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Angel, Devil, and King Martin Kutz Max-Planck Institut fr - - PowerPoint PPT Presentation

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Angel, Devil, and King Martin Kutz Max-Planck Institut fr Informatik, Saarbrcken, Germany Attila Pr CASE Western Reserve University, Cleveland, USA Martin Kutz: Angel, Devil, and King p. 1 max planck institut informatik Martin Kutz:

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max planck institut informatik

Angel, Devil, and King

Martin Kutz

Max-Planck Institut für Informatik, Saarbrücken, Germany

Attila Pór

CASE Western Reserve University, Cleveland, USA

Martin Kutz: Angel, Devil, and King – p. 1

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 2

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

Theorem [Berlekamp] The chess king can be caught on an infinite checkers board.

Martin Kutz: Angel, Devil, and King – p. 3

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max planck institut informatik

The Angel Problem

Definition [Berlekamp, Conway, Guy] A k-Angel can “fly” in one move to any unblocked square at distance at most k.

Martin Kutz: Angel, Devil, and King – p. 4

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max planck institut informatik

The Angel Problem

Definition [Berlekamp, Conway, Guy] A k-Angel can “fly” in one move to any unblocked square at distance at most k.

Martin Kutz: Angel, Devil, and King – p. 4

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max planck institut informatik

The Angel Problem

Definition [Berlekamp, Conway, Guy] A k-Angel can “fly” in one move to any unblocked square at distance at most k. Open Problem Can some k-Angel of some finite power k escape his opponent, the Devil, forever.

Martin Kutz: Angel, Devil, and King – p. 4

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max planck institut informatik

Only Fools Rush in

Definition A Fool is an Angel who commits himself to increasing his y-coordinate in every move.

Martin Kutz: Angel, Devil, and King – p. 5

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max planck institut informatik

Only Fools Rush in

Definition A Fool is an Angel who commits himself to increasing his y-coordinate in every move. Theorem [Conway] The Devil catches any k-Fool of finite power k.

Martin Kutz: Angel, Devil, and King – p. 5

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max planck institut informatik

Between 1-Angel and 2-Angel

Only the destiny of the 1-Angel (= chess king) is known. For all other k-Angels, k ≥ 2, the outcome is open.

Martin Kutz: Angel, Devil, and King – p. 6

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max planck institut informatik

Between 1-Angel and 2-Angel

Only the destiny of the 1-Angel (= chess king) is known. For all other k-Angels, k ≥ 2, the outcome is open. We don’t even know whether the chess knight can be caught.

Martin Kutz: Angel, Devil, and King – p. 6

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max planck institut informatik

Between 1-Angel and 2-Angel

Only the destiny of the 1-Angel (= chess king) is known. For all other k-Angels, k ≥ 2, the outcome is open. We don’t even know whether the chess knight can be caught. Observation: The 2-Angel is actually 4× stronger than the 1-Angel. (double speed and double-width obstacles)

Martin Kutz: Angel, Devil, and King – p. 6

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Between 1-Angel and 2-Angel

Only the destiny of the 1-Angel (= chess king) is known. For all other k-Angels, k ≥ 2, the outcome is open. We don’t even know whether the chess knight can be caught. Observation: The 2-Angel is actually 4× stronger than the 1-Angel. (double speed and double-width obstacles) We modify the problem to have speed as the only parameter.

Martin Kutz: Angel, Devil, and King – p. 6

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Angels With Broken Wings

Deprive Angels of their ability to fly across obstacles. Definition A k-King is a k-Angel who can only run, not fly. In each turn he makes k ordinary chess-king moves.

Martin Kutz: Angel, Devil, and King – p. 7

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Angels With Broken Wings

Deprive Angels of their ability to fly across obstacles. Definition A k-King is a k-Angel who can only run, not fly. In each turn he makes k ordinary chess-king moves. Proposition If the k-Angel can escape forever then so can the 99k

  • King.

Martin Kutz: Angel, Devil, and King – p. 7

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max planck institut informatik

The Main Result

Theorem The Devil can catch any α-King with α < 2.

Martin Kutz: Angel, Devil, and King – p. 8

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max planck institut informatik

The Main Result

Theorem The Devil can catch any α-King with α < 2. For fractional and irrational speed α > 1 define Angel / Devil turns be means of sturmian sequences: Shoot a ray of slope α from the origin and mark crossings with the integer grid:

1 2 3 4 5 6 1 2 3 4 5 6 7 8

α = √ 2

Martin Kutz: Angel, Devil, and King – p. 8

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max planck institut informatik

The Main Result

Theorem The Devil can catch any α-King with α < 2. For fractional and irrational speed α > 1 define Angel / Devil turns be means of sturmian sequences: Shoot a ray of slope α from the origin and mark crossings with the integer grid: horizontal line → King step vertical line → Devil move

1 2 3 4 5 6 1 2 3 4 5 6 7 8

α = √ 2

Martin Kutz: Angel, Devil, and King – p. 8

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max planck institut informatik

The Main Result

Theorem The Devil can catch any α-King with α < 2. For fractional and irrational speed α > 1 define Angel / Devil turns be means of sturmian sequences: Shoot a ray of slope α from the origin and mark crossings with the integer grid: horizontal line → King step vertical line → Devil move

1 2 3 4 5 6 1 2 3 4 5 6 7 8

α = √ 2 K D K D K K D K D K K

Martin Kutz: Angel, Devil, and King – p. 8

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max planck institut informatik

The Main Result

Theorem The Devil can catch any α-King with α < 2. For fractional and irrational speed α > 1 define Angel / Devil turns be means of sturmian sequences: Shoot a ray of slope α from the origin and mark crossings with the integer grid: horizontal line → King step vertical line → Devil move “Lemma.” This distribution is “fair” and shifting of the grid / origin does not affect winning and losing.

1 2 3 4 5 6 1 2 3 4 5 6 7 8

α = √ 2 K D K D K K D K D K K

Martin Kutz: Angel, Devil, and King – p. 8

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves.

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t Lemma Against an s t-King there exist dynamic fences of density s − t s = 1 − t s

Martin Kutz: Angel, Devil, and King – p. 9

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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t Lemma Against an s t-King there exist dynamic fences of density s − t s = 1 − t s < 1 2

  • for s

t < 2

  • Martin Kutz: Angel, Devil, and King – p. 9
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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t Lemma Against an s t-King there exist dynamic fences of density s − t s = 1 − t s < 1 2

  • for s

t < 2

  • Martin Kutz: Angel, Devil, and King – p. 9
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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t Lemma Against an s t-King there exist dynamic fences of density s − t s = 1 − t s < 1 2

  • for s

t < 2

  • Martin Kutz: Angel, Devil, and King – p. 9
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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t Lemma Against an s t-King there exist dynamic fences of density s − t s + ε = 1 − t s + ε < 1 2

  • for s

t < 2

  • Martin Kutz: Angel, Devil, and King – p. 9
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Dynamic Fences

For speed α = s t we have exactly s King moves per t Devil moves. s t s t Lemma Against an s t-King there exist dynamic fences of density s − t s + ε = 1 − t s + ε < 1 2

  • for s

t < 2

  • Martin Kutz: Angel, Devil, and King – p. 9
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Building a Box

Encircle the King with a box of fences before he can reach the boundary.

Martin Kutz: Angel, Devil, and King – p. 10

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Building a Box

Encircle the King with a box of fences before he can reach the boundary. This only works with fences of very low density.

Martin Kutz: Angel, Devil, and King – p. 10

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Building a Box

Encircle the King with a box of fences before he can reach the boundary. This only works with fences of very low density. A first result: For α < 9/8 we get fences of density < 1/9, which the Devil builds 9 9/8 = 8 times faster than the King runs.

Martin Kutz: Angel, Devil, and King – p. 10

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Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities.

Martin Kutz: Angel, Devil, and King – p. 11

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Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences The slots are wider than they are deep, so the total density lies below that of the small fences. (works only for density < 1/2)

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik

Smaller Densities

Against King speed 2 − ε, fence density

  • − ε′ is not enough.

Need smaller densities. Solution: a fence of fences The slots are wider than they are deep, so the total density lies below that of the small fences. (works only for density < 1/2) Iteration yields thinner and thinner and thinner and thinner fences . . .

Martin Kutz: Angel, Devil, and King – p. 11

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 12

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max planck institut informatik Martin Kutz: Angel, Devil, and King – p. 12

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max planck institut informatik

Conclusion

We introduced α-Kings (with any α ∈ R

  • ) to focus on speed as the

essential parameter in the Angel Problem. Theorem The Devil cathches any α-King with α < 2.

Martin Kutz: Angel, Devil, and King – p. 13

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max planck institut informatik

Conclusion

We introduced α-Kings (with any α ∈ R

  • ) to focus on speed as the

essential parameter in the Angel Problem. Theorem The Devil cathches any α-King with α < 2. Question Can he also catch the 2-King?

Martin Kutz: Angel, Devil, and King – p. 13