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Introduction to Artificial Intelligence LECTURE 11 : Nonmonotonic Reasoning

Introduction to Artificial Intelligence LECTURE 11 : Nonmonotonic Reasoning. Motivation: beyond FOL + resolution Closed-world assumption Default rules and theories Ref: “Logical Foundations of AI”, Genesereth and Nilsson, Morgan Kauffman, 1987.

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Introduction to Artificial Intelligence LECTURE 11 : Nonmonotonic Reasoning

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  1. Introduction to Artificial IntelligenceLECTURE 11: Nonmonotonic Reasoning • Motivation: beyond FOL + resolution • Closed-world assumption • Default rules and theories Ref: “Logical Foundations of AI”, Genesereth and Nilsson, Morgan Kauffman, 1987.

  2. Knowledge representation with FOL + resolution • FOL + resolution have limitations in the kind of sentences and deductions we can make. • cannot express uncertainty • cannot make unsound but likely deductions • cannot revise conclusions in light of new knowledge • cannot make conclusions based on the entire state of the KB

  3. Motivation: incompleteness • If we cannot prove P or ~P from KB, what should we conclude? KB:neighbor(israel,jordan) neighbor(israel,egypt) neighbor(lebanon,syria) … Query: neighbor(israel,morroco) • Cannot conclude anything unless there is an explicit statement (negation) in KB!

  4. Motivation: exceptions • All rules have exceptions! For each, we must forsee all of them and explicitly state them: “all birds fly” Xbird(X)=> flies(X) “except ostriches” Xbird(X)/\~ostrich(X) => flies(X) “except newborns” Xbird(X) /\ ~ostrich(X) /\ ~newborn(X) => flies(X) • We wouldlike toconclude flies(X) from bird(X) unless something is abnormal with X Xbird(X)/\ ~Abnormal(X)=> flies(X)

  5. Motivation: changes • We assumed that all clauses in KB are true and remain true. What if we later discover that this is not the case? How do we revise conclusions already made? Xcitizen(X)/\income(X,Y) => pay_tax(X,Y) • As the rules change, we need to revise all the intermediate conclusions! • We would like to identify only those that indeed need revision

  6. Possible extensions • Language: make it more expressive without loosing some of its computational properties • Semantics: revise the concept of truth value • Inferencing: design new inference rules to deal with exceptions, uncertainty, etc Minimal extensions to FOL+new inference rules!

  7. Nonmonotonic logics (NML) • FOL + resolution is monotonic:if KB|=P then (KB U {Q})|= P for all consistent KB and all Q obtained from KB by applying resolution. • The number of statements known to be true is strictly increasing over time • Non-monotonic: “jump” to conclusions that can later be withdrawn (defeasible conclusions)

  8. Consistency in NML • A deduction rule R is consistent iff: KR |=Rc KR U {c} |= Ø iff KR |= Ø • TR(KB) = the set of all conclusions from KB using inference rule R (transitive closure). Note: R is applied in parallel!

  9. Nonmonotonic frameworks 1. Closed World Assumption if you cannot prove P or ~P from KB, add ~P to KB. 2. Default Rules new inference rules on how to augment KB 3. (Predicate completion -- Circumscription) compute a formula which says how KB should be satisfied 4. (Truth Maintenance Systems) methods to maintain consistency in a KB where statements are constantly added and deleted

  10. 1. Closed-world assumption (CWA) if you cannot prove P or~P from KB, add ~P to KB KB |= c and KB |= ~c then add ~c • Idea:if you cannot prove P, assume it is false. This means you assume you know everything there is to know about the world (e.g. the world is closed). • This is the semantics of databases and of PROLOG.

  11. Complete theories • A theory T is a set of sentences closed under logical implication (like transitive closure) • T is complete if either every ground literal in the language or its negation is in the theory KB: neighbor(israel,jordan) neighbor(israel,egypt) XYneighbor(X,Y) <=> neighbor(Y,X) T(KB) is not complete because neither neighbor(egypt,jordan) or ~neighbor(egypt,jordan) is in T(KB).

  12. Theory completion • Given an incomplete KB, include the negation of a ground literal when the ground literal does not follow from KB KB: neighbor(israel,jordan) neighbor(israel,egypt) XY neighbor(X,Y) <=> neighbor(Y,X) The atom ~neighbor(egypt,jordan) will be added • Is this always consistent? NO!

  13. Completion inconsistency • Completion can lead to inconsistent theories: KB: p(a)\/p(b) neither KB |= ~p(a), p(a) nor KB |= ~p(b), p(b) follow So we add ~p(a) and ~p(b) to KB: KB’: p(a)\/p(b) ~p(a) ~p(b) KB’ is inconsistent! • Modify the completion rule for consistency

  14. CWA theorem (1) • Augment a consistent KB with a new set of sentences (beliefs), to obtain a new consistent set CWA(KB). • Theorem: CWA(KB) is consistent iff for every disjunction p1\/p2\/ …. \/pn, that followsfrom KB, where pi is a positive-ground literal, there is at least one pisuch that KB |= pi Eq: CWA(KB) is inconsistent iff there are positive ground literals p1, … pnsuch that KB |= p1\/p2\/…. \/pnbut for all i, KB |pi.

  15. CWA theorem (2) Intuition: add all of ~ pi except one, so no contradiction occurs! Proof: Let KBassumed be the set of all conclusions ~p derived with CWA rule: ~p is in KBassumed iff KB |= pandKB |= ~p CWA(KB) is inconsistent only if KB U KBassumed is. Then, there is a finite subset of KBassumed that contradicts KB. Let this subset be L = {~p1,…,~pn}. Then KB |= p1\/p2\/ …. \/pn,the negation of L. Since each ~pi is in KBassumed by thedefinition of KBassumed KB |= pi contradiction!

  16. Consistent CWA rule • Complete a KB by including all ground literals that do not contradict the theorem. Important: define the constant atoms first • Ex1: KB: p(a)\/p(b) is not consistent • Ex2: KB: Xp(X)\/q(X) p(a) q(b)for atoms a,b, the augmentation is ~q(a) and~p(b) OK for atom c, the augmentation is inconsistent: NOT OK (p(X)\/ q(X))| p(c) or (p(X)\/ q(X))| q(c)

  17. CWA consistency for Horn clauses • In general, testing for consistency to see what negated ground literals to add to KB can be expensive! • Not so for Horn clauses: Theorem:CWA(KB) isalwaysconsistent when KB is a consistent set of Horn clauses. • Follows from the fact that Horn clauses have a single positive literal. • Variant: define CWA for a subset of clauses only.

  18. Restricted CWA • Define the predicates on which CWA is applied KB: Xq(X) => p(X) q(a) p(b) \/ r(b) If we apply CWA to p(X), we will conclude only ~p(b), which keeps consistency (~r(b) cannot be inferred)

  19. Other assumptions • Domain closure assumption: Limit the constant terms in the language to be those that can be named using constant and function symbols in KB. Strong assumption: allows replacing universal quantifiers by finite conjunctions and disjunctions. Xp(X) is (X=a1 \/ X = a2…) /\p(X) • Unique names assumption:if ground terms cannot be proved equal, assume they can be assumed unequal. p(f1(a)) = p(f2(a)) where f1 and f2 are Skolem functions

  20. 2. Default rules and theories • Define a nonstandard, nonmonotonic set D of inference rules to augment the basic KB. The augmentation of KB with D, denotedE(KB,D) is the theorythatcontains the usualconclusions + those obtained by applying D on KB. • The default rules in D are of the form: bird(X) : flies(X) flies(X) “if X is a bird, and it is consistent to believe that X can fly, then it can be believed that X can fly”.

  21. Default rules semantics (X): (X) (X) • If there is an instance x of X for which the ground instance (x) follows fromE(KB,D) and for which(x) is consistent withE(KB,D), then include (x) inE(KB,D). • Default rules are useful to express beliefs that are usually but not necessarily true • In general, E(KB,D) is not unique!

  22. Example 1 • KB:bird(tweety) Xostrich(X) => ~flies(X) D: bird(X): flies(X) flies(X) thenflies(tweety)is in E(KB,D). • If we add ostrich(tweety), then we cannot deduceflies(tweety)because it is not consistent with KB.

  23. Example 1 (continued) KB: feathers(tweety) D: bird(X) : flies(X) feathers(X): bird(X) flies(X) bird(X) Default proof that flies(tweety). If we add ostrich(tweety) ostrich(X) => ~flies(X) ostrich(X) => feathers(X) we cannot (as expected) prove that flies(tweety).

  24. Example 2 • Is the default rule :~p(X) ~p(X) the same as CWA? No! • KB: p \/ q D: :~p and:~q ~p ~q • CWA(KB) is inconsistent • E(KB,D) can be either {p \/ q, ~p} or{p \/ q,~q} • However, the union of both is inconsistent!

  25. Properties of default theories • Default theories might have more than one augmentation (see previous example) • There are default theories with no augmentations KB = {p(X)}, D is:p(X)/~p(X) • Every normal default theory (only D statements of the form (X): (X)/ (X))hasan augmentation • IfD’ Dare sets of normal rulesthen for anyE(KB,D’) there is aE(KB,D) such thatE(KB,D’) E(KB,D). • Normal default rules are semi-monotonic.

  26. Example 3: anomalities (1) • Typically, drug dealers are adults • Typically, adults are employed dealer(X): adult(X) adult(X):employed(X) adult(X) employed(X) dealer(joe) adult(joe) (from default rule 1) employed(joe) (from default rule 2) Question: how to fix this anomality?

  27. Example 3: anomalities (2) • Exchange the second rule by: adult(X) : ~dealer(X) /\ employed(X) employed(X) but it is not in normal form! • Consider instead the new rules: dealer(X): adult(X) adult(X) /\ ~dealer(X) :employed(X) adult(X) employed(X) adult(X): ~dealer(X) ~dealer(X)

  28. Default rules: observations • The difference between CWA(KB) and E(KB,D) CWA: add ~p if consistent with KB Default: add ~p if consistent with E(KB,D) => the order matters!! • Inference with normal defaults: (X): (X) (X) Forward: check (X) at the time of the application Backward: two passes. First, ignore consistency checks and them verify them on the resulting chain

  29. More on nonmonotonic logics • Many different formalisms to deal with inferences of this type • circumscription: extension of predicate completion • Truth Maintenance Systems • To learn more, see advanced courses • deduction systems • advanced logics and AI courses

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