Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ - PowerPoint PPT Presentation

M UDDY C HILDREN P LAYGROUND Nina Gierasimczuk Jakub Szymanik Institute for Logic, Language and Computation, University of Amsterdam Department of Philosophy, Stockholm University LoRI @ ESSLLI’10 Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 1 / 27

1 M UDDY C HILDREN 2 E PISTEMIC POWER OF VARIOUS QUANTIFIERS 3 E PISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 B RIEF DISCUSSION Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 2 / 27

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 3 / 27

T HE PUZZLE At least one of you has mud on your forehead. 1 Can you tell whether or not you are muddy? 2 Repeating the question makes children know the answer. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 4 / 27

T HE F IRST A NNOUNCEMENT : Q UANTIFIER General form: ‘Q of you have mud on your forehead’, where Q is a generalized quantifier. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 5 / 27

Q UANTIFIERS OF TYPE (1) M = ( U , A ) After the announcement: { M : M | = Q U ( A ) } Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 6 / 27

E XAMPLES E XAMPLE ∃ = { ( U , A ) : A ⊆ U & A � = ∅} D n = { ( U , A ) : A ⊆ U & card ( A ) = k × n } most = { ( U , A ) : A ⊆ U & card ( A ) > card ( U − A ) } Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 7 / 27

N UMBER TRIANGLE Viewing finite models as pairs of integers. — (0, 0) - + (1, 0) (0, 1) - + + (2, 0) (1, 1) (0, 2) - + + + (3, 0) (2, 1) (1, 2) (0, 3) - + + + + (4, 0) (3, 1) (2, 2) (1, 3) (0, 4) Extensively studied in GQT. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 8 / 27

1 M UDDY C HILDREN 2 E PISTEMIC POWER OF VARIOUS QUANTIFIERS 3 E PISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 B RIEF DISCUSSION Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 9 / 27

T HE B ACKGROUND A SSUMPTION : Q UANTIFIER General form: ‘Q of you have mud on your forehead’, where Q is a generalized quantifier. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 10 / 27

V ARYING Q UANTIFIERS IN M UDDY C HILDREN P UZZLE ‘At least one’ and ‘At least two’ 0 1 2 3 4 0 1 2 3 4 1 x 1 x x x 1 x x x x x 2 x 1 2 x x 2 x x 1 x x 3 x 1 2 3 x 3 x x 1 2 x 4 x 1 2 3 4 4 x x 1 2 3 5 x 1 2 3 4 5 x x 1 2 3 6 x 1 2 3 4 . . . 6 x x 1 2 3 . . . Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 11 / 27

V ARYING Q UANTIFIERS IN M UDDY C HILDREN P UZZLE ‘Even’ and ‘Most’ 0 1 2 3 4 0 1 2 3 4 5 1 1 x 1 x 1 1 x 1 x x x x 2 1 x 1 x 1 2 x x 1 x x x 3 1 x 1 x 1 3 x x 1 2 x x 4 1 x 1 x 1 4 x x x 1 2 x 5 1 x 1 x 1 5 x x x 1 2 3 6 1 x 1 x 1 . . . 6 x x x x 1 2 . . . Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 12 / 27

P ATTERNS : E XEMPLARY OBSERVATION P ROPOSITION Assume n children, m ≤ n muddy children. The Muddy Children Puzzle with the background assumption ‘At least k of you have mud on your forehead’ can be solved in m − ( k − 1 ) steps, where k ≤ m. In the paper we do it systematically for various Qs. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 13 / 27

W HERE DO THOSE PATTERNS COME FROM ? ‘At least one’ 0 1 2 3 4 5 1 x 1 x x x x 2 x 1 2 x x x 3 x 1 2 3 x x 4 x 1 2 3 4 x 5 x 1 2 3 4 5 6 x 1 2 3 4 5 . . . Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 14 / 27

T HEY COME FROM THE QUANTIFIER ITSELF — - 1 - 1 2 - 1 2 3 - 1 2 3 4 - 1 2 3 4 5 Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 15 / 27

S OLVABILITY CHARACTERIZATION T HEOREM Let n be the number of children, m ≤ n the number of muddy children, and Q be the background assumption. Muddy Children situation is solvable iff ( n − m , m ) ∈ Q and there is an l ≤ n such that ( n − l , l ) �∈ Q. — ? ? ? ? ? - 3 2 1 - - 3 2 1 ‘At most 2’ Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 16 / 27

S OLVABILITY CHARACTERIZATION O BSERVATION The number assigned to a point in the number triangle is the ‘distance’ to the closest model outside of the quantifier. — ? ? ? ? ? - 3 2 1 - - 3 2 1 ‘At most 2’ Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 17 / 27

1 M UDDY C HILDREN 2 E PISTEMIC POWER OF VARIOUS QUANTIFIERS 3 E PISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 B RIEF DISCUSSION Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 18 / 27

R EPRESENTATION (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) O BSERVATION Every agent’s observation is encoded by one of at most two neighboring states in the observational level. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 19 / 27

R EPRESENTATION (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) O BSERVATION Every agent’s observation is encoded by one of at most two neighboring states in the observational level. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 19 / 27

I NITIAL M ODEL (2, 0) (1, 1) (0, 2) a, b a, b c c (3, 0) (2, 1) (1, 2) (0, 3) Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 20 / 27

S TEP 1: Q UANTIFIER A NNOUNCEMENT (2, 0) (1, 1) (0, 2) a, b a, b c c (3, 0) (2, 1) (1, 2) (0, 3) Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 21 / 27

S TEP 2: E PISTEMIC R EASONING (2, 0) (1, 1) (0, 2) a, b c c (3, 0) (2, 1) (1, 2) (0, 3) Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 22 / 27

S TEP 3: E PISTEMIC R EASONING (2, 0) (1, 1) (0, 2) a, b c (3, 0) (2, 1) (1, 2) (0, 3) Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 23 / 27

1 M UDDY C HILDREN 2 E PISTEMIC POWER OF VARIOUS QUANTIFIERS 3 E PISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 B RIEF DISCUSSION Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 24 / 27

M UDDY C HILDREN M ODELING DEL VS NT (C OG S CI ) (1,1,1) (1,1,0) (1,0,1) (0,1,1) (1,0,0) (0,1,0) (0,0,1) (2, 0) (1, 1) (0, 2) (0,0,0) (3, 0) (2, 1) (1, 2) (0, 3) Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 25 / 27

Tak Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 26 / 27

D ISCUSSION /O UTLOOK Comparison with DEL-perspective. Isomorphism and symmetry. Associate our representations with automata. Logic for public announcements with GQs. Other epistemic puzzles. Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 27 / 27