introduction to artificial intelligence lecture 11 : nonmonotonic reasoning

Intro to AI Fall 2002 © L. Joskowicz 1

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.


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


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!


Motivation: exceptions• All rules have exceptions! For each, we must

forsee all of them and explicitly state them: “all birds fly” X bird(X) => flies(X) “except ostriches”

X bird(X) /\ ~ostrich(X) => flies(X) “except newborns”

X bird(X) /\ ~ostrich(X) /\ ~newborn(X) => flies(X)

• We would like to conclude flies(X) from bird(X) unless something is abnormal with X

X bird(X) /\ ~Abnormal(X) => flies(X)


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


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!


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• Non-monotonic: “jump” to conclusions that can later be withdrawn (defeasible conclusions)


Consistency in NML

• A deduction rule R is consistent iff:

KR |=R c

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!


Nonmonotonic frameworks1. Closed World Assumption

if you cannot prove P or ~P from KB, add ~P to KB.

2. Default Rulesnew 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


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.


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 theoryKB: 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).


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!


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


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 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.


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 |= p and KB |= ~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 the definition of KBassumed KB |= picontradiction!


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: X p(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)


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) is always consistent 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.


Restricted CWA

• Define the predicates on which CWA is applied

KB: X q(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)


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”.


Default rules semantics

(X): (X) (X)

• 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

• In general, E(KB,D) is not unique!


Example 1• KB: bird(tweety)

X ostrich(X) => ~flies(X)

D: bird(X): flies(X)

flies(X)

then flies(tweety) is in E(KB,D).

• If we add ostrich(tweety), then we cannot deduce flies(tweety) because it is not consistent with KB.


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).


• 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!

Example 2


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)) has an augmentation

• If D’ D are sets of normal rules then for any E(KB,D’) there is a E(KB,D) such that E(KB,D’) E(KB,D).

• Normal default rules are semi-monotonic.


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?


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)


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


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

introduction to artificial intelligence lecture 11 : nonmonotonic reasoning

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