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

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 max planck institut informatik

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

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

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

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

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

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

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

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

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

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

Martin Kutz: Angel, Devil, and King – p. 2 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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) We modify the problem to have speed as the only parameter. Martin Kutz: Angel, Devil, and King – p. 6 max planck institut informatik

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 max planck institut informatik

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 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 max planck institut informatik

The Main Result Theorem The Devil can catch any α -King with α < 2 . 8 For fractional and irrational speed α > 1 define Angel / Devil turns be means of √ 7 α = 2 sturmian sequences: 6 Shoot a ray of slope α from the origin and 5 mark crossings with the integer grid: 4 3 2 1 0 0 1 2 3 4 5 6 Martin Kutz: Angel, Devil, and King – p. 8 max planck institut informatik

The Main Result Theorem The Devil can catch any α -King with α < 2 . 8 For fractional and irrational speed α > 1 define Angel / Devil turns be means of √ 7 α = 2 sturmian sequences: 6 Shoot a ray of slope α from the origin and 5 mark crossings with the integer grid: 4 horizontal line → King step 3 vertical line → Devil move 2 1 0 0 1 2 3 4 5 6 Martin Kutz: Angel, Devil, and King – p. 8 max planck institut informatik

The Main Result Theorem The Devil can catch any α -King with α < 2 . 8 For fractional and irrational speed α > 1 define Angel / Devil turns be means of √ 7 α = 2 sturmian sequences: K D K D K K D K D K K 6 Shoot a ray of slope α from the origin and 5 mark crossings with the integer grid: 4 horizontal line → King step 3 vertical line → Devil move 2 1 0 0 1 2 3 4 5 6 Martin Kutz: Angel, Devil, and King – p. 8 max planck institut informatik

The Main Result Theorem The Devil can catch any α -King with α < 2 . 8 For fractional and irrational speed α > 1 define Angel / Devil turns be means of √ 7 α = 2 sturmian sequences: K D K D K K D K D K K 6 Shoot a ray of slope α from the origin and 5 mark crossings with the integer grid: 4 horizontal line → King step 3 vertical line → Devil move 2 1 “Lemma.” This distribution is “fair” and shifting of the grid / origin does not affect 0 0 1 2 3 4 5 6 winning and losing. Martin Kutz: Angel, Devil, and King – p. 8 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t s Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t s Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t t s Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t s Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t t s Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik

Dynamic Fences s King moves α = s For speed we have exactly per t t Devil moves. s t t s Against an s Lemma t -King there exist dynamic fences of density s − t = 1 − t s s Martin Kutz: Angel, Devil, and King – p. 9 max planck institut informatik