Lecture Notes in Computer Science 3472

590 Therese Berg and Harald Raffelt
• SD, ED ⊆ Σ∗ are nonempty finite sets of access strings and suffixes, respectively,
• t is an SD ∪ ED-labeled binary tree where,
– non-leaf nodes are labeled with suffixes in ED, and
– leaves are labeled with access strings in SD.
The information about whether a string is accepted or not is contained in
the structure of a discrimination tree. The computation of the function γ(w)
forastringw - recall that γ(w) mapsw to the only access string w is like -
becomes simple with the use of discrimination trees; traverse the tree on w in
the following manner. Query wv formembershipineachnodelabeledv and enter
the right child node on a positive answer and left otherwise. The calculation of
γ(w) stops in a leaf which is labeled with an access string. The function γ is
total since the computation can be done for any input string w. LetSift be the
function that traverses a given discrimination tree for input string w in the just
described manner, starting at the root.
A discrimination tree is initialized with the root labeled by ε and two leaves,
one labeled by ε and the other one with a string which answers the opposite to
ε on a membership query.
The discrimination tree is always closed and consistent, therefore escaping
strings can only be obtained through counterexamples. Given a discrimination
tree DT , the escaping string is taken via the shortest prefix ui for
which siaivi+1 ∈ U ⇐⇒ si+1vi+1 �∈ U holds, where si = Sift(ui, DT ), see
Section 19.4.1 on how to handle counterexamples. The leaf si+1 is replaced by
an internal node labeled vi+1 which will separate the old leaf representing state
si+1from the new one siai. The algorithm can now start its next iteration with
another equivalence query. Altogether, note that equivalence queries are used to
modify the tree in order to derive a hypothesis of the automaton to learn, while
membership queries are used in the translation from the tree to an automaton
and for processing a counterexample.
The Learning Algorithm The discrimination tree algorithm consists of the
main function Discrimination-Tree and auxiliary helper functions; Sift, Hypothesis,
andUpdate-Tree. The complete algorithm is shown in Algorithm 17, [KV94,
BDGW97].
The main function Discrimination-Tree (line 1) first initializes the discrimination
tree. It performs a membership query for ε and if the answer to this query
is positive, meaning ε ∈ U, then the first hypothesis A has one accepting state,
otherwise one non-accepting. This state is the initial state. All the actions from
this state will loop back to this state.
The next step for the Learner is to execute an equivalence query with this
conjecture. With the information about the counterexample, which the Learner
receives, the discrimination tree will be initialized, the root labeled with ε and
the leaves labeled with ε and the counterexample, t, fromtheOracle.