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Inference in first-order logic

Inference in first-order logic
Inference in firstorder logic Chapter 9 (Please turn your mobile devices to silent. Thanks) CS3243 Inference 1 Last Time  First Order Logic  Reasons about objects, predicates I  ntroduces equality and quantifiers  Brief excursion into Prolog T   o be finished and related to more in depth today CS3243 Inference 2 Outline  Reducing firstorder inference to propositional inference  Unification  Generalized Modus Ponens  Forward chaining  Backward chaining  Resolution CS3243 Inference 3 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(Father(John)) ∧ Greedy(Father(John)) ⇒ Evil(Father(John)) … CS3243 Inference 4 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 ) ∧ OnHead(C ,John) 1 1 provided C is a new constant symbol, called a 1 Skolem constant CS3243 Inference 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. CS3243 Inference 6 Blank spaces to fill in on this slide Propositionalization   Every FOL KB can be propositionalized so as to preserve entailment   Convert it to propositional logic   A ground sentence is entailed by new KB iff entailed by original KB   Idea: propositionalize KB and query, apply resolution, return result   But there’s a problem: with function symbols, there are infinitely many ground terms:  e.g., Father(Father(Father(John))) CS3243 Inference 7 Blank spaces to fill in on this slide Propositionalization, continued Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB Remind you of any other Idea: For n = 0 to ∞ do algorithm create a propositional KB by instantiating with depthn 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 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.) CS3243 Inference 8 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 k   With p kary predicates and n constants, there are p·n instantiations. 2   E.g., p=1 k=2 n=3, Rel(,).  3 × 3 = 3 = 9 possibilities CS3243 Inference 9 Blank spaces to fill in on this slide 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,Mother(y)) y/John,x/Mother(John) Knows(John,x) Knows(x,OJ) fail   Standardizing apart eliminates overlap of variables, e.g., Knows(z ,OJ) 17 CS3243 Inference 10 Unification  To unify Knows(John,x) and Knows(y,z), θ = y/John, x/z or θ = y/John, x/John, z/John  The first unifier is more general than the second.  There is a single most general unifier (MGU) that is unique up to renaming of variables. MGU = y/John, x/z CS3243 Inference 11 The unification algorithm CS3243 Inference 12 The unification algorithm CS3243 Inference 13 Blank spaces to fill in on this slide Let’s do one together Knows(John,x) Knows(y,Mother(y)) CS3243 Inference 14 Generalized Modus Ponens (GMP) p ', p ', … , p ', ( p ∧ p ∧ … ∧ p ⇒q) 1 2 n 1 2 n where p 'θ = p θ for all i i i qθ p ' is King(John) p is King(x) 1 1 p ' is Greedy(y) p is Greedy(x) 2 2 θ is x/John,y/John q is Evil(x) q θ is Evil(John)   GMP used with KB of definite clauses (exactly one positive literal)   n.b. recall Horn form allows at most one positive literal (less restrictive)   All variables assumed universally quantified CS3243 Inference 15 Let’s do an example with a KB   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 Colonel West is a criminal CS3243 Inference 16 The example KB in FOL ... 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, i.e., ∃x Owns(Nono,x) ∧ Missile(x): Owns(Nono,M ) and Missile(M ) 1 1 … 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) CS3243 Inference 17 Forward chaining algorithm CS3243 Inference 18 Forward chaining proof CS3243 Inference 19 Forward chaining proof CS3243 Inference 20 Forward chaining proof CS3243 Inference 21 Soundness of GMP   Need to show that p ', …, p ', (p ∧ … ∧ p ⇒ q) ╞ qθ 1 n 1 n provided that p 'θ = pθ for all I i i   Lemma: For any sentence p, we have p ╞ pθ by UI 1.  (p ∧ … ∧ p ⇒ q) ╞ (p ∧ … ∧ p ⇒ q)θ = (p θ ∧ … ∧ p θ ⇒ qθ) 1 n 1 n 1 n 2.  p ', …, p ' ╞ p ' ∧ … ∧ p ' ╞ p 'θ ∧ … ∧ p 'θ 1 n 1 n 1 n 3.  From 1 and 2, qθ follows by ordinary Modus Ponens CS3243 Inference 22 Properties of forward chaining   Sound and complete for firstorder definite clauses   Datalog = firstorder definite clauses + no functions   FC terminates for Datalog in finite number of iterations   May not terminate in general if α is not entailed   This is unavoidable: entailment with definite clauses is again semidecidable CS3243 Inference 23 Definite clause Equivalence to CSPs Diff(wa,nt) ∧ Diff(wa,sa) ∧ Diff(nt,q) ∧ Diff(nt,sa) ∧ Diff(q,nsw) ∧ Diff(q,sa) ∧ Diff(nsw,v) ∧ Diff(nsw,sa) ∧ Diff(v,sa) ⇒ Colorable() Diff(Red,Blue) Diff (Red,Green) Diff(Green,Red) Diff(Green,Blue) Diff(Blue,Red) Diff(Blue,Green) Some facts   Each conjunct can be viewed as a constraint on a variable.   Every finite CSP can be expressed as a single definite clause together with some facts. CS3243 Inference 24 Improving efficiency The algorithm presented earlier isn’t efficient. Let’s make it better. 1.  Matching itself is expensive, Database indexing allows O(1) retrieval of known facts  e.g., query a table where all instantations of p(x) are stored; Missile(x) retrieves Missile(M ) 1 For predicates with many subgoals, the conjunct ordering problem applies  e.g., for Missile(x) ∧ Owns(Nono,x) ⇒ Sells(West, x, Nono) if there are many things owned by Nono, perhaps better to start with Missile(x) conjunct CS3243 Inference 25 Improving efficiency, continued 2.  Incremental forward chaining: Only match rules on iteration k if a premise was added on iteration k1 ⇒ Original algorithm discards partially matched rules ⇒ Instead, keep track of conjuncts matched to avoid duplicate work ⇒ Match each rule whose premise contains a newly added positive literal Leads to the development of Rete (“ReeTee”) networks in real world production systems 3.  Irrelevant Facts: several ways to address … let’s segue to Backward Chaining. CS3243 Inference 26 Backward chaining algorithm What type of search algorithm is this SUBST(COMPOSE(θ , θ ), p) = 1 2 SUBST(θ , SUBST(θ , p)) 2 1 CS3243 Inference 27 Backward chaining example CS3243 Inference 28 Backward chaining example CS3243 Inference 29 Backward chaining example CS3243 Inference 30 Backward chaining example CS3243 Inference 31 Backward chaining example CS3243 Inference 32 Backward chaining example CS3243 Inference 33 Backward chaining example CS3243 Inference 34 Properties of backward chaining  Depthfirst recursive proof search: space is linear in size of proof  Incomplete due to infinite loops ⇒ fix by checking current goal against every goal on stack  Inefficient due to repeated subgoals (both success and failure) ⇒ fix using caching of previous results (extra space)  Widely used for logic programming CS3243 Inference 35 Logic programming: Prolog   Backward chaining with Horn clauses + bells whistles   Program = set of clauses = head : literal , … literal . 1 n criminal(X) : american(X), weapon(Y), sells(X,Y,Z), hostile(Z).   Depthfirst, lefttoright backward chaining   Builtin predicates for arithmetic etc., e.g., X is YZ+3   Builtin predicates that can have side effects (e.g., input and output predicates, assert/retract predicates)   Closedworld assumption / database semantics ("negation as failure")   e.g., given alive(X) : not dead(X).   alive(joe) succeeds if dead(joe) fails   No checks for infinite recursion   No occurs check for unification CS3243 Inference 36 Resolution: recap and look at FOL   Full firstorder version: l ∨ ··· ∨ l , m ∨ ··· ∨ m 1 k 1 n (l ∨ ··· ∨ l ∨ l ∨ ··· ∨ l ∨ m ∨ ··· ∨ m ∨ m ∨ ··· ∨ m )θ 1 i1 i+1 k 1 j1 j+1 n where Unify(l , ¬m ) = θ. i j   The two clauses are assumed to be standardized apart so that they share no variables.   For example, ¬Rich(x) ∨ Unhappy(x), Rich(Ken) Unhappy(Ken) with θ = x/Ken   Apply resolution steps to CNF(KB ∧ ¬α); complete for FOL CS3243 Inference 37 Conversion to CNF  Everyone who loves all animals is loved by someone: Is this the same y ∀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) CS3243 Inference 38 Conversion to CNF, continued 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: Why do we need a ∀x Animal(F(x)) ∧ ¬Loves(x,F(x)) ∨ Loves(G(x),x) function and not a variable 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) CS3243 Inference 39 Resolution proof: definite clauses CS3243 Inference 40 Resolution proof: definite clauses CS3243 Inference 41 Resolution proof: definite clauses CS3243 Inference 42 Resolution proof: definite clauses CS3243 Inference 43 Resolution proof: definite clauses CS3243 Inference 44 Refutation completeness  Resolution can say yes to any entailed sentence but cannot be used to generate all entailed sentences E.   g., won’t generate Animal(x) ∨ ¬Animal(x) CS3243 Inference 45 Blank spaces to fill in on this slide Resolution special cases  Factoring: may need  To handle equality to remove redundant x=y, need to use literals (literals that demodulation (sub x are unifiable) for y in some clause that has x).  L(x) ∨ G(a,b)  B = Son(A)  ¬L(x) ∨ G(K,L)  Property(B)     CS3243 Inference 46 Summary  Examined our three strategies for logic inference in FOL:  Forward Chaining Ba   ckward Chaining (what Prolog uses) R   esolution  To think about: when is each of the three systems the most appropriate CS3243 Inference 47
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