Inference in first-order logic

Inference in first-order logic Chapter 9 Chapter 9 1

Outline Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward and backward chaining Logic programming Resolution Chapter 9 2

Universal instantiation (UI) Every instantiation of a universally quantified sentence is entailed by it: v α Subst({v/g},α) for any variable v and ground term g E.g., x King(x) Greedy(x) Evil(x) yields King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(F ather(john)) Greedy(F ather(john)) Evil(F ather(john)). Chapter 9 3

Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: v α Subst({v/k},α) E.g., x Crown(x) OnHead(x, John) yields Crown(C 1 ) OnHead(C 1,John) provided C 1 is a new constant symbol, called a Skolem constant Another example: from x d(x y )/dy=x y we obtain d(e y )/dy=e y provided e is a new constant symbol Chapter 9 4

Existential instantiation contd. UI can be applied several times to add new sentences; the new KB is logically equivalent to the old EI can be applied once to replace the existential sentence; the new KB is not equivalent to the old, but is satisfiable iff the old KB was satisfiable Chapter 9 5

Reduction to propositional inference Suppose the KB contains just the following: x King(x) Greedy(x) Evil(x) King(John) Greedy(John) Brother(Richard, John) Instantiating the universal sentence in all possible ways, we have King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(John) Greedy(John) Brother(Richard, John) The new KB is propositionalized: proposition symbols are King(John), Greedy(John), Evil(John),King(Richard)etc. Chapter 9 6

Reduction contd. Claim: a ground sentence is entailed by new KB iff entailed by original KB Claim: every FOL KB can be propositionalized so as to preserve entailment Idea: propositionalize KB and query, apply resolution, return result Problem: with function symbols, there are infinitely many ground terms, e.g., F ather(f ather(f ather(john))) Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositional KB Idea: For n = 0 to do create a propositional KB by instantiating with depth-n terms see if α is entailed by this KB Problem: works if α is entailed, loops if α is not entailed Theorem: Turing(1936), Church(1936), entailment in FOL is semidecidable Chapter 9 7

Problems with propositionalization Propositionalization seems to generate lots of irrelevant sentences. E.g., from x King(x) Greedy(x) Evil(x) King(John) y Greedy(y) Brother(Richard, John) it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant With p k-ary predicates and n constants, there are p n k instantiations With function symbols, it gets nuch much worse! Chapter 9 8

Unification We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/john, y/john} works Unify(α,β) = θ if αθ=βθ p q θ Knows(John, x) Knows(John, Jane) Knows(John, x) Knows(y, OJ) Knows(John, x) Knows(y, M other(y)) Knows(John, x) Knows(x, OJ) Chapter 9 9

Unification We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/john, y/john} works Unify(α,β) = θ if αθ=βθ p q θ Knows(John, x) Knows(John, Jane) {x/jane} Knows(John, x) Knows(y, OJ) Knows(John, x) Knows(y, M other(y)) Knows(John, x) Knows(x, OJ) Chapter 9 10

Unification We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/john, y/john} works Unify(α,β) = θ if 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, M other(y)) Knows(John, x) Knows(x, OJ) Chapter 9 11

Unification We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/john, y/john} works Unify(α,β) = θ if 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, M other(y)) {y/john, x/m other(john)} Knows(John, x) Knows(x, OJ) Chapter 9 12

Unification We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/john, y/john} works Unify(α,β) = θ if 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, M other(y)) {y/john, x/m other(john)} Knows(John, x) Knows(x, OJ) f ail Standardizing apart eliminates overlap of variables, e.g., Knows(z 17,OJ) Chapter 9 13

Generalized Modus Ponens (GMP) p 1, p 2,..., p n, (p 1 p 2... p n q) qθ where p i θ=p i θ for all i p 1 is King(John) p 1 is King(x) p 2 is Greedy(y) p 2 is Greedy(x) θ is {x/john, y/john} q is Evil(x) qθ is Evil(John) GMP used with KB of definite clauses (exactly one positive literal) All variables assumed universally quantified Chapter 9 14

Soundness of GMP Need to show that p 1,..., p n, (p 1... p n q) = qθ provided that p i θ=p i θ for all i Lemma: For any definite clause p, we have p = pθ by UI 1. (p 1... p n q) = (p 1... p n q)θ=(p 1 θ... p n θ qθ) 2. p 1,..., p n = p 1... p n = p 1 θ... p n θ 3. From 1 and 2, qθ follows by ordinary Modus Ponens Chapter 9 15

Example knowledge base 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. Prove that Col. West is a criminal Chapter 9 16

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations: Chapter 9 17

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles Chapter 9 18

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West Chapter 9 19

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missiles are weapons: Chapter 9 20

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missiles are weapons: Missile(x) Weapon(x) An enemy of America counts as hostile : Chapter 9 21

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West x 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(W est) The country Nono, an enemy of America... Enemy(N ono, America) Chapter 9 22

Forward chaining algorithm function FOL-FC-Ask(KB, α) returns a substitution or false repeat until new is empty new {} for each sentence r in KB do (p 1... p n q) Standardize-Apart(r) for each θ such that (p 1... p n )θ = (p 1... p n )θ for some p 1,...,p n in KB q Subst(θ,q) if q is not a renaming of a sentence already in KB or new then do add q to new φ Unify(q,α) if φ is not fail then return φ add new to KB return false Chapter 9 23

Forward chaining proof American(West) Missile(M1) Owns(Nono,M1) Enemy(Nono,America) Chapter 9 24

Forward chaining proof Weapon(M1) Sells(West,M1,Nono) Hostile(Nono) American(West) Missile(M1) Owns(Nono,M1) Enemy(Nono,America) Chapter 9 25

Forward chaining proof Criminal(West) Weapon(M1) Sells(West,M1,Nono) Hostile(Nono) American(West) Missile(M1) Owns(Nono,M1) Enemy(Nono,America) Chapter 9 26

Properties of forward chaining Sound and complete for first-order definite clauses Datalog = first-order definite clauses + no functions (e.g., crime KB) FC terminates for Datalog in poly iterations: at most p n k literals May not terminate in general if α is not entailed This is unavoidable: entailment with definite clauses is semidecidable Chapter 9 27

Efficient forward chaining In the basic algorithm, there are three possible sources of inefficiency: The inner loop of the algorithm involves finding all possible unifiers such that the premise of a rule unifies with a suitable set of facts in the knowledge base. The algorithm rechecks every rule on every iteration to see whether its premises are satisfied, even if very few additions are made to the knowledge base on each iteration. The algorithm might generate many facts that are irrelevant to the goal. Chapter 9 28

Matching rules against known facts M issile(x) W eapon(x): Easy, in a suitably indexed knowledge base, this can be done in constant time per fact. Missile(x) Owns(Nono,x) Sells(West,x,Nono): conjunct ordering problem: minimum-remaining-values heuristic, ordering the conjuncts to look for missiles first if fewer missiles than objects are owned by Nono. In general, matching a definite clause against a set of facts is NP-hard. Bad news: in the algorithm, it is inside the inner loop. But: It is common to assume that both the sizes of rules and the arities of predicates are bounded by a constant. We can consider subclasses of rules for which matching is efficient. Similar to what we have seen in CSP. Eliminate redundant rule-matching attempts: Incremental forward checking Chapter 9 29

Incremental forward chaining AccordingtotheFCalforithm,ontheseconditeration,theruleMissile(x) Weapon(x) matches against Missile(M1) (again), and of course the conclusion matw eapon(m 1) is already known so nothing happens. Rule: Every new fact inferred on iteration t must be derived from at least one new fact inferred on iteration t 1. Incremental forward-chaining algorithm: at iteration t, we check a rule only if its premise includes a conjunct p i that unifies with a fact p i newly inferred at iteration t 1 (With suitable indexing, it is easy to identify all the rules that can be triggered by any given fact). Chapter 9 30

Irrelevant facts An intrinnsic problem of the FC approach: FC makes all allowable inferences based on the known facts, even if they are irrelevant to the goal at hand. Solution: Backward chaining. Chapter 9 31

Backward chaining algorithm Depth-first recursive proof search: space is linear in size of proof Chapter 9 32

Backward chaining example Criminal(West) American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 33

Backward chaining example Criminal(West) {x/west} Weapon(y) American(x) Sells(x,y,z) Hostile(z) American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 34

Backward chaining example Criminal(West) {x/west} American(West) { } Weapon(y) Sells(x,y,z) Hostile(z) American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 35

Backward chaining example Criminal(West) {x/west} American(West) Weapon(y) Sells(West,M1,z) Sells(x,y,z) Hostile(Nono) Hostile(z) { } Missile(y) American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 36

Backward chaining example Criminal(West) {x/west, y/m1} American(West) Weapon(y) Sells(West,M1,z) Sells(x,y,z) Hostile(Nono) Hostile(z) { } Missile(y) { y/m1} American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 37

Backward chaining example Criminal(West) {x/west, y/m1, z/nono} American(West) Weapon(y) Sells(West,M1,z) Hostile(z) { } { z/nono } Missile(y) { y/m1} Missile(M1) Owns(Nono,M1) American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 38

Backward chaining example Criminal(West) {x/west, y/m1, z/nono} American(West) { } Weapon(y) Sells(West,M1,z) { z/nono } Hostile(Nono) Missile(y) Missile(M1) Owns(Nono,M1) Enemy(Nono,America) { y/m1} { } { } { } American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono,M 1 ), Missile(M 1 ), Missile(x) Weapon(x) x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Enemy(x, America) Hostile(x), American(W est), Enemy(N ono, America) Chapter 9 39

Full first-order version: Resolution: brief summary l 1 l k, m 1 m n (l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n )θ where Unify(l i, m j )=θ. For example, Rich(x) U nhappy(x) Rich(Ken) U nhappy(ken) with θ = {x/ken} Apply resolution steps to CNF(KB α); complete for FOL Chapter 9 40

Conversion to CNF Every sentence of first-order logic can be converted into an inferentially equivalent CNF sentence. In particular, the CNF sentence will be unsatisfiable just when the original sentence is unsatisfiable, so we have a basis for doing proofs by contradiction on the CNF sentences. Example: Everyone who loves all animals is loved by someone: x [ y Animal(y) Loves(x,y)] [ y Loves(y,x)] 1. Eliminate biconditionals and implications x [ y Animal(y) Loves(x,y)] [ y Loves(y,x)] 2. Move 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)] Chapter 9 41

Conversion to CNF contd. 3. Standardize variables: each quantifier should use a different one x [ y Animal(y) Loves(x,y)] [ z Loves(z,x)] 4. Skolemize: a more general form of existential instantiation. Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables: x [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x) (Note the difference with EI: x [Animal(A) Loves(x, A)] Loves(B, x)) 5. Drop universal quantifiers: [Animal(F(x)) Loves(x, F(x))] Loves(G(x), x) 6. Distribute over : [Animal(F(x)) Loves(G(x), x)] [ Loves(x, F(x)) Loves(G(x), x)] Chapter 9 42

Resolution proof: definite clauses American(West) Missile(M1) Missile(M1) Owns(Nono,M1) Enemy(Nono,America) Enemy(Nono,America) Criminal(x) LHostile(z) LSells(x,y,z) LWeapon(y) LAmerican(x) > > > > Weapon(x) LMissile(x) > Sells(West,x,Nono) LMissile(x) LOwns(Nono,x) > > Hostile(x) LEnemy(x,America) > LSells(West,y,z) LWeapon(y) LAmerican(West) > > LHostile(z) > LSells(West,y,z) LWeapon(y) > LHostile(z) > LSells(West,y,z) > LHostile(z) > LMissile(y) LHostile(z) > LSells(West,M1,z) > > LHostile(Nono) LOwns(Nono,M1) LMissile(M1) > LHostile(Nono) LOwns(Nono,M1) LHostile(Nono) LCriminal(West) Chapter 9 43