Perception in Update Logic

Transcription

1 Perception in Update Logic Jan van Eijck CWI, Amsterdam, Uil-OTS, Utrecht, NIAS, Wassenaar June 25, 2007 I see nobody on the road, said Alice. I only wish I had such eyes, the King remarked in a fretful tone. To be able to see Nobody! And at that distance too! Abstract Three key ways of updating one s knowledge are Lewis Carroll, Alice in Wonderland. 1. perception of states of affairs, e.g., seeing with one s own eyes that reception of messages, e.g., being told that drawing new conclusions from known facts. If one represents knowledge by means of Kripke models, the implicit assumption is that drawing conclusions is immediate. This assumption of logical omniscience is a useful abstraction. It leaves the distinction between (1) and (2) to be accounted for. In current versions of Update Logic (Dynamic Epistemic Logic, Logic of Communication and Change) perception and message reception are not distinguished. I will look at what is needed to distinguish them. 1 The Riddle of the Caps Imagine four people standing in line, with three of them looking to the left, and one looking to the right. These fellows, let us call them a, b, c, d, each wear a cap. The leftmost guy, a, can see no-one; b can see a; c can see a and b, and d, who has his head turned in the other direction, can see no-one. 1

2 Assume it is common knowledge what the perceptive capabilities of the agents are. So it is common knowledge that a can see no-one, that b can see a but no others, that c can see a and b but no others, and that d, can see no-one. In particular it is commonly known that they all wear a cap, and that no-one can see the colour of his own cap. Now let there be a public announcement there are two white caps and two black caps. After this, c knows the colour his cap, for he can reason: I see two white caps in front of me. There are only two white caps. So my own cap must be black. If c publicly announces I know the colour of my ca, then as a result b gets to know the colour of this cap as well. For b can reason as follows. If the guy behind me knows the colour of his cap, then this means that he sees two caps of the same colour in front of him. I see that a has a white cap, so my own cap must be white as well. Now suppose the initial situation is like this: the participants all face in the same directions as before, but now b and d have swapped caps. In this case, obviously, c does not know the colour of his cap. But if c publicly announces I do not know the colour of my cap, then this announcement reveals to b that the colour of his cap must be black. The ingredients that are used here are public announcements ( There are two white caps and two black caps, I know/do not know the colour of my cap ) and common knowledge about the perceptive capabilities of the participants. 2 Common Knowledge, Public Announcement and Perception Common knowledge is a term coined by David Lewis in [Lewis, 1969]. It is sometimes called mutual knowledge ([Schiffer, 1972; Clark and Marshall, 1978; 1981]), and it is essentially different from general knowledge. If a group of people say the group of NIAS fellows get separate messages that there will be dinner party on Thursday, this generates general knowledge: we all know that there is a dinner, but I do not know that you know there is a 2

3 dinner, you do not know that I know that there is a dinner,... If, instead, we all get the same about the dinner, and we see that we are all on the addressee list, this generates common knowledge. Given individual accessibilities R i for a i I, where I is a group of agents, then I-common knowledge has accessibility relation ( ) R i. i I The semantics of common knowledge is expressed in terms of a fixpoint operation, so it seems natural to assume that common knowledge is something that emerges in the limit of a series of steps of knowledge aggregation, that it is some idealized version of what happens when we acquire knowledge about knowledge in real life. Nothing could be further from the truth. Common knowledge is ubiquitous, and it comes about in a single step, in one of the following ways: by public announcement [Plaza, 1989]. by common witnessing of an event, when it is already common knowledge that all can perceive the event [Clark and Marshall, 1978] by variations and combinations of the above ( indirect co-presence, cultural co-presence [Clark and Marshall, 1978]). Since common knowledge emerges in no other ways than these, it would seem that creating common knowledge about perceptive abilities prepares the ground for creating common knowledge about perceived events. The philosophical literature on perception in particular the problems caused by perceptual illusion or hallucination is vast. See [Austin, 1962; Davidson, 1982; Dretske, 1969; 1981; Rock, 1985] for a small sample. Rather than take all of this on board, I will draw some methodological limitations from the outset, in that I will assume that perceptions are limited (not all that is known is perceived) but accurate (no illusion; all that is perceived is known). The first of these is uncontroversial. It merely states that there is more to knowledge than perception. The second allows us to get rid of puzzles about hallucination. These limitations can easily be lifted, but in this paper I see no need to do so. 3

4 Analyzing the example puzzle with this in mind, can we see what common knowledge about perceptive abilities is needed? In the example case, the following should somehow become common knowledge: that the leftmost person can see no-one, that the second person can see the first, that the third can see the first and the second, and that the rightmost guy sees no-one. To tackle this, I will use a logic of public announcement and common knowledge with the following syntax (let i range over a set of agents I, and p over a set of propositions P ): φ ::= p φ φ 1 φ 2 [α]φ [φ i ]φ 2 α ::= i?φ α 1 α 2 α 1 ; α 2 α, Assume the usual abbreviations. In particular, φ 1 φ 2 is shorthand for ( φ 1 φ 2 ), φ 1 φ 2 for (φ 1 φ 2 ), α φ for [α] φ. Finally, [i]φ is abbreviated as K i φ, and [( i I i) ]φ as Cφ. This language is interpreted in the usual multimodal Kripke models (or labeled transition systems). Such a model M is a triple (W, V, R) where W is a nonempty set of worlds, V : W P(P ) assigns a valuation to each world w W, and R : I P(W 2 ) assigns an accessibility relation i to each agent i I. If M = (W, V, R) be given, we refer to its components as W M, V M and R M. Let M be given, and let w W M. Then the truth definition is given by: M = w always M = w p : p V M (w) M = w φ : not M = w φ M = w φ 1 φ 2 : M = w φ 1 and M = w φ 2 M = w [α]φ : for all w with w α w M = w φ M = w [!φ 1 ]φ 2 : if M = w φ 1 then M φ 1 = w φ 2, with M φ 1 given by W M φ1 = {w W M M = w φ 1 }, and V M φ1 and R M φ1 the restrictions of V M and R M to W M φ1 4

5 w i w : wr M (i)w w?φ w : w = w and M = w φ w α 1;α 2 w w α 1 α 2 w : w with w α 1 w and w α 2 w : w α 1 w or w α 2 w w α w : w 1,..., w n with w = w 1, w α = w n, w i wi+1 for 1 i < n. Use [φ] M for {w W M M = w φ}. This is the public announcement restriction of the format of Logics of Communication and Change or LCC [Benthem et al., 2006b], which is in turn a version of BMS style update logic [Baltag et al., 1999]. Using b i for the cap of the i-th guy (counting from left to right) is black and D i (p) for K i p K i p, we get that update with!d i (p) expresses the public announcement: i can (perceptually) distinguish p from p. Here are the public announcements that make the relevant perceptive abilities common knowledge: The first guy cannot see the colour of any hat. D 1 (b 1 ) D 1 (b 2 ) D 1 (b 3 ) D 1 (b 4 ) The second guy can see the colour of hat 1, but not the colours of hats 2,3,4. D 2 (b 1 ) D 2 (b 2 ) D 2 (b 3 ) D 2 (b 4 ) The third guy can see the colours of hats 1,2, but not the colours of hats 3,4. D 3 (b 1 ) D 3 (b 2 ) D 3 (b 3 ) D 3 (b 4 ) The fourth guy cannot see the colour of any hat. D 4 (b 1 ) D 4 (b 2 ) D 4 (b 3 ) D 4 (b 4 ) In the semantics above, we have given a semantics of knowledge, not perception. One possible way to go in order to incorporate perception would be to extend each R i knowledge relation to an S i perception relation, add an operation for individual perception, say î, stipulate that î is interpreted by S i, and impose suitable conditions on the relations, in particular R i S i. An example would look like this (black for knowledge, red for perception): 5

6 0:bb 1:bw 2:wb 3:ww Unfortunately, there are problems with this approach. In the first place, this does not quite fit the update logic framework, for it obscures the fact that epistemic update logic is already a logic of observations [Benthem, 2005]. The main problem, however, is that this is an attempt to accommodate perception as part of the statics of what goes on. Given that perception is always perception of phenomena in a fleeting world, this is not a natural way to go. What is eternal and unchanging cannot be perceived. To be perceivable is always the hallmark of being part of the world of phenomena, the world described by esse est percipi. Perception is simply change in the world that gets noticed. Given this, the distinction between knowledge and perception reveals itself when the world itself changes. If the changes in the world go unnoticed, knowledge may be lost. If the changes in the world get noticed, we can say that change is perceived, and as a consequence knowledge gets updated. To flesh this out, what is needed is an account of perceptive abilities: Which changes can be observed by which agents? Which agents are aware of the perceptive abilities of others? Which perceptive abilities are common knowledge? 6

7 Systematic frameworks for answering these questions bring in the aspect of perspective in a natural way. There is an important distinction between individual awareness and common knowledge. Consider the example again. Imagine that the four guys with the hats are standing in line again, or rather, that someone has put them in line with their eyes closed. Now the first guy opens his eyes, and realizes: I am in a position where I see nobody. Next, the second guy opens his eyes, and realizes: I am in a position where I can see 1 in front of me. Then, the third guy opens his eyes, and realizes: I am in a position where I can see 1 and 2 in front of me. Finally, the fourth guy opens his eyes, and realizes: I am in a position where I see nobody. The important point now is that unless they all publicly announce their perception capabilities the reasoning about knowledge and ignorance scenario of the example never gets off the ground. 3 Change in the World Suppose the world changes from to That is: the second and third guy swap caps. To model this in our logic, we add substitutions. Substitutions as a mechanism for modelling actual change were first explored in [Eijck, 2004], and subsequently adopted in [Benthem et al., 2006b], and later in [Kooi, 2007]. Let a language L P be given, where P is the set of basic (factual) propositions of the language. Then an L P -substitution is a function of type L P L P that distributes over all language constructs, and that maps all but a finite number of basic propositions to themselves. L P substitutions can be represented as sets of bindings {p 1 φ 1,..., p n φ n } 7

8 where all the p i are different. If σ is an L P substitution, then the set {p P σ(p) p} is called its domain, notation dom(σ). It is customary to omit from a substitution all bindings p φ with p = φ (p not in the domain). Use ɛ for the identity substitution. Let Sub LP be the set of all L P substitutions. If M = (W, V, R) is an epistemic model and σ is an L P substitution, then VM σ is the valuation given by λp [σ(p)] M. In other words, VM σ assigns to w the set of worlds w in which σ(p) is true. For M = (W, V, R), call M σ the model given by (W, VM σ, R). Then we can extend the language with a construct [σ]φ, with the following semantic clause: M = w [σ]φ : M σ = w φ. Since applying a substitution is a functional operation we immediately get: M = w [σ]φ iff M = w [σ] φ. Using the obvious abbreviation, this says that [σ]φ is equivalent to σ φ. Now we can handle perception and change as follows. Suppose it is common knowledge that all agents can perceptively discriminate between b and b. This means that the following formula holds: C i I D i b. Suppose we change the world by swapping the truth value of b. This means we apply the following substitution: {b p}. Then if b was true before the substitution, then after it it is common knowledge that b, for the following principle holds: b C i I D i b [{b p}]c b. This illustrates the creation of common knowledge without public announcement. Another kind of change is the world is a perspective shift. Suppose the world changes from 8

9 to Again, assume that the perceptive abilities in the caps example are common kwowledge. Suppose the third guy then turns around. He is now looking to the right. Notice that nobody can see this! But the effect is dramatic. Subsequent factual changes may map S5 models to non-s5 models, for if the first guy gets a different colour cap, then the second guy will still believe that the third guy sees this, and draw the wrong conclusions. It follows that to prevent knowledge degeneration, in a system of knowledge and perception, loss of perceptive ability due to perceptive shifts must be made common knowledge. 4 Models for Epistemic/Perceptive PDL M = (W, {R i i I}, V, F ), where W is a set of worlds, the R i are binary (equivalence) relations on W, V is a valuation on W, F : P (P(I), P(I)) is an awareness function. If F (p) = (J, K), then J are the agents that are privately attuned to p, K are the agents that are publicly attuned to p, assume J K =. 9

10 5 A Language for Public Announcement, Perception, and Change Epistemic/Perceptive PDL with public announcement, factual change, and three kinds of perceptive changes: Abbreviations: K i φ for [i]φ, φ ::= p φ φ 1 φ 2 [π]φ [!φ 1 ]φ 2 [p := φ 1 ]φ 2 [Pr i p]φ [Pu i p]φ [Er i p]φ π ::= i φ? π 1 ; π 2 π 1 π 2 π C i,...,k φ for [(i k) ]φ (common knowledge) 6 Action Model for Public Announcement Everyone is aware of the announcement of p: p I Updating with this action model has the effect of removing all the worlds where p is false, while leaving the accessibility relations the same. 7 Action Model for Private Perception of State of p Assume a private observer i and suppose j, k are outsiders. 10

11 p -p jk jk p jk p jk -p -p 8 Action Model for Public Perception of State of p Assume a public observer i and suppose j, k are outsiders. p jk -p 11

12 9 Action Model for Perception of State of p Assume a private observer i and a public observer j, and suppose k is an outsider. p -p jk k p k p jk ik ik -p -p 10 Action Model for Perception of Change of State of p Assume a private observer i and a public observer j, and suppose k is an outsider. 12

13 p := phi jk p := phi k ik T [ ] M = [p] M = {w W M p V M (w)} [ φ] M = W M [φ] M [φ 1 φ 2 ] M = [φ 1 ] M [φ 2 ] M [[π]φ] M = {w W M v( if (w, v) [π ] M then v [φ] M )} [[!φ 1 ]φ 2 ] M = [φ 2 ] M φ 1 [[p := φ 1 ]φ 2 ] M = {w W M if w [pre(e)] M then (w, e) [φ] M A } with (A, e) the appropriate action model [[Pr i p]φ] M = {w W M e E s.t. [[Pu i p]φ] M = similar... [[Er i p]φ] M = [φ] M if w [pre(e)] M then (w, e) [φ] M A } with (A, E) the appropriate action model and M the result of adjusting the awareness function with M the result of adjusting the awareness function 13

14 11 Denotation of π [i] M = R i [?φ] M = {(w, w) W M W M w [φ] M } [π 1 ; π 2 ] M = [π 1 ] M [π 2 ] M [π 1 π 2 ] M = [π 1 ] M [π 2 ] M [π ] M = ([π ] M ) 12 Expressive Power Epistemic/Perceptive PDL with change and public announcement has the same expressive power as Epistemic PDL. Reduction in two stages: translate into LCC. Example clause: T F [Pr i p]φ = [(A, E)]T F φ, where (A, E) is the appropriate action model, and F is the adjusted awareness function. reduce LCC to PDL (use approach of [Benthem et al., 2006b]). Example clause: ([(A, E)][π]φ) = [π ]([(A, E)]φ), where π is the appropriately transformed PDL program π. Reduction axioms + PDL axioms give a complete axiomatisation. 13 Further Work: Modelling Limitations of Perception How does one express limitations of perceptive ability? Need to get around paradox of the heap (approach of [Shoham and del Val, 1991] not fully adequate). Distinction between actual perception and capability of perception (logic of sensors of [del Val et al., 1997]) Logic of perceptibilia [Bell, 2000]... If one represents uncertainty in perception with probabilities: link with [Benthem et al., 2006a]. 14

15 14 Conclusions Present framework demonstrates how common perception of change creates common knowledge. Common knowledge about limitations to perception can also create common knowledge. If we can all see that a person is blind, then her resulting lack of perception of change will become common knowledge. Awareness of basic propositions allows for the modelling of communication channels in DEL: if i is aware that j can perceive changes in p, then p is a communication channel between i and j. References [Austin, 1962] J.L. Austin. Sense and Sensibilia. Oxford University Press, [Baltag et al., 1999] A. Baltag, L.S. Moss, and S. Solecki. The logic of public announcements, common knowledge, and private suspicions. Technical Report SEN-R9922, CWI, Amsterdam, [Bell, 2000] J.L. Bell. Continuity and the logic of perception. Transcendent Philosophy, 1(2):1 7, [Benthem et al., 2006a] J. van Benthem, J. Gerbrandy, and B. Kooi. Dynamic update with probabilities, [Benthem et al., 2006b] J. van Benthem, J. van Eijck, and B. Kooi. Logics of communication and change. Information and Computation, 204(11): , [Benthem, 2005] Johan van Benthem. Open problems in logic and games. Technical Report PP , ILLC, Amsterdam, [Clark and Marshall, 1978] H.H. Clark and C.R. Marshall. diaries. Technical report, Stanford University, Reference [Clark and Marshall, 1981] H.H. Clark and C.R. Marshall. Definite reference and mutual knowledge. In A.K. Joshi, B. Webber, and I. Sag, editors, Elements of discourse understanding, pages Cambridge University Press,

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