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Lecture Notes in Computer Science 3472

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2 State Identification<br />

Moez Krichen<br />

VERIMAG<br />

moez.krichen@imag.fr<br />

2.1 Introduction<br />

In this chapter, we deal with the problem of state identification of f<strong>in</strong>ite-state<br />

Mealy mach<strong>in</strong>es. Before def<strong>in</strong><strong>in</strong>g the problem formally, let us illustrate it with a<br />

few examples and po<strong>in</strong>t out its differences with the similar problem of f<strong>in</strong>d<strong>in</strong>g a<br />

hom<strong>in</strong>g sequence.<br />

Informally, the state identification problem is the follow<strong>in</strong>g. We are given a<br />

(determ<strong>in</strong>istic) Mealy mach<strong>in</strong>e for which we know its behavior but not its current<br />

state. We want to f<strong>in</strong>d an <strong>in</strong>put/output experiment (i.e., an experiment dur<strong>in</strong>g<br />

which we apply <strong>in</strong>puts on the considered mach<strong>in</strong>e and observe the correspond<strong>in</strong>g<br />

outputs it produces) such that at the end of the experiment we know which<br />

state the mach<strong>in</strong>e occupied at the beg<strong>in</strong>n<strong>in</strong>g of the experiment. In other words,<br />

we know the transition and output functions of the given mach<strong>in</strong>e, but we do<br />

not know its current state and we want to f<strong>in</strong>d the latter by perform<strong>in</strong>g the<br />

experiment.<br />

Notice the difference with the problem of f<strong>in</strong>d<strong>in</strong>g a hom<strong>in</strong>g sequence, presented<br />

<strong>in</strong> the previous chapter. There, we want to know the state occupied by<br />

the mach<strong>in</strong>e after the experiment, whereas <strong>in</strong> state identification we are <strong>in</strong>terested<strong>in</strong>thestatebefore<br />

the experiment.<br />

A solution to the state identification problem is also a solution to the hom<strong>in</strong>g<br />

problem: s<strong>in</strong>ce the mach<strong>in</strong>e is determ<strong>in</strong>istic, if we are able to determ<strong>in</strong>e its state<br />

before the experiment then we also know its state after the experiment. However,<br />

as we shall see later, a solution to the hom<strong>in</strong>g problem is not necessarily a solution<br />

to the identification problem.<br />

Let us consider a simple example. Consider the mach<strong>in</strong>e M1 shown<strong>in</strong>Fig.2.1.<br />

Accord<strong>in</strong>g to this figure, M1 has two states s1 and s2. M1 has two <strong>in</strong>puts, a and<br />

b, and two outputs, 0 and 1. The transition and output functions of the mach<strong>in</strong>e<br />

are illustrated by directed arrows labeled with correspond<strong>in</strong>g <strong>in</strong>put and output<br />

symbols. For <strong>in</strong>stance, apply<strong>in</strong>g <strong>in</strong>put b on M1 when it is at state s1 will cause<br />

the mach<strong>in</strong>e to output 1 and move to state s2.<br />

Now,supposewedonotknowwhichstateM1 is currently occupy<strong>in</strong>g. We can<br />

determ<strong>in</strong>e this by apply<strong>in</strong>g <strong>in</strong>put a and observ<strong>in</strong>g the response. If the observed<br />

output symbol is 0 then we can deduce that the mach<strong>in</strong>e was <strong>in</strong>itially at state<br />

s1 otherwise the <strong>in</strong>itial state has been s2. The <strong>in</strong>put sequence a (consist<strong>in</strong>g of a<br />

s<strong>in</strong>gle symbol, <strong>in</strong> this case) has allowed us to dist<strong>in</strong>guish between the two states<br />

s1 and s2 of M1. It is a dist<strong>in</strong>guish<strong>in</strong>g sequence for this mach<strong>in</strong>e.<br />

Of course, the situation is not always as easy, and more complicated experiments<br />

may be necessary. Indeed, the state identification problem comes <strong>in</strong><br />

M. Broy et al. (Eds.): Model-Based Test<strong>in</strong>g of Reactive Systems, LNCS <strong>3472</strong>, pp. 35-67, 2005.<br />

© Spr<strong>in</strong>ger-Verlag Berl<strong>in</strong> Heidelberg 2005

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