Logic and Artificial Intelligence Lecture 13

Transcription

1 Logic and Artificial Intelligence Lecture 13 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit October 13, 2011 Logic and Artificial Intelligence 1/23

2 Epistemic Plausibility Models P P w v Epistemic-Plausibility Model: M = W, { i } i A, { i } i A, V Language: ϕ := p ϕ ϕ ψ K i ϕ B ϕ ψ [ i ]ϕ B s ϕ Truth: [[ϕ]] M = {w M, w = ϕ} M, w = K i ϕ iff for all v W, if w i v then M, v = ϕ M, w = B ϕ i ψ iff for all v Min i ([[ϕ]] M [w] i ), M, v = ψ M, w = [ i ]ϕ iff for all v W, if v i w then M, v = ϕ M, w = B s ϕ iff [[ϕ]] M [w] i and [[ϕ]] M i [[ ϕ]] M Logic and Artificial Intelligence 2/23

3 Grades of Doxastic Strength v 0 v 1 w v 2 Suppose that w is the current state. Knowledge (KP) Belief (BP) Safe Belief ( P) Strong Belief (B s P) Logic and Artificial Intelligence 3/23

4 Grades of Doxastic Strength v 0 v 1 w v 2 Suppose that w is the current state. Knowledge (KP) Belief (BP) Robust Belief ( P) Strong Belief (B s P) Logic and Artificial Intelligence 3/23

5 Grades of Doxastic Strength p p p p v 0 v 1 w v 2 Suppose that w is the current state. Belief (Bp) Robust Belief ( P) Strong Belief (B s P) Knowledge (KP) Logic and Artificial Intelligence 3/23

6 Grades of Doxastic Strength p p p p v 0 v 1 w v 2 Suppose that w is the current state. Belief (Bp) Robust Belief ([ ]p) Strong Belief (B s P) Knowledge (KP) Logic and Artificial Intelligence 3/23

7 Grades of Doxastic Strength p p p p v 0 v 1 w v 2 Suppose that w is the current state. Belief (Bp) Robust Belief ([ ]p) Strong Belief (B s p) Knowledge (KP) Logic and Artificial Intelligence 3/23

8 Grades of Doxastic Strength p p p p v 0 v 1 w v 2 Suppose that w is the current state. Belief (Bp) Robust Belief ([ ]p) Strong Belief (B s p) Knowledge (Kp) Logic and Artificial Intelligence 3/23

9 Agents may differ in precisely how they incorporate new information into their epistemic states. These differences are based, in part, on the agents perception of the source of the information. For example, an agent may consider a particular source of information infallible (not allowing for the possibility that the source is mistaken) or merely trustworthy (accepting the information as reliable, though allowing for the possibility of a mistake). Logic and Artificial Intelligence 4/23

10 Hard and Soft Updates M = W, { i } i A, { i } i A, V Find out that ϕ M = W, { i } i A, { i } i A, V W Logic and Artificial Intelligence 5/23

11 T 1, H 2 w 2 H 1, H 2 T 1, T 2 w 1 H 1, T 2 w 4 w 3 Min ([w 1 ]) = {w 4 }, so w 1 = B(H 1 H 2 ) Min ([w 1 ] [[T 1 ]] M ) = {w 2 }, so w 1 = B T 1 H 2 Min ([w 1 ] [[T 1 ]] M ) = {w 3 }, so w 1 = B T 2 H 1 Logic and Artificial Intelligence 6/23

12 T 1, H 2 w 2 H 1, H 2 T 1, T 2 w 1 H 1, T 2 w 4 w 3 Suppose the agent finds out that T 1 is/may be true. Logic and Artificial Intelligence 6/23

13 T1, H2 w 1 T1, T2 w 2 H1, T2 w 3 w 4 H1, H2!(T 1 ) = T 1, T 2 w 1 T 1, H 2 w 2 Suppose the agent finds out that T 1 is/may be true. (T 1 ) = T 1, T 2 H 1, T 2 Logic and Artificial Intelligence 6/16 w 1 B T 2 H 1 w 3 w 4 H 1, H 2 w 2 T 1, H 2 (T 1 ) = H 1, T 2 w 3 B T 2 T 1 w 4 H 1, H 2 w 1 T 1, T 2 w 2 T 1, H 2 Logic and Artificial Intelligence 6/23

14 Informative Actions E D B A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Logic and Artificial Intelligence 7/23

15 Informative Actions E D B A ϕ C Incorporate the new information ϕ(!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Logic and Artificial Intelligence 7/23

16 Informative Actions E D B A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i B M!ϕ = W!ϕ, {!ϕ i } i A, V!ϕ W!ϕ = [[ϕ]] M!ϕ i = i (W!ϕ W!ϕ )!ϕ i = i (W!ϕ W!ϕ ) Logic and Artificial Intelligence 7/23

17 Informative Actions E D B A ϕ C Radical Upgrade: ( ϕ): A i B i C i D i E, M ϕ = W, { i } i A, { ϕ i } i A, V Let [[ϕ]] w i = {x M, x = ϕ} [w] i for all x [[ϕ]] w i and y [[ ϕ]] w i, set x ϕ i y, for all x, y [[ϕ]] w i, set x ϕ i y iff x i y, and for all x, y [[ ϕ]] w i, set x ϕ i y iff x i y. Logic and Artificial Intelligence 7/23

18 Informative Actions E D B A ϕ C Conservative Upgrade: ( ϕ): A i C i D i B E Conservative upgrade is radical upgrade with the formula best i (ϕ, w) := Min i ([w] i {x M, x = ϕ}) 1. If v best i (ϕ, w) then v ϕ i x for all x [w] i, and 2. for all x, y [w] i best i (ϕ, w), x ϕ i y iff x i y. Logic and Artificial Intelligence 7/23

19 Reduction Axioms [ ϕ]b ψ χ (L(ϕ [ ϕ]ψ) B ϕ [ ϕ]ψ [ ϕ]χ) ( L(ϕ [ ϕ]ψ) B [ ϕ]ψ [ ϕ]χ) Logic and Artificial Intelligence 8/23

20 Reduction Axioms [ ϕ]b ψ χ (L(ϕ [ ϕ]ψ) B ϕ [ ϕ]ψ [ ϕ]χ) ( L(ϕ [ ϕ]ψ) B [ ϕ]ψ [ ϕ]χ) [ ϕ]b ψ χ (B ϕ [ ϕ]ψ B [ ϕ]ψ [ ϕ]χ) ( B ϕ [ ϕ]ψ B ϕ [ ϕ]ψ [ ϕ]χ) Logic and Artificial Intelligence 8/23

21 Composition [!ϕ][!ψ]χ [!(ϕ [!ϕ]ψ)]χ Logic and Artificial Intelligence 9/23

22 Composition p, q p, q w 1 w 2 (p q) w 3 p, q p, q p, q p, q w 2 w 3 (p) w 1 p, q p, q w 3 w 2 w 1 p, q Logic and Artificial Intelligence 9/23

23 What happens as beliefs change over time (iterated belief revision)? Logic and Artificial Intelligence 10/23

24 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new!ϕ 1!ϕ 2!ϕ 3!ϕ n M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Logic and Artificial Intelligence 11/23

25 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new ϕ 1 ϕ 2 ϕ 3 ϕ n M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Logic and Artificial Intelligence 11/23

26 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new!ϕ 1 ϕ 2 ϕ 3 ϕ n M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Logic and Artificial Intelligence 11/23

27 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new τ(ϕ 1 ) τ(ϕ 2 ) τ(ϕ 3 ) τ(ϕ n ) M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Logic and Artificial Intelligence 11/23

28 Iterated Updates!ϕ 1,!ϕ 2,!ϕ 3,...,!ϕ n always reaches a fixed-point p p p Contradictory beliefs leads to oscillations ϕ, ϕ,... Simple beliefs may never stabilize ϕ, ϕ,... Simple beliefs stabilize, but conditional beliefs do not A. Baltag and S. Smets. Group Belief Dynamics under Iterated Revision: Fixed Points and Cycles of Joint Upgrades. TARK, Logic and Artificial Intelligence 12/23

29 r n d w 1 w 2 w 3 (r (d Bd) ( d Bd) r d n w 1 w 3 w 2 (r (d Bd) ( d Bd) r n d w 1 w 2 w 3 Logic and Artificial Intelligence 13/23

30 Let ϕ be (r (B r q p) (B r p q)) w 3 p w 2 q w 3 p w 2 q ϕ = w 3 p ϕ = w 2 q ϕ = w 1 r w 1 r w 1 r M 1 M 2 M 3 Logic and Artificial Intelligence 14/23

31 Suppose that you are in the forest and happen to a see strange-looking animal. Logic and Artificial Intelligence 15/23

32 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. Logic and Artificial Intelligence 15/23

33 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. The guidebook says that the animal is a type of bird, so that is what you conclude: The animal before you is a bird. After looking more closely, you also notice that the animal is also red. Logic and Artificial Intelligence 15/23

34 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. The guidebook says that the animal is a type of bird, so that is what you conclude: The animal before you is a bird. After looking more closely, you also notice that the animal is also red. So, you also update your beliefs with that fact. Logic and Artificial Intelligence 15/23

35 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. The guidebook says that the animal is a type of bird, so that is what you conclude: The animal before you is a bird. After looking more closely, you also notice that the animal is also red. So, you also update your beliefs with that fact. Now, suppose that an expert (whom you trust) happens to walk by and tells you that the animal is, in fact, not a bird. Logic and Artificial Intelligence 15/23

36 b, r b, r b, r b, r b b, r b, r b, r b, r r b, r b, r b, r b, r M 0 M 1 b M 2 b, r b, r b, r b, r M 3 Logic and Artificial Intelligence 16/23

37 Note that in the last model, M 3, the agent does not believe that the bird is red. Logic and Artificial Intelligence 17/23

38 Note that in the last model, M 3, the agent does not believe that the bird is red. The problem is that there does not seem to be any justification for why the agent drops her belief that the bird is red. This seems to result from the accidental fact that the agent started by updating with the information that the animal is a bird. Logic and Artificial Intelligence 17/23

39 Note that in the last model, M 3, the agent does not believe that the bird is red. The problem is that there does not seem to be any justification for why the agent drops her belief that the bird is red. This seems to result from the accidental fact that the agent started by updating with the information that the animal is a bird. In particular, note that the following sequence of updates is not problematic: Logic and Artificial Intelligence 17/23

40 b, r b, r b, r b, r r b, r b, r b, r b, r b b, r b, r b, r b, r M 0 M 1 b M 2 b, r b, r b, r b, r M 3 Logic and Artificial Intelligence 18/23

41 t 0 b r (b r) t 1 t 2 t 3 r b t 4 t 5 Logic and Artificial Intelligence 19/23

42 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Logic and Artificial Intelligence 20/23

43 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Logic and Artificial Intelligence 20/23

44 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Logic and Artificial Intelligence 20/23

45 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Logic and Artificial Intelligence 20/23

46 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Logic and Artificial Intelligence 20/23

47 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Logic and Artificial Intelligence 20/23

48 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Logic and Artificial Intelligence 20/23

49 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Logic and Artificial Intelligence 20/23

50 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Logic and Artificial Intelligence 20/23

51 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Logic and Artificial Intelligence 20/23

52 Many of the recent developments in this area have been driven by analyzing concrete examples. This raises an important methodological issue: Implicit assumptions about what the actors know and believe about the situation being modeled often guide the analyst s intuitions. In many cases, it is crucial to make these underlying assumptions explicit. The general point is that how the agent(s) come to know or believe that some proposition p is true is as important (or, perhaps, more important) than the fact that the agent(s) knows or believes that p is the case Logic and Artificial Intelligence 21/23

53 meta-information: information about how trusted or reliable the sources of the information are. Logic and Artificial Intelligence 22/23

54 meta-information: information about how trusted or reliable the sources of the information are. This is particularly important when analyzing how an agent s beliefs change over an extended period of time. For example, rather than taking a stream of contradictory incoming evidence (i.e., the agent receives the information that p, then the information that q, then the information that p, then the information that q) at face value (and performing the suggested belief revisions), a rational agent may consider the stream itself as evidence that the source is not reliable Logic and Artificial Intelligence 22/23

55 meta-information: information about how trusted or reliable the sources of the information are. This is particularly important when analyzing how an agent s beliefs change over an extended period of time. For example, rather than taking a stream of contradictory incoming evidence (i.e., the agent receives the information that p, then the information that q, then the information that p, then the information that q) at face value (and performing the suggested belief revisions), a rational agent may consider the stream itself as evidence that the source is not reliable procedural information: information about the underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. A protocol describes what the agents can or cannot do (say, observe) in a social interactive situation or rational inquiry. Logic and Artificial Intelligence 22/23

56 Discussion A key aspect of any formal model of a (social) interactive situation or situation of rational inquiry is the way it accounts for the...information about how I learn some of the things I learn, about the sources of my information, or about what I believe about what I believe and don t believe. If the story we tell in an example makes certain information about any of these things relevant, then it needs to be included in a proper model of the story, if it is to play the right role in the evaluation of the abstract principles of the model. (Stalnaker, pg. 203) R. Stalnaker. Iterated Belief Revision. Erkentnis 70, pgs , Logic and Artificial Intelligence 23/23