Lecture Notes in Computer Science 3472

19 Model Checking 587
• se = s ′ e ′ implies TR(s, e) =TR(s ′ , e ′ )fors, s ′ ∈ SR ∪ SR · Σ and for all
e, e ′ ∈ ER, and
• row(s) =row(s ′ ) implies s = s ′ for all s, s ′ ∈ SR.
Thus, a reduced observation table differs from an observation table in two
ways. First, SR does not need to be prefix-closed. Second, every row appears
only once in the upper part of the table.
Since there cannot be two row labels in SR that map to the same row, there
is no need to check a reduced observation table for consistency. In other words,
there cannot be any inconsistency.
The reduced observation table algorithm contains only the access strings,
recall Section 19.4.1, which results in a smaller table than Angluin’s. This in
turn is the cause for using less membership queries. The definition of a closed
ROT is as for the observation table. From a closed ROT we can can construct
an automaton in the same manner as in Angluin’s algorithm. Besides containing
a smaller observation table, a second source of efficiency, compared to Angluin’s
algorithm, is the faster processing of counterexamples.
Processing a counterexample t = uivi of length m, recall Section 19.4.1,
means finding a breakpoint i such that sivi ∈ U ⇐⇒ si+1vi+1 �∈ U, where
the si = δ(row(ε), ui) are the states visited by t along the automaton and
vi are the corresponding suffixes of t. Some such breakpoint must exist since
s0v0 ∈ U ⇐⇒ smvm �∈ U, so that a sequential search will find, say, the first one
with m membership queries. Rivest and Schapire show how a binary search finds
a breakpoint with logm queries.
The reduced table is kept small by not adding all prefixes of the counterexample
as rows. This means that the new automaton may still classify the previous
counterexample incorrectly, so the same counterexample can potentially be used
to answer several equivalence queries. Two equal counterexamples can also occur
in an algorithm which uses so called discrimination trees, to be discussed in
Section 19.4.4.
The Learning Algorithm The reduced observation table algorithm is basically
constructed in the same manner as the observation table algorithm. The
difference is that since only unique rows are contained in the upper part of the
table of the reduced observation table algorithm there is no need to check for consistency.
The handling of a counterexample is different to Angluin’s algorithm.
In this algorithm we search for a breakpoint in the counterexample and add only
one row to the upper part of the table, not every prefix of the counterexample
as in the case of the other algorithm. The Reduced Observation Table algorithm
is given in Algorithm 16.
Example of Reduced Observation Table We will here present an example
how the reduced observation table evolves when learning the same example as in
Section 19.4.2 shown in Figure 19.8. The table is initialized in the same manner
as in Angluin’s algorithm so SR and ER is set to ε (line 3). For all actions in