Lecture Notes in Computer Science 3472

• SA ⊆ Σ ∗ is a nonempty finite prefix-closed set,
• EA ⊆ Σ ∗ is a nonempty finite suffix-closed set, and
• TA :((SA ∪ SA · Σ) × EA) →{+, −} is a (finite) function
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satisfying the property that se = s ′ e ′ implies TA(s, e) =TA(s ′ , e ′ )fors, s ′ ∈
SA ∪ SA · Σ and for all e, e ′ ∈ EA.
The words in SA ∪ SA · Σ are called row labels and the words in EA are called
column labels. The entries consists of signs (+/−) representing whether a word
is accepting or not.
The observation table is divided into an upper part and a lower part. The
upper part of the observation table is indexed by row labels in SA. Theyplaya
role similar to access strings in the observation pack algorithm, see Section 19.4.1.
The lower part’s row labels are indexed by all strings of the form sa, a ∈ Σ and
s ∈ SA, unless they already appear in the upper part. Moreover the table is
indexed column wise by a suffix-closed set EA of strings. The suffixes are used
in the same fashion as in Section 19.4.1, to distinguish a access string/row from
another. The function TA maps a row label s and a column label e, i.e. TA(s, e),
to the set {+/−}, it is defined to be + if se ∈ U and − otherwise. (Note that
TA is total.)
A function row(s) for every s ∈ (SA ∪ SA · Σ) denotes the finite function
f : EA →{+, −} defined by f (e) =TA(s, e). In other words, row(s) istherow
of entries in the observation table for index s.
It is possible that there exists an entry on a string in several places in the
table due to the fact that a string can be divided into different suffixes and
prefixes, i.e. row and column labels. Of course, these labels have to agree. This
is required by Definition 19.16.
A distinct row in OT characterizes a state in the automaton which can be
constructed from OT . All the row labels to unique rows must be kept in SA.
The rows labeled by elements of SA · Σ are used to create the transition function
for the automaton.
To construct a DFA from the observation table it must fulfill two criteria. It
has to be closed and consistent.
Definition 19.17. An observation table OT is closed if for each s ∈ SA · Σ
there exists an s ′ ∈ SA such that row(s) =row(s ′ ).
Definition 19.18. An observation table is consistent if whenever row(s) =
row(s ′ )fors, s ′ ∈ SA then row(sa) =row(s ′ a) for all a ∈ Σ.
When the observation table (SA, EA, TA) is closed and consistent it is possible
to construct the corresponding DFA A =(Σ,Q,δ,q0, F ) as follows:
• Q = {row(s) | s ∈ SA},
• q0 = row(ε),
• F = {row(s) | s ∈ SA and TA(s,ε)=+},
• δ(row(s), a) =row(sa).