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1 LSE-Groningen Workshop I 1 Iterated Dynamic Belief Revision Sonja Smets, University of Groningen website: Joint work with Alexandru Baltag, COMLAB, Oxford University

2 LSE-Groningen Workshop I 2 Belief Revision Standard Belief Revision (BR) theory: the AGM (Alchourrón, Gärdenfors, Makinson) axioms; the KM (Katsuno-Mendelzon) postulates; semantics (Grove sphere models); extensions and alternatives (see e.g. H. Rott).

3 LSE-Groningen Workshop I 3 The Success Postulate BR assumes as given: 1. theories ( belief sets ) T 2. new information (a formula) ϕ 3. a revision operator T ϕ I mention here only one of the AGM axioms, also accepted by KM, but which poses problems when revising with doxastic sentences: Success : ϕ T ϕ

4 LSE-Groningen Workshop I 4 Interpretation T ϕ is supposed to represent the new theory after learning ϕ: the agent s new set of beliefs, given that the initial set of beliefs was T and that the agent has learned ϕ (and only ϕ). But, beliefs about what? Beliefs have to have a content and a referent: you believe something about something. The content is a sentence in some language L. So theories are subsets T L. The referent is reality : you believe the sentence to hold in the real world.

5 LSE-Groningen Workshop I 5 Reality There always is a real state of the world s that our theories are about. A theory T is true in state s iff s = T.

6 LSE-Groningen Workshop I 6 Problems: Classical belief revision, as well as its more dynamic KM version, encounter a number of problems: 1. iterating belief revision; 2. dealing with higher-order beliefs (beliefs about other beliefs); 3. dealing with multi-agent beliefs; 4. the multiplicity of possible belief revision policies: there is no unique. In this talk we ll focus more on the first two problems.

7 LSE-Groningen Workshop I 7 Higher-Order Beliefs: No Success Take a Moore sentence: ϕ := p Bp After ϕ is learned, ϕ obviously becomes false! But the Success Postulate asks us to believe (after learning ϕ) that ϕ is true! In other words, it forces us (as a principle of rationality!) to acquire false beliefs!

8 LSE-Groningen Workshop I 8 The usual way to deal with this: simply accept that AGM cannot deal with higher-order beliefs, so limit the language L to formulas that express only factual, non-doxastic properties of the world. But the interpretation we will give to shows that AGM does not need to be limited in this way: it is perfectly natural and coherent as a belief revision theory about a fixed state of the world (the original state s 0 ).

9 LSE-Groningen Workshop I 9 The Ramsey Test The Ramsey Test asks for some notion of conditional such that, for all the possible belief sets T, we have: (ϕ ψ) T iff ψ T ϕ. T ϕ ψ ϕ T ϕ ψ

10 LSE-Groningen Workshop I 10 Impossibility theorem A key result in the subject is Gärdenfors Impossibility Theorem: there is no notion of conditional satisfying the Ramsey test with respect to some (non-trivial) operation of revision on belief sets that satisfies the AGM postulates.

11 LSE-Groningen Workshop I 11 Changing beliefs about an unchanging world The assumption underlying AGM theory is that the world that our beliefs are about is not changed by our changes of belief. But the world the higher-order beliefs are about includes the beliefs themselves. So (as the example of Moore sentences shows) the world, in this sense, is always changed by our changes of belief!

12 LSE-Groningen Workshop I 12 Saving AGM Nevertheless, we can reinterpret the AGM postulates to make them applicable to doxastic sentences: If T is the belief set at a given moment about the real state s at that moment, then T ϕ should be understood as a belief set about the same state s, as it was before the learning took place. In other words, T ϕ captures the agent s beliefs after learning ϕ about what was the case before the learning.

13 LSE-Groningen Workshop I 13 T ψ,... ϕ ψ,... T ϕ s Bψ,... ϕ B < before > ψ,... s ϕ Here, s ϕ is the state of the world after the agent learns that ϕ was true in the (initial) state s of the world. Also, < before > θ is a past tense operator, saying that θ was true before the last action.

14 LSE-Groningen Workshop I 14 Static AGM revision Let th(s) =: {ψ L : s = Bψ} be the agent s original belief set (theory) in state s about the (original) state s. Then Instead, th(s) ϕ th(s ϕ). th(s) ϕ := {ψ : < before > ψ th(s ϕ) } So AGM belief revision is static in an essential sense!

15 LSE-Groningen Workshop I 15 Iteration This is exactly what poses problems for iterating revision: what state is (T ϕ) ψ about? Well, the answer is: it is again about the same state s that T was about.

16 LSE-Groningen Workshop I 16 Initial(0) N ext(1) N ext(2) Belief T 0 ϕ 0 T 1 = T 0 ϕ 0 ϕ 1 T 2 = T 1 ϕ 1 Reality s 0 ϕ 0 s 1 = s 0 ϕ 0 ϕ 1 s 2 = s 1 ϕ 1

17 LSE-Groningen Workshop I 17 Static versus dynamic revision Compare this with a notion of dynamic revision, defined as th(s) ϕ := th(s ϕ) This operator will NOT satisfy the Success postulate. But it does something better: it keeps up with the change of reality.

18 LSE-Groningen Workshop I 18 Iterated dynamic revision T 0 = th(s 0 ), T i+1 = T i ϕ i = ( (th(s 0 ) ϕ 0 ) ) ϕ i Initial(0) N ext(1) N ext(2) Belief T 0 ϕ 0 T 1 = T 0 ϕ 0 ϕ 1 T 2 = T 1 ϕ 1 Reality s 0 ϕ 0 s 1 = s 0 ϕ 0 ϕ 1 s 2 = s 1 ϕ 1

19 LSE-Groningen Workshop I 19 So what s wrong with the Ramsey test? Suppose Ramsey test holds for static revision, i.e. there exists a conditional such that (ϕ ψ) T iff ψ T ϕ. holds for all belief sets T. Then in particular, it must hold for theory T 0 = th(s 0 ), capturing the agent s original beliefs in the original state of the world:

20 LSE-Groningen Workshop I 20 T 0 ϕ ψ ϕ ψ T 1 = T 0 ϕ s 0??? ϕ??? s 1 But then what is the semantics of? Is ϕ ψ true in the original world s 0? Obviously, the above clause implies that ϕ ψ cannot be a factual property of the world. It is a doxastic property, that expresses a feature of the agent s belief revision policy: if given information ϕ, the agent would come to believe that ψ was the case. In other words, ϕ ψ is in fact a conditional belief statement B ϕ ψ.

21 LSE-Groningen Workshop I 21 Conditional beliefs But a fully introspective agent knows his conditional beliefs (or his belief revision policy)! So, B ϕ ψ is believed at s 0 iff it is true at s 0 : T 0 B ϕ ψ ϕ ψ T 1 = T 0 ϕ s 0 B ϕ ψ ϕ??? s 1 This gives us the desired truth clause for the doxastic conditional B ϕ ψ: s = B ϕ ψ iff ψ th(s) ϕ

22 LSE-Groningen Workshop I 22 Weak Ramsey test We succeeded to define a conditional operator that satisfies a weak version of the Ramsey test, stated only for the original theory th(s) at any state s: B ϕ ψ th(s) iff ψ th(s) ϕ

23 LSE-Groningen Workshop I 23 Can this possibly satisfy the Ramsey test in its original (strong) version? This would require it to be true at all iterated theories T i = ( (th(s 0 ) ϕ 0 ) ) ϕ i 1. T 0 T i T i+1 s 0... B ϕ ψ ϕ B ϕ ψ T i iff (??) ψ T i ϕ ψ

24 LSE-Groningen Workshop I 24 Ramsey test must fail for iterated theories But recall these theories are all about the initial state s 0! But, by introspection, the agent already knows whether or not B ϕ ψ holds at s 0, and so this belief is not subject to revision: B ϕ ψ T i iff B ϕ ψ T 0 iff s 0 = B ϕ ψ iff ψ T 0 ϕ In other words, the presence of B ϕ ψ in any of the iterated revised theories T i is equivalent to its presence in T 0, and has only to do with the one-step revision T 0 ϕ of T 0, but not with the revision T i ϕ of T i!

25 LSE-Groningen Workshop I 25 Dynamic Ramsey test However, the Ramsey test can hold for dynamic revision: there does exist a conditional such that (ϕ ψ) T iff ψ T ϕ. T ϕ ψ ϕ T ϕ ψ

26 LSE-Groningen Workshop I 26 Dynamic Ramsey test - continued Take the dynamic learning modality, given by s = [ϕ]ψ iff s ϕ = ψ whenever s = ϕ Then Ramsey test for dynamic revision holds if we take as our conditional the updated belief operator [ϕ]bψ: [ϕ]bψ T iff ψ T ϕ.

27 LSE-Groningen Workshop I 27 Static and Dynamic Conditionals The conditional B ϕ ψ is static w.r.t. the actual state of the system: the state doesn t change, only the agent s belief about it changes. On the other hand, [ϕ]bψ is a dynamic conditional: the state changes (the agent learns ϕ), and after that the agent beliefs ψ. In fact, we can understand conditional beliefs dynamically: B ϕ ψ means that, after learning ϕ, the agent believes that ψ was true (in the original state) before the learning: B ϕ ψ = [ϕ]b < before > ψ

28 LSE-Groningen Workshop I 28 SEMANTICS: Plausibility (Grove) Models I only present the finite-state, single-agent case. The infinite-model, multiple-agent case is more complex. A (finite) plausibility frame is a finite set S of states (or possible worlds ) together with a connected preorder S S, called plausibility relation. Preorder : reflexive and transitive. Connected (or complete ): for all states s, t S, we have either s t or t s. This is essentially the same as a finite Grove sphere model.

29 LSE-Groningen Workshop I 29 (Conditional) Belief in Plausibility Models A sentence ϕ is believed in (any state of) a plausibility model (S ) if ϕ is true in all the most plausible worlds; i.e. in all minimal states in the set Min S := {s S : s t for all t S}. More generally, a sentence ϕ is believed conditional on P (in which case we write B P ϕ) if ϕ is true at all most plausible worlds satisfying P; i.e. in all the states in the set Min P := {s P : s t for all t S}.

30 LSE-Groningen Workshop I 30 Example 1 A coin is flipped in front of an agent (let s call her Alice), landing on the table face-up but being immediately covered, so that Alice cannot see the face. Nevertheless, Alice believes that the upper face is Heads (say, because she had a brief glimpse at the coin before it was covered, and thought that she may have seen it Heads up). And in fact, she s right: although she doesn t know it for sure, the coin is actually lying Heads up.

31 LSE-Groningen Workshop I 31 A Model for Example 1 Drawing Convention: We use labeled arrows to represent the converse plausibility relation (going from less plausible to more plausible states), but for convenience we skip all the loops (since is reflexive). A model S for Example 1: H The actual state s is the one on the left. The agent truthfully believes Heads is up; but she doesn t know that Heads is up. T

32 LSE-Groningen Workshop I 32 Dynamic Belief Revision: upgrades From a semantic point of view, dynamic belief revision is about revising the whole relational structure: changing the plausibility order (or the models). An upgrade is a model-changing operation α, that takes a plausibility model S = (S ) and returns a new model α(s) = (S, ), having the same set of states(but possibly a different order relation). An upgrade essentially encodes a belief-revision policy. Any upgrade gives a possible semantics to both the static and the dynamic revision operators ( and ).

33 LSE-Groningen Workshop I 33 Examples of Upgrades (2) Lexicographic upgrade ϕ: all ϕ-worlds become better (more plausible) than all ϕ-worlds, and within the two zones, the old ordering remains. (3) Conservative upgrade ϕ: the best ϕ-worlds become better than all other worlds, and in rest the old order remains.

34 LSE-Groningen Workshop I 34 Explanation After a conservative upgrade, the agent only comes to believe that ϕ (was the case). The lexicographic upgrade has a more radical effect: the agent comes to accept ϕ with such a conviction that she considers all ϕ-possibilities more plausible than all non-ϕ ones.

35 LSE-Groningen Workshop I 35 Dynamic Modalities The dynamic learning modality [ϕ]ψ introduced above in an an informal way can now be defined precisely, but its meaning (and notation) will depend on the belief revision policy: For Lexicographic Upgrades: [ ϕ]ψ For Conservative Upgrades: [ ϕ]ψ

36 LSE-Groningen Workshop I 36 Example 2: An Upgrade Alice learns from a trusted informer that the coin lies Tails up, but without being shown the face. Alice believes him. But, no matter how trusted is this informer, he is not infallible. (In fact, in this case he is wrong: the coin was Heads up!) Taking this as a (lexicographic or conservative) upgrade, the new, upgraded model is: H T

37 LSE-Groningen Workshop I 37 Fixed Points A plausibility model S = (S, ) is a fixed point for an upgrade α iff S is left unchanged by α: i.e. the changed model α(s) is the same as S. (Technically, this means that α(s) is bisimilar to S.) Informally, S is a fixed of an upgrade α iff α is not an informative upgrade of S: it is a redundant one, leaving unchanged the agent s conditional belief structure.

38 LSE-Groningen Workshop I 38 Truthfulness If we fix a state s S, called the actual world, in a plausibility model (S, ), then we say that an upgrade ( ϕ or ϕ) is truthful if ϕ is true in the actual world s.

39 LSE-Groningen Workshop I 39 Iterating Upgrades To study iterated belief revision, consider a finite model S 0 = (S, 0 ), and an ((infinite) sequence of upgrades, either of the conservative kind or the lexicographic kind ϕ 1,..., ϕ n,... ϕ 1,..., ϕ n,...

40 LSE-Groningen Workshop I 40 This leads to an infinite succession of upgraded models S 0, S 1,..., S n,... defined by: S i = α i (S i 1 ), where α i = ϕ i in the first case, and α i = ϕ i in the second case.

41 LSE-Groningen Workshop I 41 Iterated Upgrades Do Not Necessarily Stabilize! Proposition For some initial finite models, there exist infinite cycles of truthful upgrades (that never stabilize the model). Even worse, this still holds if we restrict to iterations of the same truthful upgrade (with one fixed sentence): no fixed point is reached. Moreover, when iterating conservative upgrades, even the (simple, unconditional) beliefs may never stabilize, but may keep oscillating forever.

42 LSE-Groningen Workshop I 42 Iterating a Truthful Conservative Upgrade Consider a pollster (Charles) with the following beliefs about how a given voter (Mary) will vote: s : R t : N w : D As before, we skip the loops and the arrows that can be obtained by transitivity. We assume the real world is s, so Mary will vote Republican (R)! But Charles believes that she will vote Democrat (D); and in case this turns out wrong, he d rather believe that she won t vote (N) than accepting that she may vote Republican.

43 LSE-Groningen Workshop I 43 A trusted informer studied Mary s voting behavior and tells Charles the following true statement ϕ: R (D BD) ( D BD) Either Mary will vote Republican or else your beliefs about whether or not she votes Democrat are wrong. The sentence ϕ is true in states s and t but not in w: s : R t : N w : D

44 LSE-Groningen Workshop I 44 Infinite Oscillations by Truthful Upgrades Let s suppose that Charles conservatively upgrades his beliefs with this new true information ϕ. The most plausible state satisfying ϕ was t, so this becomes now the most plausible state overall: s : R w : D t : N In this new model, the sentence ϕ is again true at the real world (s), as well as at the world w. So this sentence can again be truthfully announced.

45 LSE-Groningen Workshop I 45 However, if Charles conservatively upgrades again with this new true information ϕ, he will promote w as the most plausible state, reverting to the original model! Moreover, not only the whole model (the plausibility order) keeps changing, but Charles (simple, un-conditional) beliefs keep oscillating forever (between D and N)!

46 LSE-Groningen Workshop I 46 Iterating Truthful Lexicographic Upgrades Consider the same model with the same real state s: s : R t : N w : D But now consider the sentence R (D B R D) ( D B R D) If you d truthfully learn that Marry won t vote Republican, then your resulting belief about whether or not she votes Democrat would be wrong. This sentence is true in the real state s and in t but not in w, so a truthful lexicographic upgrade will give us:

47 LSE-Groningen Workshop I 47 w : D s : R t : N The same sentence is again true in (the real world) s and in w, so it can be again truthfully announced, resulting in: t : N w : D s : R Another truthful upgrade with this sentence produces w : D t : N s : R then another truthful upgrade with the same sentence gets us back to t : N w : D s : R

48 LSE-Groningen Workshop I 48 Stable Beliefs in Oscillating Models Clearly from now on the last two models will keep reappearing, in an endless cycle: as for conservative upgrades, the process never reaches a fixed! However, unlike in the conservative upgrade example, in this example the simple (unconditional) beliefs eventually stabilize: from some moment onwards, Charles correctly believes that the real world is s (vote Republican) and he will never lose this belief again! This is a symptom of a more general phenomenon:

49 LSE-Groningen Workshop I 49 Beliefs Stabilize in Iterated Lexicographic Upgrades Proposition: In any infinite sequence of truthful lexicographic upgrades { ϕ i } i on an initial (finite) model S 0, the set of most plausible states stabilizes eventually, after finitely many iterations. From then onwards, the simple (un-conditional) beliefs stay the same (despite the possibly infinite oscillations of the plausibility order).

50 LSE-Groningen Workshop I 50 Upgrades with Un-conditional Doxastic Sentences Moreover, if the infinite sequence of lexicographic upgrades { ϕ i } i consists only of sentences belonging to the language of basic doxastic logic (allowing only for simple, un-conditional belief operators) then the model-changing process eventually reaches a fixed point: after finitely many iterations, the model will stay unchanged. As we saw, this is not true for conservative upgrades.

51 LSE-Groningen Workshop I 51 Conclusions Iterating static belief revision suffers from the failure of the (strong) Ramsey Test. Dynamic belief revision satisfies the Ramsey Test. Iterated upgrades may never reach a fixed point: conditional beliefs may remain forever unsettled. When iterating truthful lexicographic upgrades, simple (non-conditional) beliefs converge to some stable belief. Truthful conservative upgrades do not have this last property.

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