Techniques for Decidability and Undecidability of Bisimilarity

In this tutorial we describe general approaches to deciding bisimilarity between vertices of (infinite) directed edge-labelled graphs. The approaches are based on a systematic search following the definition of bisimilarity. We outline (in decreasing levels of detail) how the search is modified to solve the problem for finite graphs, BPP graphs, BPA graphs, normed PA graphs, and normed PDA graphs. We complete this by showing the technique used in the case of graphs generated by onecounter machines. Finally, we demonstrate a general reduction strategy for proving undecidability, which we apply in the case of graphs generated by state-extended BPP (a restricted form of labelled Petri nets). This paper was written during a visit by the first author to Uppsala University supported by a grant from the Swedish STINT Fellowship Programme. He is also partially supported by the Grant Agency of the Czech Republic, Grant No. 201/97/0456. yThe second author is supported by Swedish TFR grants No. 221-98-103 “Verification of Infinite State Automata” and 221-97-275 “Games for Processes”. 1 Bisimulation Equivalence The bisimulation game is played on a “board” which consists of a (generally infinite) directed edge-labelled multigraph (several edges can lead between two vertices), simply called a graph in the following. We assume that this graph is labelled from a finite set of labels, and that it is finite-branching: that every vertex has finite out-degree. In particular, it is image-finite: for every vertex E and every label a, the set succa(E) = fF : E a ! Fg is finite. We also asume that these successor sets are effectively constructible. A game is defined by two vertices E0 and F0 of a graph, as well as a predetermined time limit n 2 N [ f!g (where N = f0; 1; 2; 3; : : :g); we specify such a game by Gn(E0; F0). We have two players competing in the game whom we refer to as Alice (the “Attacker”) and Bob (the “Bisimulator”). Their individual goals are as follows: 1. Alice wants to show that E0 and F0 are in some sense “different”. 2. Bob wants to show that E0 and F0 are in the same sense “the same”. The sense in which two vertices are deemed to be the same is given by the rules for playing the game. A play of the game is a sequence of pairs (E0; F0) (E1; F1) of length (1+n), with the next pair in the sequence after (Ei; Fi) arising as follows: 1. Alice chooses an edge Ei a ! Ei+1 or Fi a ! Fi+1; 2. Bob chooses a matching edge Fi a ! Fi+1 or Ei a ! Ei+1. Alice is thus acting as an attacker, trying to choose an edge leading out of one of the vertices which she believes cannot be matched by any edge (with the same label) leading out of the other vertex; Bob on the other hand is defending his thesis that the vertices are equal, that any edge leading out of either of the vertices has a matching edge leading out of the other vertex. Alice wins a play of the game if Bob ever gets stuck (that is, if he cannot respond to a move by Alice); and Bob wins any play of length n (that is, any “timed-out” play in which the players have exchanged nmoves) as well as any play in which Alice finds herself with no move possible (that is, if there are no edges leading out of either of the two specified vertices). We are interested in knowing if Bob has a winning (i.e., defending) strategy for the game Gn(E0; F0), that is, if he is able to win any play of the game regardless of the moves made by Alice. To this end, we make the following definitions. For n2N , we say that E0 and F0 are n-game equivalent (written E0 n F0) iff Bob has a winning strategy for the game Gn(E0; F0); and we say that E0 and F0 are game equivalent (written E0 F0) iff Bob has a winning strategy for the game G!(E0; F0). The relation is referred to as bisimulation equivalence, or bisimilarity. Before proceeding, it is worth recording the following straightforward facts. Fact 1 n and are equivalence (reflexive, symmetric, transitive) relations.