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. - PowerPoint PPT Presentation
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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?X citizen(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
1. Closed-world assumption (CWA)if you cannot prove P or ~P from KB,
add ~P to KBKB |= 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.
• 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 follows from KB, where pi is a positive-ground literal, there is at least one pi such that KB |= pi
Eq: CWA(KB) is inconsistent iff there are positive ground literals p1, … pn
such that KB |= p1 \/ p2 \/ …. \/ pn but for all i, KB | pi.
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.
X p(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
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, denoted E(KB,D) is the theory that contains the usual conclusions + 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”.
• If there is an instance x of X for which the ground instance (x) follows from E(KB,D) and for which (x) is consistent with E(KB,D), then include (x) in E(KB,D).
• Default rules are useful to express beliefs that are usually but not necessarily true
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)