Approximate Analyzing of Labeled Transition Systems

As the most important formal semantic model, labeled transition systems are widely used, which can describe the general concurrent systems or control systems without disturbance. However, under normal circumstance, transition systems are complex and difficult to use due to large amount of calculation and the state space explosion problems. In order to overcome these problems, approximate equivalent labeled transition systems are proposed by means of incomplete low-up matrix decomposition factorization.This technique can reduce the complexity of computation and calculate under the allowing errors. As for continuoustime linear systems, we develop a modeling method of approximated transition system based on the approximate solution of matrix, which provides a facility for approximately formal semantic modeling for linear systems and to effectively analyze errors. An example of application in the context of linear systems without disturbances is studied.


Introduction
Transition system is an important formal model of concurrent systems, which is widely used in concurrency theory, such as petri nets and process algebras.Petri nets can be used as a graphical tool to simulate the dynamic behavior and concurrent activities of the systems and also can be used as a mathematical tool to establish mathematical model describing the behaviors of the systems.It is a widely transition system and gives a method to describe Petri net using classical transition system [1].Apart from Petri net theory, there is a process algebra, which is used to describe and study concurrent or distributed system behavior theory by algebraic methods.Thus, many process algebraic systems are proposed.Among them, the first is to propose the following three process algebra systems: CCS (Calculus of Communicating Systems) proposed by Milner [2,3], CSP (Communicating Sequential Processes) proposed by Hoare [4], and ACP (Algebra of Communicating Processes) proposed by Baeten and Weijland [5].In recent years, some new process calculus systems have been proposed.For example, probabilistic process algebra [6][7][8][9][10][11][12] and stochastic process algebra [13] are proposed to describe possible or random information; real-time process algebra [14][15][16][17] is proposed to describe the action time.In the field of process algebra, as the most important semantic model, the transition system is widely used, especially labeled transition system [18], which can describe the general behavior of concurrent systems.
Approximation of transition systems has recently been introduced as a powerful tool for the approximation of continuous systems.The notion of simulation is one such formal notion of abstraction that has been used for reducing the complexity of finite state systems such as labeled transition systems.Approximation of purely discrete systems has traditionally been based on language inclusion and equivalence with notions such as simulation or bisimulation relations [19].These concepts have been useful for simplifying problems such as safety verification or controller synthesis.However, in the cases that the quality requirements of simulation or bisimulation are so exacting sometimes it is so difficult for us to construct finite abstracts with the original system simulation or bisimulation.Although finite abstract with the original system approximate simulation or approximate bisimulation is easy to construct, these methods can only ensure that the system approximately satisfies the given range and is not able to calculate the allowable error of transition between states.To solve this problem, this paper puts forward a design method for transition system-approximate simulation method based on ILU.
As a particular class of preconditioners, incomplete factorizations can be thought of as approximating the exact  factorization of a given matrix (e.g., computed via Gaussian elimination) by disallowing certain filling of this.As opposed to other PDE-based preconditioners such as multigrid and domain decomposition, this class of preconditioners is primarily algebraic in nature and can be applied to any matrices in principle.However, one of the basic methods of preconditioned method is the use of incomplete  decomposition.The most well known is the incomplete triangular decomposition ILU0 [20] that has nothing more than zero-element filling.Since ILU0 unchanged the structure of sparse matrix, the application of ILU0 is very convenient.However, because of this, it is so rough approximation of the coefficient matrix that the effectiveness of ILU0 is limited.To improve the approximation, filling with nonzero elements is allowed [21,22].While the ILU preconditioner works quite well for many problems, it does not perform well for some PDE problems [23].Then, the generalization of this modified ILU (MILU) preconditioner is proposed by Gustafsson [24].MILU is to add the fill-ins abandoned by ILU back to the diagonal type, which makes up for inaccuracies caused by discarded.And its accuracy can be calculated.However, mathematically, the transition between states can be described by matrix, and then the allowable error of the approximate transition is needed to be characterized.Furthermore, the allowable error can be defined by the MILU accuracy.
In the rest of this paper, Section 2 reviews the approximate simulation relations of transition systems.In Section 3, the approximate and incomplete factorizations are recalled.In Section 4, the approximate equivalent of labeled transition systems is proposed in detail.Section 5 proposes the theory of the approximate labeled transition systems.In Section 6, a case is studied.Then Section 7 draws the conclusion.

Approximation of Transition Systems
2.1.Labeled Transition Systems.We know that labeled transition systems allow modeling, in a unified framework, discrete, continuous, and hybrid systems.The main results [25] are reviewed here.Definition 1.A labeled transition system with observation is a tuple  = (, Σ, → ,  0 , Π, (⋅)) that consists of (i) a set  of states, (ii) a set Σ of labels, (iii) a transition relation → ⊆  × Σ × , (iv) a set  0 ⊆  of initial states, (v) a set Π of observations, (vi) an observation map (⋅) :  → Π.
For a given initial state and sequence of labels, there may exist several state trajectories of .Thus, the systems we consider are possibly nondeterministic.The associated external trajectory where describes the evolution of the observations under the dynamics of the labeled transition system.The set of external trajectories of the labeled system  is called the language of  and is denoted by ().The subset of Π reachable by the external trajectories of  is denoted by Reach(): where N is the set of positive integers.An important problem for transition systems is the safety verification problem which consists in checking whether the reachable set Reach() intersects a set of observations Π  associated with unsafe states.

Approximate Simulation Relations.
Exact simulation relations between two labeled transition systems require that their observations are (and remain) identical.Approximate simulation relations are less rigid since they only require that the observations of both systems are (and remain) arbitrarily close. Let 2 ) be two labeled transition systems with the same set of labels (Σ 1 = Σ 2 = Σ) and the same set of observations (Π 1 = Π 2 = Π) (i.e.,  1 and  2 are elements of (Σ, Π)).Let us assume that the set of observation Π is a metric space;  Π denotes the metric of Π. Definition 2. A metric of a set  is a positive function  :  ×  → R ∪ +∞, where R is the set of real numbers, such that the three following properties hold: for all , ,  ∈  If the second property is replaced by  =  ⇒ (, ) = 0 then  is called a pseudometric.If the third property is dropped, then  is called a directed metric.
Note that, in there, → 1 describes a transition relation of there exists an external trajectory of  2 with the same sequence of labels such that, for all  ∈ N,  Π (  1 ,   2 ) ≤ .
In Theorem 5, if  2 approximately simulates  1 with the precision , there must exist a state of  2 approximate simulation the state of  1 .But it is very difficult for us to calculate the degree of approximation.

Approximate and Incomplete Factorizations
The general problem of finding a preconditioner for a large sparse linear system  =  is to find a matrix  (the preconditioner).However, the matrix  should be with the following properties: (i)  is a nonsingular matrix, (ii)  is a good approximation to  in some sense, (iii) the system  =  is much easier to solve than the original system  = , (iv) the construction of  has some memory and CPU needs protection, (v) the condition number of the  −1  is much less than that of , and its minimum singular value becomes larger.
That is, we are looking for a nearby problem which is easier to solve than the given one.The idea is to look for a matrix  such that the original linear system  =  is transformed into an equivalent linear system  −1  =  −1 .Mathematically, we will solve the preconditioned system  −1  =  −1  (or the symmetric version ( −1/2  −1/2 )( 1/2 ) =  −1/2  when both  and  are symmetric positive definite) by standard iterative methods (such as the conjugate gradient method), in which only the actions of  and  −1 are needed.
There are several preconditions: matrix splitting preconditioner, polynomial preconditioners, incomplete factorization preconditioners, approximate inverse preconditioners, and multilevel preconditioners.Here, we mainly focus on incomplete factorization preconditioners, which can be applied to a general purpose.They can be thought of as modifications of Gaussian Elimination, in which some sparse mode is required as the decomposition of matrices, such as making some of the special position of the elements be zero and also abandoning some absolute value in the process of decomposition of minor elements in order to ensure that sparse decomposition.

Incomplete Low-Up Matrix Decomposition (ILU).
The most common type of incomplete factorization is based on taking a set  of matrix positions and keeping all positions outside this set equal to zero during the factorization.The resulting factorization is incomplete in the sense that fill is suppressed.Definition 6.A matrix , if  = , where  () is the lower (upper) triangular sparse matrix, is called a complete decomposition of the matrix , that is,  [26].Definition 7. A matrix , if  =  + , where  () is the lower (upper) triangular sparse matrix, is called an incomplete decomposition of the matrix , that is, ILU.Here  is called error matrix [27].
A commonly used strategy is to define  by That is, the only nonzeros allowed in the  factors are those for which the corresponding entries in  are nonzero.Let the preconditioner  be defined by the product of the resulting  factors, that is,  = .If  becomes a good preconditioner, it must be a good approximation to  in some measure. typical strategy is to require the entries of  to match those of  on the set : (3)  , =  , if (, ) ∈ .
However, the incomplete  factors of , respectively, with the lower and upper triangular part of the matrix , have the same nonzero structure.
The constructing process of ILU precondition is in the process of Gaussian elimination by discarding partially filled elements to get sparse triangular matrixes  and  such that  = .From the decomposition of ILU we can see that the preconditioners  and  have the same nonzero structure.Therefore, the error matrix  =  −  =  −  and the main difference from the usual Gaussian elimination lie in how to discard fill-ins in the Gaussian elimination.

Modified Incomplete Low-Up Matrix Decomposition (MILU).
While the ILU preconditioner works quite well for many problems, it does not perform well for some problems.We will next describe the generalization of this modified ILU (MILU) preconditioner to a general matrix  [28].
Again, for certain classes of matrices, conditions (4) and ( 1) are sufficient to determine the  factors in MILU directly.However, in practice it is easier to compute these  factors by a modification of the ILU algorithm: instead of dropping the disallowed fill-ins in the ILU algorithm, these terms are added to the diagonal of the same row.Furthermore, the idea of MILU is to make up for inaccuracies caused by discarded by using fill-ins plus backup to cover the diagonal type.
Incomplete factorization preconditioners can be portrayed accuracy and stability.Here, accuracy refers to the degree of preconditioner  and matrix  can be measured by the size of ‖ − ‖  .And stability refers to the degree of preconditioner  and the unit matrix  can be measured by the size of ‖ −  −1 ‖  (for the left preconditioner).Below we give algorithm ILU and algorithm MILU [29].
From the analysis of Algorithms 1 and 2, the difference between algorithm ILU and MILU is that certain fill-ins given up in algorithm ILU are back to the main diagonal.

Equivalent Labeled Transition Systems
In labeled transition systems, we can assume that Σ is a finite set of labels; a label  ∈ Σ describes an action.→ ⊆ ×Σ× is a transition relation, where  is a set of states.Each transition → is a tuple ⟨, ,   ⟩, where  and   represent the pre-and poststates of the transition, respectively.A state trajectory of labeled transition systems is a sequence of transitions: where  0 ∈  0 is the initial state and  0 ⊆  is the initial set of states.For a given initial state and sequence of labels, there may be several state trajectories of labeled transition systems.Thus, the systems we consider are possibly nondeterministic.
Here, we mainly consider the set Σ of labels.Let label  ∈ Σ has the following form: It can be abbreviated as ẋ = (), where ẋ = ( ẋ ) .
The labeled transition system can only be used for symbolic reasoning.Two labels are equivalent if they have the same form, syntax, and semantics.Here, Definition 8 gives the equivalent of two labels.It has the characteristic that labels' equations are equivalent instead of having the same form, syntax, and semantics.
For all labels   ∈ Σ,  ∈ N, where N is the set of positive integers,   : ẋ =   (); its equivalent label is  A labeled transition system with observation is a tuple  = (, Σ, → ,  0 , Π, (⋅)).Let   = (, Σ  , → ,  0 , Π, (⋅)).Definition 10.Labeled transition system with observation  and labeled transition system with observation   are equivalent if set Σ of labels and set Σ  of labels are equivalent.One labeled transition system is called equivalent system of another labeled transition system.
Here, we only study labels of labeled transition systems.

Approximate Labeled Transition System
In general, we know that a labeled transition system with observations is used to prescribe control systems.In this section, our goal is to study ẋ =  (the transition relationship between states).
The purpose of this paper is to characterize approximate similar transition systems that are generated by continuoustime linear systems, which are conceptually similar to the transition systems generated by discrete-time systems.We begin with studying continuous-time linear systems whose dynamics are closer to transition systems due to the existence of an atomic time step [30].Consider a continuous-time linear system without disturbances where with time  ∈ R + , state () ∈ R  and matrix  of approximate dimension, given an initial condition  0 .
Definition 11.Consider continuous-time system  given by ẋ =  and observation set Π =   .The transition system   +  = (, Σ, → , Π, (⋅)) generated by  and Π consists of (i) state space  = R  , (ii Usually, the linear systems are presented by ordinary differential equations, partial differential equations, and difference equations. For  ∈ R + , the transition relation is given by    →   if and only if there exists a function (⋅) such that (0) = , () =   , and for almost all  ∈ [0, ], ẋ () ∈ (()), where  is a valued map.(11) and state value of the initial state  is

Approximate Similar
The state is observed through the variable  1 () =  1 () and the transition relation is expressed as → 1 .
We know that the equation is the linear differential equations.We introduce the following mark: ) , where () is an  by  matrix.The above equations can be expressed as where However, () is a large sparse matrix for most mathematical models of practical problems in the real world, so it is very difficult for us to obtain the solution of the equation ẋ = ().Then we use ILU to calculate the approximate solution.The preconditioner  = , where  is a triangular matrix and  is an upper triangular matrix.Hence, the original linear system ẋ = () is transformed into an equivalent linear system  −1  =  −1 ẋ (or the symmetric version ( −1/2  −1/2 )( 1/2 ) =  −1/2 ẋ when both  and  are symmetric positive definite).Because system ẋ = () is a portrait of the transition process from the state  to another state   , the elements of () present the dynamic changes from state  to another state   .The preconditioner  is the approximate dynamic changed from the state  to another state   .Hence,  − , an error matrix, has the meaning that state  can be translated into another state   in the error, and its degree of approximation is ‖ − ‖  .Then we can define the approximate distance between the two states.
Definition 13. ∀,   ∈ , the approximate distance between  and   is Let us assume that  1 ,  2 ,  3 are elements of  and the state trajectory is The mathematical relationship of  1 →  3 is respective ẋ =  1 (), ż =  2 ().This system is a continuous transition, so a transition from state  1 to state  3 can be expressed as  1  2  1   →  3 .This can be represented by using Figure 1.A line with an arrow indicates state transition, where the end of a line without arrow points to the prestate and the end of a line with arrow points to the poststate, a label marks on the line.
Call   (,   ) = ‖ − ‖  as the degree of approximation of state  translated into state   .Furthermore, because (0) = , ( 0 ) =   , the solution  of the equivalent linear system  −1  =  −1 ẋ is the approximate solution of the original where The state is observed through the variable  2 () =  1 () and the transition relation is expressed as → 2 .Therefore, the accuracy between the exact solution  and approximate solution  of the linear systems is So ẏ = () simulates approximately ẋ = ().And therefore, we can say the state   simulates approximately the state   .
Here, approximate simulation of states is controlled by labels, where labels are special.Furthermore, the degree of approximate similar states can be obtained by the approximate solution of labels' mathematical models.

Approximate Similar Linear Systems.
We have given an approximate simulation method for a state.Now we consider approximate simulation of linear systems.

Mathematical Problems in Engineering
In the example, we use ILU method to approximate simulation states, and the precise of approximate simulation states can be calculated.That is,  −  = ( So the precision of approximate simulation states is  = ‖‖  ‖−‖  = 4 √ 14 ≈ 14.9666, where the precision retains four significant numbers after the decimal point.If we want to use MILU to solve the problem of approximate simulation states, we just deal with , that is, using Algorithm 2 to decomposition .

Conclusions
In this paper, we extend the notion of approximate simulation relations.The semantic analysis of equivalent labeled transition systems is developed by equivalent labels.And an effective characterization of approximate simulation relations of states based on ILU is also developed, where error about the state set approximate is controlled.Furthermore, the technique can be used to approximately simulate labeled transition systems generated by continuous-time system.Finally, a case study of application in the context of linear systems without disturbances is shown.
The approximate simulation relation is a key performance consideration of systems.Our future work will focus on these two items in two steps: firstly, developing methods to approximate simulation linear systems with disturbances; secondly, solving the nonlinear systems.