First Order Logic - Inference

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1 First Order Logic - Inference

2 Logistics qcompensatory Classes ooption 1 ØMarch 18 th (Saturday) and April 1 st ooption 2 (Saturday) ØApril 1 st (Saturday) and April 9 th (Sunday) qquizzes oq5 March 30th oq6 April 20 th qmid-semester Feedback for Course Correction First Order Logic CSL302 - ARTIFICIAL INTELLIGENCE 2

3 Reasoning and Inference in FOL qinstantiation qtechniques opropositionalization ounification oforward Chaining obackward Chaining oresolution FOL CSL302 - ARTIFICIAL INTELLIGENCE 3

4 Universal Instantiation quniversally quantified sentence oall students in the AI class are smart x Student AICourse, x Smart(x) qintuitively, x can be anything: o Student AICourse, Tom Smart Tom o Student AIcourse, Chair Smart Chair o Student AICourse, LeftLeg John qformally: x Subst S x/p, S Smart LeftLeg John ox is replaced with p (ground term) in S, and the quantifier is removed. FOL CSL302 - ARTIFICIAL INTELLIGENCE 4

5 Existential Instantiation qexistentially Quantified sentence othere is smart student in the AI class x Student AICourse, x Smart x qintuitively, x must name something. But what? ocan we conclude:student AICourse, Tom Smart Tom??? ono! The sentence might not be true for Tom! qinstead, Use a Skolem constant and draw the conclusion, Formally x S Subst x/k, S ok is called a Skolem constant and is a completely new symbol you created. FOL CSL302 - ARTIFICIAL INTELLIGENCE 5

6 Inference I: Propositionalization qsuppose KB contains just the following: x King x Greedy x Evil(x) King(John) Greedy(Richard) Brother(Richard, John) qinstantiating will result in King John Greedy John Evil John King Richard Greedy Richard Evil Richard King John Greedy Richard Brother(Richard, John) q The new KB is now propositionalized. What are the proposition symbols? FOL CSL302 - ARTIFICIAL INTELLIGENCE 6

7 Inference I: Propositionalization qsuppose KB contains just the following: x King x Greedy x Evil(x) King(John) Greedy(Richard) Brother(Richard, John) qinstantiating will result in King John Greedy John Evil John King Richard Greedy Richard Evil Richard King John Greedy Richard Brother(Richard, John) q The new KB is now propositionalized. What are the proposition symbols? King John, Greedy John, King Richard, FOL CSL302 - ARTIFICIAL INTELLIGENCE 7

8 Propositionalization (2) qevery FOL KB and query can be propositionalized in such a way that entailment is preserved. opropositionalize the KB and query oapply resolution! qproblems can occur with Function Symbols. oe.g., Father Father John qherbrand Theorem: If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositional KB. qidea: For n = 0 to do ocreate a propositional KB by instantiating with depth nterms. osee if α is entailed by this KB. Works fine if α is entailed, but loops otherwise. Similar to halting problem in Turning machines - Semidecidable. FOL CSL302 - ARTIFICIAL INTELLIGENCE 8

9 Problems with Propositionalization qcan generate lot of irrelevant sentences qe.g., suppose we have x King x Greedy x Evil x King John y Greedy y Brother(Richard, John) qit seems obvious that Evil John, but propositionalization produces lots of facts such as Greedy Richard that are irrelevant qwith p k-ary predicates and n constants there are p R n S instantiations qwith function symbols, it gets much worse!! FOL CSL302 - ARTIFICIAL INTELLIGENCE 9

10 Propositionalization (3) qwhat if we want to use modus ponens from propositional logic? α β, α β γ γ qin FOL? x Student AICourse, x Smart(x) Student AICourse, Ram???? qmust unify x with Ram: oneed to substitute x/ram in Student AICourse, x Smart(x)to infer Smart(Ram) FOL CSL302 - ARTIFICIAL INTELLIGENCE 10

11 Unification qgoing back, suppose we have x King x Greedy x Evil x King John Greedy John Brother(Richard, John) qwe can get the inference immediately if we can find a substitution θ such that oking(x) and Greedy(x) match with King(John) and Greedy John oθ = x/john works FOL CSL302 - ARTIFICIAL INTELLIGENCE 11

12 Unification qgoing back, suppose we have x King x Greedy x Evil x King John y Greedy y Brother(Richard, John) qwe can still find a substitution θ such that oking(x) and Greedy(x) match King(John) and Greedy y oθ = x/john, y/john works qunify α, β = θ if αθ = βθ FOL CSL302 - ARTIFICIAL INTELLIGENCE 12

13 Unification - Most General Unifier qmatch up expressions by finding variable values that make the expressions identical Unify Student AICourse, x and Student(AICourse, Ram) using {x/ram} qunify(a, b) returns the most general unifier. othat places fewest restrictions on values of variables Unify Knows Ram, x, Knows y, z returns {y Ram, x/z} or y Ram, x Ram, z Ram MGU qunification vs Substitution? FOL CSL302 - ARTIFICIAL INTELLIGENCE 13

14 Unification Examples qunify Knows John, x, Knows John, Jane - x/jane FOL CSL302 - ARTIFICIAL INTELLIGENCE 14

15 Unification Examples (2) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y - x/richard, y/john FOL CSL302 - ARTIFICIAL INTELLIGENCE 15

16 Unification Examples (3) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) FOL CSL302 - ARTIFICIAL INTELLIGENCE 16

17 Unification Examples (4) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) otwo sentences happen to use the same variable ostandardizing apart - renaming the variables in one of the sentences to avoid name clashes. Unify Brother x, Richard, Brother(John, y) x/john, y/richard FOL CSL302 - ARTIFICIAL INTELLIGENCE 17

18 Unification Examples (5) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) qunify f g x, dog, y, f g cat, y, dog x/cat, y/dog FOL CSL302 - ARTIFICIAL INTELLIGENCE 18

19 Unification Examples (6) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) qunify f g x, dog, y, f g cat, y, dog qunify f g x, f(x) FOL CSL302 - ARTIFICIAL INTELLIGENCE 19

20 Unification Examples (7) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) qunify f g x, dog, y, f g cat, y, dog qunify f g x, f(x) oa variable may not contain itself in a substitution ooccur-check FOL CSL302 - ARTIFICIAL INTELLIGENCE 20

21 Unification Examples (8) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) qunify f g x, dog, y, f g cat, y, dog qunify f g x, f(x) qunify f g cat, y, y, f x, dog x/g(cat, y), y/dog FOL CSL302 - ARTIFICIAL INTELLIGENCE 21

22 Unification Examples (9) qunify Knows John, x, Knows John, Jane qunify Brother x, John, Brother Richard, y qunify Brother x, Richard, Brother(John, x) qunify f g x, dog, y, f g cat, y, dog qunify f g x, f(x) qunify f g cat, y, y, f x, dog qunify f g y, f x x/g(y) FOL CSL302 - ARTIFICIAL INTELLIGENCE 22

23 Unification Algorithm function UNIFY(x, y, θ) returns asubstitutiontomakex and y identical inputs: x,avariable,constant,list,orcompoundexpression y, avariable,constant,list,orcompoundexpression θ, thesubstitutionbuiltupsofar(optional,defaultstoempty) 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 function UNIFY-VAR(var, x, θ) returns asubstitution 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 FOL CSL302 - ARTIFICIAL INTELLIGENCE 23

24 Generalized Modus Ponens (GMP) Student AIcourse, Ram x Student AICourse, x Smart(x)???? qunify x with Ram - x/ram in Student AICourse, x Smart(x) qapply MP to infer Smart(Ram) qgeneralized Modus Ponens (GMP) o Lifted version of MP (lifts MP from ground Pl to FOL) p a b, p c b,, p d b, (p a p c p d q) Subst θ, q qwhere p g b θ = p g θ for all i GMP used with KB of clauses, all variables assumed universally quantified. Prove GMP is sound! FOL CSL302 - ARTIFICIAL INTELLIGENCE 24

25 Inference II: Forward Chaining The algorithm: Start with the KB Add any fact you can generate with GMP (unify + GMP) Repeat until goal is reached or generation halts. FOL CSL302 - ARTIFICIAL INTELLIGENCE 25

26 Forward Chaining Example qit is a crime for an Indian to sell weapons to hostile nations. The country Nono, an enemy of India, has some missiles. All of its missiles were sold to it by Traitor, who is an Indian. qis Traitor a criminal? FOL CSL302 - ARTIFICIAL INTELLIGENCE 26

27 Forward Chaining Example qit is a crime for an Indian to sell weapons to hostile nations. The country Nono, an enemy of India, has some missiles. All of its missiles were sold to it by Traitor, who is an Indian. qis Traitor a criminal? Criminal Traitor? qkb of definite clauses (exactly one positive literal) o Indian x Weapon y Sells x, y, z Hostile z Criminal(x) o Enemy(Nono, India) o Owns Nono, M a M a is a Skolem constant o Missile M a o Missile x Owns Nono, x Sells(Traitor, x, Nono) o Indian Traitor o Missle x Weapon x o Enemy x, India Hostile(x) FOL CSL302 - ARTIFICIAL INTELLIGENCE 27

28 Forward Chaining Example Missile x Owns Nono, x Sells Traitor, x, Nono Missle x Weapon x Enemy x, India Hostile(x) Indian Traitor Missile M a Owns Nono, M a Enemy(Nono, India) Initial Facts in the KB FOL CSL302 - ARTIFICIAL INTELLIGENCE 28

29 Forward Chaining Example Missile x Owns Nono, x Sells Traitor, x, Nono Missle x Weapon x Enemy x, India Hostile(x) Weapon M a Sells Traitor, M a, Nono Hostile Nono 3 x/m a 2 1 x/nono x/m a Indian Traitor Missile M a Owns Nono, M a Enemy(Nono, India) Initial Facts in the KB FOL CSL302 - ARTIFICIAL INTELLIGENCE 29

30 Forward Chaining Example Indian x Weapon y Sells x, y, z Hostile z Criminal(x) Therefore, Traitor is a criminal Criminal Traitor x Traitor, y M a, z Nono Weapon M a Sells Traitor, M a, Nono Hostile Nono 3 x/m a 2 1 x/nono x/m a Indian Traitor Missile M a Owns Nono, M a Enemy(Nono, India) Initial Facts in the KB FOL CSL302 - ARTIFICIAL INTELLIGENCE 30

31 Inference II: Forward Chaining The algorithm: Start with the KB Add any fact you can generate with GMP (unify + GMP) Repeat until goal is reached or generation halts. FOL CSL302 - ARTIFICIAL INTELLIGENCE 31

32 Inference II: Forward Chaining qsound oyes, Because GMP is sound qcomplete oyes, if KB contains only definite clauses qefficiency ounification via exhaustive pattern matching orule rechecking for premise satisfaction at every iteration oirrelevant fact generation Check pg. 331 in textbook Check section for possible strategies FOL CSL302 - ARTIFICIAL INTELLIGENCE 32

33 Inference III: Backward Chaining Start with KB and goal Find all rules whose results unify with goal: Add the premises of these rules to the goal list Remove the corresponding result from the goal list Stop when goal list is empty(success) or progress halts (failure) FOL CSL302 - ARTIFICIAL INTELLIGENCE 33

34 Backward Chaining Example Goal Criminal Traitor FOL CSL302 - ARTIFICIAL INTELLIGENCE 34

35 Backward Chaining Example qindian x Weapon y Sells x, y, z Hostile z Criminal(x) Criminal Traitor x Traitor FOL CSL302 - ARTIFICIAL INTELLIGENCE 35

36 Backward Chaining Example qdepth-first Traversal Criminal Traitor x Traitor Indian x Weapon y Sells x, y, z Hostile z FOL CSL302 - ARTIFICIAL INTELLIGENCE 36

37 Backward Chaining Example qdepth-first Traversal Criminal Traitor x Traitor Indian Traitor Weapon y Sells x, y, z Hostile z FOL CSL302 - ARTIFICIAL INTELLIGENCE 37

38 Backward Chaining Example qdepth-first Traversal Criminal Traitor x Traitor Indian Traitor Weapon y Sells x, y, z Hostile z New Subgoal FOL CSL302 - ARTIFICIAL INTELLIGENCE 38

39 Backward Chaining Example qdepth-first Traversal qkb: Missile y Weapon y ; Missile M a Criminal Traitor x Traitor, y M a Indian Traitor Weapon y Sells x, y, z Hostile z Missile y y M a FOL CSL302 - ARTIFICIAL INTELLIGENCE 39

40 Backward Chaining Example qdepth-first Traversal Criminal Traitor x Traitor, y M a Indian Traitor Weapon y Sells x, y, z Hostile z New Subgoal Missile y y M a z? FOL CSL302 - ARTIFICIAL INTELLIGENCE 40

41 Backward Chaining Example qdepth-first Traversal qmissile y Owns Nono, y Sells Traitor, y, Nono qmissile M a qowns Nono, M a Criminal Traitor x Traitor, y M a, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono Missile y y M a Missile M a Owns Nono, M a FOL CSL302 - ARTIFICIAL INTELLIGENCE 41

42 Backward Chaining Example qdepth-first Traversal Criminal Traitor x Traitor, y M a, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono New Subgoal Missile y y M a Missile M a Owns Nono, M a FOL CSL302 - ARTIFICIAL INTELLIGENCE 42

43 Backward Chaining Example qdepth-first Traversal qkb: Enemy z, India Hostile z ; qenemy(nono, India) Criminal Traitor x Traitor, y M a, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono Missile y Missile M a Owns Nono, M a Enemy(Nono, India) y M a FOL CSL302 - ARTIFICIAL INTELLIGENCE 43

44 Backward Chaining Example qdepth-first Traversal Criminal Traitor x Traitor, y M a, z Nono Indian Traitor Weapon y Sells x, y, z Hostile z z Nono Missile y Missile M a Owns Nono, M a Enemy(Nono, India) y M a FOL CSL302 - ARTIFICIAL INTELLIGENCE 44

45 Inference III: Backward Chaining qdepth-first recursive search: space is linear in size of proof. qincomplete due to infinite loops(e.g., repeated states) ofix by checking current goal against all goals on stack ocannot fix infinite paths though qwidely used for Logic Programming oprolog (Check Section for more information). FOL CSL302 - ARTIFICIAL INTELLIGENCE 45

46 Inference IV: Resolution qrecall from propositional logic p q, p r s q q r s qliteral in one clause qits negation in the other clause qresult is the disjunction of the remaining literals. FOL CSL302 - ARTIFICIAL INTELLIGENCE 46

47 Inference IV: Resolution qin general p x q A, p B r x s y q q A r B s y l a l S, m a m d (l a l gra l gsa l S m a m tra m tsa m d )θ qwhere Unify l g, m t = θ qthe two clauses are assumed to be standardized apart so that they share no variables. qapply resolution steps to CNF(KB α). Substitute MGU x/b in all literals FOL CSL302 - ARTIFICIAL INTELLIGENCE 47

48 FOL: Conversion to CNF qeveryone who loves all animals is loved by someone x y Animal y Loves x, y y Loves(y, x) qsteps to convert to CNF oeliminate biconditionals and implications omove inwards x, p x p, x p x p ostandardize the variables: each quantifier should use a different variable oskolemize: A more general form of existential instantiation ØEach existential variable is replaced by a Skolem function of the enclosing universally quantified variables odrop universal quantifiers odistribute over FOL CSL302 - ARTIFICIAL INTELLIGENCE 48

49 Skolemize qremove existential quantifiers by elimination o xp x into P A, where A is a new constant qhowever, this cannot be applied to general sentences. For example o x y Animal y Loves x, y z Loves z, x o x Animal A Loves x, A Loves B, x qintroduce Skolem Functions F and G o x Animal F x Loves x, F x Loves G x, x Logical Agents CSL452 - ARTIFICIAL INTELLIGENCE 49

50 FOL: Conversion to CNF qit is a crime for an Indian to sell weapons to hostile nations. The country Nono, an enemy of India, has some missiles. All of its missiles were sold to it by Traitor, who is an Indian. o Indian x Weapon y Sells x, y, z Hostile z Criminal(x) o Enemy(Nono, India) o Owns Nono, M a o Missile M a o Missile x Owns Nono, x Sells(Traitor, x, Nono) o Indian Traitor o Missle x Weapon x o Enemy x, India Hostile(x) qresolution uses proof by contradiction o Show KB α by showing KB α is unsatisfiable. Variables are not standardized here. qto prove, Traitor is a criminal. Add Criminal Traitor to KB and derive empty clause. FOL CSL302 - ARTIFICIAL INTELLIGENCE 50

51 Indian x Weapon y Sells x, y, z Hostile z Criminal(x) Criminal Traitor Indian Traitor Weapon y Sells Traitor, y, z Hostile z Weapon y Sells Traitor, y, z Hostile z Missile y Sells Traitor, y, z Hostile z Sells Traitor, M a, z Hostile z Missile M a Owns Nono, M a Hostile(Nono) Indian Traitor Missle x Weapon x Missile M a Missile x Owns Nono, x Sells(Traitor, x, Nono) Missile M a Hostile(Nono) Owns Nono, M a Missile M a Hostile(Nono) Enemy Nono, India Enemy x, India Hostile(x) Enemy(Nono, India) FOL CSL302 - ARTIFICIAL INTELLIGENCE 51

52 FOL Resolution Example (2) qeveryone who loves all animals is loved by someone x y Animal y Loves x, y y Loves(y, x) qanyone who kills an animal is loved by no one x z Animal z Kills x, z qjack loves all animals qeither Jack or Curiosity killed the cat, who is named Tuna Kills Jack, Tuna Kills Curiosity, Tuna qdid Curiosity kill the cat? y Loves(y, x) x Animal x Loves Jack, x Kills Curiosity, Tuna? x Cat x Animal x Cat(Tuna) FOL CSL302 - ARTIFICIAL INTELLIGENCE 52

53 x y Animal y Loves x, y x y Animal y Loves x, y y Loves(y, x) y Loves(y, x) x y Animal y Loves x, y x y Animal y Loves x, y x y Animal y Loves x, y y Loves(y, x) y Loves(y, x) z Loves(z, x) x Animal F x Loves x, F x Loves G x, x Animal F x Loves x, F x Loves G x, x Animal F x Loves G x, x Loves x, F x Loves G x, x x z Animal z Kills x, z y Loves(y, x) x z Animal z Kills x, z y Loves(y, x) x z Animal z Kills x, z y Loves(y, x) x z Animal z Kills x, z y Loves(y, x) Animal z Kills x, z Loves(y, x) x Animal x Loves Jack, x x Animal x Loves Jack, x Animal x Loves Jack, x x Cat x Animal x x Cat x Animal x Cat x Animal x FOL CSL302 - ARTIFICIAL INTELLIGENCE 53

54 Cat(Tuna) Cat x Animal x Kills Jack, Tuna Kills Curiosity, Tuna Animal Tuna Kills Curiosity, Tuna Animal z Kills x, z Loves(y, x) Kills Jack, Tuna Kills x, Tuna Loves(y, x) Loves x, F x Loves G x, x Animal x Loves Jack, x Animal F Jack Loves G(Jack), Jack Animal F x Loves G x, x Loves(y, Jack) Loves G(Jack), Jack FOL CSL302 - ARTIFICIAL INTELLIGENCE 54

Propositional Logic. üdeclarative - pieces of syntax correspond to facts üallows partial/disjunctive/negated information

Propositional Logic. üdeclarative - pieces of syntax correspond to facts üallows partial/disjunctive/negated information First Order Logic Propositional Logic üdeclarative - pieces of syntax correspond to facts üallows partial/disjunctive/negated information ücompositional (B "" P "% is derived from B "" and P "% ) ücontext