The Scholten/Dijkstra Pebble Game Played Straightly, Distributedly, Online and Reversed Wolfgang Reisig Department of Computer Science, Humboldt-Universit¨at zu Berlin With pleasure I remember the visit of Boaz at GMD in Bonn in the early 1980s and at TU Munich in late 1980s as well as long discussions with him in Dagstuhl, and several meetings in Tel Aviv. What a rich source of inspiration! Abstract. The Scholten/Dijkstra “Pebble Game” is re-examined. We show that the algorithm lends itself to a distributed as well as an online version, and even to a reversed variant. Technically this is achieved by exploiting the local and the reversible nature of Petri Net transitions. Furthermore, these properties allow to retain the verification arguments of the algorithm. 1 Introduction University Video Communications [1] distributes a “Distinguished Lecture Se- ries” of “Leaders in Computer Science and Electrical Engineering” where in an “Academic Honour Presentation” Edsger W. Dijkstra talked about Reasoning about programs. As an example, Dijkstra presents a “Pebble Game” as an ex- ample of a nondeterministic algorithm. Gries in [2] refers the problem to Carl Scholten, due to a letter from Dijkstra in fall 1979. Scholten plays the game with black and white beans in a coffee can. Dijkstra models this algorithm as a guarded command program and proves its decisive properties. Section 2.1 of this paper recalls Dijkstra’s oral presentation of the algorithm as well his program. A Petri Net model of the algorithm is given in Sect. 2.2, and verified in Sect. 2.3. Part 3 of this paper presents three variants that provide more insight into the nature of the algorithm. In particular, a distributed and an online version exploit the local nature of the algorithm’s steps. Furthermore, it is shown that the algorithm can be played ”backwards”, exploiting the reversible nature of the algorithm’s steps. Interesting enough, the decisive verification arguments remain valid in all three variants. 2 The Algorithm’s Basic Version 2.1 Dijkstra’s Algorithm and Model We quote Dijkstra’s oral presentation of [1]: A. Avron et al. (Eds.): Trakhtenbrot/Festschrift, LNCS 4800, pp. 589–595, 2008. c  Springer-Verlag Berlin Heidelberg 2008