Int. J. Communications, Network and System Sciences, 2012, 5, 638-643 Published Online September 2012 (
Local Search Heuristics for NFA State
Minimization Problem*
Andrey V. Tsyganov
Department of Higher Mathematics, Ulyanovsk State Pedagogical University, Ulyanovsk, Russia
Received July 2, 2012; revised July 26, 2012; accepted August 6, 2012
In the present paper we introduce new heuristic methods for the state minimization of nondeterministic finite automata.
These methods are based on the classical Kameda-Weiner algorithm joined with local search heuristics, such as sto-
chastic hill climbing and simulated annealing. The description of the proposed methods is given and the results of the
numerical experiments are provided.
Keywords: Nondeterministic Finite Automata; State Minimization; Heuristics; Local Search; Parallelism
1. Introduction
Finite automata (FA) are widely used in various fields
and especially in the theory of formal languages. We
suppose that the reader is familiar with the basics of
automata theory (see, for example, [1]) and provide only
some necessary definitions.
Let be an alphabet and n where
i be a word. A set of words* is called
a language
waa a
The nondeterministic finite automaton (NFA) is a
QIF Q
, where is a finite set of
states, is a finite alphabet,
 is a transi-
tion relation,
Q and
Q are respectively the
sets of initial and final states. Transitions of the automa-
are often described by the transition function
. The NFA is called deterministic (DFA)
1I and
,:a ,1qa
qQ (or
Finite automata may be used to recognize and define
languages. Two automata are called equivalent if they
recognize one and the same language. For each NFA the
equivalent DFA may be constructed using the powerset
construction process (each state of such DFA is a subset
of states of the original NFA).
For a word the reverse word is
12 n
waa a
waa a
, for a language L the reverse language is
,,,, and for an automaton
the reverse automaton is
QFI where
ij iff
For a given language L a DFA which recognizes it and
has the minimum possible number of states is called the
canonical automaton and a DFA which recognizes
and has the minimum possible number of states is called
the reverse canonical automaton (these automata are
unique for L up to isomorphism).
The NFA state minimization problem is formulated as
follows: for a given NFA A find an automaton which is
equivalent to it and has the minimum possible number of
states. Note that solution of this problem may not be
unique. As it is shown in [2] the state minimization
problem for NFA is PSPACE-complete. The worst case
complexity for the same problem for DFA is
logOQ Q .
All known exact NFA state minimization methods use
different types of exhaustive search and are computa-
tionally hard. Very often they become impractical even
for relatively small automata. This is one of the reasons
why they are not implemented in software tools that deal
with finite automata and related structures, such as
AMoRE [3], FSM [4], Vaucanson [5], JFlap [6]. More-
over, only few of such tools provide heuristic NFA state
minimization algorithms.
In the present paper we propose new heuristic methods
for NFA state minimization problem which are based on
the classical Kameda-Weiner algorithm [7] and well-
known local search heuristics (metaheuristics). The nov-
elty of these methods is that the most time consuming
part of the exact algorithm is replaced with fast heuristic
procedures. The obtained methods are not exact but they
allow to reduce minimization time.
*The work is supported by the Ministry of Education and Science o
the Russian Federation (grant n umber 1.919.2011).
opyright © 2012 SciRes. IJCNS
The remainder of the paper has the following structure.
In Sections 2 and 3 a brief description of the Ka-
meda-Weiner algorithm and local search heuristics is
given, in Section 4 the proposed algorithm is described
and in Section 5 the results of some numerical experi-
ments are provided.
2. Kameda-Weiner Algorithm
Lets us consider the brief description of the Kameda-
Weiner algorithm (for detailed description see [7]).
Suppose that NFA A is given. The algorithm searches
for the minimum state NFA(s) equivalent to A using bi-
nary matrix RAM (Reduced Automaton Matrix) which is
constructed as follows.
First, canonical automaton B and reverse canonical
automaton C for A are constructed. Note that each state
of these automata is a subset of states of the automaton A.
Then, for each nonempty states i of
and p
1, ,Bi m
q of the element of the RAM
is defined by the following formula
1,Cj n
Let X be a subset of rows and Y a subset of columns of
the RAM. Then
1,3 2,3
1,2,3 2
is called a (prime) grid if it sat-
isfies the following conditions: 1) all intersections of its
rows and columns contain 1 s; and 2) neither X nor Y can
be enlarged without v iolating the fist condition.
The set of grids cover RAM if each 1 in it belongs to
at least one grid in the set. A minimum cover of RAM is
a cover which consists of the minimum number of grids.
Let us consider the NFA A with transitio n table shown
in Table 1 (the example is taken from [7]).
Tables 2 and 3 show the canonical automaton B and
the reverse canonical automaton C of A respectively.
RAM of A is presented in Table 4.
There are 4 grids in RAM: 1)
, 2)
, 3)
, 4)
. It is
easy to see that the first two grids make the minimum
cover of RAM.
3 1,2
 
Given a cover of RAM one can construct an NFA
which may be equivalent to the original NFA A (in this
case the cover is called legitimate). This is done by the
means of the special intersection rule. The number of
states in the constructed NFA equals to the number of
grids in the cover. In the considered example the
minimum cover of RAM is legitimate and yields the
minimum state NFA shown in Table 5.
The general schema of the Kameda-Weiner minimiza-
tion technique is described by Algorithm 1. Note that
steps 1, 3 and 4 of this algorithm theoretically have ex-
ponential complexity. On practice the construction of the
canonical automata usually performed rather quick and
the most time consuming parts of the algorithm are steps
Table 1. NFA A.
1 1,3 2
2 1 2,3
3 1 3
Table 2. Canonical automaton B.
Table 3. Reverse canonical automaton C.
21, 2,3c2
Table 4. RAM of A.
1,2,3 2,3
2,3 0 1 1
1 1 1 0
1,3 1 1 1
Table 5. Minimum state NFA equivalent to A.
p 2
Algorithm 1. Kameda-Weiner algorithm.
Require: NFA A
1:Cons truct canonical automata B and C
2:Const ruct RAM
3:Find all g rids of RAM
4: Find minimum legiti mate c ove r(s) of RAM and construct
minimum state NFA(s) using intersec tion rule
Ensure: Minimum state NFA(s) equivalent to A
Copyright © 2012 SciRes. IJCNS
3 and 4 which perform the exhaustive search for grids
and covers respectively.
3. Local Search Heuristics
Local Search (LS) is a group of metaheuristic optimiza-
tion techniques which are widely used especially in com-
binatorial optimization. Its general schema is described
by Algori thm 2.
Each LS algorithm starts from some initial solution
(line 1) and then iteratively updates it (lines 2 - 4) until
stop condition is satisfied (in the most simple case it is
the maximum number of steps). The Neighbor() function
finds neighbors of the current solution Solution and the
Update() function changes the current solution depending
on the found neighbors. The quality of the solutions is
compared using the special Cost() function.
The simplest LS algorithm is called Hill Climbing
(HC). In HC the Update() function always selects the
best neighbor of the current solution (the steepest move).
The Stochastic Hill Climbing (SHC) is a variant of HC
which randomly chooses one of the neighbors and decides
whether to move to it or to consider another neighbor.
The disadvantage of the HC algorith ms is that th ey can
easily stuck in the local optimum. The Simulated An-
nealing (SA) is a more complicated LS algorithm which
tries to avoid this problem. Algorithm 3 shows in detail
the minimization process using SA.
The distinctive featur e of the algorithm is the usage of
the control parameter T (temperature) which slowly de-
creases as the number of iterations k increases. The
Neighbor() function generates a neighbor of the current
solution and the Update() function accepts it with the
if 0;
e,if 0,
ost Solution
where .
Cost NewSolutionC 
As it follows from (1) the algorithm always accepts the
generated neighbor if it is not worse than the current so-
lution. If the neighbor is worse than the current solution
it is accepted with some probability which decreases as
the temperature decreases. So, the worse moves are made
more often in the beginning of the optimization process.
In classical SA the best solution is not stored (lines 2, 3,
8 - 12 are missing) and the algorithm returns the current
More details on heuristic optimization algorithms can
be found in [8,9].
4. Combinig Kameda-Weiner Algorithm
with Local Search Heuristics
First of all let us consider in more detail the last step of
Algorithm 2. Local search.
Require: InitialSolution
1: Solution := InitialSolution
2: repeat
3: Solution := Update(Neigh bo r(Solution))
4: until StopCondition()
Ensure: Solution
Algorithm 3. Simulated annealing.
Require: InitialSolution
1: Solution := InitialSolution
2: BestSolution := Solution
3: BestCost := Cost(Solution)
4: T := InitialTemperatue()
5: k := 0
6: while not StopCondition() do
7: NewSolution := Neighbor(Solution)
8: NewCost := Cost(NewSolution)
9: if NewCost < BestCost then
10: BestSolution := NewSolution
11: BestCost := NewCost
12: end if
13: Solution := Update(Solution, NewSolution, T)
14: k := k + 1
15: T := Update Temperature(T, k)
16: end while
Ensure: BestSolution
the Kameda-Weiner algorithm, i.e. the exhaustive search
for minimum legitimate covers (see Algorithm 4). Here
M is a set of minimum state NFAs. IsLegitimateCover()
function tests whether the set of grids is a legitimate
cover and IntersectionRule() constructs NFA using the
cover. The bounds and of the main loop may
be calculated as follows:
min 2
N is
the number of states in canonical automaton B,
min, 1, 1iNNN
max GA B
N where G
N is the number
of grids in RAM, A is the number of states in A and
is the number of states in B. NFor each step of the outer loop in the inner loop (lines
5 - 10) all possible i-combinations of grids have to be
analyzed. The idea of the heuristic methods proposed in
this paper is to replace this computationally hard process
Copyright © 2012 SciRes. IJCNS
Algorithm 4. Exhaustive search for minimum legitimate
1: Calculate and
2: :M
3: for i from to do
4: found := false
5: for all i -combinations of grids Comb do
6: if IsLegitimateCover(Comb) then
7: {IntersectionRule(Comb)}
8: found := true
9: end if
10: end for
11: if found = true then
12: break
13: end if
14: end for
by non-exhaustive LS procedures (of course, this may
result in obtaining approximate solutions, i.e. reduced
NFAs). This approach is described by Algorithm 5.
So, we use LS in the Kameda-Weiner algorithm to
find minimum covers of RAM and then analyze their size
and legitimacy. Now let us consider the details of this
process. The solution of both LS methods (SHC and SA)
for the considered problem is a cover which is coded by a
binary vector where 1 in i-th position means that i-th grid
is included in a cover and 0 means that it is no t included.
In the considered example vector means that
only first two grids are included in the cover.
To start search LS algorithms need the initial so lution.
The simplest way to setup the initial solution is to use the
trivial solution with all bits set to 1. To obtain nontrivial
initial solution Algorithm 6 may be used.
The Cost() function simply counts the number of 1s in
the vector and the Neighbor() function inverts several
bits in it. After creating a neighbor of the current solutio n
we need to check its feasibility (i.e. to check whether the
obtained set of grids covers all 1s in RAM). If the con-
structed neighbor is not a feasible solution then we add
one or several 1s to it using algorithm similar to Algo-
rithm 6. (e.g., in the considered example the solution
is not feasible and it needs to be corrected ).
To ensure the diversity of minimum covers one may
run LS several times or use parallel versions of LS where
each thread starts from its own initial solution. Note also
that if the minimum legitimate covers not found then
both exact end heuristic methods return the canonical
automaton B if it has less number of states than the given
Algorithm 5. Heuristic search for minimum legitimate cov-
1: Calculate and i
2:Find minimum cover(s) of RAM using LS
3: if cover(s) of size less or equal to found then
4: Selec t mi n i mum le g i t i ma t e cover(s) and construct
minimum state NFA(s) using intersection rule
5: end if
Algorithm 6. Construction of initial solution for LS algo-
1: repeat
2: Take next 1 in the RAM
3: Randomly choose a grid that covers this 1
4: Set to 1 the corresponding bit of the InitialSolution
5: Eliminate all 1s that are covered by the chosen grid
from the RAM
6: until there are 1s in the RAM
automaton A.
5. Numerical Experiments
We have implemented the proposed NFA minimization
methods in the ReFaM project. ReFaM (Rational Ex-
pressions and Finite Automata Minimization) is a part of
the HeO (Heuristic Optimization) library. This library is
a cross-platform open source project written in C++ that
provides several parallel metaheuristic optimization
methods such as Genetic Algorithm (GA), Simulated
Annealing (SA), Stochastic Hill Climbing (SHC) and
Branch and Bound (BnB). These methods are imple-
mented as algorithmic skeletons using metaprogramming,
pattern design and different parallelization techniques
(OpenMP and MPI). The latest version of the library may
be obtained via SVN (the homepage of the project:
In the exact version of the Kameda-Weiner algorithm
the search for grids of the RAM and the search for
minimum legitimate covers were parallelized using
OpenMP and MPI techniques. In the heuristic versions of
this algorithm the parallel versions of SHC and SA algo-
rithms of the HeO library were used. Each version of the
algorithm is implemented in a separate solver.
Let us compare the performance of the exact and heu-
ristic solvers for the random sample of the 100 pairwise
inequivalent trim NFAs generated with the following
parameters: number of states 6Q, number of initial
states 1I
, number of final states 2F, alphabet
Copyright © 2012 SciRes. IJCNS
size 2 , transition density 20.3
D, where
T is the number of automaton transitions. The computa-
tional experiments were conducted using the following
SMP system: Intel Core 2 Quad Q6600 2.4 Ghz, 4 Gb
RAM, MS Wind o ws XP P r of es si onsl SP 3.
First of all let us consider the results of the exact
solver which are presen ted in Table 6. Here, m and n are
the number of rows and number of columns in RAM
respectively; d is the density of ones in RAM; G is
the number of grids in RAM;
N is the number of the
minimum state NFAs;
Q is the number of states in
the mini mum state NFAs; min and max are minimal and
maximal values;
is the mean value;
is the stan-
dard deviation. The results were obtained using 4 threads.
The average minimization time is 1.1 seconds and the
total number of the minimum state automata found for
the whole sample is 268, 9 automata have no equivalent
minimum state NFA. As it can be seen from the table
some of the automata in the sample have multiple mini-
mum state NFAs (the mean value is 2.95) and the aver-
age number of states in the minimum state NFAs is 3.52.
Now let us consider the results of the heuristic solvers
obtained with different number of threads N = 1, 2, 4, 8,
16, 32, 64 which are presented in Tables 7 and 8. Since
LS is used only at the last stage of the Kameda-Weiner
algorithm the first 4 columns of Table 6 will be the same
for heuristic solvers and we replace them with the fol-
lowing columns: T—the total number of minimum
(reduced) state NFAs found for the sample, U— num-
ber of unminimized automata, T—average minimization
time in seconds (for columns
N and
the mean
values are provided).
As it can be seen from these tables the total number of
the minimum state NFAs increases and the number of
unminimized automata decreases as the number of
threads grows. The average minimization time is very
small and remains almost constant up to 4 threads and
then increases proportionally to the number of threads
because the hardware used for experiments supports si-
multaneous execution only of 4 threads.
6. Conclusion
In the present paper we have considered new heuristic
algorithms for NFA state minimization problem which is
known to be computationally hard. These algorithms are
a combination of the classical Kameda-Weiner algorithm
and local search heuristics which are widely used in
combinatorial optimization. The essential feature of the
proposed algorithm is that the most time consuming part
of the exact algorithm is replaced with fast local search
procedures. Numerical experiments have shown that such
combination is much less time consuming and allows to
Table 6. Exact algorithm results.
m n d
min 1 1 0.63 1 0 1
max 16 14 1.00 92 32 5
5.70 6.00 0.73 15.40 2.95 3.52
2.83 2.85 0.05 17.74 5.21 0.95
Table 7. SA algorithm results.
1 69 31 1.00 3.48 0.0071
2 76 28 1.06 3.43 0.0071
4 98 25 1.31 3.47 0.0078
8 114 20 1.43 3.49 0.0148
16 122 20 1.53 3.41 0.0258
32 137 19 1.69 3.41 0.0491
64 157 16 1.87 3.45 0.0917
Table 8. SHC algorithm results.
1 69 31 1.00 3.52 0.0057
2 75 28 1.04 3.42 0.0056
4 95 26 1.28 3.46 0.0059
8 112 21 1.42 3.52 0.0111
16 128 24 1.68 3.42 0.0192
32 148 16 1.76 3.46 0.0364
64 161 16 1.92 3.45 0.0674
obtain acceptable results. In the future we plan to con-
centrate on the other time consuming part of the Ka-
meda-Weiner algorithm, i.e. the exhaustive search for
grids of the RAM.
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