Inference in first-order logic

Transcription

1 Inference in first-order logic CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2014 Soleymani Artificial Intelligence: A Modern Approach, 3 rd Edition, Chapter 9

2 Outline Reducing first-order inference to propositional inference Potentially great expense Lifting inference (direct inference in FOL) Unification Generalized Modus Ponens KB of Horn clauses Forward chaining Backward chaining General KB Resolution 2

3 FOL to PL FOL to PL conversion First order inference by converting the knowledge base to PL and using propositional inference. How to remove universal and existential quantifiers? Universel Instantiation Existential Instantiation 3

4 Universal instantiation (UI) Every instantiation of a universally quantified sentence is entailed by it: v α Subst( v g, α) Example, α: x King(x) Greedy(x) Evil(x) Subst x Jon, α King(Jon) Greedy(Jon) Evil(Jon) Subst( x Ricard, α) King(Ricard) Greedy(Ricard) Evil(Ricard) Subst( x Fater(Jon), α) King(Fater(Jon)) Greedy(Fater(Jon)) Evil(Fater(Jon)) 4

5 Existential instantiation (EI) For any sentence α, variable v, and a new constant symbol k: v α Subst( v k, α) Example, α: x Crown(x) OnHead(x, Jon) Subst x C 1, α Crown(C 1 ) OnHead(C 1, Jon) C 1 is a new constant symbol, called a Skolem constant EI can be applied one time (existentially quantified sentence can be discarded after it) 5

6 Reduction to propositional inference: Example KB of FOL sentences: x King x Greedy x Evil x King Jon Greedy Jon Broter Ricard, Jon x Crown(x) OnHead(x, Jon) A universally quantified sentence is replaced by all possible instantiations & an existentially quantified sentence by one instantiation: King(Jon) Greedy(Jon) Evil(Jon) King(Ricard) Greedy(Ricard) Evil(Ricard) King(Jon) Greedy(Jon) Broter(Ricard, Jon) Crown C 1 OnHead C 1, Jon 6

7 Propositionalization Every FOL KB and query can be propositionalized Algorithms for deciding PL entailment can be used Problem: infinitely large set of sentences Infinite set of possible ground-term substitution due to function symbols Solution: e.g., Fater( Fater(Fater(Jon))) Theorem (Herbrand, 1930): If a sentence α is entailed by an FOL KB, it can be entailed by a finite subset of its propositionalized KB 7 for n = 0 to do Generate all instantiations with depth n nested symbols if α is entailed by this KB then return true

8 Propositionalization (Cont.) Problem: When procedure go on and on, we will not know whether it is stuck in a loop or the proof is just about to pop out Theorem (Turing-1936, Church-1936): Deciding entailment for FOL is semidecidable Algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every non-entailed sentence. 8

9 Propositionalization is inefficient Generates lots of irrelevant sentences Example: King Ricard Greedy Ricard Evil Ricard is irrelevant. KB x King x Greedy x Evil x King Jon Greedy Jon Broter Ricard, Jon Query Evil(x) p k-ary predicates and n constants p n k instantiations 9

10 Lifting & Unification Lifting: raising inference rules or algorithms from ground (variable-free) PL to FOL. E.g., generalized Modus Ponens Unification: Lifted inference rules require finding substitutions that make different expressions looks identical. Instantiating variables only as far as necessary for the proof. All variables are assumed universally quantified. 10

11 Unification: Simple example Example: Infer immediately when finding a substitution θ such that King(x) and Greedy(x) match King(Jon) and Greedy(y) KB x King x Greedy x Evil x King Jon y Greedy y Broter Ricard, Jon Query Evil(x) 11

12 Unification: examples UNIFY p, q = θ where Subst θ, p = Subst(θ, q) p Knows(Jon, x) Knows(Jon, x) Knows(Jon, x) Knows(Jon, x) q Knows(Jon, Jane) Knows(y, Bill) Knows(y, Moter(y)) Knows(x, Elizabet) θ *x/jane+ *x Bill, y/jon+ *x Moter(Jon), y/jon+ fail 12

13 Unification: examples UNIFY p, q = θ where Subst θ, p = Subst(θ, q) p Knows(Jon, x) Knows(Jon, x) Knows(Jon, x) Knows(Jon, x) q Knows(Jon, Jane) Knows(y, Bill) Knows(y, Moter(y)) Knows(x, Elizabet) θ *x/jane+ *x Bill, y/jon+ *x Moter(Jon), y/jon+ fail 13

14 Unification: complications Complications Two sentences using the same variable name Standardizing apart: renaming the variables causing name clashes in one of sentences More than one unifier Most General Unifier (MGU) p Knows(Jon, x) Knows(Jon, x) q Knows(x, Elizabet) Knows(x 2, Elizabet) θ fail *x Elizabet, x 2 /Jon+ 14

15 Unification (MGU) UNIFY( Knows Jon, x, Knows y, z ) θ 1 = *y/jon, x/z + θ 2 = *y/jon, x/jon, z/jon+ θ 1 is more general There is a single most general unifier (MGU) that is unique up to renaming of variables. 15

16 Generalized Modus Ponens (GMP) For atomic sentences p i, p i, q & Subst θ, p i = Subst θ, p i for all i p 1, p 2,, p n, (p 1 p 2 p n q) Subst(θ, q) p 1 : King(Jon) p 1 : King(x) p 2 : Greedy(y) p 2 : Greedy(x) q: Evil(x) θ = *x/jon, y/jon+ Subst(θ, q) is Evil(Jon) KB King x Greedy x Evil x King Jon Greedy y Query Evil(x) 16

17 GMP is sound p 1, p 2,, p n, (p 1 p 2 p n q) Subst(θ, q) Universal instantiation: p Subst(θ, p) 1) p 1 p n Subst θ, p 1 Subst(θ, p n ) 2) p 1 p 2 p n q Subst θ, p 1 Subst θ, p n Subst(θ, q) 3) Subst(θ, q) follows by Modus Ponens. 17

18 Unification algorithm function UNIFY(x, y, θ) returns a substitution to make x and y identical inputs: x, a variable, a constant, list, or compound expression y, a variable, a constant, list, or compound expression θ, the substitution built up so far (optional, defaults to empty) if θ = failure then return failure else if x = y then return θ else if VARIABLE? (x) then return UNIFY_VAR(x, y, θ) else if VARIABLE? (y) then return UNIFY_VAR(y, x, θ) else if COMPOUND? (x) and COMPOUND? (y) then return UNIFY(x. ARGS, y. ARGS, UNIFY(x. OP, y. OP, θ)) else if LIST? (x) and LIST? (y) then return UNIFY(x. REST, y. REST, UNIFY(x. FIRST, y. FIRST, θ)) else return failure 18

19 Unification algorithm function UNIFY_VAR(var, x, θ) returns a if *var/val+ θ then return UNIFY(val, x, θ) else if *x/val+ θ then return UNIFY(var, val, θ) else if OCCUR_CHECK? (var, x) then return failure else return add *var/x+ to θ 19

20 KB of definite clauses Definite clause: disjunctions of literals of which exactly one is positive. Atomic: P 1, P 2, Implication: P 1 P 2 P k P k+1 First order definite clause examples: King x Greedy x Evil x King(Jon) Broter(Jon, Ricard) 20

21 KB of first order definite clauses: Example 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. Question: Prove that Colonel West is a criminal 21

22 KB of first order definite clauses: Example 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. American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) 22

23 KB of first order definite clauses: Example 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. Enemy(Nono, America) 23

24 KB of first order definite clauses: Example 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. x Owns(Nono, x) Missile(x) Owns(Nono, M 1 ) Missile(M 1 ) 24

25 KB of first order definite clauses: Example 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. Missile(x) Owns(Nono, x) Sells(West, x, Nono) 25

26 KB of first order definite clauses: Example 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. American(West) 26

27 KB of first order definite clauses: Example 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. We need to know missiles are weapons: We need to know an enemy of America counts as hostile : 27 Missile(x) Weapon(x) Enemy(x, America) Hostile(x)

28 KB of first order definite clauses: Example American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Enemy(Nono, America) Owns(Nono, M 1 ) Missile(M 1 ) Missile(x) Owns(Nono, x) Sells(West, x, Nono) American(West) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) 28

29 FC algorithm function FOL_FC_ASK(KB, α) returns a substitution or false 29 inputs: KB, a set of first-order definite clauses α, the query, an atomic sentence repeat until new = *+ new = *+ for each rule in KB do (p 1 p n q) STANDARDIZE_VARS(rule) for each θ such that SUBST θ, p 1 p n = SUBST(θ, p 1 p n ) q SUBST(θ, q) for some p 1,, p n in KB if q does not unify with some sentence already in KB or new then add new to KB return false add q to new φ UNIFY(q, α) if φ is not fail then return φ

30 FC: example Known facts 1) American West 2) Owns Nono, M 1 Missile M 1 3) Enemy(Nono, America) 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 30

31 FC: example 1) Missile x Owns Nono, x Sells West, x, Nono *x/m 1 + 2) American x Weapon y Sells x, y, z Hostile z Criminal x 3) Missile x Weapon x 4) Enemy(x, America) Hostile(x) *x/m 1 + *x/nono+ 31

32 FC: example 1) Missile x Owns Nono, x Sells West, x, Nono 2) American x Weapon y Sells x, y, z Hostile z Criminal x 3) Missile x Weapon x 4) Enemy(x, America) Hostile(x) *x/m 1 + *x/west, y/m 1,z/Nono+ *x/m 1 + *x/nono+ 32

33 Properties of FC Sound Complete (for KBs of first-order definite clauses) For Datalog KBs, proof is fairly easy. Datalog KBs contain first-order definite clauses with no function symbols FC reaches a fix point for Datalog KBs after finite number of iterations It may not terminate in general if α is not entailed (query has no answer) entailment with definite clauses is also semi-decidable 33

34 More efficient FC Heuristics for matching rules against known facts Conjunct ordering Incremental forward chaining to avoid redundant rule matching Every new fact inferred on iteration t must be derived from at least one new fact inferred on iteration t 1. Check a rule only if its premise includes a newly inferred fact (at iteration t 1) Avoid drawing irrelevant facts (to the goal) 34

35 Backward Chaining (BC) Works backward from the goal, chaining through rules to find known facts supporting the proof AND-OR search OR: any rule ls goal in KB can be used to prove the goal AND: all conjuncts in ls must be proved. It is used in logic programming (Prolog) 35

36 BC algorithm (depth-first recursive proof search) function FOL_BC_ASK(KB, query) returns a generator of substitutions return FOL_BC_OR(KB, query, *+) generator FOL_BC_OR(KB, goal, θ) yields a substitution 36 for each rule (ls rs) in FETCH_RULES_FOR_GOAL(KB, goal) do (ls, rs) STANDARDIZE_VARS(ls, rs) for each θ in FOL_BC_AND(KB, ls, UNIFY(rs, goal, θ) do yield θ generator FOL_BC_AND(KB, goals, θ) yields a substitution if θ = failure then return else if LENGTH goals else do first FIRST goals rest REST goals = 0 then yield θ for each θ in FOL_BC_OR(KB, SUBST θ, first, θ) do for each θ in FOL_BC_AND(KB, rest, θ ) do yield θ

37 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 37

38 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 38

39 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 39

40 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 40

41 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 41

42 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 42

43 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 43

44 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 44

45 Backward chaining example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) 45

46 Properties of BC Space is linear in the size of the proof Suffers from repeated states and incompleteness Repeated subgoals (both success and failure) Solution: caching of previous results Incompleteness due to infinite loops Solution: checking current goal against every goal on stack 46

47 Resolution algorithm General theorem proving Apply resolution rule to CNF(KB α) Refutation-complete for FOL 47

48 Resolution PL resolution: l i and m j are complement l 1 l 2 l k, m 1 m 2 m n l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n First-order resolution UNIFY l i, m j = θ Clauses are assumed to be standardized apart (sharing no variables). l 1 l 2 l k, m 1 m 2 m n SUBST(θ, l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n ) Example: Ric x Unappy x, Ric(Ken) Unappy(Ken) θ = *x/ken+ 48

49 Converting FOL sentence to CNF Example: everyone who loves all animals is loved by someone x, y Animal(y) Loves(x, y)-,y Loves(y, x)- 1) Eliminate implications (P Q P Q) x, y Animal y Loves(x, y)-,y Loves(y, x)- 2) Move inwards: x p to x p, x p to 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)- 3) Standardize variables (avoiding same names for variables) x, y Animal(y) Loves(x, y)-,z Loves(z, x)- 49

50 Converting FOL sentence to CNF 4) Skolemize: a more general form of existential instantiation. Each existentially quantified 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) 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)- 50

51 Properties of resolution Resolution Complete when used in combination with factoring Extend factoring to FOL Factoring in PL: reduces two literals to one if they are identical Factoring in FOL: reduces two literals to one if they are unifiable. The unifier must be applied to the entire clause 51

52 Resolution: definite clauses example 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy(x, America) Hostile(x) Convert to CNF 1) Enemy(Nono, America) 2) Owns Nono, M 1 Missile M 1 3) American West 4) Missile x Owns Nono, x Sells West, x, Nono 5) American x Weapon y Sells x, y, z Hostile z Criminal x 6) Missile x Weapon x 7) Enemy x, America Hostile(x) 52

53 Resolution: definite clauses example KB α 53

54 Resolution: general example Every one who loves all animals is loved by someone. x, y Animal(y) Loves(x, y)-,y Loves(y, x)- Anyone who kills an animal is loved by no one. x, z Animal(z) Kills(x, z)-, y Loves(y, x)- Jack loves all animals. x Animal(x) Loves(Jack, x) Either Jack or Curiosity killed the cat, who is named Tuna. Kills Jack, Tuna Kills(Curiosity, Tuna) Cat(Tuna) Query: Did Curiosity kill the cat? Kills(Curiosity, Tuna) 54

55 Resolution: general example CNF (KB α): Animal(F(x)) Loves(G(x), x) Loves(x, F(x)) Loves(G(x), x) Loves(y, x) Animal(z) Kills(x, z) Animal(x) Loves(Jack, x) Kills Jack, Tuna Kills(Curiosity, Tuna) Cat(Tuna) Kills(Curiosity, Tuna) Query: Who killed the cat? α: w Kills(w, Tuna) The binding *w/curiosity+ in one of steps 55

56 Resolution: general example Query: Who killed the cat? α: w Kills(w, Tuna) The binding *w/curiosity+ in one of steps 56

57 Prolog (Programming in logic) Basis: backward chaining with Horn clauses Depth-first, left-to-right BC Program = set of clauses Fact e.g., american(west). Rule: head :- literal 1,,literal n. e.g., criminal(x) :- american(x), weapon(y), sells(x,y,z), hostile(z). 57

58 Prolog: Example 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=[] 58

59 Summary Lifting and unification Inference on KB of definite clauses Forward chaining Backward chaining Inference on general KB of FOL Resolution 59