State identification

Abstract

In this chapter, we addressed the state identification problem for Mealy machines. It consists in checking whether it is possible to determine the initial state of a given Mealy machine by applying inputs on it and observing the corresponding outputs. A solution of this problem is called a distinguishing sequence. Not all Mealy machines have distinguishing sequences. In particular, nonminimal machines have no distinguishing sequences, since there is no way for distinguishing their equivalent states. In this chapter, we have only considered Mealy machines which are minimal, deterministic and fully specified. Distinguishing sequences can be either preset or adaptive. A PDS is a sequence of inputs whereas an ADS is a decision tree where inputs may be different depending on the observed outputs during the experiment. If a machine has a PDS then it has an ADS (because a PDS is an ADS), however, the converse is not true. Main results about PDSs: it is PSPACE-complete to test whether a given FSM has a PDS. In [LY94], Lee and Yannakakis show that this remains true even for Mealy machines with binary input and output alphabets. Checking the existence of a PDS can be reduced to a reachability analysis in the super graph of the considered machine. The size of this graph is exponential with respect to the number of states of the corresponding machine. It is possible to compute a shortest PDS of a given machine by performing a breadth-first search throughout the super graph of the machine. In [LY94], Lee and Yannakakis also show that there are machines for which the shortest PDS has exponential length. Main results about ADSs: it can be checked whether a given Mealy machine has an ADS in time O(pn 2), where n and p are the number of states and inputs of the considered machine, respectively. This can be done by executing an algorithm similar to the classical minimization algorithm. O(pn2) can be reduced to O(pnlogn) by executing an algorithm inspired by Hopcroft's minimization algorithm [Hop71]. For computing "optimal" ADSs, in [LY94], Lee and Yannakakis define the so-called splitting tree. The latter provides for some subset of states an input sequence which allows to reduce the uncertainty about this subset. A given machine has an ADS if and only if it has a complete splitting tree. The algorithm (Algorithm 7) for checking the existence and computing a splitting tree takes time 0(pn2). The size of the splitting tree it results in is O(n2). Finally, if a complete splitting tree is found, then an ADS for the corresponding machine can be deduced from it in time O(n2). Moreover, the obtained ADS has O(n2) nodes and length at most n(n - 1)/2. © Springer-Verlag Berlin Heidelberg 2005.