Default Reasoning - Motivation

Transcription

1 Default Reasoning - Motivation The most what we know about the world is only almost always true. We never have total knowledge about the world. / Typically a bird can fly. Unless it is a penguin ostrich kiwi etc. Possible formalization in PL1 (first order predicate logic) cx Bird(x) o lpenguin(x) o lostrich(x)... 0 Flies(x) / Problems 1) Can we ever specify... completely? 2) We cannot conclude for any bird that it can fly if we do not know that it is not a penguin not a ostrich not a kiwi and not a.

2 pproach / We would like to have something like typically birds can fly. Possibility dditional (non-logical) inference rules Bird(x) Flies(x) Flies(x) Intended meaning If x is a bird and there is no evidence that Flies(x) might be false then assume Flies(x). Exceptions are represented by simple implications cx Penguin(x) 0 lflies(x) cx Ostrich(x) 0 lflies(x) cx Kiwi(x) 0 lflies(x) Monotonicity of PL1 if X is a consequence of S then it is also a consequence of any set containing S. One cannot preempt conclusions by adding new premises.

3 Formal Framework for Non-Monotonic Reasoning PL1 with w traditional provability and Cn consequence operation. We consider only sets of closed formulas. Default rules α( x) β 1 (x)... β (x) m γ(x) in which α( x) β ( x)... β ( x) ( x) 1 m γ are formulas with free variables x = o α(x) Prerequisite / must be provable o β 1 ( x)... β (x) Justification / consistent assumptions m o γ(x) Consequent default rule is called closed if it does not have any free variables. x1... x n Default theory tuple (D W) in which D is a countable set of default rules and W is a countable set of PL1 formulas. Closed default theory ll default rules are closed. / From now on we consider closed default theories.

4 Properties of Default Reasoning Non-Monotonic Reasoning ssume there is exactly one default and nothing more is known. B Then we can conclude B. If the fact l is added B cannot be concluded! Non-Determinism Spouse(xy) hometown(y) = z hometown(x) = z hometown(x) = z Employer(xy) location(y) = z hometown(x) = z hometown(x) = z Furthermore Spouse(mary tom) hometown(tom) = toronto Employer(mary univ) location(univ) = vancouver / Default rules are used to complement incomplete theories. Thereby several complements can be possible multiple extensions (credulous strategy). / lternatively the intersection over all possbile extensions can be used. (skeptical strategy).

5 dditional Examples Default assumptions in frames and semantic networks. Example By default the hometown of a person is Palo lto. [Person UNIT Basic... <hometown {(a city) Palo lto; DEFULT}>... ] Person(x) hometown(x) hometown(x) = y = y Closed World ssumption. For each base relation R in a DB R( x R( x x... x n n ) ) Frame Default (as an alternative to frame axioms). For each relation R with a situation argument s and transition function f R(xs) R(xf(xs)) R(xf(xs))

6 Problems in Default Logic Case nalysis in Default Logic Example Emu( x) Fly( x) Fly( x) Ostrich(x) Fly(x) Fly(x) We cannot infer from (Emu(Tweety) n Ostrich(Tweety)) lfly(tweety). Two solutions 1) Combine the default rules to a single default Big Bird(x) Fly(x) Fly(x) and add Emu(x) n Ostrich(x) 0 Big-Bird(x) to the facts. 2) Reformulate the defaults (no prerequiste) Emu(x) Fly(x) Ostrich(x) Fly(x) Emu(x) Fly(x) Ostrich(x) Fly(x)

7 Extensions of Default Theories The default rules D extend the PL1 theory that is given by W Extensions of (D W). Desirable properties of an extension E of (D W) includes the facts W 4 E is deductively closed E = Cn(E) all applicable defaults fired If α β1... β γ 1) m 2 D α 2) 2 E 3) β 1... β m ; E γ then 2 E. E only consists of those facts that must be in E according to these prerequisites.

8 Formal Definition of Extensions of Default Theories Let = (D W) be a closed default theory. For each set of closed formulas S (S) is the smallest set that meets the following requirements D1 W 4 (S) D2 Cn( (S)) = (S) (Fixpoint) D3 If α β1... β γ 1) 2 D α m 2) 2 (S) and 3) β 1... β m ; S then γ2 (S). set of closed formulas E is an extension of if E = (E).

9 1) W = {B 0 l o lc} E 1 = Cn(W { C}) E 2 = Cn(W {B}) Examples = C C B B D 2) W = { } E 1 = Cn({lC}) E 2 = Cn({lD}) = C D D C D 3) W = { } E 1 = Cn({l}) E 2 = Cn({ dx P(x)}) = x P x x P x D ) ( ) (

10 Semi-Normal Defaults Besides normal defaults the semi-normal defaults are useful in I systems αβ γ γ Important for interacting normal defaults dult(x)employed(x) Dropout(x) dult(x) Employed(x) dult(x) Dropout(x) Employed(x) Employed(x) better semi-normal default dult (x)employed(x) Dropout(x) Employed(x) For so called ordered default theories a proof theory exists [Etherington 88]. But this kind of conflict resolution is not simple. Especially for more complex theories.