Lecture Notes in Computer Science 3472

19 Model Checking 595
queries. At this point we can implement the observation pack algorithm in two
different ways:
(1) Check for closedness (and rebuild the automaton) from scratch every time.
This means n(k +1)n 2 queries.
(2) Use the fact that access strings are never removed from the pack. This means
that the set of queries asked in one closedness check is a subset of the queries
to be asked in the next one. So, the total number of different queries over
all checks is at most (k +1)n 2 . We can avoid repeating queries by recording
all answers to membership queries, at the expense of using more memory.
Now consider the queries used to process the counterexample. If we do not
insist on obtaining the shortest distinguishing experiment, we can use Rivest
and Shapire’s binary search. This means using O(log m) queries for each counterexample,
hence O(n log m) for the at most n counterexamples.
In total, the algorithm that records all answers uses at most O(kn 2 +n log m)
membership queries.
This is also the cost of the reduced observation table algorithm, if precisely
the data structure recording all answers to membership queries is employed.
The discrimination tree algorithm, as described in [KV94], rebuilds the automaton
from scratch every time and processes the counterexample sequentially,
so it uses O(kn 3 + nm) membership queries. It is not difficult, however, to make
it record previous queries and use binary search to process the counterexample.
This modified version will have cost of O(kn 2 + n log m).
In the observation table algorithm, the number of columns in the table is at
most n, but the number of rows can be as large as O(knm) because all prefixes
of counterexamples are added as rows. Consequently, the number of queries can
be up to O(kn 2 m).
It can be shown that for any algorithm, making only O(n) equivalence
queries, at least Ω(kn log n) membership queries have to be made. Further results
on lower bounds can be found in [BDGW97].
Note however, that the results are worst-case estimations. One might might
in practice trade membership queries for equivalence queries. Experiences with
learning algorithms are given in Section 19.4.8.
19.4.7 Domain-Specific Optimizations
The number of queries can be expected to be a limiting factor in practice. Let
us study optimizations for learning that are possible when certain further information
about the system to learn is provided. The rationale of the presented approach
is that in practice, one is often concerned with learning a certain reactive
system that can be understood as a special deterministic finite state automaton
[HNS03].
The general concept of the optimizations presented here is that instead of
the Teacher, an Assistant is queried that might either answer a query by consulting
the Teacher, or, when possible, deduces the answer to the query using