Decidability of Reachability Problems for Classes of Two Counters Automata

We present a global and comprehensive view of the properties of subclasses of two counters automata for which counters are only accessed through the following operations: increment (+1), decrement (-1), reset (c := 0), transfer (the whole content of counter c is transfered into counter c′), and testing for zero. We first extend Hopcroft-Pansiot's result (an algorithm for computing a finite description of the semilinear set post*) to two counters automata with only one test for zero (and one reset and one transfer operations). Then, we prove the semilinearity and the computability of pre* for the subclass of 2 counters automata with one test for zero on c1, two reset operations and one transfer from c1 to c2. By proving simulations between subclasses, we show that this subclass is the maximal class for which pre* is semilinear and effectively computable. All the (effective) semilinearity results are obtained with the help of a new symbolic reachability tree algorithm for counter automata using an Acceleration function. When Acceleration has the so-called stability property, the constructed tree computes exactly the reachability set.