Following up on the Dynamic Epistemic Logic treatment of Cheryl’s Birthday Puzzle, Malvin Gattinger (ILLC Amsterdam) has formalized the problem in DEMO_S5, a Dynamic Epistemic Logic model checker written in Haskell by Jan van Eijck (CWI Amsterdam and ILLC). The original DEMO system was described in:
Jan van Eijck: “DEMO—a demo of epistemic modelling” In: Johan van Benthem, Dov Gabbay, and Benedikt Löwe, eds., Interactive Logic. Selected Papers from the 7th Augustus de Morgan Workshop, London. Vol. 1. Amsterdam University Press, 2007.
module CHERYL where import Data.List import Data.Function import DEMO_S5 import KRIPKEVIS
We first define the set of all possibilities:allpos :: [(Int, String)] allpos = [ (15,"May"), (16,"May"), (19,"May"), (17,"June"), (18,"June"), (14,"July"), (16,"July"), (14,"August"), (15,"August"), (17,"August") ]
This forms the set of worlds in our initial model. Moreover, also the set of actual worlds is the full set, hence
allposoccurs twice in the definition below. The two elements of
relsdefine the epistemic relations of Albert and Bernard. Instead of listing explicitly which possibilities they can distinguish we use haskell functions to say that they confuse the same day and the same month, respectively.initCheryl :: EpistM (Int,String) initCheryl = Mo allpos [a,b]  rels allpos where rels = [ ( a, groupBy ((==) `on` snd) allpos ), ( b, groupBy ((==) `on` fst) (sortBy (compare `on` fst) allpos) ) ]
This is the initial model with all possibilities:
The formula saying that i knows Cheryl’s birthday is defined as the disjunction over all statements of the form “Agent i knows that the birthday is s”:knWhich :: Agent -> Form (Int, [Char]) knWhich i = Disj [ Kn i (Info s) | s <- allpos ]
Now we update the model three times, using the function
upd_pafor public announcements.
First with Albert: I don’t know when Cheryl’s birthday is and I know that Bernard does not know.model2 = upd_pa initCheryl (Conj [Ng $ knWhich a, Kn a $ Ng (knWhich b)])
The second announcement by Bernard: “Now I know when Cheryl’s birthday is.”model3 = upd_pa model2 (knWhich b)
Finally, Albert says: “Now I also know when Cheryl’s birthday is.”model4 = upd_pa model3 (knWhich a)
Lastly, this helper function uses
texModelfrom our modified
KRIPKEVISmodule to generate the drawings:myTexModel :: EpistM (Int,String) -> String -> IO String myTexModel (Mo states _ _ rels pointed) fn = texModel showState showAg showVal "" (VisModel states rels [(s,0)|s<-states] pointed) fn where showState (n,string) = (show n) ++ string showVal _ = "" showAg i = if i==a then "Albert" else "Bernard"