Reducing the Complexity of Checking the Existence and Derivation of
Adaptive Synchronizing Experiments for Nondeterministic FSMs
Natalia Kushik
1,2
, Nina Yevtushenko
2
and Husnu Yenigun
3
1
Telecom SudParis, Evry, France
2
Tomsk State University, Tomsk, Russia
3
Sabanci University, Istanbul, Turkey
Keywords: Nondeterministic Finite State Machines, Adaptive Sequences, Synchronizing Experiments.
Abstract: In this paper, we address the problem of setting a discrete system specified as a Finite State Machine (FSM)
to a known initial state. As the system behavior can be nondeterministic, we discuss the complexity of
problems related to deriving synchronizing sequences for nondeterministic FSMs and propose a way for
decreasing such complexity. In fact, we suggest to use adaptive synchronizing sequences instead of preset
ones, and prove that for non-initialized nondeterministic FSMs the problem of checking the existence of an
adaptive synchronizing sequence is in P.
1 INTRODUCTION
As the complexity of hardware/software systems
increases, new methods and techniques for efficient
checking of their functional and non-functional
requirements need to be developed. Such checking
can be effectively performed when the formal
description of a system under test is provided as a
finite state model. Finite state models are widely
used to derive tests with the guaranteed fault
coverage for checking functional requirements for
various types of digital systems (see, for example,
Gill, 1961; Chow, 1978).
A Finite State Machine (FSM) has finite non-
empty sets of states, inputs and outputs; when an
input is applied, the FSM moves to the next state
producing an output. In other words, FSMs include a
‘natural reactivity’ and that is the reason why they
are widely used when modeling systems working in
request-response mode. The behavior of such a
machine is described as a set of available
input/output sequences (traces). In general, the set of
traces is infinite. However, under certain conditions,
it is possible to identify a finite subset of traces such
that the application of input sequences from this
subset and the observation of the expected output
responses from an implementation under test (IUT)
allows one to conclude that the system meets the
corresponding requirements. Such a finite set of
traces is used to form, as it is called in the literature,
a checking sequence or a test suite.
Complex systems can usually have a
nondeterministic behavior, i.e. there can be several
output responses specified for an FSM at a given
state when an input sequence is applied. Meanwhile,
test generation techniques against nondeterministic
FSMs have higher complexity (Petrenko,
Yevtushenko, and Bochmann, 1996; Hierons, 1998)
and they mostly remain non-applicable to ‘real-life’
digital systems. Moreover, most of the test
derivation techniques for nondeterministic FSMs,
including the derivation of a single checking
sequence applied to an IUT rely on the fact that the
initial state of an IUT is known. Nevertheless, it is
not always the case and thus, effective methods and
techniques for setting up an IUT to a known
(current) state still remain one of the important
research topics for machines that can have a
nondeterministic behavior. Usually this ‘set up’ is
made by an application of an input sequence such
that after observing the output response (in which
case the sequence is called a homing sequence) or
without observing the output response (in which
case the sequence is called a synchronizing
sequence), one can conclude about the current state
of an IUT. Homing and synchronizing sequences
can either serve as preambles of checking sequences
or they can be used to set up a system into a
particular critical state. After such a set up, the
Kushik, N., Yevtushenko, N. and Yenigun, H.
Reducing the Complexity of Checking the Existence and Derivation of Adaptive Synchronizing Experiments for Nondeterministic FSMs.
DOI: 10.5220/0005854500830090
In Proceedings of the International Workshop on domAin specific Model-based AppRoaches to vErificaTion and validaTiOn (AMARETTO 2016), pages 83-90
ISBN: 978-989-758-166-3
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
83

required input sequences can be applied for checking
an IUT against appropriate (functional or non-
functional) test purposes such as safety, security,
robustness, etc. However, for nondeterministic
FSMs the length of homing and synchronizing
sequences can be exponential w.r.t. the number of
states (Ito and Shikishima-Tsuji, 2004; Kushik and
Yevtushenko, 2013). Correspondingly, additional
research should be performed for efficient derivation
of homing and synchronizing sequences for
nondeterministic FSMs.
In order to simplify the derivation of homing and
synchronizing sequences researchers turn their
attention to so called adaptive sequences that in
many cases can be shorter than the ordinary (preset)
ones. A sequence is adaptive if the next input to be
applied to an IUT is chosen based on the previously
observed outputs, and a sequence is preset if the
outputs need to be observed only after the
application of the whole sequence, or need not to be
observed at all.
Homing and synchronizing sequences are well
studied for deterministic FSMs and deterministic
automata in which outputs are omitted. Both types of
sequences allow to determine the current state of the
machine after the application of the input sequence.
In the case of a homing sequence, the conclusion
about the current FSM state is made based on the
observed output response, whereas for a
synchronizing sequence, the final state is unique
independently of the initial state of the FSM and the
observed output sequence.
For deterministic FSMs as well as for
deterministic automata the length of both homing
and synchronizing sequences is polynomial
(Klyachko, Rystsov and Spivak, 1987; Cerny, 1964;
Hibbard, 1961) when at each state a transition under
each input is defined. Moreover, it has been shown
that for deterministic FSMs the length of a homing
sequence cannot be decreased when considering
adaptive sequences instead of preset (Hibbard,
1961). For nondeterministic machines, preset
homing and synchronizing sequences are known to
have exponential length (Ito and Shikishima-Tsuji,
2004; Kushik and Yevtushenko, 2013).
Nevertheless, it has been also shown that the length
of a homing sequence can be decreased up to
polynomial when considering complete
nondeterministic FSMs (Kushik et al., 2014).
The authors are not aware of any results
regarding the existence check or the derivation of
adaptive synchronizing sequences for
nondeterministic FSMs, except for an idea presented
in (Kushik and Yevtushenko, 2012). In this paper, a
method is presented for checking the existence and
for deriving such adaptive synchronizing sequences
for a nondeterministic FSM that can have an
arbitrary number of initial states. In this case, an
adaptive synchronizing sequence is represented as a
corresponding single-input output-complete acyclic
FSM, introduced in (Petrenko and Yevtushenko,
2005) and hereafter referred to as a Test Case.
In this paper, necessary and sufficient conditions
for the existence of a synchronizing test case are
established. The obtained criterion allows to assess
the complexity of the corresponding procedure as
well as to determine the maximal length (height) of a
corresponding synchronizing test (if it exists). As
this bound is exponential, we show how this
complexity can be decreased and show that for non-
initialized FSMs the problem of checking the
existence of an adaptive synchronizing sequence is
in P. Furthermore, we propose an algorithm for
deriving a synchronizing test case for a
nondeterministic non-initialized FSM with the
polynomial length. The results listed above form the
main contribution of the paper.
The rest of the paper is organized as follows.
Section 2 contains preliminaries. The necessary and
sufficient conditions for the existence of an adaptive
synchronizing test case for a weakly initialized
nondeterministic FSM are established in Section 3.
The complexity of the corresponding procedure is
given in the same section. As this complexity is
shown to be exponential, a novel method is proposed
in Section 4 for nondeterministic non-initialized
FSMs for which the problem of checking the
existence of a synchronizing test case can be solved
in polynomial time. Section 5 concludes the paper.
2 PRELIMINARIES
A weakly initialized Finite State Machine (FSM) S
is a 5-tuple (S, I, O, h
S
, S
in
), where S is a finite set of
states with the set S
in
⊆ S of initial states; I and O are
finite non-empty disjoint sets of inputs and outputs,
respectively; h
S
⊆ S × I × O × S is a transition
relation, where a 4-tuple (s, i, o, s′) ∈ h
S
is a
transition. If | S
in
| = 1 then the FSM S is an
initialized FSM. If S
in
= S the machine is called a
non-initialized machine. An input i is defined at a
state s if there exists a transition (s, i, o, s′) ∈ h
S
.
FSM S = (S, I, O, h
S
, S
in
) is complete if for each pair
(s, i) ∈ S × I there exists a pair (o, s′) ∈ O × S such
that (s, i, o, s′) ∈ h
S
; otherwise, the machine is
partial. FSM S is nondeterministic if for some pair
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84

(s, i) ∈ S × I, there exist at least two transitions (s, i,
o
1
, s
1
), (s, i, o
2
, s
2
) ∈ h
S
, such that o
1
≠ o
2
and/or s
1
≠
s
2
. FSM S is observable if for each two transitions
(s, i, o, s
1
), (s, i, o, s
2
) ∈ h
S
it holds that s
1
= s
2
. Note
that in this paper, we consider complete observable
nondeterministic FSMs if the contrary is not
explicitly stated.
FSM S is single-input if at each state there is at
most one defined input at the state, i.e., for each two
transitions (s, i
1
, o
1
, s
1
), (s, i
2
, o
2
, s
2
) ∈ h
S
it holds
that i
1
= i
2
, and FSM S is output-complete if for each
pair (s, i) ∈ S × I such that the input i is defined at
state s, there exists a transition from s under i for
every output in O (Petrenko and Yevtushenko,
2005).
A trace of S at state s is a sequence of
input/output pairs of consecutive transitions starting
from state s. Given a trace i
1
o
1
… i
k
o
k
at state s, the
input projection i
1
… i
k
of the trace is a defined input
sequence at state s. For an observable
nondeterministic FSM, if γ = i
1
o
1
… i
k
o
k
is a trace at
a state s, then there exists a unique sequence of
consecutive transitions (s, i
1
, o
1
, s
1
)(s
1
, i
2
, o
2
,
s
2
)…(s
k-1
, i
k
, o
k
, s
k
). As usual, for state s and a
sequence γ ∈ (IO)* of input/output pairs, the γ-
successor of state s is the set of all states that are
reached from s by trace γ. If γ is not a trace at state s
then the γ-successor of state s is empty or we simply
say that the γ-successor of state s does not exist. For
an observable FSM S, for any string γ ∈ (IO)*, the
cardinality of the γ-successor of state s is at most
one. Given a subset S′ of states, the γ-successor of S′
is the union of γ-successors over all states of the set
S′.
Given an input alphabet I and an output alphabet
O, a test case TC(I, O) is an initially connected
single-input output-complete observable initialized
FSM P = (P, I, O, h
P
, {p
0
}) with an acyclic transition
graph. Given a complete FSM S over alphabets I and
O, a test case TC(I, O) represents an adaptive
experiment with the FSM S (Kushik et al., 2014).
If |I| > 1 then a test case is a partial FSM. A state
p ∈ P is a deadlock state of the FSM P if there are
no defined inputs at this state. In general, given a
test case P, the length (height) of the test case P is
defined as the length of a longest trace from the
initial state to a deadlock state of P and it specifies
the length of the longest input sequence that can be
applied to an FSM S during the experiment. As
usual, for complexity reasons, one is interested in
deriving a test case with minimal length.
A test case P is a homing test case for an FSM S
if for every trace γ of P from the initial state to a
deadlock state, the γ-successor of S
in
has at most one
state. A homing test case is a synchronizing test case
for an FSM S, if there exists a state s such that for
every trace γ of P from the initial state to a deadlock
state, γ-successor of S
in
is either {s} or the empty set.
If there exists a homing/synchronizing test case for
the FSM S then the set S
in
is a homing/synchronizing
set and the test case P is a homing/synchronizing test
case for the set S
in
. Otherwise, the set S
in
is not
homing/synchronizing.
We further discuss how to check whether there
exists a synchronizing test case for a given complete
observable nondeterministic FSM in polynomial
time. When there exists such a synchronizing test
case, we propose a technique to derive one. We also
provide the upper bound on the length of a
synchronizing test case. As this technique relies on
the presence of so called definitely-reachable state
(Petrenko and Yevtushenko, 2011) in the equivalent
initialized FSM, we repeat this notion here and
briefly sketch the procedure to check the existence
of such state in a nondeterministic FSM.
Given a complete weakly initialized FSM S = (S,
I, O, h
S
, S
in
), state s′ ∈ S is definitely-reachable (d-
reachable) from state s ∈ S if there exists a test case
P(s, s′) over alphabets I and O such that for every
trace γ of P(s, s′) from the initial state to a deadlock
state, the γ-successor of state s is either the empty set
or is the set {s′}. We hereafter refer to such a test
case as a d-transfer test case.
If the machine is initialized and state s′ ∈ S is
definitely reachable from the initial state then we
simply say that state s′ ∈ S is definitely-reachable.
In (Petrenko and Yevtushenko, 2011), necessary
and sufficient conditions are established that allow
to check if state s′ ∈ S is definitely reachable from
state s ∈ S. In particular, it is proven that state s′ of
an FSM S is definitely reachable from state s if and
only if S has a single-input acyclic submachine S
′
with the initial state s and the only deadlock state s′
such that for each input defined in some state of S′,
the state has all the transitions of S labeled with this
input. Moreover, in the same paper, an efficient
method is proposed for checking whether state s′ is
definitely reachable from a state s, and if it is, then it
is proposed how to derive a corresponding test case
(see Procedure 1 given below).
Note that since any d-transfer test case P(s, s′) is
an acyclic submachine of the machine S, then the
length of any trace in P(s, s′) does not exceed the
number n of states of S; in other words, one needs at
most n – 1 inputs to adaptively transfer the possibly
nondeterministic machine from state s to state s′.
Reducing the Complexity of Checking the Existence and Derivation of Adaptive Synchronizing Experiments for Nondeterministic FSMs
85

Therefore, the length of a longest trace in a shortest
test case P(s, s′) is polynomial and is at most n – 1.
Procedure 1 for constructing a d-transfer test case
P(s, s′)
Input: An FSM S = (S, I, O, h
S
, S
in
), states s, s′ ∈ S,
s′≠ s.
Output: a d-transfer test case P(s, s′) if the state s′ is
definitely reachable from s or a message “the state s′
is not definitely reachable from s”.
Construct an FSM P(s, s′) = (P, I, O, h
P
, {p
0
}) as
follows
Step 1 P := {s′}; h
P
:= ∅;
Step 2
While there exist a state ŝ ∉ P and a set of inputs I
ŝ
,
such that for each input i ∈ I
ŝ
, (ŝ, i, o, s'') ∈ h
S
,
s'' ∈ P for every possible output o to an input i at
state ŝ in FSM S
P : = P ∪ {ŝ}; h
P
:= h
P
∪ {(ŝ, i, o, s'')}
EndWhile
If s ∉ P
Then return the message “the state s′ is not
definitely reachable from s”.
Step 3 p
0
:= s;
In the FSM P(s, s′), remove from each state with
several defined inputs all outgoing transitions with
the same input until each such state has a single
defined input.
Delete states which are unreachable from the initial
state, add transitions to the designated deadlock state
D for each missing output at each state (if any).
Return P(s, s′).

3 DERIVING SYNCHRONIZING
TEST CASES FOR
NONDETERMINISTIC FSMs
In this section, we discuss how a synchronizing test
case can be derived for a complete observable
nondeterministic FSM. In fact, the corresponding
procedure is based on necessary and sufficient
conditions for the existence of such test case that are
stated in Proposition 1. The conditions rely on
checking the existence of definitely-reachable states
in the corresponding initialized machine. Such an
equivalent initialized machine E
S
can be obtained
through the determinization of the underlying
automaton. The set of traces of E
S
is the union of
sets of traces over all states of the set S
in
.
Proposition 1. There exists a synchronizing test
case for a complete observable FSM S = (S, I, O, h
S
,
S
in
), if and only if the FSM E
S
has a definitely-
reachable state {s} for some s ∈ S.
Proof. There is a trace γ at the initial state of the
equivalent initialized machine E
S
if and only if such
a trace is a trace at some initial state of FSM S.
Correspondingly, the γ-successor of the initial state
of E
S
is the γ-successor of the set S
in
in FSM S. If the
machine E
S
has a definitely-reachable state {s} then
each trace of the FSM P(e
0
, {s}) from the initial
state to a deadlock state where e
0
= S
in
is the initial
state of E
S
, takes the FSM S from any initial state to
state s. On the other hand, let there exist some state
s ∈ S and a synchronizing test case P for FSM S
such that each trace γ of the FSM P from the initial
state to a deadlock state takes the FSM S from any
initial state to state s. In this case, the γ-successor of
the set S
in
is the set {s} and the state {s} is a
definitely-reachable state in the FSM E
S
with a
corresponding d-transfer test case P.

Proposition 1 establishes necessary and sufficient
conditions for the existence of a synchronizing test
case P for a nondeterministic complete FSM S.
Moreover, it also gives a hint for a procedure to
derive such a test case (when it exists). The first step
of this procedure is the derivation of an initialized
complete FSM E
S
that is equivalent to the machine S
under experiment. At the second step, Procedure 1 is
called, and for each state {s} of the equivalent FSM
E
S
, it is checked if {s} is definitely-reachable.
As the number of states of the machine E
S
does
not exceed 2
n
– 1 when | S | = n, and by construction,
the FSM E
S
is always observable, the following
statement holds.
Proposition 2. For each synchronizing FSM S =
(S, I, O, h
S
, S
in
), | S | = n, there exists a synchronizing
test case with the length that does not exceed 2
n
– n –
1.
As an example, consider an FSM S with a flow
table in Table 1 (Kushik and Yevtushenko, 2012).
Table 2 represents the flow table for an initialized
observable FSM E
S
. Note, that for the sake of
simplicity we denote a subset {s
1
, …, s
k
} of FSM
states as
s
1
,..., s
k
.
By direct application of Procedure 1, one can
check that state
3
is definitely-reachable in the
initialized FSM E
S
(Table 2).
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86

Table 1: The flow table of an FSM S.
i/s 1 2 3
a 2/1 3/0; 2/1 1/1
b 3/0, 1 3/0; 1/2 2/0, 1
c 2/0, 1 3/0; 1/2 3/0, 1
Table 2: The flow table of the equivalent FSM E
S
.
i
/
s
1, 2, 3
1, 2
2, 3
2
3 1
a
1, 2
/1
3
/0
3
/0
2
/1
3
/0
1, 2
/1
3
/0
2
/1
1
/1
2
/1
b
2, 3
/0
2, 3
/1
1
/2
3
/0
3
/1
1
/2
2, 3
/0
2
/1
1
/2
3
/0
1
/2
2
/0
2
/1
3
/0
3
/1
c
2, 3
/0
2, 3
/1
1
/2
2, 3
/0
2
/1
1
/2
3
/0
3
/1
1
/2
3
/0
1
/2
3
/0
3
/1
2
/0,1
In fact, at the first iteration the set P = {
3
} is
updated with state
1
due to input b. Then state
2
is
added to the set P = {
3
,
1
} due to and input b or c,
etc. In the end, the set P equals the set of states of
the FSM E
S
. Therefore, there exists a synchronizing
test case for the FSM S, and this test case is
represented in Fig. 1. Note, that in this example the
length of the test case equals three, i.e. the worst
complexity case is not reached.
Figure 1: A synchronizing test case P for an FSM S
(Method 1).
In general, the upper bound on the length of a
synchronizing test case is exponential, and we
further discuss how this complexity can be
decreased.
4 REDUCING THE LENGTH OF
SYNCHRONIZING TEST
CASES FOR
NONDETERMINISTIC FSMs
In this section, we discuss how the length of
synchronizing test cases can be reduced for
nondeterministic FSMs. We show that proposed
conditions are necessary and sufficient for non-
initialized FSMs, i.e., for complete and observable
machines where each state can be initial state.
However, for weakly initialized machines the
conditions are only sufficient when checking the
existence of a synchronizing test case.
As each synchronizing test case is a homing test
case with additional constraints, the existence check
as well as the derivation of such test cases relies on
such procedures for homing test cases. In (Kushik
and Yevtushenko, 2015), it has been shown that
when each state of a complete observable
nondeterministic FSM can be initial, the existence of
a homing test case can be checked in polynomial
time. The procedure for deriving such test cases and
potential heuristics improving the performance of
the corresponding algorithm are discussed in
(Kushik and Yenigun, 2015). We further briefly
sketch this procedure (Procedure 2) as it is used for
deriving a synchronizing test case.
Not every homing test case is a synchronizing
test case for an FSM S, and moreover, even if a
homing test case exists for S, a synchronizing test
case may not exist for it. However, if for an
adaptively homing non-initialized FSM S there
exists a state s′ such that for any state s ∈ S, there
exists a d-transfer test case P(s, s′), then a
synchronizing test case can always be derived (and
vice versa).
Proposition 3. There exists a synchronizing test
case for a complete observable FSM
S = (S, I, O, h
S
, S
in
), S
in
= S, if and only if FSM S is
homing and there exists a state s′ ∈ S such that for
each state s ∈ S state s′ is definitely reachable from
s.
Proof. Indeed, given a homing FSM S with a
homing test case R, consider a trace γ that takes R
from the initial state to a deadlock state. Since R is a
homing test case, the γ-successor of each state of the
set S either does not exist or contains a unique state
s. Since for any state s ∈ S, there exists a d-transfer
test case P(s, s′), then each trace σ that takes P(s, s′)
from the initial state to a deadlock state takes the
FSM S from state s to state s′, i.e., the γσ-successor
Reducing the Complexity of Checking the Existence and Derivation of Adaptive Synchronizing Experiments for Nondeterministic FSMs
87

of each state of the set S either does not exist or
contains a unique state s′. On the other hand, if a
synchronizing test case P exists for the FSM S then
there exists a state s′ such that for each state s ∈ S
the test case P is a d-transfer test case P(s, s′).

Procedure 2 for deriving a homing test case for a
complete observable FSM
Input: A non-initialized complete observable FSM
S = (S, I, O, h
S
, S)
Output: A homing test case P for the FSM S or a
message “FSM S is not homing”
Step 1 Derive a test case R
jk
for each pair {s
j
, s
k
} of
different states of S such that the set of states of
different test cases do not intersect except for the
designated deadlock state. If they do, without loss of
generality, rename them. These test cases can be
derived by calling a corresponding procedure from
(Kushik and Yevtushenko, 2015). Represent them as
as tree-like FSMs (where only the leaf nodes are
allowed to have an indegree greater than 1) with
terminal nodes r
1
, …, r
n
and designated deadlock
state D, such that there exists a trace γ from the
initial state of R
jk
to state r
j
if and only if {r
j
} is the
γ-successor of the pair {s
j
, s
k
}.
If at least one pair of states is not homing
Then Return the message “FSM S is not homing”.
Step 2 k := 3
While k ≤ n
For each trace γ = α(io) that takes R
12...(k-1)
from the
initial state to the deadlock state r
a
, a ∈ {1, …, n},
while taking S from state s
k
to state s
t
;
1. Replace in R
12..(k-1)
the transition (r, i, o, r
a
)
with the transition (r, i, o, D
ta
).
2. Append the R
12..(k-1)
with R
ta
at state D
ta
always
checking that the set of states of a test case
under construction is different from the set of
each R
jk
except of the designated deadlock
state D; if that is not the case, without loss of
generality, rename the corresponding states.
EndFor
k++
EndWhile
Step 3 Delete each state r
a
that is not reachable from
the initial state. Minimize FSM R
12...n
in a usual way
(if necessary).
Return the test case R
12...n
.

Note, that for a weakly initialized FSM the
conditions of Proposition 3 become only sufficient.
The reason is that when a synchronizing test case P
exists for a weakly initialized FSM S, there exists a
state s′ such that for each initial state s ∈ S
in
the test
case P is a d-transfer test case P(s, s′). However, the
latter does not necessarily hold for each state of
FSM S.
Checking the existence of a homing test case R
for a complete observable FSM S = (S, I, O, h
S
, S)
can be performed in polynomial time (Kushik and
Yevtushenko, 2015). On the other hand, the
complexity of checking the existence of a d-transfer
test case P(s, s′) for a state pair {s, s′} is polynomial
as well. Indeed, this complexity is ‘hidden’ in the
maximal number of iterations at Step 2 (Procedure
1). In the worst case, at each iteration, only one state
is added to the set P. Therefore, after at most (n – 1)
iterations, either a message “the state s′ is not
definitely reachable from s” will be produced or a
test case P(s, s′) will be returned. As the number of
state pairs is polynomial, the problem of checking
the existence of a state s′ ∈ S such that for each state
s ∈ S, state s′ is definitely reachable from s, can be
solved in polynomial time. In other words, the
following proposition holds.
Proposition 4. For complete observable non-
initialized FSMs checking the existence of a
synchronizing test case can be performed in
polynomial time.
When an adaptive synchronizing test case exists
it can be derived with the use of Procedure 2 (for a
homing test case) and application of Proposition 3.
The corresponding algorithm is presented as
Procedure 3.
Proposition 5. For each synchronizing FSM S =
(S, I, O, h
S
, S), S
in
= S, there exists a synchronizing
test case with the length of the order O(n
3
).
Proof. In fact, the length of the homing test
R
12...n
(if it exists) does not exceed (n – 1)
(
2
n
)
,
where
(
2
n
)
is the number of different state pairs of
the FSM S. The length of a d-transfer test case P(s
k
,
r
a
) does not exceed (n – 1) if it exists. Therefore, the
overall length of a synchronizing test case is at most
((n – 1)
2
n)/2 + (n – 1).

In the running example, we first derive a homing
test case R
1,2,3
for the set S
in
= {1, 2, 3}. A homing
test case that can be returned by Procedure 2 is
illustrated in Fig. 2.
AMARETTO 2016 - International Workshop on domAin specific Model-based AppRoaches to vErificaTion and validaTiOn
88

Figure 2: A homing test case R
1,2,3
for an FSM S.
Procedure 3 for deriving a synchronizing test
case for a complete observable FSM
Input: A non-initialized complete observable FSM
S = (S, I, O, h
S
, S)
Output: A synchronizing test case P for the FSM S
or a message “FSM S is not synchronizing”
Step 1 Derive a homing test case R
12...n
for the FSM
S by calling Procedure 2.
If Procedure 2 returns a message “FSM S is not
homing”
Then Return the message “FSM S is not
synchronizing”
Step 2 k := 1
While k ≤ n
For each terminal state r
a
of the homing test case
R
12...n
, such that r
a
≠ s
k
Call Procedure 1 to derive a d-transfer test case P(r
a
,
s
k
)
EndFor
If P(r
a
, s
k
) is derived for each terminal state r
a
of the
test case R
12...n
, i.e. each state r
a
is adaptively
transferred to state s
k
,
Then
For each trace γ = α(io) that takes R
12...n
from the
initial state to state r
a
, a ∈ {1, …, n}
Replace in R
12...n
the transition (r, i, o, r
a
) with the
transition (r, i, o, D
a
).
Append the R
12...n
with P(r
a
, s
k
) at state D
a
.
EndFor
Minimize the obtained FSM in a usual way (if
necessary) and Return the reduced FSM R
synch
.
k++
EndWhile
Step 3
Return the message “FSM S is not synchronizing”.

As state
1
can be transferred to state
3
by an
application of a single input b, the resulting
synchronizing test case still has length 3 (Fig. 3).
Figure 3: A synchronizing test case R
synch
for an FSM S
(Method 2).
5 CONCLUSIONS
In this paper, the notion of an adaptive
synchronizing test case for nondeterministic finite
state machines has been proposed. A method for
checking the existence of such test cases and a
procedure for their derivation are also proposed
whenever it is possible. As the height of the
corresponding test case in general can be
exponential, we discussed a possibility of reducing
the complexity. In particular, we have proposed the
necessary and sufficient conditions for non-
initialized FSMs for checking the existence of a
synchronizing test case in polynomial time. The
conditions are only sufficient when checking the
existence of a synchronizing test case for weakly
initialized FSMs. As a future work, we plan to
perform experimental evaluation in order to
investigate how ‘realistic’ is the FSM class with the
reduced complexity of adaptive synchronizing test
cases. We also would like to investigate more FSM
classes for which the complexity of related problems
is polynomial, as well as to study adaptive
synchronizing test cases for partial machines, i.e. the
problem of careful adaptive FSM synchronization.
ACKNOWLEDGEMENTS
This work is partially supported by RFBR
Grant# 15-58-46013 CT_a and TÜBİTAK Grant#
114E921.
Reducing the Complexity of Checking the Existence and Derivation of Adaptive Synchronizing Experiments for Nondeterministic FSMs
89

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