setiap FOL KB dapat di propositional kan
sebagai entailment
• Klaim: Kalimat ground di entail dengan KB baru jika dan
hanya jika entail terhadap original KB
• Idea: propositionalize KB dan query, lalu gunakan
resolution
• Problem: dengan adanya function, ground terms menjadi
infinite.
• Contoh: Father(Father(Father(John)))
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Masalah dengan propositionalization • Propositionalization menghasilkan banyak kalimat irrelevant.
• Contoh: x King(x) Greedy(x) Evil(x)
King(John)
y Greedy(y)
Brother(Richard,John)
manusia bisa cepat mengerti kalau Evil(John), namun
propositionalization menghasilkan:
King(Richard) Greedy(Richard) Evil(Richard)
Greedy(Richard)
yang irrelevant
• Dengan p predicate k-ary dan n constant, ada p x nk instantiation!
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Pokok Bahasan
• Mengubah First Order Logic ke Propotional
Logic
• Unification
• Generalized Modus Ponens
• Forward chaining
• Backward chaining
• Resolution
• Ringkasan
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Unification • Inference : Substitution θ sedemikian hingga King(x) dan Greedy(x)
cocok dengan King(John) dan Greedy(y)
θ = {x/John,y/John}
• Unify(α,β) = θ jika αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane)
Knows(John,x) Knows(y,OJ)
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
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Unification • Inference : Substitution θ sedemikian hingga King(x) dan Greedy(x)
cocok dengan King(John) dan Greedy(y)
θ = {x/John,y/John}
• Unify(α,β) = θ jika αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane) {x/Jane}}
Knows(John,x) Knows(y,OJ)
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
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Unification • Inference : Substitution θ sedemikian hingga King(x) dan Greedy(x)
cocok dengan King(John) dan Greedy(y)
θ = {x/John,y/John}
• Unify(α,β) = θ jika αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane) {x/Jane}}
Knows(John,x) Knows(y,OJ) {x/OJ,y/John}}
Knows(John,x) Knows(y,Mother(y))
Knows(John,x) Knows(x,OJ)
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Unification • Inference : Substitution θ sedemikian hingga King(x) dan Greedy(x)
cocok dengan King(John) dan Greedy(y)
θ = {x/John,y/John}
• Unify(α,β) = θ jika αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane) {x/Jane}}
Knows(John,x) Knows(y,OJ) {x/OJ,y/John}}
Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}}
Knows(John,x) Knows(x,OJ)
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Unification • Inference : Substitution θ sedemikian hingga King(x) dan Greedy(x)
cocok dengan King(John) dan Greedy(y)
θ = {x/John,y/John}
• Unify(α,β) = θ jika αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane) {x/Jane}}
Knows(John,x) Knows(y,OJ) {x/OJ,y/John}}
Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}}
Knows(John,x) Knows(x,OJ) {fail}
• Standardizing apart variable menghilangkan overlap, contoh
Knows(z17,OJ)
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Pokok Bahasan
• Mengubah First Order Logic ke Propotional
Logic
• Unification
• Generalized Modus Ponens
• Forward chaining
• Backward chaining
• Resolution
• Ringkasan
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Generalized Modus Ponens
(GMP)
p1' is King(John) p1 is King(x)
p2' is Greedy(y) p2 is Greedy(x)
θ is {x/John,y/John} q is Evil(x)
q θ is Evil(John)
• GMP dengan KB yang berisi definite clauses: p1 p2 … pn q
• Semua variabel diasumsikan universally quantified
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p1', p2', … , pn', ( p1 p2 … pn q)
qθ
dimana pi'θ = pi θ untuk semua i
Inference Rule GMP
Soundness pada GMP
• Perlu menunjukkan bahwa
p1', …, pn', (p1 … pn q) ╞ qθ
dimana pi'θ = piθ untuk semua i
• Lemma: Untuk setiap definite clause p, terdapat p ╞ pθ dengan Universal instantiation (UI)
1. (p1 … pn q) ╞ (p1 … pn q)θ = (p1θ … pnθ qθ)
2. p1', …, pn' ╞ p1' … pn' ╞ p1'θ … pn'θ
3. Dari 1 dan 2, qθ mengikuti Modus Ponens
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Contoh Knowledge Base
Buktikan bahwa Colonel West is a criminal !!
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“ The law says that it is a crime for an American to sell weapons to
hostile nations. The country Nono, an enemy of America, has
some missiles, and all of its missiles were sold to it by Colonel
West, who is American. “
Contoh Knowledge Base • ... it is a crime for an American to sell weapons to hostile nations:
American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)
• Nono … has some missiles: x Owns(Nono,x) Missile(x):
Owns(Nono,M1) and Missile(M1)
• … all of its missiles were sold to it by Colonel West
Missile(x) Owns(Nono,x) Sells(West,x,Nono)
• Missiles are weapons:
Missile(x) Weapon(x)
• An enemy of America counts as "hostile“:
Enemy(x,America) Hostile(x)
• West, who is American …
American(West)
• The country Nono, an enemy of America …
Enemy(Nono,America)
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Pokok Bahasan
• Mengubah First Order Logic ke Propotional
Logic
• Unification
• Generalized Modus Ponens
• Forward chaining
• Backward chaining
• Resolution
• Ringkasan
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Forward chaining FOL
dengan GMP
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Algoritma forward chaining
Contoh Forward chaining FOL
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Contoh Forward chaining FOL
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Contoh Forward chaining FOL
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Forward chaining properti
• Sound dan complete untuk first-order definite clauses
• Datalog = first-order definite clauses + tanpa function
Time complexity FC pada Datalog polynomial
• Tapi pada kasus umum, bisa infinite loop kalau tidak di-
entail.
• Proses pattern matching pada premise NP-hard.
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Pokok Bahasan
• Mengubah First Order Logic ke Propotional
Logic
• Unification
• Generalized Modus Ponens
• Forward chaining
• Backward chaining
• Resolution
• Ringkasan
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Algoritma backward chaining
Backward chaining FOL
dengan GMP
SUBST(COMPOSE(θ1, θ2), p) = SUBST(θ2, SUBST(θ1, p))
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Contoh Backward chaining FOL
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Backward chaining properti
• Depth-first search:
• linear space complexity
• incomplete (infinite loops)
• inefficient (repeated subgoals)
• Prinsip dasar logic programming
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Logic programming: Prolog
Logic programming Ordinary programming
1. Identify problem Identify problem
2. Assemble information Assemble information
3. Tea break Figure out solution
4. Encode information in KB Program solution
5. Encode problem instance as facts Encode problem instance as data
6. Ask queries Apply program to data
7. Find false facts Debug procedural errors
(Harusnya) lebih mudah men-debug Capital(NewYork;US) daripada x = x + 2 !
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Prolog
• Appending two lists to produce a third:
append([],Y,Y).
append([X|L],Y,[X|Z]) :- append(L,Y,Z).
• query: append(A,B,[1,2]) ?
• answers: A=[] B=[1,2]
•
A=[1] B=[2]
A=[1,2] B=[]
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Konversi ke CNF
• Everyone who loves all animals is loved by someone: x [y Animal(y) Loves(x,y)] [y Loves(y,x)]
• 1. Eliminasi biconditionals dan implikasi x [y Animal(y) Loves(x,y)] [y Loves(y,x)]
• 2. Pindah inwards: x p ≡ x p, x p ≡ x p x [y (Animal(y) Loves(x,y))] [y Loves(y,x)]
x [y Animal(y) Loves(x,y)] [y Loves(y,x)]
x [y Animal(y) Loves(x,y)] [y Loves(y,x)]
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Konversi ke CNF.
3. Standarisasi variabel: x [y Animal(y) Loves(x,y)] [z Loves(z,x)]
4. Skolemize: bentuk umum pada existential instantiation. Setiap variabel existential diganti dengan Skolem function pada universally
quantified variables:
x [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x)
5. Hapus semua universal quantifiers: [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x)
6. Distribusi over : [Animal(F(x)) Loves(G(x),x)] [Loves(x,F(x)) Loves(G(x),x)]
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Resolution proof: definite clauses
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Sumber :
1.Slide perkuliahan Stuart Russell's (Berkeley) http://aima.cs.berkeley.edu/
2.Slide perkuliahan Sistem Cerdas Ruli Manurung (Universitas Indonesia)
http://www.cs.ui.ac.id/WebKuliah/IKI30320/
