Lecture Notes in Computer Science 3472

19 Model Checking 577
otherwise it returns a counterexample to A. The counterexample is in the
format of a string which is accepted by M but not by A or vice versa.
In the following sections we will describe different algorithms for the Learner,
all using the setting described above. The first algorithm, Observation Packs, abstracts
away the data structure in which a Learner stores the gathered information,
using sets instead. In the subsequent section, Section 19.4.2, we present Angluin’s
algorithm, in which the observed information is stored in an Observation
Table. TheReduced Observation Table algorithm, presented in Section 19.4.3,
is similar to the Observation Table algorithm except it stores less information.
Finally, Section 19.4.4 describes the Discrimination Tree algorithm which stores
information in a binary tree.
19.4.1 Observation Packs
Balcázar et al. abstract from different data structures and present a unified
view on the learning problem studied here, storing information in several, called
observation packs sets [BDGW97]. An observation pack can be seen as a way of
storing pieces of information the Learner has about an unknown regular set.
One piece of information is a so called observation. Anobservation is a pair
of the form (s, +) or (s, −) for a word s. Thelabel +/− signifies the answer to
a membership query on string s. The observations in a set must be consistent,
i.e., the same word does not appear with different +/− labels.
The observations are organized in finite sets called components, whichare
not necessarily disjoint. A component is denoted Ck ,wherek ∈ N is the component’s
index. An observation pack O, or pack for short, is a finite sequence
of components, O =(C0,...,Cn−1) forsomen ∈ N, for which two conditions
must hold:
OP1 Let sk ∈ Ck be the shortest word in Ck ;thensk is a prefix of all other words
in Ck .
OP2 For each two components Ck and Cl with l �= k, thereexistswkl such that
both sk wkl ∈ Ck and slwkl ∈ Cl but (Ck , +) ⇐⇒ (Cl, −), i.e., they have
different labels.
The string sk is a word that identifies Ck . The set of suffixes for each sk is
defined as Ek = {w | sk w ∈ Ck }. Sothewordwkl mentioned above is in Ek ∩ El.
Furthermore sk = sk ε implies ε ∈ Ek by the definition of Ek .
We collect in the set S all the shortest words, sk , from each component
and call them access strings. The access strings are then used to index both
components and sets of suffixes, so the component Ck is Cs and Ek is Es where
s = sk for some s ∈ S. An observation pack can be identified with the finite set
S of access strings and a mapping from S associating to each s the finite set Es.
The set Cs is the set of words sw, for an access string s ∈ S and suffix w ∈ Es.
Definition 19.10. A language Uagreeswith an observation pack O if all +
labels mark words in U, while all − labels mark words not in U.