Lecture Notes in Computer Science 3472

586 Therese Berg and Harald Raffelt
The algorithm returns (line 8) to check again that the observation table is
closed and consistent. This time it will discover an inconsistency in A 1 due to
row(εa) �= row(aa), the lefthand side being + and the right hand side being −.
Anewsuffixa, which distinguishes the two inconsistent rows a and ε, are added
to EA (line 12). The empty entries in the new columns are filled and the result
is A 2 in Table 19.2.
Table A 2 is next checked that it is closed (line 15). Since no row label in the
lower part of the table does not already exist in the upper part it is closed. It is
now possible to form the automaton A 2 , showed in Figure 19.10.
Next the algorithm performs an equivalence query to the Oracle with the
hypothesis A 2 (line 22). The response given is again a counterexample, this time
t = bb, sincebb �∈ L(Mex ) but bb ∈L(A 2 ). The string bb and its prefixes are
added to SA. The lower part of the table is extended by adding the new row
labels ba, bba and bbb. The algorithm fills all the empty entries by executing
membership queries. This yields table A 3 in Table 19.2.
In the last step the algorithm finds one more inconsistency, due to row(εb) �=
row(bb). Solving the inconsistency yields the new column label b, which is added
to EA. The resulting table A 4 , see Table 19.2, is closed and consistent and
the corresponding hypothesis, A 4 in Figure 19.10, returns a ’yes’ in the final
equivalence check. The algorithm returns A 4 and halts.
19.4.3 Reduced Observation Tables
We have so far seen two proposals of learning algorithms, the observation pack
and the observation table (or Angluin’s) algorithms. The next algorithm we will
present is a Learner closely related to Angluin’s algorithm.
In the setting of the observation table algorithm, see Section 19.4.2, the
observation table is likely to contain several rows representing one state. The
algorithm presented here is based on the observation table algorithm but contains
a smaller version of the table. We will refer to this algorithm by the name Reduced
Observation Table Algorithm, introduced by Rivest and Schapire [RS93].
Many notions of Angluin’s algorithm can directly be transfered to the reduced
observation table algorithm. In view of how a table is constructed the sets SA,
EA, and the table function TA correspond directly to SR, ER, andTR, respectively.
The entries, row labels, column labels and rows are also to be interpreted
as in Angluin’s algorithm. As in the case of the observation table algorithm, the
information the Learner accumulates is a finite collection of observations, which
is organized into a reduced observation table, denoted ROT . The table is defined
as follows:
Definition 19.19. A tuple ROT =(SR, ER, TR) over a given alphabet Σ is a
Reduced Observation Table, where
• SR, ER ⊆ Σ ∗ are nonempty finite sets,
• TR :((SR ∪ SR · Σ) × ER) →{+, −} is a (finite) function,