CO-EVOLUTION PRESERVING MODEL REDUCTION FOR

UNCERTAIN CYBER-PHYSICAL SYSTEMS

Towards a Framework for Nanoscience

Manuela L. Bujorianu and Marius C. Bujorianu

School of Mathematics, University of Manchester, U.K.

Keywords:

Cyber-physical systems, Adaptive bisimulation, Co-evolution, Stochastic model checking, Qualitative model

reduction, Nanoscience.

Abstract:

The problem of abstracting computational relevant properties from sophisticated mathematical models of

physical environments has become crucial for cyber-physical systems. We approach this problem using

Hilbertean formal methods, a semantic framework that offers intermediate levels of abstractions between the

physical world described in terms of differential equations and the formal methods associated with theories

of computation. Although, Hilbertean formal methods consider both deterministic and stochastic physical

environments, in this paper, we focus on the stochastic case. The abstraction method can be used for veriﬁ-

cation, but also to improve the controller design and to investigate complex interactions between computation

and physics. We deﬁne also a computational equivalence relation called adaptive model reduction, because

it considers the co-evolution between a computation device environment and its physical environment during

abstraction.

1 INTRODUCTION

The interaction between physics and computation can

be very subtle. The research experience from ar-

eas like nanoscience (Hornyak e.a. 2008) and quan-

tum computing (Accardi e.a. 2006), or from smart

dust, shows that common principles can be distilled

from these different worlds. At a larger scale, the

general system theory provides a systematic reper-

toire of common properties of the physical and digital

dynamical systems. This experiences give hope for

a sound semantic framework for cyber-physical sys-

tems (CPS). The manifestos on CPS - see, for exam-

ple(Tabuada 2006) - emphasize the need for a funda-

mentally new theoretical foundation. This foundation

should be interdisciplinary and at the right level of ab-

straction: it should offer analytical tools to investigate

physical models, and, at the same time, to be abstract

enough to give semantics for models of computation.

In this paper, we consider Hilbertian For-

mal Methods (HFM) (Bujorianu, Bujorianu 2007a,

2007b) as a semantic framework for CPS model-

ing. HFM represent a logical framework that uses

functional and stochastic analysis to construct logics

for reasoning about qualitative properties of physi-

cal phenomena. These logics can be easily integrated

with speciﬁcation logics for automata. In this work,

we focus more on the method part of HFM, and less

on the formal aspects. In the HFM framework, we

use hybrid systems to design an abstraction method

that simpliﬁes the physical models whilst the com-

putational properties are simulated. Intuitively, the

computational discrete steps are preserved, while the

mathematical models of the continuous phenomena in

the environment are drastically simpliﬁed.

The qualitative model reductions method we pro-

pose is a fundamental step towards stochastic model

checking (SMC) (Bujorianu, Bujorianu 2006) for un-

certain CPS. Stochastic model checking coincides

with probabilistic model checking (Bujorianu, Katoen

2008) for Markov chains. In the case of continuous

or hybrid stochastic dynamical systems, the SMC is

a specialization of the stochastic reachability analy-

sis (Bujorianu 2004) by means of computer science

inspired abstraction (Bujorianu, Lygeros, Bujorianu

2005a) or bisimulation methods (Bujorianu, Lygeros,

Bujorianu 2005b) , (Bujorianu, Bujorianu 2008b).

In the context of uncertain cyber-physical sys-

tems, we introduce a new concept of behavior equiv-

alence called adaptive bisimulation. In the theory

of concurrent discrete processes, bisimulation is a

method for reducing the state space, while the tran-

Bujorianu M. and Bujorianu M. (2009).

CO-EVOLUTION PRESERVING ABSTRACT MODEL REDUCTION FOR UNCERTAIN CYBER-PHYSICAL SYSTEMS - Towards a Framework for

Nanoscience.

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 39-46

DOI: 10.5220/0002217800390046

 SciTePress

sitions are preserved. Using category theory the con-

cept of bisimulation was deﬁned for continuous and

hybrid dynamical systems (Haghverdi, Tabuada, Pap-

pas 2005). Based on the same categorical machin-

ery, in (Bujorianu, Lygeros, Bujorianu 2005b), bisim-

ulation has been deﬁned for stochastic hybrid sys-

tems. However, in the context of uncertain CPS,

the classical concept of bisimulation seems to be too

strong (i.e., systems that are considered equivalent

by a designer or by an observer, fail to be bisimi-

lar). More appropriate concepts of behavioral equiv-

alence, like approximate bisimulation and behavioral

bisimulation have been proposed in (Bujorianu, Bu-

jorianu, Blom 2008) and (Bujorianu, Lygeros, Bujo-

rianu 2005a). Under approximate bisimulation, the

trajectories of two randomized hybrid systems differ

with a small distance, the measurement being done

according with a suitable metric. For the behavioral

bisimulation, two equivalent systems have the same

probabilities of reaching some speciﬁc state sets. Al-

though these bisimulation concepts are better in de-

scribing properties of systems that operate in physical

environments, they do not imply the preservation of

the interaction between computation and physics. The

key point in deﬁning such a bisimulation consists in

modeling this interaction. In this paper, we model this

interaction using an abstract measure called energy,

which is a basic concept of HFM. The energy char-

acterizes globally the cyber physical process, but also

it can discriminate continuous (physical) evolutions,

discrete (computational) transitions and control (the

process killing, in order to start another one). This

last aspect makes the difference between a CPS and

a classical automaton: a computation device has the

capability to inﬂuence its physical environment (and

achieving co-evolution in this way). Naturally, the

CPS bisimulation should be related to energy preser-

vation. An intuitive illustration of adaptive bisimu-

lation is given by the following scenario. During its

evolution, a CPS may produce a change of its envi-

ronment. Suppose that for the new dynamical system

modeling the environment is classically bisimilar with

the former one. Then, for an adaptive bisimilar CPS

the computational component will exhibit a equiva-

lent behavior.

The paper road map can be described as follows.

The following section contains the mathematical set-

ting. In Section 3 we formulate the stochastic model

checking problem and we prove two results that make

the problem solvable. In Section 4 we investigate the

qualitative model reductions and bisimulations. The

ﬁnal section contains some short conclusions.

2 THE MATHEMATICAL

FRAMEWORK

2.1 Uncertain Cyber-physical Systems

The theory of hybrid systems is a well-established

modeling paradigm for embedded systems. Similarly,

the theory of concurrent embedded hybrid systems

(Bujorianu, Lygeros, Bujorianu 2005a) constitutes a

suitable modeling framework for CPS. In the follow-

ing an uncertain cyber-physical system is modeled as

a randomized embedded hybrid system.

There are two major ways to randomize a contin-

uous or hybrid dynamical system: In one approach,

the concept of noise is used to model small random

perturbations. The randomized system has trajecto-

ries that closely resemble those of the deterministic

initial system. The noise based randomization is car-

ried out using stochastic differential equations. When

the inﬂuence of the random perturbation changes dra-

matically the system evolution, the randomization is

carried out using stochastic kernels that replace the

concept of reset maps from deterministic hybrid sys-

tem models.

A Ucps U = (Q,X ,F,R,λ) consists of

• a ﬁnite set of discrete variables Q;

• a map X : Q → R

d(.)

that sends each q ∈ Q into a

mode (an open subset) X

of R

d(q)

, where d(q) is the

Euclidean dimension of the corresponding mode;

• a map F : Q → 2

SDE

which speciﬁes the continu-

ous evolution of the automaton in terms of stochastic

differential equations (SDE) over the continuous state

for each mode;

• a family of stochastic kernels R = (R

)

q∈Q

× (∪B (X

)| j ∈ Q\{q}) → [0, 1];

• a transition rate function

λ : (∪X

| j ∈ Q) → R

, (1)

which gives the distributions of the jump times.

The executions of a Ucps can be described as fol-

lows: start with an initial point x

∈ X

, follow a

solution of the SDE associated to X

, jump when

this trajectory hits the boundary or according with

the transition rate λ (the jump time is the minimum

of the boundary hitting time and the time, which is

exponentially distributed with the transition rate λ).

Under standard assumptions, for each initial condi-

tion x ∈ j ∈ Q∪X

, the possible trajectories starting

from x, form a stochastic process. Moreover, for all

initial conditions x, the executions of a Ucps form

the semantics, which can be thought of as a Markov

process in a general setting. Let us consider M =

(Ω,F , F

) be the semantics of U. Under mild

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

assumptions on the parameters ofU, M can be viewed

as a family of Markov processes with the state space

(X,B ), where X is the union of modes and B is its

Borel σ-algebra. Let B

(X) be the lattice of bounded

positive measurable functions on X. The meaning of

the elements of M can be found in any source treat-

ing continuous-parameter Markov processes (see, for

example, (Davis 1993)). Suppose we have given a σ-

ﬁnite measure µ on (X,B ).

In the following we give some operator character-

izations of stochastic processes, which are employed

in this paper to deﬁne a qualitative model reduction

for Ucps.

2.2 Hilbertean Formal Methods

The HFM abstract away the analytical properties of

deterministic and stochastic differential operators us-

ing the so called kernel operator (deﬁned in the fol-

lowing). Using methods of functional analysis HFM

elegantly generalize both deterministic and stochas-

tic systems. In this work we focus on the stochastic

case. Let us describe brieﬂy the mathematical appara-

tus that is usually employed to study continuous time

continuous space Markov processes.

The transition probability function is p

(x,A) =

∈ A), A ∈ B . This is the probability that, if

= x, x

will lie in the set A.

The operator semigroup P is deﬁned by

f(x) =

f(y)p

(x,dy) = E

f(x

),∀x ∈ X,

where E

is the expectation w.r.t. P

The operator resolvent V = (V

)

α≥0

associated with

P is

f(x) =

∞

−αt

f(x)dt,

x ∈ X. Let denote by V the initial operator V

of V ,

which is known as the kernel operator of the Markov

process M. The operator resolvent (V

)

α≥0

is the

Laplace transform of the semigroup.

The strong generator L is the derivative of P

at t = 0.

Let D(L ) ⊂ B

(X) be the set of functions f for which

the following limit exists (denoted by L f):

lim

tց0

f − f).

In the HFM, there is developed a semantic frame-

work for concurrent embedded systems constructed

using energy forms. We specialize this theory for

function spaces, reaching in this way the theory of

Dirichlet forms (Ma, Rockner 1990).

A quadratic form E can be associated to the gen-

erator of a Markov process in a natural way.

Let L

(X,µ) be the space of square integrable

µ-measurable extended real valued functions on

X, w.r.t. the natural inner product < f, g >

f(x)g(x)dµ(x).

The quadratic form E :

E ( f, g) = − < L f,g >

, f ∈ D(L ),g ∈ L

(X,µ) (2)

deﬁnes a closed form. This leads to another way of

parameterizing Markov processes. Instead of writing

down a generator one starts with a quadratic form. As

in the case of a generator it is typically not easy to

fully characterize the domain of the quadratic form.

For this reason one starts by deﬁning a quadratic form

on a smaller space and showing that it can be extended

to a closed form in subset of L

(µ). When the Markov

process can be initialized to be stationary, the measure

µ is typically this stationary distribution (see (Davis

1993) p.111). More generally, µ does not have to be a

ﬁnite measure.

A coercive closed form is a quadratic form

(E ,D(E )) with D(E ) dense in L

(X,µ), which sat-

isﬁes the: (i) closeness axiom, i.e. its symmetric part

is positive deﬁnite and closed in L

(X,µ), (ii) con-

tinuity axiom. E is called bilinear functional en-

ergy (BLFE) if, in addition, it satisﬁes the third ax-

iom: (iii) contraction condition, i.e. ∀u ∈ D(E ),

∗

= u

∧ 1 ∈ D(E ) and E (u ± u

∗

,u∓ u

∗

) ≥ 0.

For a the general theory of closed forms associated

with Markov processes see (Ma, Rockner 1990).

Let (L ,D(L )) be the generator of a coercive form

(E ,D(E )) on L

(X,µ), i.e. the unique closed linear

operator on L

(X,µ) such that 1− L is onto, D(L ) ⊂

D(E ) and E (u,v) =< −L u,v > for all u ∈ D(L )

and v ∈ D(E ). Let (T

)

t>0

be the strongly continu-

ous contraction semigroup on L

(X,µ) generated by

L and (G

)

α>0

the corresponding strongly continu-

ous contraction semigroup (which exist according to

the Hille-Yosida theorem).

A right process M with the state space X is as-

sociated with a BLFE (E ,D(E )) on L

(X,µ) if the

semigroup (P

) of the process M is a µ-version

the form semigroup (T

). It has been proved (Al-

beverio, Ma, Rockner 1993) and (Ma, Rockner 1990)

that only those BLFEs, which satisfy some regularity

conditions can be associated with some right Markov

processes and viceversa (Th.1.9 of (Albeverio, Ma,

Rockner 1993)).

Prop. 4.2 from (Albeverio, Ma, Rockner 1993)

states that two right Markov processes M and M

′

with

state space X associated with a common quasi-regular

BLFE (E , D(E )) are stochastically equivalent (Ma,

Rockner 1990). That means a quasi-regular BLFE

I.e., for all f ∈ L

(X, µ) the function P

f is a µ-version

(differs on a set of µ-measure zero) of T

f for all t > 0.

CO-EVOLUTION PRESERVING ABSTRACT MODEL REDUCTION FOR UNCERTAIN CYBER-PHYSICAL

SYSTEMS - Towards a Framework for Nanoscience

characterizes a class of stochastically equivalent right

Markov processes.

Let M = (Ω,F , F

) be a right Markov process

with the state space X. Now assume that X is a Lusin

space (i.e. it is homeomorphic to a Borel subset of

a compact metric space) and B (X) or B is its Borel

σ-algebra. Assume also that µ is a σ-ﬁnite measure

on (X, B ) and µ is a stationary measure of the process

M. Let X

another Lusin space (with B

its Borel

σ-algebra) and F : X → X

be a measurable function.

Let σ(F) be the sub-σ-algebra of B generated by F. If

µ is a probability measure then the projection operator

between L

(X,B ,µ) and L

(X,σ(F),µ) is the condi-

tional expectation E

[·|F]. Recall that E

is the expec-

tation deﬁned w.r.t. P

and that P

(A) =

(A)dµ,

A ∈ F . We denote by µ

the image of µ under F,

i.e. µ

) = µ(F

−1

)), for all A

∈ B

. In gen-

eral, anything associated with X

will carry the #-

superscript symbol in this section.

Let E be the BLFE on L

(X,µ) associated to M.

F induces a form E

on L

,µ

) by

) = E (u

◦ F, v

◦ F); (3)

for u

∈ D[E

], where

D[E

] = {u

∈ L

,µ

)|u

◦ F ∈ D[E ]}. (4)

It can be shown (see Prop.1.4 in (Iscoe, McDonald

1990)), under a mild condition on the conditional ex-

pectation operator E

[·|F] that E

is a BLFE. If, in

addition, E

is quasi-regular then we can associate it

a right Markov process M

= (Ω,F , F

) with

the state space X

. The process M

is called the in-

duced Markov process w.r.t. to the proper map F. If

the image of M under F is a right Markov process

then x

= F(x

). The process M

might have some

different interpretations like a reﬁnement of discrete

transitions structure, or an approximation of continu-

ous dynamics or an abstraction of the entire process.

It is difﬁcult to ﬁnd a practical condition to impose

on F, which would guarantee that E

, as deﬁned by

(3) and (4), is also quasi-regular. To circumvent this

problem, it is possible to restrict the original domain

D[E

] and impose some regularity conditions on F

(for more details, see (Iscoe, McDonald 1990)).

Assumption 1. Suppose that E

is a quasi-regular

BLFE.

3 THE STOCHASTIC MODEL

CHECKING PROBLEM

Let us consider M = (Ω,F ,F

) a strong

Markov process, which is the semantics of a UCPS.

For this strong Markov process we address a veriﬁca-

tion problem consisting of the stochastic reachability

problem deﬁned as follows. Given a set A ∈ B (X)

and a time horizon T > 0, let us to deﬁne (Bujorianu

2004):

Reach

(A) = { ω ∈ Ω | ∃t ∈ [0, T] : x

(ω) ∈ A}

Reach

∞

(A) = { ω ∈ Ω | ∃t ≥ 0 : x

(ω) ∈ A}. (5)

These two sets are the sets of trajectories of M, which

reach the set A (the ﬂow that enters A) in the interval

of time [0,T] or [0,∞).

The reachability problem consists of determining the

probabilities of such sets. The reachability problem

is well-deﬁned, i.e. Reach

(A), Reach

∞

(A) are in-

deed measurable sets. Then the probabilities of reach

events are

P(T

< T) or P(T

< ∞) (6)

where T

= inf{t > 0|x

∈ A} and P is a probability on

the measurable space (Ω,F ) of the elementary events

associated to M. P can be chosen to be P

(if we want

to consider the trajectories, which start in x) or P

(if

we want to consider the trajectories, which start in x

given by the distribution µ).

Usually a target set A in the state space is a level

set for a given function F : X → R, i.e. A = {x ∈

X|F(x) > l} (F can be chosen as the Euclidean norm

or as the distance to the boundary of E). The proba-

bility of the set of trajectories, which hit A until time

horizon T > 0 can be expressed as

P(supF(x

)|t ∈ [0, T]) > l. (7)

Our goal is to deﬁne a new stochastic process M

such

that the probabilities (6) are preserved.

Ideally, since (6) can be written as (7), F(x

)

would represent the best candidate for deﬁning a pos-

sible qualitative model reduction for M, which pre-

serves the reach set probabilities. The main difﬁculty

is that F(x

) is a Markov process only for special

choices of F (Rogers, Pitman 1981). The problem

is how to choose F well.

Note, if A

is open in X

and A = F

−1

), then

we consider the two ﬁrst hitting times T

(w.r.t. M)

and T

(w.r.t. M

) of A and A

, respectively. Recall

that T

= inf{t > 0|x

∈ A}.

The following results show that the stochastic

model checking problem is solvable for uncertain cps.

Proposition 1. Under the assumption.1, if µ is a

probability measure and ξ = +∞ (M has no killing),

then

exp(−T

) ≤ E

exp(−T

) (8)

where E

(resp. E

) is the expectation deﬁned w.r.t.

(resp. P

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

If M is the semantics of a UCPS U, given a tar-

get state set A ∈ B (X), then the goal in the stochas-

tic reachability analysis is to compute the probability

≤ T) for a ﬁnite horizon time T > 0. We now

translate the relation (8) in terms of probability of the

reachable sets.

Proposition 2. Under the assumption.1, if µ is a

probability measure, then

≤ T) ≤ eK min{TE

) + (9)

< u

∈ D(E

), u

≥ 1, (10)

− a.e. on A

} (11)

where K > 0 is the sector constant of E .

4 ADAPTING VERIFICATION TO

CO-EVOLUTION

The idea is to apply a “state space reduction” tech-

nique based on the general ‘induced BLFEs’ method

to achieve qualitative model reductions for Ucps.

With this technique, the semantics of Ucps are ‘ap-

proximated’ by a one-dimensional stochastic process

with a much smaller state space.

4.1 Qualitative Model Reduction

The stochastic reachability deﬁnition gives the idea to

introduce the following concept of qualitative model

reduction for Ucps.

Deﬁnition 1. Given a right Markov process M de-

ﬁned on the Lusin state space (X,B ), and F :

X → R a measurable weight function, suppose that

assumption.1 is fulﬁlled. The process M

associated

to the induced BLFE E

under function F is called a

qualitative model reduction of M.

Let U be a UCPS and M its semantics. Suppose

that M is a right Markov process deﬁned on the Lusin

state space (X,B ).

Deﬁnition 2. Any UCPS U

whose semantics is a

qualitative model reduction of M is called a qualita-

tive model reduction of U.

Let U be a Ucps and M its semantics (that is a

right Markov process, with the state space X).

Proposition 3. If M is a diffusion then any qualitative

model reduction M

of M is a diffusion.

Proposition 4. If M is a jump process then any qual-

itative model reduction M

of M is again a jump pro-

cess.

Proof. This statement can be obtained as a conse-

quence of the abstract version of the Kolmogorov

backward equations (Davis 1993)

∂

∂t

f(x) = LP

f(x), P

f = f, f ∈ D(L ) (12)

and the equality (14). If the equations (12) are as-

sociated to an initial diffusion process (resp. jump

process) then the relation (14) allow to obtain the fact

that the transition probabilities of the induced process

satisfy a similar equation, such that the induced pro-

cess is still a diffusion process (resp. jump process).

The same conclusion can be obtain using the stochas-

tic calculus of BLFEs (Iscoe, McDonald 1990).

Since the semantics of a Ucps is, in most cases, a

stochastic process, which can be viewed an interleav-

ing between some diffusionprocesses and a jump pro-

cess (see (Bujorianu, Lygeros 2004) for a very gen-

eral model for Ucps and its semantics as a Markov

string), we can write the following result as a corol-

lary of Prop.3.

Proposition 5. Any qualitative model reduction of a

Ucps is again a Ucps.

Let (L ,D(L )) and (L

,D(L

)) be the generators

of E and E

, respectively. For the following results

we suppose that the Ass.1 is fulﬁlled.

Proposition 6. The generators L and L

are related

as follows

L (u

◦ F) = L

◦ F, ∀u

∈ D(L

) (13)

Theorem 7. For all A

∈ B

) and for all t > 0 we

have

(Fx, A

) = p

(x,F

−1

)) (14)

where (p

) and (p

) are the transition functions of M

and M, respectively.

Proof. Let F

be deﬁned as F

: B

) → B

(X);

= u

◦ F. Then (13) becomes (L ◦ F

◦ L

,∀u

∈ D(L

) (∗∗). For a strong Markov

process, the opus of the kernel operator is the inverse

operator of the inﬁnitesimal generator of the process.

Now, from (∗∗) we get a similar relation between the

kernel operatorsV and V

of the processes M and M

i.e. F

◦V

= V ◦ F

on B

), or

◦ F = V(u

◦ F),∀u

∈ B

) (15)

since if u

∈ B

) thenV

∈ D(L

). For u

= 1

(the indicator function of A

), by the kernel operator

integral deﬁnition, we obtain (14).

Relation (15) implies the following corollary:

Corollary8. The semigroups (P

) and (P

) of M

and

M are related by

◦ F = P

◦ F),∀u

∈ B

). (16)

CO-EVOLUTION PRESERVING ABSTRACT MODEL REDUCTION FOR UNCERTAIN CYBER-PHYSICAL

SYSTEMS - Towards a Framework for Nanoscience

4.2 Adaptive Bisimulation

In this subsection we deﬁne a new concept of adap-

tive bisimulation for cps. This concept is deﬁned as

measurable relation, which induces equivalentBLFEs

on the quotient spaces. In deﬁning adaptive bisimula-

tion, we do not impose the equivalence of the quotient

processes, which might not have Markovian proper-

ties (Rogers, Pitman 1981), but we impose the equiv-

alence of the qualitative model reductions (that can

differ from the quotient processes) associated with the

induced BLFEs, with respect to the projection maps.

Let (X,B (X)) and (Y,B (Y)) be Lusin spaces and

let R ⊂ X × Y be a relation such that Π

(R ) = X

and Π

(R ) = Y. We deﬁne the equivalence relation

on X that is induced by the relation R ⊂ X × Y, as

the transitive closure of {(x, x

′

)|∃y s.t. (x,y) ∈ R and

′

,y) ∈ R }. Analogously, the induced (by R ) equiv-

alence relation on Y can be deﬁned. We write X/

and Y/

for the sets of equivalence classes of X and

Y induced by R . We denote the equivalence class of

x ∈ X by [x]. Let

(X) = B (X) ∩ {A ⊂ X| if x ∈ A and [x] = [x

′

] then

′

∈ A}

be the collection of all Borel sets, in which any equiv-

alence class of X is either totally contained or totally

not contained. It can be checked that B

(X) is a σ-

algebra. Let π

: X → X/

be the mapping that maps

each x ∈ X to its equivalence class and let

B (X/

) = {A ⊂ X/

|π

−1

(A) ∈ B

(X)}.

Then (X/

,B (X/

)), which is a measurable space,

is called the quotient space of X w.r.t. R . The quo-

tient space of Y w.r.t. R is deﬁned in a similar way.

We deﬁne a bijective mapping ψ : X/

→ Y/

ψ([x]) = [y] if (x, y) ∈ R for some x ∈ [x] and some

y ∈ [y].

We say that the relation R is measurable if X and Y

if for all A ∈ B (X/

) we have ψ(A) ∈ B (Y/

) and

vice versa, i.e. ψ is a homeomorphism. Then the real

measurable functions deﬁned on X/

can be identi-

ﬁed with those deﬁned on Y/

through the homeo-

morphism ψ. We can write B

(X/

)

∼

(Y/

Moreover, these functions can be thought of as real

functions deﬁned on X or Y measurable w.r.t. B

(X)

or B

(Y).

Assumption 2. Suppose that X/

and Y/

with the

topologies induced by projection mappings are Lusin

spaces.

Suppose we have given two right Markov processes

M and W with the state spaces X and Y. Assume

that µ (resp. ν) is a stationary measure of the pro-

cess M (resp. W). Let µ/

(resp. ν/

) the image of

µ (resp. ν) under π

(resp. π

). Let E (resp. F ) the

quasi-regular BLFE corresponding to M (resp. W).

The equivalence between the induced processes can

be used to deﬁne a new bisimulation between Markov

processes, as follows.

Deﬁnition 3. Under assumptions 1 and 2, a measur-

able relation R ⊂ X × Y is a bisimulation between

M and W if the mappings π

and π

deﬁne the same

induced BLFE on L

(X/

,µ/

) and L

(Y/

,ν/

respectively.

This deﬁnition states that M and W are bisimilar if

E /

= F /

. Here, E /

(resp. F /

) is the induced

BLFE of E (resp. F ) under the mapping π

(resp.

). Clearly, this can be possible iff µ/

= ν/

Assumption 3. Suppose that E /

and F /

are

quasi-regular BLFE.

Denote the Markov process associated to E /

(resp. F /

) by M/

(resp. W/

Proposition 9. Under assumptions 1, 2 and 3, M and

W are stochastic bisimilar under R iff their qualita-

tive model reductions M/

and W/

with respect to

and, respectively π

are µ/

-equivalent.

LetU andU

′

be two UCPSs, with the semantics M

and W, strong Markov processes deﬁned on the state

spaces (X,B (X)) and (Y,B (Y)), respectively.

Deﬁnition 4. U and U

′

are called bisimilar if there

exist a bisimulation relation under which their seman-

tics M and W are bisimilar

4.3 An Example

Let us recall the chemically reacting system case

study from (Singh, Hespanha 2005), where it is in-

vestigated using the theory of polynomial stochastic

hybrid systems. Consider a system of n species X

j = 1,..,n, inside a ﬁxed volume V involved in K re-

actions of the form

+... + u

−→ ν

+... + ν

,∀i ∈ {1,...,K}

where the species A

have a constant number of

molecules. The meaning and the assumptions about

the coefﬁcients of the reaction equation are given in

(Singh, Hespanha 2005). c

is a reaction parameter

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

which is used in deﬁning the probability that a par-

ticular reaction takes place on (t,t + dt). The sys-

tem is characterized by the trivial dynamics

= 0, x =

,...,x

]

, a family of K reset maps x = φ

−

: R

→ R

, and a corresponding family of tran-

sition intensities λ

: R

→ [0,∞), ∀i = 1,.., K. For

each i = 1,..,K, the reset map φ

and the correspond-

ing λ

is uniquely deﬁned by the i

reaction equation

and given by x 7→ φ

(x), φ

(x) = x + [ν

− u

,ν

−

,...,ν

−u

]

; λ

(x) = c

(x), whereU

represents

the number of distinct molecular reactant combina-

tions present in V at time t for the reaction R

. The

executions of such a system are deﬁned in (Singh,

Hespanha 2005).

Now we apply the method of qualitative model re-

duction to this process. We can show that executions

of this cps form a particular kind of right Markov

process called jump process (Davis 1993). The ex-

tended generator (Th.1 (Singh, Hespanha 2005)) is

(Lψ)(x) =

∑

i=1

(ψ(φ

(x)) − ψ(x))λ

(x), ψ ∈ D(L).

Let us consider a proper map F : R

→ R and write

the generator of the induced process for

◦ F, ψ

∈ D(L

L(ψ

◦ F)(x) =

∑

i=1

(ψ

(F(φ

(x))) − ψ

(F(x)))λ

(x)

Deﬁne φ

: ImF → R by φ

(Fx) = F(φ

(x)) and

: ImF → R by λ

(Fx) = λ

(x). In order to have

these two function well-deﬁned we need to impose

some compatibility conditions between F and reset

maps φ

and their corresponding transition intensities

as follows: Fx = Fx

′

⇒ F(φ

(x)) = F(φ

′

)) and

(x) = λ

′

). This means that F preserves the jumps

(reset maps and transition intensities), i.e. the pre-

jump locations have the same image under F then the

intensities of transition should be equal and the post-

jump locations have the same image under F. Using

(13), the generator of the induced process is

) =

∑

i=1

(ψ

(φ

)) − ψ

))λ

);

= Fx; x ∈ X.

For simplicity, we suppose that the reactions R

are

reversible in time. Then the generator is self-adjoint

(or Hermitian). The (symmetric) quasi-regular energy

bilinear form on L

,µ) associated to the given pro-

cess (with µ a stationary distribution) can be written

E (ψ,ϕ) =

∑

i=1

(ψ(φ

(x)) − ψ(x))(ϕ(φ

(x)) −

ϕ(x))λ

(x)µ(dx)

Then the induced energy bilinear form E

(R,µ

) (where µ

is the image of µ under F) w.r.t.

F is

(ψ

,ϕ

) =

∑

i=1

[ψ

(φ

(Fx)) − ψ

(Fx)]

[ϕ

(φ

(Fx)) − ϕ

(Fx)]λ

(Fx)

µ(dx)

∑

i=1

[ψ

(φ

)) − ψ

)][ϕ

(φ

)) − ϕ

)]λ

)µ

(dx

Clearly, E

is associated to a jump process - thequal-

itative model reduction of the given process. In this

particular case, the induced process is exactly the im-

age under F of the initial jump process.

5 CONCLUSIONS

In this paper, we have used the concept of energy,

which is a key ingredient of Hilbertean formal meth-

ods, to deﬁne qua;itative model reduction and behav-

ioral equivalence for cyber-physical systems operat-

ing in random environments. Energy is a versatile an-

alytical concept that characterizes in a subtle way the

interaction between computation and physics, as well

as their co-evolution.

Adaptive bisimulation means the energy preser-

vation of the stochastic processes generated by the

cyber-physical system evolutions. The energy con-

cept can be also used to deﬁne qualitative model re-

ductions for cyber-physical systems. Given an quali-

tative model reduction function that reduces the state

space, we have deﬁned a standard construction that

associates a qualitative model reduction (called stan-

dard) on the reduced state space. The mathemati-

cal results from Section 4.1 show that the qualitative

model reduction method preserves important analytic

properties (related to HFM). Two uncertain CPS are

adaptive bisimilar if they have the same energy. The

theorem from Section 4.2 shows that two uncertain

CPS are adaptive bisimilar iff their standard qualita-

tive model reductions are equivalent as Markov pro-

cesses.

We have formulated the stochastic model check-

ing problem (a subproblem of stochastic reachabil-

ity analysis, corresponding to the probabilistic model

checking of Markov chains). We proved two results

that show that the problem is solvable for uncertain

cyber-physical systems. The mathematical results

from Section 3 provide a upper bound for the reach

set probabilities. In this way, one can prove that the

probability of reaching a state in a certain set can be

small enough.

CO-EVOLUTION PRESERVING ABSTRACT MODEL REDUCTION FOR UNCERTAIN CYBER-PHYSICAL

SYSTEMS - Towards a Framework for Nanoscience

The most closely related model is that of stochas-

tic hybrid automata (Bujorianu 2004). These au-

tomata are not necessarily embedded systems and

their hybrid behavior is often an internal feature (as

for cars, aircraft, mobile robots and so on) rather than

the interaction with a physical environment (a feature

of embedded systems). Cyber-physical systems are

also networked.

In following work we will reﬁne the formal

framework presented in this paper to be used for

nanoscience.

ACKNOWLEDGEMENTS

This work was funded by the EPSRC project

EP/E050441/1 CICADA.

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