Lecture Notes in Computer Science 3472

66 Moez Krichen
The two authors also argue a more general problem, similar to the problem
of checking the existence of an ADS for some given machine. It consists of identifying
the initial state of a given machine with the extra assumption that this
initial state belongs to a subset of the set of states. It is shown that this problem
is harder than the initial problem of checking the existence of an ADS as stated
by the following.
Theorem 2.25. ([LY94]) Given an FSM M and a set of possible initial states
Q, It is PSPACE-complete to tell whether there is an (adaptive) experiment that
identifies the initial state of M .
2.5 Summary
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
PDSthenithasanADS(becauseaPDSisanADS),however,theconverseis
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
anADSintimeO(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(pn 2 ) 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