Classes of Pebble Games and Complete Problems

Graph Searching and Related Problems

“…Pebble games on graphs and digraphs have been studied by both mathematicians and computer scientists [18,47,52,85,97,110,111]. Pebbling is a method of analyzing computational situations, especially situations in which notions such as time (number of operations) and space (number of memory locations) are of interest [51,86,91,94,98,102]. We focus on the relationship between graph searching and pebbling [97].…”

On the Complexity of Finding Narrow Proofs

“…We let [k] be the set of k pebbles in the game. A rule is of the form (u, v, w, c, d), with c = d, u = v = w = u and the intended meaning that if pebble c is on u and pebble d is on v and there is no pebble on w, then one player can move pebble c from u to w. This is a slight more wasteful notion as originally used in [15], where the set of rules is R ′ ⊆ U 3 , but it is useful in our reduction to specify the pebbles c and d in the rules. A position of the KAI-game is an injective mapping The KAI-game is played by two players and proceeds in rounds.…”

“…The variables A i,c j,l say "pigeon A i,c j sits in hole l". It is ensured by the clauses (13), (14) and (15) that if A i,c j is arriving, then it will sit in some hole. The clauses (16) state that in every hole there is at most one pigeon.…”

Selective search in games of different complexity

“…Many games are proven to be PSPACE-complete (Papadimitriou, 1994;Van Emde Boas, 2002), such as Go-Moku (Reisch, 1980), Hex (Reisch, 1981), Othello (Iwata and Kasai, 1994), Scotland Yard (Sevenster, 2006), Connect6 (Hsieh and Tsai, 2007) and Amazons (Hearn, 2009). Games that are proven to be EXPTIME-complete include Chinese Checkers (Kasai, Adachi, and Iwata, 1979), chess (Fraenkel and Lichtenstein, 1981), Go (Robson, 1983), checkers (Robson, 1984) and Shogi (Adachi, Kamekawa, and Iwata, 1987). For an overview of a large number of games and their complexity classes we refer to Demaine and Hearn (2001).…”