Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008
Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set of subsets of S contanng, whch s closed under complements and countable unons and ntersectons. 3 µ : A [0, 1] s a probablty measure, that s µ(s) = 1 and µ( ) = 0 If {A 1, A 2,...} s a countable set of parwse dsjont elements of A, then µ( j=1 A j) = j=1 µ(a j). (Countable addtvty) If A = P(S), then the probablty space s called dscrete, and the probablty functon µ can be vewed as mappng each element of S nto [0, 1].
How the σ-algebra helps us The σ-algebra lets us restrct the doman of the probablty measure. Restrctng the doman of the probablty measure lets us reflect uncertanty about the probablty of ndvdual elements of S. Probablstc Epstemc Logc offers a dfferent way of handlng uncertanty about probabltes; t lets us express uncertanty about probablty spaces.
Probablstc Epstemc Model Defnton Let Φ be a set of proposton letters, and I be a set of agents. A probablstc epstemc model s a tuple M = (X, { } I,, {P,x } I,x X ), where X s a fnte set (a subset of X 2 ) s an epstemc relaton for each agent I, that s x y f consders y possble from x s a functon assgnng to each proposton letter p the set of states where t s true. for each agent and state x, the probablty space P,x s defned as the tuple (S,x, A,x, µ,x ), where S,x X s the sample space (fnte because X s fnte) A,x s a σ-algebra µ,x : A,x [0, 1] s a probablty measure over S,x
Addng probablty formulas Add to epstemc logc probablty formulas of the form P (ϕ) q for a ratonal number q. Evaluate the truth of such a formula at a ponted model (M, x) (where M s a probablstc epstemc model and x s a state n M) n the followng way. (M, x) P (ϕ) q ff µ,x ([[ϕ]] S,x ) q, where [[ϕ]] s the set of states n M where ϕ s true. Of course, ths defnton only works f [[ϕ]] S,x A,x. We can ensure ths by ether 1 requrng each A,x be large enough so that [[ϕ]] S,x A,x s guaranteed for all ϕ, 2 extendng the functon µ,x to all subsets. Inner and outer measures are two such extensons, and they need not obey all condtons of a measure.
Inner and Outer Probabltes Let (S, A, µ) The nner and outer measures are defned on any set Y P(S) by (outer measure) µ (Y ) = nf{µ(b) : Y B, B A} (nner measure) µ (Y ) = sup{µ(b) : B Y, B A} When the σ-algebra A s fnte, these are equvalent to (outer measure) µ (Y ) = µ( {B : Y B, B A}) (nner measure) µ (Y ) = µ( {B : B Y, B A}) The nner and outer measures are related by µ (Y ) = 1 µ (Y ) Nether the nner nor the outer measure s n general a measure.
Observaton about fnte σ-algebras Any fnte σ-algebra A can be characterzed by an equvalence relaton. The equvalence relaton s such that the equvalence classes are the sets {A A : x A} for each x S. Some of these sets produced are dentcal for dfferent x s. Although S may be nfnte, the there wll only be fntely many equvalence classes. A σ-algebra can be generated by the equvalence classes of any equvalence relaton.
Fagn, Halpern, and Tuttle example Suppose there are two agents and k. 1 k s frst gven a bt 0 or 1. k learns he has ths bt, s aware that k receved a bt, but does not know what bt k receved. 2 k flps a far con and looks at the result. sees k look at the result, but does not what the result s. 3 k performs acton s f the con agrees wth the bt (gven that heads agrees wth 1 and tals agrees wth 0), and performs acton d otherwse. Ths example s from R. Fagn & J. Halpern (1994) Reasonng about Knowledge and Probablty. Journal of the ACM 41:2, pp. 340 367.
Dscusson There are four possble sequences of events: (1, H), (1, T ), (0, H), (0, T ) (note that the acton s or d s determned from the frst two). Untl k performs the acton s or d, agent consders any of these four states possble. (1, H) (1, T ) (0, H) (0, T ) We ndcate s uncertanty between two states usng a double arrow between the two states. In partcular, an arrow from state x to state y ndcates that consders y possble f x s the actual state. (Before the bt s gven, k s epstemc relaton wll be the same).
Here s a possblty for s probablty spaces. The sample space enclosed n a box, and the σ-algebra equvalence classes are enclosed n the dotted ovals. (1, H) (1, T ) (0, H) (0, T ) M 1 The sample space s the same as the set of states consders possble. Indvdual states cannot be measurable (otherwse 0 or 1 must be assgned a probablty).
Another possblty has a sample space contanng only the states wth the correct bt (but recall that consders all states possble and both sample spaces possble). (1, H) (1, T ) (0, H) (0, T ) M 2 Wthout assgnng probablty to the bt, can now assgn a probablty to the actons s and d.
Here s uncertan among 4 probablty spaces. (1, H) (1, T ) (0, H) (0, T ) M 3
Modelng a sequence of events It s suggested that each of these models may reasonably represent s probablty spaces at a certan stage n the sequence of events (but to make to make better sense of the transton, we add a lttle more n parentheses that was not n the orgnal statement of the example): M 1 wth the tme before the bt s gven to k (suppose does not yet know that k wll perform acton s or d). M 2 wth the tme after the bt s gven to k, (after k tells he wll do ether s or d dependng on the con toss,) but before the con s flpped. M 3 wth the tme after the con s tossed, (after k spontaneously offers a bet about what acton he wll take,) but before k performs hs acton.
acton model Defnton (Acton Model) An acton model (Σ, { }, {P,σ }, pre) s a probablstc epstemc model wth the valuaton functon replaced by a functon pre whch assgns to each σ Σ a class of ponted probablstc epstemc models. Each element σ Σ s called an acton type.
Update Product We defne a mechancal procedure called the update product for transformng one model nto another model gven an acton (represented by an acton model). We defne the update product n two stages. 1 The frst, called the unrestrcted product, s to take a product that does not make use of the pre functon. 2 The second, called the relatvzaton, s to restrct the unrestrcted product to a set of states characterzed by the pre functon.
fnte product measure Defnton (Fnte product space) The product space of probablty spaces (S 1, A 1, µ 1 ) and (S 2, A 2, µ 2 ) s (S 3, A 3, µ 3 ), where 1 S 3 = S 1 S 2 s the Cartesan product. 2 A 3 s the smallest σ-algebra contanng {A B : A A 1, B A 2 } 3 The probablty measure s defned as µ 3 (A) = n µ 1 (B k )µ 2 (C k ) k=1 where B k A 1, C k A 2, and A = n =1 B k C k
unrestrcted product Defnton (unrestrcted product) The unrestrcted product between a probablstc epstemc model M and an acton model Σ s M U Σ wth the followng components: 1 X = X Σ 2 (x, σ) (z, τ) ff x z and σ τ 3 p = p Σ 4 We defne P,(x,σ) to be the fnte product space between P,x and P,σ
relatvzaton Defnton (relatvzaton) The relatvzaton of a probablstc epstemc model M to Y X s gven by M R Y wth the followng components: 1 X Y = Y 2 x Y z ff x z and x, z Y 3 p Y = p Y 4 For x Y, f µ,x (Y ) = 0, then defne P,x to be the trval probablty space on the sngleton x. Otherwse 1 S Y,x = S,x Y 2 A Y,x s the σ-algebra generated by {A Y : A A,x } 3 The probablty measure s defned by µ Y,x (A) = µ,x (A Y ) µ,x (Y )
Establshng Addtvty Theorem Let (S, A, µ) be a probablty space, such that A s a fnte σ-algebra. If A, B A and Y S, then µ (A B Y ) = µ (A Y ) + µ (B Y ) µ (A B Y ) = µ (A Y ) + µ (B Y ) The proof of the outer measure part rests on the fact that Ŷ = {C : Y C : C A} A, and that A B Y s a dsjont unon of  Y and B Y. The proof of the nner measure part follows smlar reasonng.
The update product Defnton Update Product Gven a probablstc epstemc model M and an acton model Σ, let Y = {(x, σ) : (M, x) pre(σ)} Then the update product between M and Σ, wrtten M Σ s M Σ = (M U Σ) R Y
What acton models should be used? From M 1 to M 2, there are two events: 1 a sem-prvate announcement of the bt to k 2 a publc announcement that k plans to do ether acton s or d. From M 2 to M 3, there are two events: 1 a sem-prvate announcement to k of the result of the con toss 2 a publc announcement regardng k s bet offer We frst consder gong from M 1 to M 2 usng just one acton model, and smlarly from M 2 to M 3 wth just one acton model. We then consder gong from M 1 to M 2 usng a sequence of two acton models, and smlarly from M 2 to M 3.
Sem-prvate announcement The relatonal structure of a sem-prvate announcement s gven by, k σ τ, k and k s probablty spaces: σ τ
M 1 to M 2 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
M 1 to M 2 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
(1H, σ) (1T, σ) (0H, τ) (0T, τ)
M 2 to M 3 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
M 2 to M 3 (1H, σ) (1T, σ) (1H, τ) (1T, τ) (0H, σ) (0T, σ) (0H, τ) (0T, τ)
(1H, σ) (1T, τ) (0H, σ) (0T, τ)
From M 1 to M 2 frst stage: sem-prvate announcement relatonal structure:, k σ τ, k s probablty space: σ τ k s probablty spaces: σ τ pre(σ) ncludes states wth 1, and pre(τ) ncludes states wth 0.
From M 1 to M 2 second stage: publc announcement relatonal structure:, k, k σ τ, k Ths s the publc announcement the precondton of σ or the precondton of τ as long as no state satsfes both precondtons. and k s probablty spaces: σ τ pre(σ) ncludes states wth 1, and pre(τ) ncludes states wth 0.
From M 2 to M 3 The sem-prvate and publc announcement acton models are the same n all components except for the precondton functon pre. Instead of 1, the precondton of σ s H Instead of 0, the precondton of τ s T.
M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
M 1, k, k, k (1, H) (1, T ), k, k, k, k (0, H) (0, T ), k, k, k
after 1st sem-prvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after 1st sem-prvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after 1st sem-prvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after 1st sem-prvate announcement, k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after frst publc announcement (M 2 ), k, k, k (1, H) (1, T ) (0, H) (0, T ), k, k, k
after 2nd sem-prvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
after 2nd sem-prvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
after 2nd sem-prvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
after 2nd sem-prvate announcement, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
M 3, k, k (1, H) (1, T ) (0, H) (0, T ), k, k
Recordng the past Here are two ways: lst hstory or statc hstory: Defne a lst hstory to be a lst (S 0, S 1,..., S n ) of probablstc epstemc model, for whch for each j, S j+1 = S j A for some acton model A. We may want to fx some underlyng structure of A for techncal reasons. Defne a statc hstory to be an augmented probablstc epstemc model (X, { } I,, {P,x }, Y ), where Y s a bnary relaton over X, and a number of condtons are mposed n order to ensure that xyz can adopt the readng x follows from z and some acton. The goal s to ensure that the statc hstory s structurally equvalent to a lst hstory. When provng completeness, we have found t easer to use the statc hstores.
Need for non-measurable sets n completeness proof? The answer to ths s stll unknown. Here are some comments: Completeness for Probablstc Epstemc Logc (whch nvolves non-measurable sets) constructs a fltraton that has dscrete measures. A smlar fltraton can be constructed for a probablstc dynamc epstemc logc wth a past-tme operator, but the fltraton wll guarantee all condtons needed to reflect update products. I suggest usng truth-preservng model transformatons to convert the fltraton nto a true statc hstory. It s yet unknown whether we would beneft from beng able to transform a model nto one where probabltes are not dscrete.