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

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

Sep 25, 2023 5 likes •285 views

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 @ ESSLLI10 Gierasimczuk, Szymanik (ILLC, SU) Muddy

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SLIDE 1

MUDDY CHILDREN PLAYGROUND

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

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SLIDE 2

1 MUDDY CHILDREN 2 EPISTEMIC POWER OF VARIOUS QUANTIFIERS 3 EPISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 BRIEF DISCUSSION

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 2 / 27

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SLIDE 3

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

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SLIDE 4

THE PUZZLE

1

At least one of you has mud on your forehead.

2

Can you tell whether or not you are muddy? Repeating the question makes children know the answer.

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 4 / 27

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SLIDE 5

THE FIRST ANNOUNCEMENT: QUANTIFIER

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

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SLIDE 6

QUANTIFIERS OF TYPE (1)

M = (U, A) After the announcement: {M : M | = QU(A)}

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 6 / 27

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SLIDE 7

EXAMPLES

EXAMPLE

∃ = {(U, A) : A ⊆ U & A = ∅} Dn = {(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

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SLIDE 8

NUMBER 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

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SLIDE 9

1 MUDDY CHILDREN 2 EPISTEMIC POWER OF VARIOUS QUANTIFIERS 3 EPISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 BRIEF DISCUSSION

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 9 / 27

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SLIDE 10

THE BACKGROUND ASSUMPTION: QUANTIFIER

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

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SLIDE 11

VARYING QUANTIFIERS IN MUDDY CHILDREN PUZZLE

‘At least one’ and ‘At least two’ 1 2 3 4 1 x 1 x x x 2 x 1 2 x x 3 x 1 2 3 x 4 x 1 2 3 4 5 x 1 2 3 4 6 x 1 2 3 4 . . . 1 2 3 4 1 x x x x x 2 x x 1 x x 3 x x 1 2 x 4 x x 1 2 3 5 x x 1 2 3 6 x x 1 2 3 . . .

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 11 / 27

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SLIDE 12

VARYING QUANTIFIERS IN MUDDY CHILDREN PUZZLE

‘Even’ and ‘Most’ 1 2 3 4 1 1 x 1 x 1 2 1 x 1 x 1 3 1 x 1 x 1 4 1 x 1 x 1 5 1 x 1 x 1 6 1 x 1 x 1 . . . 1 2 3 4 5 1 x 1 x x x x 2 x x 1 x x x 3 x x 1 2 x x 4 x x x 1 2 x 5 x x x 1 2 3 6 x x x x 1 2 . . .

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 12 / 27

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SLIDE 13

PATTERNS: EXEMPLARY OBSERVATION

PROPOSITION

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

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SLIDE 14

WHERE DO THOSE PATTERNS COME FROM?

‘At least one’ 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

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SLIDE 15

THEY 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

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SLIDE 16

SOLVABILITY CHARACTERIZATION

THEOREM

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

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SLIDE 17

SOLVABILITY CHARACTERIZATION

OBSERVATION

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

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SLIDE 18

1 MUDDY CHILDREN 2 EPISTEMIC POWER OF VARIOUS QUANTIFIERS 3 EPISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 BRIEF DISCUSSION

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 18 / 27

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SLIDE 19

REPRESENTATION

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

OBSERVATION

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

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SLIDE 20

REPRESENTATION

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

OBSERVATION

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

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SLIDE 21

INITIAL MODEL

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

a, b a, b c c

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 20 / 27

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SLIDE 22

STEP 1: QUANTIFIER ANNOUNCEMENT

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

a, b a, b c c

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 21 / 27

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SLIDE 23

STEP 2: EPISTEMIC REASONING

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

a, b c c

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 22 / 27

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SLIDE 24

STEP 3: EPISTEMIC REASONING

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

a, b c

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 23 / 27

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SLIDE 25

1 MUDDY CHILDREN 2 EPISTEMIC POWER OF VARIOUS QUANTIFIERS 3 EPISTEMIC MODELS BASED ON NUMBER TRIANGLE 4 BRIEF DISCUSSION

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 24 / 27

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SLIDE 26

MUDDY CHILDREN MODELING DEL VS NT (COGSCI)

(1,1,1) (1,1,0) (1,0,1) (1,0,0) (0,1,1) (0,1,0) (0,0,1) (0,0,0)

(3, 0) (2, 0) (2, 1) (1, 1) (1, 2) (0, 2) (0, 3)

Gierasimczuk, Szymanik (ILLC, SU) Muddy Children Playground LoRI @ ESSLLI’10 25 / 27

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SLIDE 27

Tak

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

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SLIDE 28

DISCUSSION/OUTLOOK

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