Lecture Notes in Computer Science 3472

582 Therese Berg and Harald Raffelt
A 0
s0 a, b
A 1
s0
a, b
s1
s2
a, b
a, b
A 2
Fig. 19.9. The machine’s approximations
have to create a new component, C3, based on the escaping string b, andaddit
to O. In order to distinguish the access strings s0 and s1, weaddtoE3, thesuffix
a to distinguish b from s1, and the suffix b to distinguish b from s0. (Thestring
ε distinguishes b from s2.) The new component is C3 = {(b, +), (ba, +), (bb, −)}
with corresponding suffix set E3 = {ε, a, b}, see Step 3.
We are now able to create our next hypothesis A 2 and conduct an equivalence
query for it. The answer to this query is ’yes’ and the algorithm terminates and
outputs the correct automaton A 2 , seen in Figure 19.9.
19.4.2 Angluin’s Algorithm
The learning algorithm by Balcázar et al is rather abstract, putting the selected
information in sets. In this section we discuss a learning algorithm that still
puts information into sets but uses a more concrete data structure: a table. The
observation pack algorithm does not tell us how to store exactly all information
we query for. In the algorithm we will introduce next, we see how it is possible
to store all information in an easy manner. We will now present the Angluin Algorithm,
which we will also refer to as the Observation Table Algorithm [Ang87].
The algorithm, or Learner, makes use of the same environment and plays the
same roll as described in Section 19.4. Initially the algorithm has no knowledge
of the SUT’s regular language, but the information it accumulates about the
behavior of the machine, is entered into a so called observation table. Due to the
table the algorithm has at any point in time information about a finite collection
of strings over a known finite alphabet Σ, classifying them as members or nonmembers
of some unknown regular set U.
Angluin’s algorithm will make sure that the observation table fulfills some
criteria before it constructs a deterministic finite-state machine A from information
in the observation table. This hypothesis of what the language of the SUT
is, is the Learner’s conjecture which will be sent to the Oracle.
The Observation Table The information accumulated by the algorithm is a
finite collection of observations, which is organized into an observation table.
The table is defined as follows:
Definition 19.16. An Observation Table over a given alphabet Σ is a tuple
OT =(SA, EA, TA) 1 ,where
1 The index A signifies that the sets belong to Angluin’s algorithm.
b
s0
s3
a
a
b
s1
s2
a, b
a, b