İlk sayfaya geç Many Birds Fly, Some dont Elaborations on a Quantification Approach to the Problem of Qualification

İlk sayfaya geç Content 4 Introduction 4 The Proposed Approach ; Qualification as Quantification 4 Difficulties and Limitations 4 Partial Set Inclusion 4 Epistemic Entrenchment on a Belief 4 Propositions 4 Multiple Set Inclusion 4 Conclusions

İlk sayfaya geç Introduction n Programs with common-sense -> Qualification. u Representing characteristics Û All birds except penguins, ostriches, and... roasted chickens. n First Order Logic Û All birds fly except those which are abnormal. Tweety is abnormal, Ostriches are abnormal. n Plausible conclusions without contrary evidence Û If Woody is a bird, and, if it cannot be inferred that it is abnormal, derive the conclusion that it flies

İlk sayfaya geç Introduction (cont) n Jumping to conclusion u Minimization approach u ignorance n Non-monotonic reasoning Û If now it is known that Woody is an ostrich, then abandon the previous conclusion and derive that it doesn't fly. n "Intended" conclusion If birds typically fly and ostriches typically don't, knowing that Tweety is an ostrich and a bird (and not knowing that ostriches are birds), decide what to conclude, whether it flies or not

İlk sayfaya geç The Proposed Approach ; Qualification as Quantification n Universal Quantification Û Birds fly. Û Elephants are grey. u inadequate, lacks expressiveness n "Birds fly" should be associated with a degree; Epistemic entrenchment u 0 : lowest degree u 1 : highest degree n Ignorance 0.5 of birds fly can not say if Tweety flies or not

İlk sayfaya geç Difficulties and Limitations n Assessment of degree of epistemic entrenchment u exhibiting certain characteristics u collection of objects n Depth of chaining Û If birds typically fly, and some flying objects lay eggs, (and not knowing whether birds lay eggs or not), can it be derived that birds typically lay eggs?

İlk sayfaya geç Partial Set Inclusion A formal system - an extension of First Order Logic Definition 1 : Consider sets A and B with N and M elements, respectively. The degree of partial set inclusion of A in B, I (A, B) is ( If P is the cardinality of An B ) I (A, B) ={P / N if N > 0 1 if N = 0 Trivial Conclusions ; A B I(A, B) = 1, B A I(A, B) = M/N, A B=0 I(A,B)=0. Degree of partial set inclusion epistemic entrenchment Û Men are mortal Û Elephants are grey

İlk sayfaya geç Epistemic Entrenchment on a Belief Definition 2 : Consider two sets A and B, so that I(A, B) =, and x is any element of the set A. i.e., x A. (i) the value of the epistemic entrenchment on the belief "A is included in B" is given by I(A, B). (ii) knowing that x A, the value of the epistemic entrenchment on the belief "x belongs to B", denoted by E(x, A, B), is given by I(A, B). E(x, A, A) = I(A, A) = 1

İlk sayfaya geç Epistemic Entrenchment on a Belief (cont) Birds fly.Ostriches fly. B = {b l,...,b N } be the set of birds, 0 = {O 1,...,O M } the set of ostriches and F the set of flying objects. (i) the set B in the set F, I(B, F) =, (ii) the set 0 in the set F, 1(0, F) =β. Ex ; Tweety is a bird, i.e., E(Tweety, B, B) = 1 and I(B,F) = 0.8 => the degree of epistemic entrenchment that Tweety is a flying object is 0.8. Û If it is known that Tweety is a bird and most birds fly then an agent should have a "strong belief" that Tweety is a flying object.

İlk sayfaya geç Epistemic Entrenchment on a Belief (cont) D = {(b i, 0 1 ), (b l, 02),..., (b i,O M ), (b 2, 0 1 ),..., (b N, O M )} n one of the pairs (b i, O J ) is Tweety. All the elements of B F and 0 F belong to F, u a pair (b i, O J ) with b i B F and O J 0 F is an incompatible pair for Tweety. (1 - )N β M of such pairs. u a pair (b i, O J ) with b i B n F and O J 0 F is also an incompatible pair for Tweety. N(l - β)M of such pairs. u (i) N β M is the number of pairs with b i B F, and O J 0 F (ii) (1 - )N(1 - β)M is the number of pairs with b i B F and O J 0 F.

İlk sayfaya geç Propositions : Proposition 1 Let B and O be two sets partially included in a set F with degrees of partial set inclusion I(B, F) = and I(0, F) =. respectively. Let x be an element belonging to B 0. Let also D be the cartesian product of B and 0, D = B x 0, The proportion of pairs of D compatible for x with both elements belonging to F, among the compatible pairs for x is given by = / ( + (1- )(1- )) If a=0.8 and b=0.2 epistemic entrenchment is 0.5 (Symmetrical Evidences)

İlk sayfaya geç Propositions : Proposition 2 Let B and 0 be two sets partially included in a set F, I(B, F)= and I(0,F) =. Let x be an element included in B and O. n If I(O, B) = 1 then I(B O, F) =. Û If it is known that Tweety is a bird and an ostrich, and all ostriches are birds and most ostriches do not fly, then an agent should have a "strong belief" that Tweety is not a flying object. n If E(Tweety, Y O, F) = 1 and E(Tweety, B 0, F) = 0.2 then E(Tweety, B O Y, F) = 1 Û If it is known that Tweety is a yellow ostrich, and all yellow ostriches fly, then an agent totally believes that Tweety is a flying object.

İlk sayfaya geç Multiple Set Inclusion I(A, B) = and I(B, C) = I(A, C)=? Û If most birds fly and most flying objects lay eggs can it be concluded that most birds lay eggs? Proposition 3 Consider a set A and a set B with N and M elements, respectively. Let C be another set. If I (A, B) = and I (B, C) =, I(A, C) is given by max(O, - (1 - )N) I(A, C) min(1, 1 - ( - M /N)) Û If most birds fly and most flying birds lay eggs can it be concluded that most birds lay eggs?

İlk sayfaya geç Multiple Set Inclusion (cont) Proposition 4 Consider three sets A, B and C. Assuming that the degrees I(A, B) = and I(A B, C) = are known, the interval representing the set of possible values for the value of I (A, C) is given by I(A, C) ( - 1) + 1 Ex : If 0.8 of birds are flying and 0.9 of flying birds lay eggs, then the degree of partial set inclusion of the set of birds in the set of objects laying eggs is in [0.72,0.92]

İlk sayfaya geç Multiple Set Inclusion (cont) Proposition 5 Consider three sets A, B and C. Assuming that the degrees I(A, B) =, I(A B, C) = and I(A S, C) = are known, the value I. of I(A, C) is given by I(A, C) = + (1- ) Ex : If 0.8 out of the birds are flying, 0.9 out of the flying birds lay eggs and 0.9 out of the non-flying birds also lay eggs, then the degree of partial set inclusion of the set of birds in the set of objects that lay eggs is exactly 0.9, meaning that 0.9 of the birds lay eggs.

İlk sayfaya geç Conclusions n simplicity and applicability n not a general solution, but a different perspective n enough expressive power n more informed knowledge representation n all characteristics of FOL.

İlk sayfaya geç Thank You