Lecture Notes in Computer Science 3472

2 State Identification 41
FSMs (finite-state machines). These are machines, the response of which for any
initial state and for any applied input is known and unique.
An input sequence of the machine M is a finite sequence of input symbols.
Similarly, an output sequence is a finite sequence of output symbols. An input
or output sequence may be empty. The (input or output) empty sequence is
denoted by ɛ.
We introduce the following extensions of the transition and output functions
of a Mealy machine M =(I , O, S,δ,λ). First, we extend the functions λ and
δ from input symbols to input sequences. For an initial state s0 and an input
sequence x = a1a2 ···al, wehave:
where:
λ(s0, x )=b1b2 ···bl and δ(s0, x )=sl;
bi = λ(si−1, ai) andδ(si−1, ai) =si for i =1, ···, l.
The transition function is also extended from single states to sets of blocks of
states. By definition, a block of states is a nonempty subset of states. For the
block of states B we have:
δ(B, x )={δ(s, x ) | s ∈ B}.
Furthermore, we introduce the following notations. For some given input
sequence x , δ(·, x ) denotes the mapping defined from S to S such that:
∀ s ∈ S : δ(·, x )(s) =δ(s, x ).
Similarly, λ(·, x ) denotes the mapping from S to O ∗ such that:
∀ s ∈ S : λ(·, x )(s) =λ(s, x ).
For some input or output sequence x , |x | denotes the length of the considered
sequence. In the same manner, |B| denotes the cardinality of B a given subset
of S.
More details on Mealy machines are given in Appendix 21.
2.3 Preset Distinguishing Sequences
Here, we deal with preset distinguishing sequences (PDS). We first give the
mathematical definition of a PDS. Then, we list the main properties it matches.
Finally, we deduce the way for checking the existence and constructing a PDS.
2.3.1 What Is a PDS?
Here is the formal definition of a PDS.
Definition 2.1. An input sequence x is a PDS for a Mealy machine M =
(I , O, S,δ,λ)if: