Building a new CTL model checker using Web Services

Building a new CTL model checker using Web
Services
Florin Stoica
Department of Computer Science
Faculty of Sciences, “Lucian Blaga” University of Sibiu
Sibiu, Romania
florin.stoica@ulbsibiu.ro
Laura Florentina Stoica
Department of Computer Science
Faculty of Sciences, “Lucian Blaga” University of Sibiu
Sibiu, Romania
laura.cacovean@ulbsibiu.ro
Abstract —The Computation Tree Logic (CTL) is widely used
to capture compositions of reactive systems. Model checking is
particularly well-suited for the automated verification of finite-state
systems, both for software and for hardware. A CTL model
checker tool allows designers to automatically verify that systems
satisfy specifications expressed in the language of CTL logic. In
this paper we present a new CTL model checker implemented in
client-server paradigm. CTL Designer, the client tool, allows an
interactive construction of the CTL models as state-transition
graphs. Java and C# APIs are provided for programmatic
construction of large models. The server part of our tool embeds
the core of the CTL model checker and is published as a Web
service. The performance evaluation in terms as speed and
scalability was accomplished implementing an algorithm to find a
winning strategy in the Tic-Tac-Toe game.
Keywords—Model Checking, CTL, Web services
I. INTRODUCTION
Testing and simulation can give us only confidence in the
implementation of a software system, but cannot prove that all
bugs have been found. However testing is neither exhaustive
nor very effective for software, especially concurrent software,
which is much more complex than sequential software. Thus,
there has been a tremendous push for efficient algorithms and
techniques that allow one to prove that a program satisfies
certain properties. The process of stating and proving
properties about programs is known as program verification.
Verification of a software system involves checking whether
the system in question behaves as it was designed to behave.
Design validation involves checking whether a system design
satisfies the system requirements. Both of these tasks, system
verification and design validation can be accomplished
thoroughly and reliably using model-based formal methods,
such as model checking [17]. Model checking is the formal
process through which a given specification representing a
desired behavioral property is verified to hold for a given
system (the model). Main concern of formal methods in
general, and model checking in particular, is helping to design
correct systems [1].
In the following we consider that model is a directed graph
where the nodes represent the states of the system and the
edges represents the state transitions, more specifically a
Kripke structure.
The state explosion problem and reducing the time required
to verify models are widely agreed to be the most formidable
challenges facing the application of model checking to large
and complex real-world systems. This naturally raises interest
in using parallelism to improve the performance of many
formal model checkers.
A. Related work
Two most common methods of performing model checking
are explicit enumeration of states of the model and respectively
the use of symbolic methods. Some examples of well-known
explicit-state parallel model checkers are DiViNe and PSPIN.
DiVinE [1], [2] is a distributed model checker for explicit state
LTL (Linear Temporal Logic) model checking and is able to
handle large systems consisting of as many as 419 million
states, as stated in [3]. PSPIN has also been used for
performing distributed model checking with the capability of
handling up to around 2.8 million states [13]. The basic idea
behind symbolic model checking is to use a more efficient
“symbolic” representation for the Kripke structure being
checked and for sets of states of the Kripke structure. Since the
sizes of these representations is typically the limiting factor in
applying model checking, an efficient representation can
potentially allow much larger structures to be checked.
Symbolic model checkers, such as CadenceSMV [15] and
NuSMV [5] analyze the state space symbolically using binary
decision diagrams (BDDs), which were introduced in [4]. The
binary decision diagram is a data structure for representing
Boolean functions. In contrast with explicit-state model
checking, states in symbolic model checking are represented
implicitly, as a solution to a logical equation. This approach
saves space in memory since syntactically small equations can
represent comparatively large sets of states [17]. A symbolic
model checker represents the Kripke structure itself
symbolically using BDDs to represent transition relations by
Boolean expressions and perform all calculations directly using
these Boolean expressions, rather than using the Kripke
structure explicitly.
B. Comparing Symbolic and Explicit Model Checking
In symbolic model checking, the space used to represent a
state is limited by the internal data structures. In explicit model
checking, the state size is not strictly limited, but is related with
state space size in the total memory consumption. In [8] is

presented a comparison between RULEBASE, a symbolic model
checker developed at IBM Haifa Research Laboratory and the
explicit LTL model checker SPIN [10]. The software verified
was a distributed storage subsystem software application. The
state space size handled by SPIN was 108 in a 3-process model.
Using symbolic model checking, RULEBASE keeps a
compressed representation of the state space and thus was able
to manage 10150 states. On the other hand, because of the limit
on state size, RULEBASE could not represent a state large
enough to include the information needed for more than 2-
process configuration [8]. Most hardware designs are based on
a clocked-approach and thus are synchronous. For these
systems, the symbolic model checking approach is more
appropriate [14]. On the other hand, for nondeterministic, high-level
models of hardware protocols, it has previously been
argued that explicit model checking is better than symbolic
model checking [11]; this is because the communication
mechanisms inherent in protocols tend to cause the BDDs in
symbolic model checking to blow up [3]. In their basic form,
symbolic approaches tend to perform poorly on asynchronous
models where concurrent interleaving are the main source of
explosion, and explicit-state model-checkers have been the
preferred approach for such models [3]. A detailed
experimental comparison between performance of explicit-state
model checkers and symbolic model checkers can be
found in [19]. The study follows an automata-theoretic
approach in program verification, originally proposed by Vardi
and Wolper [20]. Given a program P and a property ϕ, the task
is to check whether the program satisfies the property ϕ. If the
program P is viewed as a finite-state generator of words, and
the specification ϕ as a finite-state acceptor, the model-checking
problem is reduced to an automata-theoretic question:
whether the automaton P A Aϕ ∩ is empty. A non-deterministic
automaton A defined over a nonempty finite alphabet Σ is said
to be universal if it acceptsΣ* . The universality problem is to
check if A is universal. If AP is a universal automaton, the
model checking problem is reduced to check whether Aϕ is
also universal. The study presented in [19] uses two
approaches, explicit and symbolic, for solving the universality
problem. In order to present a full comparison between the
symbolic and the explicit algorithms, was performed a scaling
comparison of Cadence SMV, NuSMV as symbolic BDD-based
model checkers and respectively SPIN. The direct
comparison of the three model checkers shows that the explicit
one (SPIN) is much faster than the symbolic ones. In
conclusion, experimental results show that the explicit
approach scales better than the symbolic one, which was rather
surprising but confirms similar statements from [3], [11].
C. Motivation and objective
The broad goal of our research was to develop a reliable,
easy to maintain, scalable model checker tool to improve
applicability of CTL (Computation Tree Logic) model
checking in design of general-purpose computer software.
Concurrent software is asynchronous as the different
components might be running on different processors or be
interleaved by the scheduler of the operating system. Taking
into account the above considerations, in our tool we are using
an explicit-state model technique. In order to scaling its
verification ability to handle real-world applications and to
address the state explosion problem, our tool is based on an
efficient data structure for internal representation of the model
to be verified [9]. An orthogonal approach to increase the
capacity of an explicit-state model checker tool is to exploit the
memory and computational resources of multiple computers in
a distributed computing environment [3]. Following this idea,
our tool is based on Web Services technology to address the
time constraints in verification of large models.
The paper is organized as follows. In section 2 are
introduced concepts related to the CTL model checking
problem: first is defined the CTL model then are presented the
CTL syntax and semantics. In section 3 is described the
implementation of the CTL model checker in ANTLR. The
architecture of the Web service which embeds the server part
of our tool is presented in section 4. The performance
evaluation of out tool is presented in section 5. Conclusions are
presented in section 6.
II. CTL MODEL CHECKING
A temporal logic is essentially an ordinary predicate or
propositional logic with the addition of modal operators for
describing how the interpretation of symbols changes over time
[11]. The CTL logic is interpreted over Kripke structures. CTL
logic is branching-time logic, meaning that its formulas are
interpreted over all paths beginning in a given state (an
initial state) of the Kripke structure. The problem of model
checking is to verify if a finite-state abstraction of a reactive
system satisfies a temporal-logic specification expressed in the
language of CTL logic. A CTL model checker is a tool which
verifies that a given system satisfies a given CTL logic
formula, using an algorithm that determines the states of a
model that satisfy the formula.
A. The CTL model
A model is defined as a Kripke structure M=(S, Rel,
1
2
1
P:S→2AP) where S 2
1
is a finite 2
1
1
sets of states also called ϕϕϕϕϕnodes,
Rel⊆ϕϕϕS×S is a transition relation denoting a set of directed
edges, and P is a labeling function that defines for each state s
∈ S the set P(s) of all atomic propositions from AP that are
valid in s. The transition relation Rel is left-total, i.e., ∀ s ∈ S ∃
s' ∈ S such that (s,s') ∈ Rel. For each s ∈ S, the notation
succ(s)={s'∈ S |(s,s')∈Rel} is used to denote the set of
successors of s. From definition of Rel, each state from S must
have at least one successor, i.e., ∀s∈S, succ(s)≠∅. A path in M
is an infinite sequence of states (s0,s1, s2,…) such that ∀i, i≥0,
we have (si,si+1)∈Rel. We use s' ∈ succ(s) to denote that there
is a relation (s, s') in Rel. The labeling function P maps for each
state s ∈ S the set P(s) of all atomic propositions from AP that
are valid in s [12]. A CTL formula has the following syntax:
ϕ :: true|false|ap|(¬ )| ∧ | ∨ | ⇒ | AX | EX
| AG ϕ1
ϕ1
| EG ϕ1
| AF ϕ1
| EF ϕ1
| ϕ1
AU ϕ2
| ϕ1
EU ϕ2
, ∀ap∈AP.
The CTL syntax include several operators for describing
temporal properties of systems: A (for all paths), E (there is
a path), X (at the next state), F (in future), G (always) and
U (until). A CTL formula encodes properties that can occur

along a particular temporal path as well as to the set of all
possible paths. The set of all paths through a Kripke
structure is assumed to correspond to the set of all possible
computations of a system.
B. Fixed-point characterization of CTL
Let M=(S, Rel, P:S→2AP) be an arbitrary finite Kripke
structure. Given a state s in S, is defined a satisfaction relation
(M, s) ⊨ ϕ to specify that formula ϕ holds in s. We denote by
⟦ϕ⟧M = {s∈S | (M, s) ⊨ ϕ} the set of all states from S which
satisfy the formula ϕ (the set of states at which ϕ is true). ⟦ϕ⟧M
is called the denotation of ϕ in model M. Because often M is
implicit, we write ⟦ϕ⟧ rather than ⟦ϕ⟧M . Thus (M, s) ⊨ ϕ ⇔ s
∈⟦ϕ⟧. In the following 2S is used to denote the power set of the
set S. Let f : 2S →2S be a set valued function and Z ⊆ S a
subset of S. Z is called a fixed point of f if f (Z ) = Z. Z is called
the least fixed point (LFP) of f if f (Z ) = Z and ∀ U ⊆ S, f (U )
= U ⇒ Z ⊆ U. Z is called the greatest fixed point (GFP) of f if f
(Z ) = Z and ∀ U ⊆ S, f (U ) = U ⇒ U ⊆ Z.
The universal and existential pre-image functions
pre∀, pre∃ : 2S →2S are defined by:
pre∀(X) = { s∈S | succ(s) ⊆ X}
pre∃(X) = { s∈S | succ(s) ∩ X ≠ ∅}
(1)
For a CTL formula ϕ, the model checker will compute ⟦ϕ⟧
recursively, using the rules described in the following table,
where LFP and GFP represent the least fixed point and
respectively the greatest fixed point of the specified functions.
The algorithm for effective computation of ⟦EF ϕ⟧ is presented
in the following section.
TABLE I. RECURSIVELY COMPUTATION OF DENOTATION ⟦ϕ⟧ OF CTL
FORMULA ϕ.
Formula ϕ Computation of ⟦ϕ⟧
ap { s ∈ S | ap ∈P(s) }
true (false) S (∅)
¬ϕ1 S ⟦ϕ1⟧
ϕ1∧ϕ2, ϕ1∨ϕ2 ⟦ϕ1⟧ ∩ ⟦ϕ2⟧, resp. ⟦ϕ1⟧ ∪ ⟦ϕ2⟧
AX ϕ1, EX ϕ1 pre∀(⟦ϕ1⟧), resp. pre∃(⟦ϕ1⟧)
ϕ1 AU ϕ2 LFP of f(X) = ⟦ϕ2⟧ ∪(⟦ϕ1⟧ ∩ pre∀(X))
ϕ1 EU ϕ2 LFP of f(X) = ⟦ϕ2⟧ ∪(⟦ϕ1⟧ ∩ pre∃(X))
AG ϕ1 GFP of f(X) = ⟦ϕ1⟧ ∩ pre∀(X)
EG ϕ1 GFP of f(X) = ⟦ϕ1⟧ ∩ pre∃(X)
AF ϕ1 LFP of f(X) = ⟦ϕ1⟧ ∪ pre∀(X)
EF ϕ1 LFP of f(X) = ⟦ϕ1⟧ ∪ pre∃(X)
III. IMPLEMENTATION OF A MODEL CHECKER IN ANTLR
The CTL model checker is provided as a compiler
C:Ls→Lt, where Ls is the source language and Lt is the target
language. The source language Ls is the language describing
the CTL formulas and the target language Lt is a language
which describes the set of nodes from the model M where the
corresponding CTL formulas are satisfied. The compiler C
translates a formula ϕ of the CTL model to the set of nodes ⟦ϕ⟧
over which formula ϕ is satisfied. That is, C (ϕ)=⟦ϕ⟧ where
⟦ϕ⟧={s∈S| (M,s) ⊨ ϕ}. The implementation of the compiler C
is made in two steps. First, we need a syntactic parser to verify
the syntactic correctness of a given formula ϕ. Then, we should
deal with the semantics of the CTL language, respectively with
the implementation of the CTL operators presented in Table I.
Writing a translator for certain language is difficult to be
achieved, requiring time and a considerable effort [18]. For
implementation of the algebraic compiler we choose the
ANTLR (Another Tool for Language Recognition). ANTLR
[16] is a compiler generator which takes as input a grammar -
an exact description of the source language, and generates a
recognizer for the language defined by the grammar. In order
to translate a formula ϕ of a CTL model to the set of nodes
⟦ϕ⟧ over which formula ϕ is satisfied, is necessary to attach
actions to grammatical constructions within specification
grammar of CTL. The actions are written in target language of
the generated parser (in our case, Java). These actions are
incorporated in source code of the parser and are activated
whenever the parser recognizes a valid syntactic construction
in the translated CTL formula. In case of our compiler C, the
actions define the semantics of the CTL model checker, i.e., the
implementation of the CTL operators. The model checker
generated by ANTLR from our specification grammar of CTL
takes as input the model M (where are defined the sets S, Rel,
and P) and a formula ϕ, and provides as output the denotation
of ϕ – the set of states where the formula ϕ is satisfied, using
an algorithm which handles temporal operators AX, EX by
computing pre-images using expressions given in (1) and
implements the temporal operators AG, EG, AF, EF, AU, EU
by applying rules described in Table I, until a fixpoint is
reached. The algorithm for computing ⟦EF ϕ⟧ is presented in
Fig. 1 [6]. For the formal specification of the EF operator
given in Fig. 1, the corresponding action included in our
ANTLR grammar of CTL language is detailed in Fig. 2,
where PreExist() represents the implementation of function
pre∃(X) described in equation (1).
Z:=S; Z':= ⟦ϕ⟧;
while (Z≠Z') do
Z:=Z';
Z':=Z'∪ pre∃(Z');
endwhile
⟦EF ϕ⟧:=Z';
Fig. 1. Formal definition of the set expression ⟦EF ϕ⟧.
ctlFormula returns [HashSet set]
@init { }
: 'ef' e=implExpr {
HashSet rez = new HashSet(all_setS);
HashSet rez1 = new HashSet($e.set);
while (!rez.equals(rez1)) {
rez.clear();
rez.addAll(rez1);
HashSet tmp = PreExist(rez1);
rez1.addAll(tmp);
}
$set = rez1;
}
Fig. 2. Implementation of the EF operator in ANTLR
Analogue were implemented all CTL temporal operators.
For efficient representation of CTL models, our tool is based
on SingleGraph class from GraphStream package [9].

IV. ARCHITECTURE OF THE CTL MODEL CHECKER TOOL
The system architecture of the CTL model checker tool is
depicted in the following UML package diagram:
Fig. 3. The system architecture of the CTL model checker tool.
The CTL model checker tool contains the following packages:
• The algebraic compiler (CTL Compiler) embedded into
the Web Service (CTL Checker).
• The GUI client application written in C# and used for
interactive construction of the CTL models as directed
graphs (CTL Designer).
• The CTL non-GUI model package contains classes used
for internal representation of a CTL model as a directed
graph. There are available two libraries, for C# and
respectively for Java. For internal representation of a CTL
model in C#, our implementation is based on data
structures provided by [7], more precisely symmetrically
stored forward and backward adjacency lists. The Java
implementation is based on GraphStream [9].
• The XML API for CTL models package contains classes
needed to encode the CTL model into XML.
• The CTL GUI Model package is responsible with
graphical representation of the Kripke structures as
directed graphs.
We choose to publish our implementation of CTL model
checker as a Web service in order to utilize the combined
resources of distributed computers and to bring advantages of
distributed verification to various clients over the Web. For a
better understanding of the CTL model checking process, in
Fig. 4 is represented the Use Case Diagram of our model
checker. The Web service will receive from a client the XML
representation of a CTL model M and a CTL formula ϕ to be
verified. The original form of the CTL model M is then
reconstructed and passed to the algebraic compiler C generated
by ANTLR using our CTL extended grammar. For a
syntactically correct formula ϕ, the compiler will return as
result C(ϕ)={s∈ S | s ⊨ ϕ}, the set of states in which the
formula is satisfied. If as input is an erroneous formula ϕ, the
model checker will return to client an message describing the
error. In our tool is enabled a compression facility for large
CTL models, to reduce the network traffic between client and
server. Our Web service is using GlassFish or Tomcat as a
Web container. For testing purposes, the CTL model checker
described in this paper is available online via two Web services
hosted by use-it.ro and respectively by mcheck-useit.
rhcloud.com.
Fig. 4. The Use Case Diagram of the CTL model checker
V. PERFORMANCE EVALUATION OF OUR TOOL
In this section we describe the usage of our model-checker
to design a game strategy when playing Tic-Tac-Toe (called
TTT for short in the rest of this paper). First, let us describe the
(classical) game of Tic-Tac-Toe. The game is played by two
opponents, X and O, with a turn-based modality on a 3×3
board. The two players take turns to put pieces on the board. A
single piece is put for each turn and a piece once put does not
move. A player wins the game by first lining three of his or her
pieces in a straight line, no matter horizontal, vertical or
diagonal. If the entire board becomes full but no player has
formed a line, the result is a draw. In our example, player X is
played by the application and player O should be played by a
human. CTL model checking algorithm is used to return a
strategy to achieve a winning strategy for the computer. The
TTT is a turn-based synchronous game. In such a system, at
every transition there is just one agent that is permitted to make
a choice (and hence determine the future). In the following we
will show how to use the CTL formalizations in finding
winning strategies in case of TTT game.
A. Modelling the Game
We suppose that positions of the board are numbered as in
Fig. 5:
0 1 2
3 4 5
6 7 8
Fig. 5. Labeling the grids on the board
Values of the board locations are denoted by xi ∈ {0,1,2},
where i ∈ {0,1,...,8}. The value 0 means an empty position, the
value 1 denotes a previous move of the player X and the value
2 represents a move of the player O. For the sequence of values
l m n x x x we define:
min( ,1) min( Σxl xmxn = xl + xm,1)+min(xn ,1)
where l,m,n ∈{0,1,...,8}. Formally, the Kripke model of TTT
is defined as M = (S, Rel, P:S→2AP) with its structure
explained in the following. The set of atomic propositions AP
is denoted by:

AP = {( 1 2 0,3,6 3 6 0,1,2 , , l l l l l l l l x x x x x x + + = + + =
0 4 8 2 4 6 x x x , x x x , T ) | {0,1, 2} for 0,8 and kx ∈ k =
T ∈{1,2}}. The number of successors of a state is given by the
formula:
− Σ .
9 l l l
1 2
0,3,6
l
x x + x +
=
A state labeled with value T =1 signify that is turn of the
player X for making the move and if T =2 then the player O
will make the next move. The game stops (so no moves are
possible) if the board moves locations are full, i.e.:
Σ 1 2
=
0,3,6
l l l 9
l
x x + x +
=
Another situation where the game is not continuing is when
a player won. The state s is a winning state for player X if
111∈ P(s) and it is a winning state for player O if 222∈ P(s) .
Alternation to move can be formalized as follows: for a
transition (s, s ')∈Rel , there are the following cases:
T ∈ P(s) ⇒3 −T ∈ P(s ')
where T ∈{1,2} . Assuming that the game is in the state s0 ∈ S,
we denote by k the number of empty positions of the board.
The winning strategy of player X aims to determine all states
from the model satisfying the formula: (AX EX)k/2(AX 111 ), to
choose the move which favors wining of the game in the
future.
B. Experimental results
Although the game implemented is relatively simple, due to
the large size of the structure representing the CTL model at
the first moves, it represents a good opportunity to study the
effectiveness of our approach in designing and implementing a
CTL model checker. In the Table II are presented the results
showing the performance of the CTL model checker when
running on Intel Core I5, 2.5 GHz, 4Gb RAM to find a winning
strategy for Tic-Tac-Toe game.
TABLE II. EVALUATING THE PERFORMANCE OF THE CTL MODEL
CHECKER
States 2236 2307 2425 3423 3683 3732 4255 4791
Time(sec) 0.13 0.14 0.16 0.3 0.32 0.35 0.43 0.6
VI. CONCLUSIONS
In this paper was presented a new CTL model checking
tool, based on client/server architecture. In contrast to previous
approaches, our tool permits an interactive design of the CTL
models as state-transition graphs, and is based on client/server
architecture. The implementation of the CTL model checking
algorithm is based on Java code generated by ANTLR using an
original grammar for the language of CTL formulas, having
semantic actions attached to production rules. Programmatic
construction of large models is supported through two libraries
(C# and Java). The server part of our tool was published as a
Web service and is available online at two Internet locations,
use-it.ro and mcheck-useit.rhcloud.com. All components of our
CTL model checker tool can be downloaded from http://use-it.
ro.
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Building a new CTL model checker using Web Services