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Automated Deduction

Automated Deduction. resolution (Otter) backward-chaining (Prolog) forward-chaining (Rete, Clips, Jess) also called “theorem provers” or “inference engines” example application: think about using rules to infer “right-of-way” in a driving simulation. Resolution.

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Automated Deduction

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  1. Automated Deduction • resolution (Otter) • backward-chaining (Prolog) • forward-chaining (Rete, Clips, Jess) • also called “theorem provers” or “inference engines” • example application: • think about using rules to infer “right-of-way” in a driving simulation

  2. Resolution • first-order (with unification) • (ab)(a’c)ᅡ(b”c”) • where q=unifier(a,a’) and b”=apply(b,q), c”=apply(c,q) • example • X works_for(X,government) v receives_pension(X) • works_for(kate,government) v works_for(kate,walmart) • works_for(kate,walmart) v receives_pension(kate) • q={X/kate}

  3. resolution is refutation-complete for FOL • must convert all sentences to CNF • a ground instance of a clause is formed by instantiating all variables with some constant • P(X,Y)vQ(Y,Y) : {P(sam,bill)vQ(bill,bill); P(sam,joe)vQ(joe,joe)...} • Herbrand’s Theorem: if a set of sentences is unsatisfiable, then there exists a set of ground (propositional) instances that is unsatisfiable • Ground Resolution Theorem: if a set of propositional clauses is unsatisfiable, then there is a finite derivation of the empty clause • Lifting Lemma: If there is proof of the empty clause using ground instances, then there is parallel proof using the original clauses with variables (using unification)

  4. Converting FOL sentences to CNF • eliminate implications (abᅡab) • move neg. inwards (DeMorgans’ Laws, x Pᅡx P) • standardize variables apart (if X is used in multiple quantifiers, replace instance with X1, X2...) • skolemization (replace  vars with *new* constants) • drop universal quantifiers  • distribute  over  • break conjunctions into separate clauses • “Everyone who loves all animals is loved by someone.” • x (y animal(y)loves(x,y))y loves(y,x)

  5. 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 missles, and all of its missles were sold to it by Colonel West, who is an American.” • Prove that Colonel West is a criminal. • x,y,z american(x)&weapon(y)&sells(x,y,z)&hostile(z) criminal(x) • x owns(nono,x)&missle(x) • x owns(nono,x)&missle(x)sells(west,x,nono) • x missle(x)weapon(x) • x enemy(x,america)hostile(x) • american(west), enemy(nono,america) • negation of query: criminal(west)

  6. convert to CNF • x,y,z american(x)&weapon(y)&sells(x,y,z)&hostile(z) criminal(x) {american(x1),weapon(y1),sells(x1,y1,z1),hostile(z1), criminal(x1)} • x missle(x)weapon(x) {missle(x2),weapon(x2)} • x owns(nono,x)&missle(x)sells(west,x,nono) {owns(nono,x3), missle(x3),sells(west,x3,nono)} • x owns(nono,x)&missle(x) {owns(nono,M),missle(M)} – M is a skolem constant • x enemy(x,america)hostile(x) {enemy(x4,america),hostile(x4)} • american(west), enemy(nono,america),criminal(west)

  7. maybe thousands of clauses – difficult search • search strategies: which clauses to resolve? • unit preference: • one clause must be of length 1 • guarantees to reduce length of other clause (toward 0) • set of support: • one clause must be from the sos, e.g. negated query • “source” of the unsatisfiability • input resolution: • one clause must be from the input (KB or query) • linear resolution • one clause must be from negated query OR a successor derived from it • complete • Otter – a real-world resolution theorem prover

  8. Backward-chaining • recall: subgoal stack • try to reduce to facts; might have to back-track • KB must be in Horn-clause form • in FOL, use unification • for popped subgoal, try to unify with fact or head of some clause • Prolog – a practical implementation of a back-chaining theorem prover • funky syntax • can be used for many solving many problems • learn how to use back-chaining as “computational model” • widely employed for expert systems, intelligent agents, control applications...

  9. Prolog syntax • facts: predicate(args,...). • rules: • no quantifiers, variables in capital letters, • written backwards, “:-” means “” , read as “if” • “,” means conjunction; no disjunction • X dog(X)  mammal(X) • mammal(X) :- dog(X). • canPlay(Child) :- hasEaten(Child,dinner),finished(Child,homework). • also support for strings, lists, numbers...

  10. Problem solving by back-chaining • combinatorial enumeration/search (unbound variables) • often interested in variable binding of solution father(X,Y) :- parent(X,Y),male(X). mother(X,Y) :- parent(X,Y),female(X). sibling(X,Y) :- parent(Z,X),parent(Z,Y). grandparent(X,Y) :- parent(X,Z),parent(Z,Y). uncle(X,Y) :- parent(Z,Y),sibling(X,Z),male(X). male(john). male(sam). male(joe). male(bill). female(sue). female(ellen). parent(sam,john), parent(ellen,john). parent(ellen,joe). parent(sam,joe). parent(sue,bill). parent(al,sam). parent(sue,ellen). query: ?- uncle(bill,john). yes. query: ?- uncle(sue,john). fail. query: ?- grandparent(X,john). X=al; X=sue

  11. Map-Coloring in Prolog color(red). color(green). color(blue). color(yellow). valid_coloring(A,B,C,D,E) :- color(A),color(B),color(C),color(D),color(E), not A=B, not A=C, not A=D, not A=E, not B=C, not B=D, not C=D, not D=E. • effectively enumerates all combinations of colors and tests them for consistency. • will try A=red, B=red, C=red, D=red, E=red first • fail, because doesn’t satisfy not A=B. • then back-track to A=red, B=red, C=red, D=red, E=green, which fails not A=C. • and so on, until reach A=red, B=green, C=blue, D=yellow, E=green • not very efficient, but illustrative of the kind of combinatorial problem-solving that can be simulated via back-chaining

  12. Negation in Prolog • Can have negative literals in antecedents. • not strict FOL semantics • “negation-as-failure” • ...,not p(X),... means try to prove p(X) (by back-chaining with current variable bindings) • if cannot prove it, then proceed • if can prove it, then fail and back-track • very handy – allows default inference, compact KB • canFly(X) :- bird(X),not broken(wings(X)). • bird(X) :- canary (X). • bird(X) :- penguin(X). • bird(X) :- eagle(X). • canary(tweety). penguin(opus). eagle(sam). • broken(wings(opus)). ?- canFly(B). B=sam. B=tweety.

  13. Forward-Chaining • requires Horn-clause KB • combining universally-quantified rules with ground facts can generate many inferences • how to do this efficiently • Rete algorithm • generates a graph structure • firing rules to create nodes • “production system”, basis of many expert systems • e.g. XCON or R1, for configuring computers, or MYCIN for diagnosing blood diseases • also the basis of “cognitive architectures” like ACT and SOAR (Allan Newell and Herb Simon) • based on theory that brain does symbolic pattern matching, which triggers associations that activate other concepts...

  14. Rete algorithm • until quiescence... • find all rules that could fire (i.e. are “activated” by a combination of input nodes, see colors below) • pick rule with highest priority (conflict resolution) • unify antecedent with incoming edges • apply unifier to consequent; create new node • uses hash tables and many other optimizations for efficiency father(X,Y) :- parent(X,Y),male(X). mother(X,Y) :- parent(X,Y),female(X). sibling(X,Y) :- parent(Z,X),parent(Z,Y). grandparent(X,Y) :- parent(X,Z),parent(Z,Y). uncle(X,Y) :- parent(Z,Y),sibling(X,Z),male(X). male(john). male(sam). male(joe). male(bill). female(sue). female(ellen). parent(sam,john), parent(ellen,john). parent(ellen,joe). parent(sam,joe). parent(sue,bill). parent(al,sam). parent(sue,ellen). uncle(bill,john) sibling(bill,ellen) mother(sue) grandparent(sue,john)

  15. practical implementations of Rete (forward-chaining inference engines) • CLIPS • invented at NASA in 1980’s • used for many applications, especially discrete simulations, e.g. of traffic flow, shuttle operation, clock mechanisms, agents • encode rules in KB; executes forward-chaining by Rete • http://www.siliconvalleyone.com/clips.htm • JESS • re-implementation in Java at Sandia National Lab • http://www.jessrules.com/

  16. syntax: (defrule name PATTERN => ACTION) • (defrule welcome-toddlers (person ?X) (age ?X ?Y) (< ?Y 3)) => (assert (toddler ?X)) (printout t "Hello, little one!" crlf)) • (defrule library-rule-1 (book (name ?X) (status late) (borrower ?Y)) (borrower (name ?Y) (address ?Z)) => (send-late-notice ?X ?Y ?Z)) • pattern-match left-hand side • execute right-hand side • actions: assert, retract, printout... • can have structured “objects” with “slots” • can define functions... • can call invoke external procedures (e.g. graphics)

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