AachenDepartment of Computer Science

Technical Report

Compositional Modeling and

Minimization of Time-Inhomogeneous

Markov Chains

Tingting Han and Joost-Pieter Katoen and Alexandru Mereacre

ISSN 0935–3232 · Aachener Informatik Berichte · AIB-2007-21

RWTH Aachen · Department of Computer Science · December 2007

The publications of the Department of Computer Science of RWTH AachenUniversity are in general accessible through the World Wide Web.

http://aib.informatik.rwth-aachen.de/

Compositional Modeling and Minimization

of Time-Inhomogeneous Markov Chains

Tingting Han1,2, Joost-Pieter Katoen1,2, and Alexandru Mereacre1

1 RWTH Aachen University, Software Modeling and Verification Group, Germany{tingting.han,katoen,mereacre}@cs.rwth-aachen.de

2 University of Twente, Formal Methods and Tools Group, The Netherlands

Abstract. This paper presents a compositional framework for the modeling ofinteractive continuous-time Markov chains with time-dependent rates, a subclassof communicating piecewise deterministic Markov processes. A poly-time algo-rithm is presented for computing the coarsest quotient under strong bisimulationfor rate functions that are either piecewise uniform or (piecewise) polynomial.Strong as well as weak bisimulation are shown to be congruence relations forthe compositional framework, thus allowing component-wise minimization. In ad-dition, a new characterization of transient probabilities in time-inhomogeneousMarkov chains with piecewise uniform rates is provided.

1 Introduction

Modeling large stochastic discrete-event dynamic systems is a difficult task thattypically requires human intelligence and ingenuity. To facilitate this process,formalisms are needed that allow for the modeling of such systems in a composi-tional manner. This allows to construct models of simpler components—usuallyfrom first principles—that can be combined by appropriate composition oper-ators to yield complete system models. In concurrency theory, process algebra[Mil89,Hoa85] has emerged as an important framework to achieve composition-ality: it provides a formal apparatus for compositional reasoning about structureand behavior of systems, and features abstraction mechanisms allowing systemcomponents to be treated as black boxes.

Although originally aimed at purely functional behavior, process algebras forstochastic systems have been investigated thoroughly, see e.g., [HHK02,Her02].In all these approaches, the dynamics of the stochastic models is assumed tobe time-homogeneous, i.e., the probabilistic nature of mode transitions as wellas the time-driven behavior are independent of the global time. This is, how-ever, a serious drawback to adequately model random phenomena that occurin practice such as failure rates of hardware components (a bath-tub curve),software reliability (which reduces due to memory leaks and increases after arestart), and battery depletion (where the power extraction rate non-linearlydepends on the remaining amount of energy [CJH07]), to mention a few. Thispaper attempts to overcome this deficiency by providing a process algebra fortime-inhomogeneous continuous-time Markov chains (ICTMCs). This is a veryversatile class of models and is a natural stepping-stone towards more full-fledgedstochastic hybrid system models such as piecewise deterministic Markov pro-cesses (PDPs [Dav93]). We show that ICTMCs can be compositionally modeledby using a time-dependent adaptation of the framework of interactive Markov

chains (IMCs) [Her02]. To facilitate this, ICTMCs are equipped with the poten-tial for interaction, i.e., synchronization. Instrumental to this approach is thememoryless property of ICTMCs.

More importantly though, notions of strong and weak bisimulation are de-fined and shown to be congruences. Together with efficient quotienting algorithmsthis allows for the component-wise minimization of hierarchical ICTMC models.Finally, we present an axiomatization for strong and weak bisimulation whichallows to simplify models by pure syntactic manipulations as opposed to per-forming minimization on the model level 3. As a generalization of results onordinary lumpability on Markov chains [Buc94], we show that strong bisimula-tion preserves transient and long-run state probabilities in ICTMCs. This allowsto minimize symbolically ICTMCs prior to their analysis.

We present a bisimulation minimization algorithm to obtain the coarsest (andthus smallest) strong bisimulation quotient of a large class of interactive ICTMCs,viz. those that have piecewise uniform—rate Rk(t) on piece k is of the formfk(t)·R for integrable function R—polynomial, or piecewise polynomial—whereeach polynomial is of degree three—rate functions. The worst-case time and spacecomplexity is O (ma lg(n) + Mmr lg(n)) and O (ma + mr), respectively, whereM+1 is the number of pieces (or degrees of the polynomial), ma is the numberof action-labeled transitions and mr the number of rate-labeled transitions in theICTMC under consideration. This algorithm is based on the partition-refinementbisimulation algorithm for Markov chains by Derisavi et al. [DHS03] and Paige-Tarjan’s algorithm for labeled transition systems (LTS) [PT87].

Related work. ICTMCs are related to piecewise deterministic Markov processes(PDPs), a more general class of continuous-time stochastic discrete-event dy-namic systems proposed by Davis [Dav93]. The probabilistic nature of modetransitions in PDPs is as for ICTMCs; in fact, ICTMCs are a subclass of PDPswhen the global time t has a clock dynamics i.e., t = 1. The notion of parallelcomposition of ICTMCs corresponds to that for communicating PDPs (CPDPs)as introduced by Strubbe and van der Schaft [SvdS06,SJvdS03]. Alternativemodeling formalisms for PDPs are, e.g., variants of colored Petri nets [EB05]but they lack a clear notion of compositionality. Compositional modeling for-malisms for hybrid systems have been considered by, e.g., [AGLS06,AGH+00].Strong bisimulation has been proposed for several classes of (stochastic) hybridsystems, see e.g., [BLB05,HTP05,SvdS05]. Our notion of bisimulation is closelyrelated to that for CPDPs [SvdS05] but differs in the fact that the maximalprogress assumption—a race between one or more rates and a transition thatis not subject to interaction with the environment is resolved in favor of theinternal transition—is not considered in [SvdS05].

2 Inhomogeneous Continuous Time Markov Chains

Definition 1 (ICTMC). An inhomogeneous continuous-time Markov chain isa tuple C = (S,R) where: S = {1, 2, . . . , n} is a finite set of states, and R(t) =[Ri,j(t) ≥ 0] ∈ IRn×n is a time-dependent rate matrix, where Ri,j(t) is the ratebetween states i, j ∈ S at time t ∈ IR≥0.

3 The proofs for the congruence results and the axiomatization have been obtained by standardproof techniques and are omitted from the paper.

4

Let diagonal matrix E(t) = diag [Ei(t)] ∈ IRn×n, where Ei(t) =∑

j∈SRi,j(t) for

all i, j ∈ S, i 6= j i.e., Ei(t) is the total exit rate of state i at time t. Consider anon-homogeneous Poisson process {Z(t)|t ≥ 0} with rate R(t). The probabilityof k arrivals in the interval [t, t + ∆t] is:

Pr{Z(t + ∆t) − Z(t) = k} =

[

∫ t+∆tt R(ℓ)dℓ

]k

k!e−

R t+∆tt

R(ℓ)dℓ, k = 0, 1, . . . .

The probability that there will be no arrivals in the interval [t, t + ∆t] is:

Pr{Z(t + ∆t) − Z(t) = 0} = e−R t+∆t

tR(ℓ)dℓ = e−

R ∆t

0R(t+ℓ)dℓ. (1)

Let the random variable Wi,j(t) be the firing time of transition i → j (i, j ∈S) with rate Ri,j(t) at time t. From (1) we obtain the cumulative probabilitydistribution of the firing time of transition i → j:

Pr {Wi,j(t) ≤ ∆t} = 1 − Pr{Z(t + ∆t) − Z(t) = 0} = 1 − e−R ∆t

0Ri,j(t+ℓ)dℓ. (2)

Probability measures. For every ICTMC one can specify measures of interest.These measures are either related to the states or to the transitions of an ICTMC.Consider a random variable Wi(t) which denotes the waiting time in state i.

Property 1.

Pr {Wi(t) ≤ ∆t} = 1 − e−R ∆t

0Ei(t+ℓ)dℓ. (3)

An intuitive explanation of (3) is that the waiting time Wi(t) in state i is deter-mined by the minimal firing time of all k outgoing transitions from state i, i.e.,Wi(t) = min {Wi,1(t), . . . ,Wi,k(t)}. When Ri,j(t) = Ri,j and Ei(t) = Ei for allt ∈ IR≥0, i.e., the ICTMC is a CTMC, Wi(t) has the distribution 1− e−Ei∆t. Aninteresting property is that the waiting time in any state i is memoryless, i.e.:

Pr{

Wi(t) ≤ t′ + ∆t|Wi(t) > t′}

= Pr{

Wi(t + t′) ≤ ∆t}

. (4)

This can be shown as follows:

Pr{

Wi(t) ≤ t′ + ∆t|Wi(t) > t′}

=e−

R t′

0 Ei(t+ℓ)dℓ − e−R t′+∆t0 Ei(t+ℓ)dℓ

e−R t′

0Ei(t+ℓ)dℓ

=1 − e−R t′+∆t

0Ei(t+ℓ)dℓ+

R t′

0Ei(t+ℓ)dℓ = Pr

{

Wi(t + t′) ≤ ∆t}

.

Equation (4) will be of importance when we later define a calculus for ICTMCs.

Property 2. The probability Pri,j(t) to select transition i → j (i 6= j, i, j ∈ S)with rate Ri,j(t) at time t is:

Pri,j(t) =

∫ ∞

0Ri,j(t + τ)e−

R τ0 Ei(t+ℓ)dℓdτ. (5)

When rates are constant, the measure (5) takes the form Pri,j =Ri,j

Ei(Pri,j(t) =

Pri,j for all t ∈ IR≥0), which corresponds to transition probability in CTMCs. Animportant relation which always holds for CTMCs is that the probability to leavea non-absorbing state i (Ei(t) 6= 0 for all t ∈ IR≥0) equals

∑

j∈SPri,j = 1. On the

other hand, for ICTMCs,∑

j∈SPri,j(t) = 1 only holds when limτ→∞

∫ τ0 Ei(ℓ)dℓ =

∞.

5

Property 3. The cumulative probability distribution Pri,j(t,∆t) to move fromstate i to state j (i 6= j) with rate Ri,j(t) in ∆t units of time starting at time tis:

Pri,j(t,∆t) =

∫ ∆t

0Ri,j(t + τ)e−

R τ

0Ei(t+ℓ)dℓdτ. (6)

Notice that (6) is the same as (5) except that the range of the outer-most integralis [0,∆t]. For CTMCs, equation (6) results (Pri,j(t,∆t) = Pri,j(∆t) for all t ∈

IR≥0) in Pri,j(∆t) =Ri,j

Ei

(

1 − e−Ei∆t)

.

Transient probability distribution. One important measure which quantifies theprobability to be in a specific state at some time point is the transient prob-ability distribution. Consider an ICTMC described by the stochastic process{X(t)|t ≥ 0}. The transient probability distribution Pr {X(t + ∆t) = j}, denotedby πj (t + ∆t), is the probability to be in state j at time t+∆t, and is describedby the equation:

πj (t + ∆t) =∑

i∈S

Pr {X(t) = i} · Pr {X(t + ∆t) = j|X(t) = i} . (7)

Equation (7) can be expressed in matrix form as: π(t + ∆t) = π(t)Φ(t + ∆t, t),where π(t) = [π1 (t) , . . . , πn (t)] and Φ(t + ∆t, t) represents the transition prob-ability matrix. This equation represents the solution of a system of ODEs:

dπ(t)

dt= lim

∆t→0

π(t + ∆t) − π(t)

∆t= lim

∆t→0π(t)

[Φ(t + ∆t, t) − I]

∆t. (8)

For the diagonal elements qi,i(t) of the matrix lim∆t→0[Φ(t+∆t,t)−I]

∆t from (8), weobtain:

qi,i(t) = lim∆t→0

Pr {X(t + ∆t) = i|X(t) = i} − 1

∆t.

As Pr {X(t + ∆t) = i|X(t) = i} denotes the probability to stay in state i for atleast ∆t units of time or the probability to return to state i in two or more steps,it follows:

qi,i(t) = lim∆t→0

e−R ∆t

0Ei(t+ℓ)dℓ − 1 + o (∆t)

∆t= −Ei(t),

where o (∆t) denotes the probability to make two or more transitions in ∆t

units of time. Notice that lim∆t→0o(∆t)

∆t = 0. For the off-diagonal elements qi,j(t)

(i 6= j) of matrix lim∆t→0[Φ(t+∆t,t)−I]

∆t , the relation is similar:

qi,j(t) = lim∆t→0

Pr {X(t + ∆t) = j|X(t) = i}

∆t= lim

∆t→0

Pri,j(t,∆t) + o (∆t)

∆t,

which can be reduced using (6) to:

qi,j(t) = lim∆t→0

∫ ∆t0 Ri,j(t + τ)e−

R τ0 Ei(t+ℓ)dℓdτ + o (∆t)

∆t= Ri,j(t).

The resulting infinitesimal generator matrix Q(t) has the form:

Q(t) = lim∆t→0

[Φ(t + ∆t, t) − I]

∆t= R′(t) −E(t),

6

where R′ equals R except that R′

i,i(t) = 0. Plugging Q(t) into equation (8)yields the system of ODEs which describe the evolution of transient probabilitydistribution over time (Chapman-Kolmogorov equations):

dπ(t)

dt= π(t)Q(t),

n∑

i=1

πi(t0) = 1, (9)

where π(t0) is the initial condition. From the literature (see [Kai80, pages 594–631]) it is known that the solution π(t) of (9), written as:

π(t) = π(t0)Φ(t, t0) (10)

has the transition probability matrix given by the Peano-Baker series:

Φ(t, t0) = I +

∫ t

t0

Q(τ1)dτ1 +

∫ t

t0

Q(τ1)

∫ τ1

t0

Q(τ2)dτ2dτ1 + . . . . (11)

Note that if Q(τ1)∫ τ1t0

Q(τ2)dτ2 =∫ τ1t0

Q(τ2)dτ2Q(τ1) then Φ(t, t0) = eR t

t0Q(τ)dτ

.If the rate matrix R(t) is piecewise constant i.e., R(t) = Rk or Q(t) = Qk forall t ∈ [tk, tk+1) and k ≤ M ∈ IN (M + 1 is the total number of constant pieces),equation (10) can also be rewritten as (see [RWVT95]):

π(t) =

π(t0)eQ0(t−t0) if t ∈ [t0, t1)

......

π(tM )eQM (t−tM ) if t ∈ [tM ,∞)

and π(tk) = π(tk−1)eQk−1(tk−tk−1).

The general case is when the rate matrix is piecewise uniform i.e., R(t) = Rk(t) =fk(t)Rk or Q(t) = Qk(t) = fk(t)Qk for any integrable function fk(t) : IR≥0 →IR≥0 on time interval [tk, tk+1), constant matrices Rk, Qk and k ≤ M .

Theorem 1. The transient probability distribution π(t) of an ICTMC C = (S,R)with a piecewise uniform rate matrix R(t) and M+1 pieces is given by:

π(t) =

π(t0)eQ0

R t

t0f0(τ)dτ

if t ∈ [t0, t1)...

...

π(tM )eQM

R ttM

fM (τ)dτif t ∈ [tM ,∞)

where π(tk) = π(tk−1)eQk−1

R tktk−1

fk−1(τ)dτ.

Proof. When the rate matrix R(t) is piecewise uniform with M+1 pieces, theinfinitesimal generator Q(t) takes the form:

Q(t) =

Q0(t) if t ∈ [t0, t1)...

...QM (t) if t ∈ [tM ,∞)

where Q(t) = Qk(t) = fk(t)Qk for all t ∈ [tk, tk+1) and k ≤ M . Therefore, wecan describe the transient probability distribution as follows:

π(t) =

π(t0)Φ(t, t0) if t ∈ [t0, t1)...

...π(tM )Φ(t, tM ) if t ∈ [tM ,∞)

.

7

Notice that π(tk) = π(tk−1)Φ(tk, tk−1). Now the question is: how can we computeΦ? As we already know, Φ is given by the Peano-Baker series, cf. Equation (11).The characterization of Φ in terms of the Peano-Baker series does not yieldan effective computational recipe for Φ. Note however that the infinitesimalgenerator Q(t) commutes. Let us consider the k’th piece i.e., Q(t) = Qk(t) forall t ∈ [tk, tk+1). The fact that Qk(t) commutes follows from:

Qk(t)

∫ t

tk

Qk(τ)dτ =

∫ t

tk

Qk(τ)dτQk(t) ⇔

fk(t)Qk

∫ t

tk

fk(t)Qkdτ =

∫ t

tk

fk(t)Qkdτfk(t)Qk ⇔

Q2kfk(t)

∫ t

tk

fk(t)dτ =

∫ t

tk

fk(t)dτfk(t)Q2k.

Therefore, Φ(t, tk) = eQk

R t

tkfk(τ)dτ

. The transient probability distribution be-comes:

π(t) =

π(t0)eQ0

R tt0

f0(τ)dτif t ∈ [t0, t1)

......

π(tM )eQM

R ttM

fM (τ)dτif t ∈ [tM ,∞)

3 Inhomogeneous Interactive Markov Chains

In order to facilitate the compositional modeling of ICTMCs, we equip theseprocesses with the capability to allow for their mutual interaction. This is es-tablished by adding actions to ICTMCs. Let Act be the countable universe ofactions. The aim of these actions is that certain actions can only be performedtogether with other processes.

Definition 2 (I2MC). An inhomogeneous interactive Markov chain (I2MC) isa tuple I = (S, Act,→,R, s0) where S and R are as before, →⊆ S × Act × S isa transition relation and s0 ∈ S is the initial state.

The semantic model of I2MC represents the time-dependent variant of IMC[Her02].

Process algebra for I2MC. Originally developed by Hoare and Milner (see [Mil89,Hoa85]),process algebras have been developed as a compositional framework for describ-ing the functional behavior of the system. It allows for modeling complex systemsin a component-wise manner by offering a set of operators to combine componentmodels. Actions are the most elementary notions. The combination of several ac-tions using the operators forms a process. We extend this framework by stochastictiming facilities.

Definition 3. Let X be a process variable, λ(t) ∈ IR≥0 with t ∈ IR≥0, A ⊆ Actand a ∈ Act. The syntax of inhomogeneous interactive Markov language (I2ML)for I2MCs is defined as follows:

P ::= 0 | a.P | λ(t).P | P + P | P‖AP | P \ A | X.

8

a.Pa

−→P λ(t).Pλ(t)−→P

Pa

−→P ′

P+Qa

−→P ′

Pλ(t)−→P ′

P+Qλ(t)−→P ′

Pa

−→P ′

P‖AQa

−→P ′‖AQ(a /∈ A)

Pλ(t)−→P ′

P‖AQλ(t)−→P ′‖AQ

Pa

−→P ′ and Qa

−→Q′

P‖AQa

−→P ′‖AQ′(a ∈ A)

Pλ(t)−→P ′

P\Aλ(t)−→P ′\A

Pa

−→P ′

P\Aa

−→P ′\A(a /∈ A)

E[X :=E/X ]λ(t)−→E′

X :=Eλ(t)−→E′

Pa

−→P ′

P\Aτ

−→P ′\A(a ∈ A)

E[X :=E/X ]a

−→E′

X :=Ea

−→E′

Table 1. Inference rules for the operational semantics of I2ML.

Process variables are assumed to be defined by recursive equations of the formX := P , where P is an I2ML term. The null process 0 is the deadlock process andcannot perform any action. The prefix operators are a.P and λ(t).P for actionsand rates, respectively. The choice operator P + Q chooses between processesP or Q. Process P‖AQ denotes the parallel composition of processes P andQ where synchronization is required only for actions in A; actions not in A areperformed autonomously. The process P \A behaves like P except that all actionsin A become unobservable to other processes; this is established by relabeling aby the distinguished action τ ∈ Act. The operational semantics of I2ML termsis defined by the inference rules in Table 1 where for the sake of concisenesssymmetric rules are not shown.

A few remarks concerning time-prefix and choice are in order. The processλ(t).P evolves into P within ∆t time units with probability:

Prλ(t).P,P (t,∆t) =

∫ ∆t

0λ(t + τ)e−

R τ

0λ(t+ℓ)dℓdτ = 1 − e−

R ∆t

0λ(t+ℓ)dℓ,

given that λ(t).P is enabled at the global time t. The above relation can beeasily proven from (6) by taking i = λ(t).P , j = P , Ri,j(t + τ) = λ(t + τ) andEs(t+ℓ) = λ(t+ℓ). The process λ(t).P +µ(t).Q can evolve into P if the time delaygenerated by a stochastic process with rate λ(t) is smaller than that generatedby a different stochastic process with rate µ(t). By a symmetric argument it mayevolve into Q. Therefore, from (3) it follows that the distribution of time until

a choice is made is Pr{W (t) ≤ ∆t} = 1 − e−R ∆t0 λ(t+τ)+µ(t+τ)dτ . For a choice

between |J | processes (J is a finite index set), the distribution of the waiting

time becomes Pr{W (t) ≤ ∆t} = 1 − e−R ∆t0

P

i∈J λi(t+τ)dτ . If the rates λi(t) inthe process

∑

i∈J λi(t).Pi are constant (λi(t) = λi), then the waiting time isexponentially distributed with the sum of the rates λi i.e. Pr{W (t) ≤ ∆t} =1 − e−

P

i∈J λi∆t. This corresponds to the interpretation of choice in Markovianprocess algebras [HHK02]. It is important to note that when Pi = P for all i ∈ J ,the process

∑

i∈J λi(t).P will evolve into P with rate∑

i∈J λi(t).

9

0 use.PPλ1(t)

λ2(t)

use

(a) The I2MC for process P

0 use.QQµ1(t)

µ2(t)

use

(b) The I2MC for process Q

P‖{use}Q

0‖{use}Q P‖{use}use.Q P‖{use}0

λ1(t)

µ2(t)

µ1(t)

use.P‖{use}use.Q

0‖{use}0

0‖{use}use.Q use.P‖{use}0

use

µ2(t) λ2(t) λ2(t)

use.P‖{use}Q

µ2(t)

µ1(t)

λ2(t)

λ1(t)

µ1(t) λ1(t)

(c) The I2MC for the process P‖{use}Q

Fig. 2. Two faulty components.

Parallel composition. When considering just actions the asynchronous parallelcomposition has the same functionality as that from basic process calculi. On theother hand when considering stochastic delays the composition is more involved.

P‖Q

P ′‖Q P‖Q′

P ′‖Q′

λ(t)

µ(t)

µ(t)

λ(t)

Fig. 1. P‖Q.

Consider P := λ(t).P ′ and Q := µ(t).Q′. They canevolve into P ′ and Q′ after a time delay governed by adistribution with rate λ(t) and µ(t), respectively. Sincethe waiting time in any state is memoryless (4), we canshow the way by which processes P and Q are com-posed (Fig. 1). First consider that when both processesstart their execution in initial state P‖Q (the shad-owed state) they probabilistically select a time delay,say, ∆tλ for P and ∆tµ for Q. If ∆tλ < ∆tµ then

P finishes its execution first and evolves into P ′. The same applies to Q when∆tµ < ∆tλ. By intuition we could think that when it is already in P ′‖Q, ∆tλ = 0and the remaining delay for process Q until it finishes its execution is ∆tµ−∆tλ.What really happens is that on entering state P ′‖Q both delays are set to zeroi.e., ∆tλ = ∆tµ = 0. As P ′ has no transitions, ∆tλ remains 0 but for Q its delayis initialized to a new value which might be different from ∆tµ − ∆tλ due to aprobabilistic selection. Due to the memoryless property, however, the remainingdelay for Q is fully determined by µ only.

Example 1. Consider two hardware components described by the equations P :=λ1(t).0+λ2(t).use.P and Q := µ1(t).0+µ2(t).use.Q, respectively. The underlyingI2MCs for both components are depicted in Fig. 2 (a) and (b). Each of the

10

components may fail with rate λ1(t) and µ1(t), respectively. As a result of thefailure they evolve into process 0. On the other hand, the components may moveto the working state with the rate λ2(t) and µ2(t), respectively, where theycan use some resources. If one of them fails then the entire system fails. Bothcomponents can use the resources at the same time if the system is workingproperly. Fig. 2 (c) depicts the I2MC of P‖{use}Q.

4 Strong and Weak Bisimulation

In order to compare the behavior of ICTMCs (and their interactive variants) weexploit the well-studied and widely accepted notion of bisimulation [Buc94,Mil89,Her02].A classical bisimulation relation requires equivalent states to be able to mutuallymimic their stepwise behavior. In the probabilistic setting this is interpreted asrequiring equivalent states to have equal cumulative rates to move to any equiva-lence class. Bisimulation is considered as a natural notion of equivalent behavior,is equipped with quotienting algorithms, and has a clear correspondence to equiv-alence in terms of logical behavioral specifications. In this section, we will definestrong bisimulation for I2MC starting from a similar notion on ICTMCs. Somealgebraic and probabilistic properties of bisimulation are investigated. The sameapplies to weak bisimulation that allows for the abstraction of internal, i.e., τactions.

Bisimulation for ICTMCs.

Definition 4 (ICTMC strong bisimulation). An equivalence R ⊆ S × S isa strong bisimulation whenever for all (P,Q) ∈ R, t ∈ IR≥0 and C ∈ S/R:

R(P,C, t) = R(Q,C, t),

where R(P,C, t) =∑

i{|λ(t)|Pλ(t)→i P ′, P ′ ∈ C|}. P and Q are strongly bisimilar,

denoted P ∼ Q, if (P,Q) is contained in some strong bisimulation R.

Here, {| . . . |} denotes a multiset. It follows that ∼ is the largest strong bisimula-tion, i.e., it contains any strong bisimulation. To be able to compare ICTMCs bybisimulation, let us equip an ICTMC with an initial state s0 ∈ S. Two ICTMCsCP = (SP ,RP , s0

P ) and CQ = (SQ,RQ, s0Q) are bisimilar, denoted CP ∼ CQ, iff

their initial states are bisimilar, i.e., s0P ∼ s0

Q. The quotient of an ICTMC under∼ is defined in the following way.

Definition 5 (Bisimulation quotient). For the ICTMC C = (S,R, s0) and∼, the quotient C/∼ is defined by C/∼= (S/∼,R∼, s0

∼) where s0∼ = [s0]∼ and R∼

is defined by: R∼([P ]∼, [P ′]∼, t) = R(P, [P ′]∼, t) for all t ∈ IR≥0.

Note that C is strongly bisimilar to C/∼. An important property of strong bisim-ulation is that it preserves transient probabilities; in particular, this means thatthere is a strong relationship between the transient probabilities in an ICTMCand its quotient.

Theorem 2. Let C = (S,R, s0) be an ICTMC. For every C ∈ S/∼, the transientprobability distribution πC(t) of the state C in the quotient chain C/∼ is:

πC(t) =∑

s∈C

πs(t) for all t ∈ IR≥0,

11

where πs(t) is the transient probability distribution of state s ∈ S in C.

Proof. The proof is carried out in two steps. First, we characterize the transientprobability distribution by a system of ODEs for the aggregated chain C/∼ interms of equivalence classes C ∈ S/∼. Then by suitable transformation we willderive an equivalent system of ODEs for the original ICTMC with state space S.Note that if πC(t) =

∑

s∈C πs(t), then dπC(t)dt =

∑

s∈Cdπs(t)

dt due to the linearityof the differentiation operator. The system of ODEs for C/∼ is:

dπC1(t)

dt =∑

Cv∈S/∼ R∼(Cv , C1, t)πCv (t) −∑

Ck∈S/∼ R∼(C1, Ck, t)πC1(t)...dπCi

(t)

dt =∑

Cv∈S/∼ R∼(Cv, Ci, t)πCv (t) −∑

Ck∈S/∼ R∼(Ci, Ck, t)πCi(t)

...dπCn (t)

dt =∑

Cv∈S/∼ R∼(Cv, Cn, t)πCv (t) −∑

Ck∈S/∼ R∼(Cn, Ck, t)πCn(t)

where R∼(Cv, Ci, t) is the time-dependent rate between classes Cv and Ci in C/∼.The equation:

dπCi(t)

dt=

∑

Cv∈S/∼

R∼(Cv , Ci, t)πCv (t) −∑

Ck∈S/∼

R∼(Ci, Ck, t)πCi(t)

shows that the change of probability mass in class Ci is equal to the differencebetween the in-flow and the out-flow for the state Ci. We assume that the rateR∼(Ci, Ci, t) = 0 for every Ci ∈ S/∼, as for ICTMCs the presence of such ratedoes not influence the transient probability distribution of the class Ci. Theout-flow can be rewritten using Definition 5 as:

∑

Ck∈S/∼

R∼(Ci, Ck, t)πCi(t) = πCi

(t)∑

Ck∈S/∼

R(s(i)m , Ck, t)

here m = |Ci| and s(i)m is the m’th state of class Ci (in fact the choice of s

(i)m is

arbitrary; any state in Ci could be taken). By substituting πC(t) =∑

s∈C πs(t)in the above equation, we get:(

πs(i)1

(t) + · · · + πs(i)m

(t))

∑

Ck∈S/∼

R(s(i)m , Ck, t) =π

s(i)1

(t)∑

Ck∈S/∼

R(s(i)m , Ck, t) + · · ·

+ πs(i)m

(t)∑

Ck∈S/∼

R(s(i)m , Ck, t).

As all states s(i)m ∈ Ci are bisimilar we have for every equivalence class Ck the

relation R(s(i)1 , Ck, t) = · · · = R(s

(i)m , Ck, t) which gives:

πs(i)1

(t)∑

Ck∈S/∼

R(s(i)1 , Ck, t) + · · · + π

s(i)m

(t)∑

Ck∈S/∼

R(s(i)m , Ck, t).

By noting that⋃n

j=1 Cj = S for |S/∼| = n we get the out-flow equation:

∑

Ck∈S/∼

R∼(Ci, Ck, t)πCi(t) = π

s(i)1

(t)R(s(i)1 , S, t) + · · · + π

s(i)m

(t)R(s(i)m , S, t).

12

Now we proceed with the in-flow equation:

∑

Cv∈S/∼

R∼(Cv, Ci, t)πCv (t) = R∼(C1, Ci, t)πC1(t) + · · · + R∼(Cn, Ci, t)πCn(t)

Using Definition 5, the above right-hand sum results in:

R(s(1)v1

, Ci, t)πC1(t) + · · · + R(s(n)vn

, Ci, t)πCn(t),

where vℓ = |Cℓ| and s(ℓ)vℓ

is the vℓ’th states of class Cℓ. By substituting πC(t) =∑

s∈C πs(t) we get:

R(s(1)v1

, Ci, t)(

πs(1)1

(t) + · · · + πs(1)v1

(t))

+ · · · + R(s(n)vn

, Ci, t)(

πs(n)1

(t) + · · ·

+πs(n)vn

(t))

= R(s(1)v1

, Ci, t)πs(1)1

(t) + · · · + R(s(1)v1

, Ci, t)πs(1)v1

(t) + · · ·

+ R(s(n)vn

, Ci, t)πs(n)1

(t) + · · · + R(s(n)vn

, Ci, t)πs(n)n

(t).

As all states s(ℓ) in the equivalence class Cℓ are bisimilar, R(s(ℓ)1 , Ci, t) = · · · =

R(s(ℓ)vℓ

, Ci, t) which yields:

R(s(1)1 , Ci, t)πs

(1)1

(t) + · · · + R(s(1)v1

, Ci, t)πs(1)v1

(t) + · · ·+

R(s(n)1 , Ci, t)πs

(n)1

(t) + · · · + R(s(n)vn

, Ci, t)πs(n)vn

(t) =∑

s∈S

R(s,Ci, t)πs(t) =∑

s∈S

R(s, s(i)1 , t)πs(t) + · · · +

∑

s∈S

R(s, s(i)m , t)πs(t).

Combining the equations for the in-flow and out-flow we obtain:

d∑m

j=1 πs(i)j

(t)

dt=

∑

s∈S

R(s, s(i)1 , t)πs(t) + · · · +

∑

s∈S

R(s, s(i)m , t)πs(t)−

πs(i)1

(t)R(s(i)1 , S, t) − · · · − π

s(i)m

(t)R(s(i)m , S, t).

By linearity of the differential operator for the equivalence class Ci we obtain mequations:

dπs(i)1

(t)

dt =∑

s∈SR(s, s

(i)1 , t)πs(t) − π

s(i)1

(t)R(s(i)1 , S, t)

...dπ

s(i)ℓ

(t)

dt =∑

s∈SR(s, s

(i)ℓ , t)πs(t) − π

s(i)ℓ

(t)R(s(i)ℓ , S, t)

...dπ

s(i)m

(t)

dt =∑

s∈SR(s, s

(i)m , t)πs(t) − π

s(i)m

(t)R(s(i)m , S, t)

.

Forming a system of ODEs from the equations of all equivalence classes Ci ∈ S/∼we obtain the transient probability distribution for the original ICTMC.

From Theorem 2 we may conclude that the steady state probability distribu-tion (if it exists) is also preserved.

13

Corollary 1. Let C = (S,R, s0) be an ICTMC. For every C ∈ S/∼, the steady-state probability distribution πC of the state C in the quotient chain C/∼ is:

πC = limt→∞

πC(t) = limt→∞

∑

s∈C

πs(t) =∑

s∈C

πs,

where πs is the steady-state probability distribution of state s ∈ S.

In many cases it is reasonable to assume that two processes P and Q areequal up to time T . For this case we propose the finite-horizon bisimulation.

Definition 6. An equivalence R ⊆ S × S is a finite-horizon bisimulation when-ever for all (P,Q) ∈ R, t ∈ [0, T ] (T ∈ IR≥0) and C ∈ S/R:

R(P,C, t) = R(Q,C, t).

P and Q are finitely-horizon bisimilar, denoted P ∼T Q, if (P,Q) is containedin some finite-horizon bisimulation R.

Notice that the definition of finite-horizon bisimulation ∼T is the same exceptthat the time t lies in the interval [0, T ]. It is easy to see that finite-horizonbisimulation preserves the transient distribution up to time T .

Proposition 1. For 0 < t1 < · · · < T < · · · < ∞ it holds:

∼0⊆ ∼t1⊆ · · · ⊆ ∼T · · · ⊆ ∼∞=∼

Thus , P ∼ti Q implies P ∼tj Q for every tj < ti. It follows that for tj < ti, thequotient under ∼tj is coarser than under ∼ti .

Bisimulation for I2MCs. So far, we have presented bisimulation for ICTMCs.In order to define bisimulation for I2MCs, unobservable actions (i.e., τ) requirespecial care. Consider the following diagram:

P02λ(t)

//

τ

A

A

A

A

A

A

A

A

P1

∼

P2

Q0λ(t)

//

λ(t)

B

B

B

B

B

B

B

B

Q1

∼

Q2

where P1 ∼ P2 ∼ Q1 ∼ Q2. At first sight, it seems natural to consider P0 ∼ Q0

as R(P0, C, t) = R(Q0, C, t) = 2λ(t). But, state P0 can do something more.There is a transition P0

τ→ P2 which consumes no time as a τ -action is an

internal one and is not prevented by the environment (maximal progress as-

sumption). The probability that transition P02λ(t)−→ P1 will be taken in 0 time

units is PrP0,P1(t, 0) =∫ 00 2λ(t + τ)e−

R τ

02λ(t+ℓ)dℓdτ = 0. Thus, we may conclude

that P0 ≁ Q0. When specifying the definition of bisimilarity we have to treatimmediate actions (τ) in a special way. Let S be the state-space of an I2MC.

Definition 7 (I2MC strong bisimulation). An equivalence R ⊆ S × S is astrong bisimulation whenever for all (P,Q) ∈ R, t ∈ IR≥0, a ∈ Act and C ∈ S/R:

– Pa

−→ P ′ implies Qa

−→ Q′ for some Q′ and (P ′, Q′) ∈ R.– Q

a−→ Q′ implies P

a−→ P ′ for some P ′ and (P ′, Q′) ∈ R.

14

P + 0 = P (P + Q) + R = P + (Q + R)

P + Q = Q + P λ(t).P + µ(t).P = (λ(t) + µ(t)).P

a.P + a.P = a.P λ(t).P + τ.Q = τ.Q

Table 2. Sound and complete axioms for ∼ on the I2ML sequential fragment.

– Pτ9 (or Q

τ9) implies R(P,C, t) = R(Q,C, t).

P and Q are strongly bisimilar, denoted P ∼ Q, if (P,Q) is contained in somestrong bisimulation R.

Example 2. Consider the I2MC from Fig. 2 (c) and λ1(t) = µ1(t), λ2(t) = µ2(t).Its quotient under bisimulation is depicted in Fig. 3. The equivalence classesC1, C2 and C3 contain the following states C1 =

{

0‖{use}Q,P‖{use}0}

, C2 ={

P‖{use}use.Q, use.P‖{use}Q}

and C3 ={

0‖{use}use.Q, use.P‖{use}0, 0‖{use}0}

.

P‖{use}QC1

2λ1(t)

use.P‖{use}use.QC3

use

C2

λ2(t)

2λ2(t)

λ1(t) + λ2(t)

Fig. 3. Bisimulation quotient.

In a similar way as for ICTMCs,one can consider the quotient ofan I2MC. The compositional natureof I2MC, however, allows in princi-ple for obtaining such quotient ina component-wise manner, e.g., thequotient of P‖AQ can be obtainedby first constructing the quotients of

P and Q, then combine them, and quotienting the composition. The necessaryrequirement that ∼ needs to fulfill is that it is a congruence relation. The rela-tion ∼ is a congruence whenever for processes P and Q it holds: P ∼ Q impliesC[P ] ∼ C[Q] where C[·] is any context. (A context is basically a process termcontaining a hole that may be filled with any process.)

Theorem 3. ∼ is a congruence with respect to all operators in I2ML.

Finite-horizon bisimulation is a congruence with one additional property.

Proposition 2. For any processes P , P ′, Q, Q′ and intervals [0, T1] and [0, T2]with T1, T2 ∈ R≥0 we have:

P ∼T1 P ′ and Q ∼T2 Q′ implies P‖AQ ∼min(T1,T2) P ′‖AQ′ for all A ⊆ Act.

As a next step, we consider the possibility to establish bisimulation symbol-ically, i.e., on the level of the syntax of the earlier introduced language I2ML.This is facilitated by an axiomatization for ∼. The soundness of these axiomsensures that any two terms that are syntactically equal (denoted =) are bisimi-lar; formally, P = Q ⇒ P ∼ Q. Whenever the axioms are complete, in addition,any strongly bisimilar processes can be represented by the same expressions inI2ML, i.e., P ∼ Q ⇒ P = Q. Summarizing, any bisimulation can be establishedsyntactically, i.e., by just manipulating terms rather than I2MCs, provided theaxiom system is sound and complete. Let A∼ be the set of axioms listed in Table

15

2 extended with the expansion law:

P‖AQ =∑

i∈J1

λi(t). (Pi‖AQ) +∑

k∈J3

µk(t). (P‖AQk) +∑

aj=bl∈A

aj . (Pj‖AQl)+

+∑

aj /∈A∧aj∈J2

aj. (Pj‖AQ) +∑

bl /∈A∧bl∈J4

bl. (P‖AQl)

where P :=∑

i∈J1λi(t).Pi +

∑

j∈J2aj .Pj and Q :=

∑

k∈J3µk(t).Qk +

∑

l∈J4bl.Ql

with the finite index sets J1, J2, J3 and J4. Then the following holds:

Theorem 4. For any P,Q ∈ RG, A∼ ⊢ P = Q if and only if P ∼ Q.

The term RG denotes the set of all regular (no parallel composition inside re-cursion) and guarded expressions. While A∼ ⊢ P = Q means that P = Q canbe deduced from the set of sound and complete axiom system A∼. The axiomλ(t).P + µ(t).P = (λ(t) + µ(t)).P is due to the fact that the sum of two Poissonprocesses with rates λ(t) and µ(t) is a Poisson process with the rate λ(t) + µ(t),whereas the axiom λ(t).P + τ.Q = τ.Q is due to the maximal progress assump-tion. Notice that A∼ also contains all standard axioms which involve hiding andrecursion operators which are standard and omitted here.

Bisimulation minimization. The previous sections have set the stage for bisim-ulation minimization. Experiments have shown that in the traditional [FV98]as well as in the stochastic setting [KKZJ07] exponential state space savingscan be achieved. Given that ∼ is a congruence, individual processes can be re-placed by their bisimilar quotient (under ∼) and the peak memory requirementscan be reduced significantly. This all, however, requires an efficient bisimulationminimization algorithm. We adopt the partition-refinement paradigm to obtain aminimization algorithm for I2MCs. As the problem for arbitrary rate functions isundecidable, we restrict to three classes of rate matrices R(t): piecewise uniform,polynomial (R(t) = tM+1RM+1 + · · ·+ tR2 +R1, where Ri with i ≤ M +1 ∈ INare constant matrices) and piecewise polynomial (each piece is a polynomial ofdegree three). The same classes have been considered for the transient probabil-ity distribution, cf. Theorem 1. Rate comparisons and summations can easily berealized for these classes of functions. For rate matrix R, let M + 1 denote thetotal number of intervals for piecewise uniform R(t), the polynomial degree whenR(t) is polynomial, and the number of polynomial pieces when R(t) is piecewisepolynomial.

Our bisimulation minimization algorithm for I2MCs is based on a generaliza-tion of the algorithm for obtaining the coarsest quotient of a Markov chain underbisimulation by Derisavi et al. [DHS03], and Paige-Tarjan’s algorithm for LTS.The basic idea is to minimize iteratively over all pieces (or degrees of the poly-nomials). The bisimulation algorithm exploits an efficient data structure whichgroups all states with the same outgoing rate. This is in fact a binary tree whereeach node has four parameters: node.left and node.right - pointers to the leftand right child, respectively, node.sum - stores the sum of the rates and node.S- stores all states with the same node.sum. Using such data structures, the time-and space complexity of bisimulation minimization for I2MCs reduces to:

Theorem 5. The coarsest quotient under ∼ of any I2MC can be obtained ina worst-case time complexity O (ma lg(n) + Mmr lg(n)) and space complexity

16

O (ma + mr), where ma and mr is the number of action-labeled and rate-labeledtransitions, respectively.

Proof. The base algorithm for I2MC bisimulation quotient computation basedon partition refinement techniques is presented in Algorithm 1.

Algorithm 1 Bisimulation quotient computation for I2MCs.

Require: I2MC (S,Act,→,R), MEnsure: I2MC (S/∼,Act,→∼,R∼)1: Πτ := {s ∈ S|s

τ→} Π := {s ∈ S|s

τ9}

2: L := push({S}) {#Create the stack of splitters}3: while L 6= ∅ do

4: C := pop(L) {#Extract a splitter from L}5: [Π, L] := Refinea(Π,C, L)6: [Πτ , L] := Refinea(Πτ , C, L)7: [Π, L] := Refiner(Π,C, L,R, M)8: end while

9: Π := Π ∪ Πτ

10: for all B ∈ Π do

11: let s ∈ B {#Choose any state from block B}12: for all p ∈ Post(s) do

13: B′ := [p]Π

14: if (s, a, p) ⊆→ then

15: (B, a, B′) ⊆→∼ {#Transition from B to B′ labeled with a}16: end if

17: for i = 0 to M do

18: R∼(B, B′, i) := R∼(B, B′, i) + R(s, p, i)19: end for

20: end for

21: end for

22: return [Π,→∼,R∼ ]

The variables and functions used in Algorithm 1 have the following meaning:

– M - is the rate matrix R(t) parameter.

– Π - is the set of all states with no outgoing τ -transition.

– Πτ - is the set of all states with at least one outgoing τ -transition.

– L - is the stack of all current splitters.

– C - is the current splitter which splits the partition Π and Πτ .

– Refinea(Π,C,L) - is the function which refines partition Π using the currentsplitter C with respect to action-labeled transitions. It returns the refinedpartition and a new set of splitters.

– Refiner(Π,C,L,R,M) - is the function which refines partition Π using thecurrent splitter C with respect to rate-labeled transitions. It returns the re-fined partition and a new set of splitters.

– [s]Π - is the block of state s in partition Π.

The Algorithm 1 starts with two initial partitions Π and Πτ (line 1). Then itsuccessively refines them by using the functions Refinea and Refiner (line 5-7). Itstops when there are no more splitters. The lines 10-21 have the role of creatingthe rate matrix R∼ for the quotient state-space S/∼. The function Refinea canbe implemented using the adaptation of Fernandez (see [Fer89]) to Paige and

17

Tarjan’s algorithm (see [PT87]) or the more efficient implementation of Dovier etal. [DPP04]. The time and space complexity is of order O (ma lg n) and O (ma),respectively, where ma is the total number of action-labeled transitions. It isimportant to notice that in function Refinea as well as Refiner the splitter withthe largest size will be deleted from the set of potential splitters. This will give atotal of lg(n+1) iterations for the cycle 3-8. The pseudo code of function Refiner

is presented in Algorithm 2. The function Refiner uses the following variables

Algorithm 2 Refiner(Π,C,L,R,M)Require: Π,C, L, R, MEnsure: Π,L1: for i = 0 to M do

2: BΠ := ∅

3: for all p ∈ C do

4: for all s ∈ Pre(p) do

5: s.sum := 06: end for

7: end for

8: for all p ∈ C do

9: for all s ∈ Pre(p) do

10: s.sum := s.sum + R(s, p, i)11: end for

12: end for

13: for all s ∈ Pre(C) do

14: B′ := [s]Π

15: delete s from B′

16: Insert(T(B′), s)17: BΠ := BΠ ∪ {B′}18: end for

19: for all B′ ∈ BΠ do

20: Π := Π ∪ {T(B′)[1], . . . , T(B′)[k]}21: end for

22: end for

23: for all B′ ∈ BΠ do

24: Bmax := largest block of {B′, T(B′)[1], . . . , T(B′)[k]}25: push (L, {B′, T(B′)[1], . . . , T(B′)[k]} − {Bmax})26: end for

27: return [Π, L]

and functions:

– Pre(C) - is the set of predecessor states of splitter C.

– BΠ - is a set of blocks in partition Π.

– R(s, p, i) - the rate from state s to state p with parameter i (interval, poly-nomial coefficient, etc).

– T(B′) - binary tree of block B′, with T(B′)[j] being the j’th sub-block of treeT(B′).

– Insert(T(B′), s) - the function inserts the state s ∈ B′ into the binary treeT(B′) of block B′.

The Algorithm 2 has the same functionality as Split function from Derisavi et al.(see [DHS03]). The crucial difference is the line 1 which is a cycle over all time in-tervals or polynomial coefficients and the lines 19-21 which after each refinement

18

create a new partition, where the new refined blocks are {T(B′)[1], . . . ,T(B′)[k]}.As the time and space complexity of Derisavi’s base algorithm (the algorithmwhich calls the function Split) are bounded by O (mr lg n) and O (mr), respec-tively, where mr is the total number of rate-labeled transitions. We obtain inour case that the invocation of the base Algorithm 2 with Refiner function onlywill result in a time and space complexity of O (Mmr lg n) and O (mr), respec-tively. Considering also Refinea function we obtain that the total time and spacecomplexity of the bisimulation quotienting algorithm are O (ma lg n + Mmr lg n)and O (ma + mr).

Recall that ICTMCs are I2MCs that contain no action-labeled transitions. Asa side result, the above theorem yields that the coarsest bisimulation quotient ofa time-inhomogeneous CTMC can be obtained with time and space complexityO (Mmr lg(n)) and O (mr), respectively. (The time complexity for homogeneousMarkov chains is O (mr lg(n)) [DHS03]). Given the results in this paper that ∼preserves transient and steady state distributions, our algorithm can be used tominimize prior to any such analysis.

Weak bisimulation for I2MCs. Strong bisimulation requires equivalent states tosimulate their mutual stepwise behavior. While preserving the branching struc-ture, strong bisimulation also requires mimicking of immediate actions (τ). Asimmediate actions consume no time it seems reasonable that two states will beequivalent regardless of the number of τ -steps in a sequence that they make.Therefore, the equivalence which will allow for the abstraction of sequences ofimmediate actions will be denoted as weak bisimulation. Let the transition

τ=⇒ be

the reflexive and transitive closure ofτ

−→∗

anda

=⇒ a shorthand forτ

=⇒a

−→τ

=⇒(a 6= τ).

Definition 8 (I2MC weak bisimulation). An equivalence R ⊆ S × S is aweak bisimulation whenever for all (P,Q) ∈ R, t ∈ IR≥0, a ∈ Act and C ∈ S/R:

– Pa

−→ P ′ implies Qa

=⇒ Q′ for some Q′ and (P ′, Q′) ∈ R.– P

τ9 implies R(P,C, t) = R(Q′′, C, t) for some Q′′ τ

9 such that Qτ

=⇒ Q′′

and (P,Q′′) ∈ R.

For Q symmetric rules apply. P and Q are weakly bisimilar, denoted P ≈ Q, if(P,Q) is contained in some weak bisimulation R.

It seems intuitive that for the sequence Qτ

=⇒ Q′′ the rates R(P,C, t) andR(Q′′, C, t) have to be compared starting from time t′ = t + ∆t where ∆t isthe time needed to make all τ in the sequence Q

τ=⇒ Q′′. As τ transitions take

no time the result will be the same even when the rates are compared from timet.

Example 3. Consider the I2MC from Fig. 3 and its abstraction i.e. all actions aretransformed into immediate ones (τ). The quotient under ≈ is depicted in Fig. 4,with C1, C2 and C3 as in Fig. 3 and C0 = {P‖{use}Q, use.P‖{use}use.Q}. It isimportant to note that after abstraction the transition labeled with use resultsin an immediate transition which gives the possibility to put the states P‖{use}Qand use.P‖{use}use.Q in the same equivalence class. Also note that the obtainedI2MC has no transitions labeled with actions, i.e., it is an ICTMC. This shows

19

a.τ.P = a.P a.(P + τ.Q) + a.Q = a.(P + τ.Q)

P + τ.P = τ.P λ(t).τ.P = λ(t).P

Table 3. Sound and complete axioms for ⋍ on the I2ML sequential fragment.

that weak bisimulation may be an effective mechanism to turn an I2MC into anICTMC, which may be subject to analysis as discussed in Section 2.

C0C12λ1(t)

C3 C2

λ1(t) + λ2(t) 2λ2(t)λ2(t)

Fig. 4. Weak bisimulation quotient.

As in the case of strong bisimulation,weak bisimulation is also a congruencewith respect to I2ML operators. But thereis an exception. Weak bisimulation is nota congruence with respect to the choice(P + Q) operator [Mil89]. This is due tothe fact that weak bisimulation will equate

two processes whenever one can doτ

=⇒ and the other one can do nothing. Inorder to cope with the choice operator one has to differentiate between

a=⇒ and

τ=⇒

a−→

τ=⇒ when a = τ as follows:

Definition 9 (Weak congruence). Pand Q are weakly congruent, denoted byP ⋍ Q, whenever for all a ∈ Act, t ∈ IR≥0 and C ∈ RG/≈:

– Pa

−→ P ′ implies Qτ

=⇒a

−→τ

=⇒ Q′ for some Q′ and P ′ ≈ Q′.

– Qa

−→ Q′ implies Pτ

=⇒a

−→τ

=⇒ P ′ for some P ′ and P ′ ≈ Q′.

– Pτ9 (or Q

τ9) implies R(P,C, t) = R(Q,C, t).

Theorem 6. ⋍ is a congruence with respect to all operators in I2ML.

Consider the set of axioms from Table 2 and 3 together with axioms relatedto hiding and recursion operators as A⋍. As for strong bisimulation the followingalso holds for weak congruence:

Theorem 7. For any P,Q ∈ RG, A⋍ ⊢ P = Q if and only if P ⋍ Q.

Recall that P and Q are regular and guarded process terms.

5 Concluding Remarks and Future Work

This paper presented a compositional formalism for time-inhomogeneous continu-ous-time Markov chains (ICTMCs), a subclass of piecewise deterministic Markovprocesses (PDPs). The main contributions are a full-fledged process algebra forinteractive ICTMCs, congruence results for weak and strong bisimulation, anda polynomial-time quotienting algorithm. In addition, a new characterization oftransient probabilities is provided for rate functions that are piecewise uniform.In contrast to works on communicating PDPs [SvdS06,SJvdS03,SvdS05], thispaper considers weak bisimulation, congruence results and axiomatization, and,more importantly a notion of bisimulation which respects maximal progress.Current work consists of investigating improvements to the quotienting algorithmakin to [DPP04], model-checking algorithms [Kat06], and simulation relations forICTMCs.

20

Acknowledgment. This research has been performed as part of QUPES projectthat is financed by the Netherlands Organization for Scientific Research (NWO).

References

[AGH+00] Rajeev Alur, Radu Grosu, Yerang Hur, Vijay Kumar, and Insup Lee. ModularSpecification of Hybrid Systems in CHARON. In Nancy A. Lynch and Bruce H.Krogh, editors, HSCC, volume 1790 of Lecture Notes in Computer Science, pages6–19. Springer, 2000.

[AGLS06] Rajeev Alur, Radu Grosu, Insup Lee, and Oleg Sokolsky. Compositional modelingand refinement for hierarchical hybrid systems. J. Log. Algebr. Program., 68(1-2):105–128, 2006.

[BLB05] Manuela L. Bujorianu, John Lygeros, and Marius C. Bujorianu. Bisimulation forGeneral Stochastic Hybrid Systems. In Morari and Thiele [MT05], pages 198–214.

[Buc94] Peter Buchholz. Exact and ordinary lumpability in finite Markov chains. Journalof Applied Probability, 31:59–75, 1994.

[CJH07] Lucia Cloth, Marijn R. Jongerden, and Boudewijn R. Haverkort. Computing BatteryLifetime Distributions. In DSN, pages 780–789. IEEE Computer Society, 2007.

[CL06] Christos G. Cassandras and John Lygeros, editors. Stochastic Hybrid Systems:Recent Developments and Research Trends, In Christos G. Cassandras, and JohnLygeros, editors, Stochastic Hybrid Systems: Recent Developments and ResearchTrends. CRC Press, 2006.

[Dav93] Mark H. A. Davis. Markov Models and Optimization. Chapman and Hall, 1993.[DHS03] Salem Derisavi, Holger Hermanns, and William H. Sanders. Optimal state-space

lumping in Markov chains. Inf. Process. Lett., 87(6):309–315, 2003.[DPP04] Agostino Dovier, Carla Piazza, and Alberto Policriti. An efficient algorithm for

computing bisimulation equivalence. Theor. Comput. Sci., 311(1-3):221–256, 2004.[EB05] Mariken Everdij and Henk Blom. Piecewise deterministic Markov processes repre-

sented by dynamically coloured Petri nets. Stochastics, 77(1):1–29, 2005.

[Fer89] Jean-Claude Fernandez. An Implementation of an Efficient Algorithm for Bisimu-lation Equivalence. Sci. Comput. Program., 13(1):219–236, 1989.

[FV98] Kathi Fisler and Moshe Y. Vardi. Bisimulation Minimization in an Automata-Theoretic Verification Framework. In Ganesh Gopalakrishnan and Phillip J. Wind-ley, editors, FMCAD, volume 1522 of Lecture Notes in Computer Science, pages115–132. Springer, 1998.

[Her02] Holger Hermanns. Interactive Markov Chains: The Quest for Quantified Quality,volume 2428 of Lecture Notes in Computer Science. Springer, 2002.

[HHK02] Holger Hermanns, Ulrich Herzog, and Joost-Pieter Katoen. Process algebra forperformance evaluation. Theor. Comput. Sci., 274(1-2):43–87, 2002.

[Hoa85] Charles A. R. Hoare. Communicating sequential processes. Prentice-Hall, Inc., UpperSaddle River, NJ, USA, 1985.

[HTP05] Esfandiar Haghverdi, Paulo Tabuada, and George J. Pappas. Bisimulation relationsfor dynamical, control, and hybrid systems. Theor. Comput. Sci., 342(2-3):229–261,2005.

[Kai80] Thomas Kailath. Linear Systems. Prentice-Hall, Inc., 1980.

[Kat06] Joost-Pieter Katoen. Stochastic model checking. In Cassandras and Lygeros [CL06],pages 79–106.

[KKZJ07] Joost-Pieter Katoen, Tim Kemna, Ivan S. Zapreev, and David N. Jansen. Bisim-ulation Minimisation Mostly Speeds Up Probabilistic Model Checking. In OrnaGrumberg and Michael Huth, editors, TACAS, volume 4424 of Lecture Notes inComputer Science, pages 87–101. Springer, 2007.

[Mil89] Robin J. Milner. Communication and Concurrency. Prentice-Hall, Inc., 1989.[MT05] Manfred Morari and Lothar Thiele, editors. Hybrid Systems: Computation and

Control, 8th International Workshop, HSCC 2005, Zurich, Switzerland, March 9-11, 2005, Proceedings, volume 3414 of Lecture Notes in Computer Science. Springer,2005.

[PT87] Robert Paige and Robert Endre Tarjan. Three Partition Refinement Algorithms.SIAM J. Comput., 16(6):973–989, 1987.

21

[RWVT95] Andy Rindos, Steven Woolet, Ioannis Viniotis, and Kishor S. Trivedi. Exact meth-ods for the transient analysis of non-homogeneous continuous-time Markov chains.In W. J. Stewart (ed.), Numerical Solution of Markov Chains (NSMC), pages 121–134. Kluwer Academic Publishers, 1995.

[SJvdS03] Stefan N. Strubbe, Agung A. Julius, and A. J. van der Schaft. Communicatingpiecewise deterministic Markov processes. In IFAC Conf. on Analysis and Designof Hybrid Systems (ADHS), pages 349–354, 2003.

[SvdS05] Stefan N. Strubbe and A. J. van der Schaft. Bisimulation for Communicating Piece-wise Deterministic Markov Processes (CPDPs). In Morari and Thiele [MT05], pages623–639.

[SvdS06] Stefan N. Strubbe and A. J. van der Schaft. Compositional modelling of stochastichybrid systems. In Cassandras and Lygeros [CL06], pages 47–77.

22

Aachener Informatik-Berichte

This is the list of all technical reports since 1987. To obtain copies of reports

please consult

http://aib.informatik.rwth-aachen.de/ or send your request to: Informatik-

Bibliothek, RWTH Aachen, Ahornstr. 55, 52056 Aachen, Email: biblio@informatik.rwth-aachen.de

1987-01 ∗ Fachgruppe Informatik: Jahresbericht 1986

1987-02 ∗ David de Frutos Escrig, Klaus Indermark: Equivalence Relations of Non-

Deterministic Ianov-Schemes

1987-03 ∗ Manfred Nagl: A Software Development Environment based on Graph

Technology

1987-04 ∗ Claus Lewerentz, Manfred Nagl, Bernhard Westfechtel: On Integration

Mechanisms within a Graph-Based Software Development Environment

1987-05 ∗ Reinhard Rinn: Uber Eingabeanomalien bei verschiedenen Inferenzmod-

ellen

1987-06 ∗ Werner Damm, Gert Dohmen: Specifying Distributed Computer Archi-

tectures in AADL*

1987-07 ∗ Gregor Engels, Claus Lewerentz, Wilhelm Schafer: Graph Grammar En-

gineering: A Software Specification Method

1987-08 ∗ Manfred Nagl: Set Theoretic Approaches to Graph Grammars

1987-09 ∗ Claus Lewerentz, Andreas Schurr: Experiences with a Database System

for Software Documents

1987-10 ∗ Herbert Klaeren, Klaus Indermark: A New Implementation Technique

for Recursive Function Definitions

1987-11 ∗ Rita Loogen: Design of a Parallel Programmable Graph Reduction Ma-

chine with Distributed Memory

1987-12 J. Borstler, U. Moncke, R. Wilhelm: Table compression for tree automata

1988-01 ∗ Gabriele Esser, Johannes Ruckert, Frank Wagner Gesellschaftliche As-

pekte der Informatik

1988-02 ∗ Peter Martini, Otto Spaniol: Token-Passing in High-Speed Backbone

Networks for Campus-Wide Environments

1988-03 ∗ Thomas Welzel: Simulation of a Multiple Token Ring Backbone

1988-04 ∗ Peter Martini: Performance Comparison for HSLAN Media Access Pro-

tocols

1988-05 ∗ Peter Martini: Performance Analysis of Multiple Token Rings

1988-06 ∗ Andreas Mann, Johannes Ruckert, Otto Spaniol: Datenfunknetze

1988-07 ∗ Andreas Mann, Johannes Ruckert: Packet Radio Networks for Data Ex-

change

1988-08 ∗ Andreas Mann, Johannes Ruckert: Concurrent Slot Assignment Protocol

for Packet Radio Networks

1988-09 ∗ W. Kremer, F. Reichert, J. Ruckert, A. Mann: Entwurf einer Netzw-

erktopologie fur ein Mobilfunknetz zur Unterstutzung des offentlichen

Straßenverkehrs

1988-10 ∗ Kai Jakobs: Towards User-Friendly Networking

1988-11 ∗ Kai Jakobs: The Directory - Evolution of a Standard

1988-12 ∗ Kai Jakobs: Directory Services in Distributed Systems - A Survey

23

1988-13 ∗ Martine Schummer: RS-511, a Protocol for the Plant Floor

1988-14 ∗ U. Quernheim: Satellite Communication Protocols - A Performance

Comparison Considering On-Board Processing

1988-15 ∗ Peter Martini, Otto Spaniol, Thomas Welzel: File Transfer in High Speed

Token Ring Networks: Performance Evaluation by Approximate Analysis

and Simulation

1988-16 ∗ Fachgruppe Informatik: Jahresbericht 1987

1988-17 ∗ Wolfgang Thomas: Automata on Infinite Objects

1988-18 ∗ Michael Sonnenschein: On Petri Nets and Data Flow Graphs

1988-19 ∗ Heiko Vogler: Functional Distribution of the Contextual Analysis in

Block-Structured Programming Languages: A Case Study of Tree Trans-

ducers

1988-20 ∗ Thomas Welzel: Einsatz des Simulationswerkzeuges QNAP2 zur Leis-

tungsbewertung von Kommunikationsprotokollen

1988-21 ∗ Th. Janning, C. Lewerentz: Integrated Project Team Management in a

Software Development Environment

1988-22 ∗ Joost Engelfriet, Heiko Vogler: Modular Tree Transducers

1988-23 ∗ Wolfgang Thomas: Automata and Quantifier Hierarchies

1988-24 ∗ Uschi Heuter: Generalized Definite Tree Languages

1989-01 ∗ Fachgruppe Informatik: Jahresbericht 1988

1989-02 ∗ G. Esser, J. Ruckert, F. Wagner (Hrsg.): Gesellschaftliche Aspekte der

Informatik

1989-03 ∗ Heiko Vogler: Bottom-Up Computation of Primitive Recursive Tree

Functions

1989-04 ∗ Andy Schurr: Introduction to PROGRESS, an Attribute Graph Gram-

mar Based Specification Language

1989-05 J. Borstler: Reuse and Software Development - Problems, Solutions, and

Bibliography (in German)

1989-06 ∗ Kai Jakobs: OSI - An Appropriate Basis for Group Communication?

1989-07 ∗ Kai Jakobs: ISO’s Directory Proposal - Evolution, Current Status and

Future Problems

1989-08 ∗ Bernhard Westfechtel: Extension of a Graph Storage for Software Doc-

uments with Primitives for Undo/Redo and Revision Control

1989-09 ∗ Peter Martini: High Speed Local Area Networks - A Tutorial

1989-10 ∗ P. Davids, Th. Welzel: Performance Analysis of DQDB Based on Simu-

lation

1989-11 ∗ Manfred Nagl (Ed.): Abstracts of Talks presented at the WG ’89 15th

International Workshop on Graphtheoretic Concepts in Computer Sci-

ence

1989-12 ∗ Peter Martini: The DQDB Protocol - Is it Playing the Game?

1989-13 ∗ Martine Schummer: CNC/DNC Communication with MAP

1989-14 ∗ Martine Schummer: Local Area Networks for Manufactoring Environ-

ments with hard Real-Time Requirements

1989-15 ∗ M. Schummer, Th. Welzel, P. Martini: Integration of Field Bus and

MAP Networks - Hierarchical Communication Systems in Production

Environments

1989-16 ∗ G. Vossen, K.-U. Witt: SUXESS: Towards a Sound Unification of Ex-

tensions of the Relational Data Model

24

1989-17 ∗ J. Derissen, P. Hruschka, M.v.d. Beeck, Th. Janning, M. Nagl: Integrat-

ing Structured Analysis and Information Modelling

1989-18 A. Maassen: Programming with Higher Order Functions

1989-19 ∗ Mario Rodriguez-Artalejo, Heiko Vogler: A Narrowing Machine for Syn-

tax Directed BABEL

1989-20 H. Kuchen, R. Loogen, J.J. Moreno Navarro, M. Rodriguez Artalejo:

Graph-based Implementation of a Functional Logic Language

1990-01 ∗ Fachgruppe Informatik: Jahresbericht 1989

1990-02 ∗ Vera Jansen, Andreas Potthoff, Wolfgang Thomas, Udo Wermuth: A

Short Guide to the AMORE System (Computing Automata, MOnoids

and Regular Expressions)

1990-03 ∗ Jerzy Skurczynski: On Three Hierarchies of Weak SkS Formulas

1990-04 R. Loogen: Stack-based Implementation of Narrowing

1990-05 H. Kuchen, A. Wagener: Comparison of Dynamic Load Balancing Strate-

gies

1990-06 ∗ Kai Jakobs, Frank Reichert: Directory Services for Mobile Communica-

tion

1990-07 ∗ Kai Jakobs: What’s Beyond the Interface - OSI Networks to Support

Cooperative Work

1990-08 ∗ Kai Jakobs: Directory Names and Schema - An Evaluation

1990-09 ∗ Ulrich Quernheim, Dieter Kreuer: Das CCITT - Signalisierungssystem

Nr. 7 auf Satellitenstrecken; Simulation der Zeichengabestrecke

1990-11 H. Kuchen, R. Loogen, J.J. Moreno Navarro, M. Rodriguez Artalejo:

Lazy Narrowing in a Graph Machine

1990-12 ∗ Kai Jakobs, Josef Kaltwasser, Frank Reichert, Otto Spaniol: Der Com-

puter fahrt mit

1990-13 ∗ Rudolf Mathar, Andreas Mann: Analyzing a Distributed Slot Assign-

ment Protocol by Markov Chains

1990-14 A. Maassen: Compilerentwicklung in Miranda - ein Praktikum in funk-

tionaler Programmierung (written in german)

1990-15 ∗ Manfred Nagl, Andreas Schurr: A Specification Environment for Graph

Grammars

1990-16 A. Schurr: PROGRESS: A VHL-Language Based on Graph Grammars

1990-17 ∗ Marita Moller: Ein Ebenenmodell wissensbasierter Konsultationen - Un-

terstutzung fur Wissensakquisition und Erklarungsfahigkeit

1990-18 ∗ Eric Kowalewski: Entwurf und Interpretation einer Sprache zur Beschrei-

bung von Konsultationsphasen in Expertensystemen

1990-20 Y. Ortega Mallen, D. de Frutos Escrig: A Complete Proof System for

Timed Observations

1990-21 ∗ Manfred Nagl: Modelling of Software Architectures: Importance, No-

tions, Experiences

1990-22 H. Fassbender, H. Vogler: A Call-by-need Implementation of Syntax Di-

rected Functional Programming

1991-01 Guenther Geiler (ed.), Fachgruppe Informatik: Jahresbericht 1990

1991-03 B. Steffen, A. Ingolfsdottir: Characteristic Formulae for Processes with

Divergence

1991-04 M. Portz: A new class of cryptosystems based on interconnection net-

works

25

1991-05 H. Kuchen, G. Geiler: Distributed Applicative Arrays

1991-06 ∗ Ludwig Staiger: Kolmogorov Complexity and Hausdorff Dimension

1991-07 ∗ Ludwig Staiger: Syntactic Congruences for w-languages

1991-09 ∗ Eila Kuikka: A Proposal for a Syntax-Directed Text Processing System

1991-10 K. Gladitz, H. Fassbender, H. Vogler: Compiler-based Implementation

of Syntax-Directed Functional Programming

1991-11 R. Loogen, St. Winkler: Dynamic Detection of Determinism in Func-

tional Logic Languages

1991-12 ∗ K. Indermark, M. Rodriguez Artalejo (Eds.): Granada Workshop on the

Integration of Functional and Logic Programming

1991-13 ∗ Rolf Hager, Wolfgang Kremer: The Adaptive Priority Scheduler: A More

Fair Priority Service Discipline

1991-14 ∗ Andreas Fasbender, Wolfgang Kremer: A New Approximation Algorithm

for Tandem Networks with Priority Nodes

1991-15 J. Borstler, A. Zundorf: Revisiting extensions to Modula-2 to support

reusability

1991-16 J. Borstler, Th. Janning: Bridging the gap between Requirements Anal-

ysis and Design

1991-17 A. Zundorf, A. Schurr: Nondeterministic Control Structures for Graph

Rewriting Systems

1991-18 ∗ Matthias Jarke, John Mylopoulos, Joachim W. Schmidt, Yannis Vassil-

iou: DAIDA: An Environment for Evolving Information Systems

1991-19 M. Jeusfeld, M. Jarke: From Relational to Object-Oriented Integrity

Simplification

1991-20 G. Hogen, A. Kindler, R. Loogen: Automatic Parallelization of Lazy

Functional Programs

1991-21 ∗ Prof. Dr. rer. nat. Otto Spaniol: ODP (Open Distributed Processing):

Yet another Viewpoint

1991-22 H. Kuchen, F. Lucking, H. Stoltze: The Topology Description Language

TDL

1991-23 S. Graf, B. Steffen: Compositional Minimization of Finite State Systems

1991-24 R. Cleaveland, J. Parrow, B. Steffen: The Concurrency Workbench: A

Semantics Based Tool for the Verification of Concurrent Systems

1991-25 ∗ Rudolf Mathar, Jurgen Mattfeldt: Optimal Transmission Ranges for Mo-

bile Communication in Linear Multihop Packet Radio Networks

1991-26 M. Jeusfeld, M. Staudt: Query Optimization in Deductive Object Bases

1991-27 J. Knoop, B. Steffen: The Interprocedural Coincidence Theorem

1991-28 J. Knoop, B. Steffen: Unifying Strength Reduction and Semantic Code

Motion

1991-30 T. Margaria: First-Order theories for the verification of complex FSMs

1991-31 B. Steffen: Generating Data Flow Analysis Algorithms from Modal Spec-

ifications

1992-01 Stefan Eherer (ed.), Fachgruppe Informatik: Jahresbericht 1991

1992-02 ∗ Bernhard Westfechtel: Basismechanismen zur Datenverwaltung in struk-

turbezogenen Hypertextsystemen

1992-04 S. A. Smolka, B. Steffen: Priority as Extremal Probability

1992-05 ∗ Matthias Jarke, Carlos Maltzahn, Thomas Rose: Sharing Processes:

Team Coordination in Design Repositories

26

1992-06 O. Burkart, B. Steffen: Model Checking for Context-Free Processes

1992-07 ∗ Matthias Jarke, Klaus Pohl: Information Systems Quality and Quality

Information Systems

1992-08 ∗ Rudolf Mathar, Jurgen Mattfeldt: Analyzing Routing Strategy NFP in

Multihop Packet Radio Networks on a Line

1992-09 ∗ Alfons Kemper, Guido Moerkotte: Grundlagen objektorientierter Daten-

banksysteme

1992-10 Matthias Jarke, Manfred Jeusfeld, Andreas Miethsam, Michael Gocek:

Towards a logic-based reconstruction of software configuration manage-

ment

1992-11 Werner Hans: A Complete Indexing Scheme for WAM-based Abstract

Machines

1992-12 W. Hans, R. Loogen, St. Winkler: On the Interaction of Lazy Evaluation

and Backtracking

1992-13 ∗ Matthias Jarke, Thomas Rose: Specification Management with CAD

1992-14 Th. Noll, H. Vogler: Top-down Parsing with Simultaneous Evaluation on

Noncircular Attribute Grammars

1992-15 A. Schuerr, B. Westfechtel: Graphgrammatiken und Graphersetzungssys-

teme(written in german)

1992-16 ∗ Graduiertenkolleg Informatik und Technik (Hrsg.): Forschungsprojekte

des Graduiertenkollegs Informatik und Technik

1992-17 M. Jarke (ed.): ConceptBase V3.1 User Manual

1992-18 ∗ Clarence A. Ellis, Matthias Jarke (Eds.): Distributed Cooperation in

Integrated Information Systems - Proceedings of the Third International

Workshop on Intelligent and Cooperative Information Systems

1992-19-00 H. Kuchen, R. Loogen (eds.): Proceedings of the 4th Int. Workshop on

the Parallel Implementation of Functional Languages

1992-19-01 G. Hogen, R. Loogen: PASTEL - A Parallel Stack-Based Implementation

of Eager Functional Programs with Lazy Data Structures (Extended

Abstract)

1992-19-02 H. Kuchen, K. Gladitz: Implementing Bags on a Shared Memory MIMD-

Machine

1992-19-03 C. Rathsack, S.B. Scholz: LISA - A Lazy Interpreter for a Full-Fledged

Lambda-Calculus

1992-19-04 T.A. Bratvold: Determining Useful Parallelism in Higher Order Func-

tions

1992-19-05 S. Kahrs: Polymorphic Type Checking by Interpretation of Code

1992-19-06 M. Chakravarty, M. Kohler: Equational Constraints, Residuation, and

the Parallel JUMP-Machine

1992-19-07 J. Seward: Polymorphic Strictness Analysis using Frontiers (Draft Ver-

sion)

1992-19-08 D. Gartner, A. Kimms, W. Kluge: pi-Redˆ+ - A Compiling Graph-

Reduction System for a Full Fledged Lambda-Calculus

1992-19-09 D. Howe, G. Burn: Experiments with strict STG code

1992-19-10 J. Glauert: Parallel Implementation of Functional Languages Using

Small Processes

1992-19-11 M. Joy, T. Axford: A Parallel Graph Reduction Machine

1992-19-12 A. Bennett, P. Kelly: Simulation of Multicache Parallel Reduction

27

1992-19-13 K. Langendoen, D.J. Agterkamp: Cache Behaviour of Lazy Functional

Programs (Working Paper)

1992-19-14 K. Hammond, S. Peyton Jones: Profiling scheduling strategies on the

GRIP parallel reducer

1992-19-15 S. Mintchev: Using Strictness Information in the STG-machine

1992-19-16 D. Rushall: An Attribute Grammar Evaluator in Haskell

1992-19-17 J. Wild, H. Glaser, P. Hartel: Statistics on storage management in a lazy

functional language implementation

1992-19-18 W.S. Martins: Parallel Implementations of Functional Languages

1992-19-19 D. Lester: Distributed Garbage Collection of Cyclic Structures (Draft

version)

1992-19-20 J.C. Glas, R.F.H. Hofman, W.G. Vree: Parallelization of Branch-and-

Bound Algorithms in a Functional Programming Environment

1992-19-21 S. Hwang, D. Rushall: The nu-STG machine: a parallelized Spineless

Tagless Graph Reduction Machine in a distributed memory architecture

(Draft version)

1992-19-22 G. Burn, D. Le Metayer: Cps-Translation and the Correctness of Opti-

mising Compilers

1992-19-23 S.L. Peyton Jones, P. Wadler: Imperative functional programming (Brief

summary)

1992-19-24 W. Damm, F. Liu, Th. Peikenkamp: Evaluation and Parallelization of

Functions in Functional + Logic Languages (abstract)

1992-19-25 M. Kesseler: Communication Issues Regarding Parallel Functional Graph

Rewriting

1992-19-26 Th. Peikenkamp: Charakterizing and representing neededness in func-

tional loginc languages (abstract)

1992-19-27 H. Doerr: Monitoring with Graph-Grammars as formal operational Mod-

els

1992-19-28 J. van Groningen: Some implementation aspects of Concurrent Clean on

distributed memory architectures

1992-19-29 G. Ostheimer: Load Bounding for Implicit Parallelism (abstract)

1992-20 H. Kuchen, F.J. Lopez Fraguas, J.J. Moreno Navarro, M. Rodriguez

Artalejo: Implementing Disequality in a Lazy Functional Logic Language

1992-21 H. Kuchen, F.J. Lopez Fraguas: Result Directed Computing in a Func-

tional Logic Language

1992-22 H. Kuchen, J.J. Moreno Navarro, M.V. Hermenegildo: Independent

AND-Parallel Narrowing

1992-23 T. Margaria, B. Steffen: Distinguishing Formulas for Free

1992-24 K. Pohl: The Three Dimensions of Requirements Engineering

1992-25 ∗ R. Stainov: A Dynamic Configuration Facility for Multimedia Commu-

nications

1992-26 ∗ Michael von der Beeck: Integration of Structured Analysis and Timed

Statecharts for Real-Time and Concurrency Specification

1992-27 W. Hans, St. Winkler: Aliasing and Groundness Analysis of Logic Pro-

grams through Abstract Interpretation and its Safety

1992-28 ∗ Gerhard Steinke, Matthias Jarke: Support for Security Modeling in In-

formation Systems Design

1992-29 B. Schinzel: Warum Frauenforschung in Naturwissenschaft und Technik

28

1992-30 A. Kemper, G. Moerkotte, K. Peithner: Object-Orientation Axiomatised

by Dynamic Logic

1992-32 ∗ Bernd Heinrichs, Kai Jakobs: Timer Handling in High-Performance

Transport Systems

1992-33 ∗ B. Heinrichs, K. Jakobs, K. Lenßen, W. Reinhardt, A. Spinner: Euro-

Bridge: Communication Services for Multimedia Applications

1992-34 C. Gerlhof, A. Kemper, Ch. Kilger, G. Moerkotte: Partition-Based Clus-

tering in Object Bases: From Theory to Practice

1992-35 J. Borstler: Feature-Oriented Classification and Reuse in IPSEN

1992-36 M. Jarke, J. Bubenko, C. Rolland, A. Sutcliffe, Y. Vassiliou: Theories Un-

derlying Requirements Engineering: An Overview of NATURE at Gen-

esis

1992-37 ∗ K. Pohl, M. Jarke: Quality Information Systems: Repository Support for

Evolving Process Models

1992-38 A. Zuendorf: Implementation of the imperative / rule based language

PROGRES

1992-39 P. Koch: Intelligentes Backtracking bei der Auswertung funktional-

logischer Programme

1992-40 ∗ Rudolf Mathar, Jurgen Mattfeldt: Channel Assignment in Cellular Radio

Networks

1992-41 ∗ Gerhard Friedrich, Wolfgang Neidl: Constructive Utility in Model-Based

Diagnosis Repair Systems

1992-42 ∗ P. S. Chen, R. Hennicker, M. Jarke: On the Retrieval of Reusable Soft-

ware Components

1992-43 W. Hans, St.Winkler: Abstract Interpretation of Functional Logic Lan-

guages

1992-44 N. Kiesel, A. Schuerr, B. Westfechtel: Design and Evaluation of GRAS,

a Graph-Oriented Database System for Engineering Applications

1993-01 ∗ Fachgruppe Informatik: Jahresbericht 1992

1993-02 ∗ Patrick Shicheng Chen: On Inference Rules of Logic-Based Information

Retrieval Systems

1993-03 G. Hogen, R. Loogen: A New Stack Technique for the Management of

Runtime Structures in Distributed Environments

1993-05 A. Zundorf: A Heuristic for the Subgraph Isomorphism Problem in Ex-

ecuting PROGRES

1993-06 A. Kemper, D. Kossmann: Adaptable Pointer Swizzling Strategies in

Object Bases: Design, Realization, and Quantitative Analysis

1993-07 ∗ Graduiertenkolleg Informatik und Technik (Hrsg.): Graduiertenkolleg In-

formatik und Technik

1993-08 ∗ Matthias Berger: k-Coloring Vertices using a Neural Network with Con-

vergence to Valid Solutions

1993-09 M. Buchheit, M. Jeusfeld, W. Nutt, M. Staudt: Subsumption between

Queries to Object-Oriented Databases

1993-10 O. Burkart, B. Steffen: Pushdown Processes: Parallel Composition and

Model Checking

1993-11 ∗ R. Große-Wienker, O. Hermanns, D. Menzenbach, A. Pollacks, S. Repet-

zki, J. Schwartz, K. Sonnenschein, B. Westfechtel: Das SUKITS-Projekt:

A-posteriori-Integration heterogener CIM-Anwendungssysteme

29

1993-12 ∗ Rudolf Mathar, Jurgen Mattfeldt: On the Distribution of Cumulated

Interference Power in Rayleigh Fading Channels

1993-13 O. Maler, L. Staiger: On Syntactic Congruences for omega-languages

1993-14 M. Jarke, St. Eherer, R. Gallersdoerfer, M. Jeusfeld, M. Staudt: Con-

ceptBase - A Deductive Object Base Manager

1993-15 M. Staudt, H.W. Nissen, M.A. Jeusfeld: Query by Class, Rule and Con-

cept

1993-16 ∗ M. Jarke, K. Pohl, St. Jacobs et al.: Requirements Engineering: An In-

tegrated View of Representation Process and Domain

1993-17 ∗ M. Jarke, K. Pohl: Establishing Vision in Context: Towards a Model of

Requirements Processes

1993-18 W. Hans, H. Kuchen, St. Winkler: Full Indexing for Lazy Narrowing

1993-19 W. Hans, J.J. Ruz, F. Saenz, St. Winkler: A VHDL Specification of a

Shared Memory Parallel Machine for Babel

1993-20 ∗ K. Finke, M. Jarke, P. Szczurko, R. Soltysiak: Quality Management for

Expert Systems in Process Control

1993-21 M. Jarke, M.A. Jeusfeld, P. Szczurko: Three Aspects of Intelligent Co-

operation in the Quality Cycle

1994-01 Margit Generet, Sven Martin (eds.), Fachgruppe Informatik: Jahres-

bericht 1993

1994-02 M. Lefering: Development of Incremental Integration Tools Using Formal

Specifications

1994-03 ∗ P. Constantopoulos, M. Jarke, J. Mylopoulos, Y. Vassiliou: The Software

Information Base: A Server for Reuse

1994-04 ∗ Rolf Hager, Rudolf Mathar, Jurgen Mattfeldt: Intelligent Cruise Control

and Reliable Communication of Mobile Stations

1994-05 ∗ Rolf Hager, Peter Hermesmann, Michael Portz: Feasibility of Authenti-

cation Procedures within Advanced Transport Telematics

1994-06 ∗ Claudia Popien, Bernd Meyer, Axel Kuepper: A Formal Approach to

Service Import in ODP Trader Federations

1994-07 P. Peters, P. Szczurko: Integrating Models of Quality Management Meth-

ods by an Object-Oriented Repository

1994-08 ∗ Manfred Nagl, Bernhard Westfechtel: A Universal Component for the

Administration in Distributed and Integrated Development Environ-

ments

1994-09 ∗ Patrick Horster, Holger Petersen: Signatur- und Authentifikationsver-

fahren auf der Basis des diskreten Logarithmusproblems

1994-11 A. Schurr: PROGRES, A Visual Language and Environment for PRO-

gramming with Graph REwrite Systems

1994-12 A. Schurr: Specification of Graph Translators with Triple Graph Gram-

mars

1994-13 A. Schurr: Logic Based Programmed Structure Rewriting Systems

1994-14 L. Staiger: Codes, Simplifying Words, and Open Set Condition

1994-15 ∗ Bernhard Westfechtel: A Graph-Based System for Managing Configura-

tions of Engineering Design Documents

1994-16 P. Klein: Designing Software with Modula-3

1994-17 I. Litovsky, L. Staiger: Finite acceptance of infinite words

30

1994-18 G. Hogen, R. Loogen: Parallel Functional Implementations: Graphbased

vs. Stackbased Reduction

1994-19 M. Jeusfeld, U. Johnen: An Executable Meta Model for Re-Engineering

of Database Schemas

1994-20 ∗ R. Gallersdorfer, M. Jarke, K. Klabunde: Intelligent Networks as a Data

Intensive Application (INDIA)

1994-21 M. Mohnen: Proving the Correctness of the Static Link Technique Using

Evolving Algebras

1994-22 H. Fernau, L. Staiger: Valuations and Unambiguity of Languages, with

Applications to Fractal Geometry

1994-24 ∗ M. Jarke, K. Pohl, R. Domges, St. Jacobs, H. W. Nissen: Requirements

Information Management: The NATURE Approach

1994-25 ∗ M. Jarke, K. Pohl, C. Rolland, J.-R. Schmitt: Experience-Based Method

Evaluation and Improvement: A Process Modeling Approach

1994-26 ∗ St. Jacobs, St. Kethers: Improving Communication and Decision Making

within Quality Function Deployment

1994-27 ∗ M. Jarke, H. W. Nissen, K. Pohl: Tool Integration in Evolving Informa-

tion Systems Environments

1994-28 O. Burkart, D. Caucal, B. Steffen: An Elementary Bisimulation Decision

Procedure for Arbitrary Context-Free Processes

1995-01 ∗ Fachgruppe Informatik: Jahresbericht 1994

1995-02 Andy Schurr, Andreas J. Winter, Albert Zundorf: Graph Grammar En-

gineering with PROGRES

1995-03 Ludwig Staiger: A Tight Upper Bound on Kolmogorov Complexity by

Hausdorff Dimension and Uniformly Optimal Prediction

1995-04 Birgitta Konig-Ries, Sven Helmer, Guido Moerkotte: An experimental

study on the complexity of left-deep join ordering problems for cyclic

queries

1995-05 Sophie Cluet, Guido Moerkotte: Efficient Evaluation of Aggregates on

Bulk Types

1995-06 Sophie Cluet, Guido Moerkotte: Nested Queries in Object Bases

1995-07 Sophie Cluet, Guido Moerkotte: Query Optimization Techniques Ex-

ploiting Class Hierarchies

1995-08 Markus Mohnen: Efficient Compile-Time Garbage Collection for Arbi-

trary Data Structures

1995-09 Markus Mohnen: Functional Specification of Imperative Programs: An

Alternative Point of View of Functional Languages

1995-10 Rainer Gallersdorfer, Matthias Nicola: Improving Performance in Repli-

cated Databases through Relaxed Coherency

1995-11 ∗ M.Staudt, K.von Thadden: Subsumption Checking in Knowledge Bases

1995-12 ∗ G.V.Zemanek, H.W.Nissen, H.Hubert, M.Jarke: Requirements Analy-

sis from Multiple Perspectives: Experiences with Conceptual Modeling

Technology

1995-13 ∗ M.Staudt, M.Jarke: Incremental Maintenance of Externally Materialized

Views

1995-14 ∗ P.Peters, P.Szczurko, M.Jeusfeld: Oriented Information Management:

Conceptual Models at Work

31

1995-15 ∗ Matthias Jarke, Sudha Ram (Hrsg.): WITS 95 Proceedings of the 5th

Annual Workshop on Information Technologies and Systems

1995-16 ∗ W.Hans, St.Winkler, F.Saenz: Distributed Execution in Functional Logic

Programming

1996-01 ∗ Jahresbericht 1995

1996-02 Michael Hanus, Christian Prehofer: Higher-Order Narrowing with Defi-

nitional Trees

1996-03 ∗ W.Scheufele, G.Moerkotte: Optimal Ordering of Selections and Joins in

Acyclic Queries with Expensive Predicates

1996-04 Klaus Pohl: PRO-ART: Enabling Requirements Pre-Traceability

1996-05 Klaus Pohl: Requirements Engineering: An Overview

1996-06 ∗ M.Jarke, W.Marquardt: Design and Evaluation of Computer–Aided Pro-

cess Modelling Tools

1996-07 Olaf Chitil: The Sigma-Semantics: A Comprehensive Semantics for Func-

tional Programs

1996-08 ∗ S.Sripada: On Entropy and the Limitations of the Second Law of Ther-

modynamics

1996-09 Michael Hanus (Ed.): Proceedings of the Poster Session of ALP96 - Fifth

International Conference on Algebraic and Logic Programming

1996-09-0 Michael Hanus (Ed.): Proceedings of the Poster Session of ALP 96 -

Fifth International Conference on Algebraic and Logic Programming:

Introduction and table of contents

1996-09-1 Ilies Alouini: An Implementation of Conditional Concurrent Rewriting

on Distributed Memory Machines

1996-09-2 Olivier Danvy, Karoline Malmkjær: On the Idempotence of the CPS

Transformation

1996-09-3 Vıctor M. Gulıas, Jose L. Freire: Concurrent Programming in Haskell

1996-09-4 Sebastien Limet, Pierre Rety: On Decidability of Unifiability Modulo

Rewrite Systems

1996-09-5 Alexandre Tessier: Declarative Debugging in Constraint Logic Program-

ming

1996-10 Reidar Conradi, Bernhard Westfechtel: Version Models for Software Con-

figuration Management

1996-11 ∗ C.Weise, D.Lenzkes: A Fast Decision Algorithm for Timed Refinement

1996-12 ∗ R.Domges, K.Pohl, M.Jarke, B.Lohmann, W.Marquardt: PRO-

ART/CE* — An Environment for Managing the Evolution of Chemical

Process Simulation Models

1996-13 ∗ K.Pohl, R.Klamma, K.Weidenhaupt, R.Domges, P.Haumer, M.Jarke: A

Framework for Process-Integrated Tools

1996-14 ∗ R.Gallersdorfer, K.Klabunde, A.Stolz, M.Eßmajor: INDIA — Intelligent

Networks as a Data Intensive Application, Final Project Report, June

1996

1996-15 ∗ H.Schimpe, M.Staudt: VAREX: An Environment for Validating and Re-

fining Rule Bases

1996-16 ∗ M.Jarke, M.Gebhardt, S.Jacobs, H.Nissen: Conflict Analysis Across Het-

erogeneous Viewpoints: Formalization and Visualization

1996-17 Manfred A. Jeusfeld, Tung X. Bui: Decision Support Components on the

Internet

32

1996-18 Manfred A. Jeusfeld, Mike Papazoglou: Information Brokering: Design,

Search and Transformation

1996-19 ∗ P.Peters, M.Jarke: Simulating the impact of information flows in net-

worked organizations

1996-20 Matthias Jarke, Peter Peters, Manfred A. Jeusfeld: Model-driven plan-

ning and design of cooperative information systems

1996-21 ∗ G.de Michelis, E.Dubois, M.Jarke, F.Matthes, J.Mylopoulos, K.Pohl,

J.Schmidt, C.Woo, E.Yu: Cooperative information systems: a manifesto

1996-22 ∗ S.Jacobs, M.Gebhardt, S.Kethers, W.Rzasa: Filling HTML forms simul-

taneously: CoWeb architecture and functionality

1996-23 ∗ M.Gebhardt, S.Jacobs: Conflict Management in Design

1997-01 Michael Hanus, Frank Zartmann (eds.): Jahresbericht 1996

1997-02 Johannes Faassen: Using full parallel Boltzmann Machines for Optimiza-

tion

1997-03 Andreas Winter, Andy Schurr: Modules and Updatable Graph Views for

PROgrammed Graph REwriting Systems

1997-04 Markus Mohnen, Stefan Tobies: Implementing Context Patterns in the

Glasgow Haskell Compiler

1997-05 ∗ S.Gruner: Schemakorrespondenzaxiome unterstutzen die paargramma-

tische Spezifikation inkrementeller Integrationswerkzeuge

1997-06 Matthias Nicola, Matthias Jarke: Design and Evaluation of Wireless

Health Care Information Systems in Developing Countries

1997-07 Petra Hofstedt: Taskparallele Skelette fur irregular strukturierte Prob-

leme in deklarativen Sprachen

1997-08 Dorothea Blostein, Andy Schurr: Computing with Graphs and Graph

Rewriting

1997-09 Carl-Arndt Krapp, Bernhard Westfechtel: Feedback Handling in Dy-

namic Task Nets

1997-10 Matthias Nicola, Matthias Jarke: Integrating Replication and Commu-

nication in Performance Models of Distributed Databases

1997-11 ∗ R. Klamma, P. Peters, M. Jarke: Workflow Support for Failure Manage-

ment in Federated Organizations

1997-13 Markus Mohnen: Optimising the Memory Management of Higher-Order

Functional Programs

1997-14 Roland Baumann: Client/Server Distribution in a Structure-Oriented

Database Management System

1997-15 George Botorog: High-Level Parallel Programming and the Efficient Im-

plementation of Numerical Algorithms

1998-01 ∗ Fachgruppe Informatik: Jahresbericht 1997

1998-02 Stefan Gruner, Manfred Nagel, Andy Schurr: Fine-grained and

Structure-Oriented Document Integration Tools are Needed for Devel-

opment Processes

1998-03 Stefan Gruner: Einige Anmerkungen zur graphgrammatischen Spezifika-

tion von Integrationswerkzeugen nach Westfechtel, Janning, Lefering und

Schurr

1998-04 ∗ O. Kubitz: Mobile Robots in Dynamic Environments

1998-05 Martin Leucker, Stephan Tobies: Truth - A Verification Platform for

Distributed Systems

33

1998-06 ∗ Matthias Oliver Berger: DECT in the Factory of the Future

1998-07 M. Arnold, M. Erdmann, M. Glinz, P. Haumer, R. Knoll, B. Paech, K.

Pohl, J. Ryser, R. Studer, K. Weidenhaupt: Survey on the Scenario Use

in Twelve Selected Industrial Projects

1998-09 ∗ Th. Lehmann: Geometrische Ausrichtung medizinischer Bilder am

Beispiel intraoraler Radiographien

1998-10 ∗ M. Nicola, M. Jarke: Performance Modeling of Distributed and Repli-

cated Databases

1998-11 ∗ Ansgar Schleicher, Bernhard Westfechtel, Dirk Jager: Modeling Dynamic

Software Processes in UML

1998-12 ∗ W. Appelt, M. Jarke: Interoperable Tools for Cooperation Support using

the World Wide Web

1998-13 Klaus Indermark: Semantik rekursiver Funktionsdefinitionen mit Strik-

theitsinformation

1999-01 ∗ Jahresbericht 1998

1999-02 ∗ F. Huch: Verifcation of Erlang Programs using Abstract Interpretation

and Model Checking — Extended Version

1999-03 ∗ R. Gallersdorfer, M. Jarke, M. Nicola: The ADR Replication Manager

1999-04 Marıa Alpuente, Michael Hanus, Salvador Lucas, German Vidal: Spe-

cialization of Functional Logic Programs Based on Needed Narrowing

1999-05 ∗ W. Thomas (Ed.): DLT 99 - Developments in Language Theory Fourth

International Conference

1999-06 ∗ Kai Jakobs, Klaus-Dieter Kleefeld: Informationssysteme fur die ange-

wandte historische Geographie

1999-07 Thomas Wilke: CTL+ is exponentially more succinct than CTL

1999-08 Oliver Matz: Dot-Depth and Monadic Quantifier Alternation over Pic-

tures

2000-01 ∗ Jahresbericht 1999

2000-02 Jens Voge, Marcin Jurdzinski A Discrete Strategy Improvement Algo-

rithm for Solving Parity Games

2000-03 D. Jager, A. Schleicher, B. Westfechtel: UPGRADE: A Framework for

Building Graph-Based Software Engineering Tools

2000-04 Andreas Becks, Stefan Sklorz, Matthias Jarke: Exploring the Semantic

Structure of Technical Document Collections: A Cooperative Systems

Approach

2000-05 Mareike Schoop: Cooperative Document Management

2000-06 Mareike Schoop, Christoph Quix (eds.): Proceedings of the Fifth Interna-

tional Workshop on the Language-Action Perspective on Communication

Modelling

2000-07 ∗ Markus Mohnen, Pieter Koopman (Eds.): Proceedings of the 12th Inter-

national Workshop of Functional Languages

2000-08 Thomas Arts, Thomas Noll: Verifying Generic Erlang Client-Server Im-

plementations

2001-01 ∗ Jahresbericht 2000

2001-02 Benedikt Bollig, Martin Leucker: Deciding LTL over Mazurkiewicz

Traces

2001-03 Thierry Cachat: The power of one-letter rational languages

34

2001-04 Benedikt Bollig, Martin Leucker, Michael Weber: Local Parallel Model

Checking for the Alternation Free mu-Calculus

2001-05 Benedikt Bollig, Martin Leucker, Thomas Noll: Regular MSC Languages

2001-06 Achim Blumensath: Prefix-Recognisable Graphs and Monadic Second-

Order Logic

2001-07 Martin Grohe, Stefan Wohrle: An Existential Locality Theorem

2001-08 Mareike Schoop, James Taylor (eds.): Proceedings of the Sixth Interna-

tional Workshop on the Language-Action Perspective on Communication

Modelling

2001-09 Thomas Arts, Jurgen Giesl: A collection of examples for termination of

term rewriting using dependency pairs

2001-10 Achim Blumensath: Axiomatising Tree-interpretable Structures

2001-11 Klaus Indermark, Thomas Noll (eds.): Kolloquium Programmier-

sprachen und Grundlagen der Programmierung

2002-01 ∗ Jahresbericht 2001

2002-02 Jurgen Giesl, Aart Middeldorp: Transformation Techniques for Context-

Sensitive Rewrite Systems

2002-03 Benedikt Bollig, Martin Leucker, Thomas Noll: Generalised Regular

MSC Languages

2002-04 Jurgen Giesl, Aart Middeldorp: Innermost Termination of Context-

Sensitive Rewriting

2002-05 Horst Lichter, Thomas von der Maßen, Thomas Weiler: Modelling Re-

quirements and Architectures for Software Product Lines

2002-06 Henry N. Adorna: 3-Party Message Complexity is Better than 2-Party

Ones for Proving Lower Bounds on the Size of Minimal Nondeterministic

Finite Automata

2002-07 Jorg Dahmen: Invariant Image Object Recognition using Gaussian Mix-

ture Densities

2002-08 Markus Mohnen: An Open Framework for Data-Flow Analysis in Java

2002-09 Markus Mohnen: Interfaces with Default Implementations in Java

2002-10 Martin Leucker: Logics for Mazurkiewicz traces

2002-11 Jurgen Giesl, Hans Zantema: Liveness in Rewriting

2003-01 ∗ Jahresbericht 2002

2003-02 Jurgen Giesl, Rene Thiemann: Size-Change Termination for Term

Rewriting

2003-03 Jurgen Giesl, Deepak Kapur: Deciding Inductive Validity of Equations

2003-04 Jurgen Giesl, Rene Thiemann, Peter Schneider-Kamp, Stephan Falke:

Improving Dependency Pairs

2003-05 Christof Loding, Philipp Rohde: Solving the Sabotage Game is PSPACE-

hard

2003-06 Franz Josef Och: Statistical Machine Translation: From Single-Word

Models to Alignment Templates

2003-07 Horst Lichter, Thomas von der Maßen, Alexander Nyßen, Thomas

Weiler: Vergleich von Ansatzen zur Feature Modellierung bei der Soft-

wareproduktlinienentwicklung

2003-08 Jurgen Giesl, Rene Thiemann, Peter Schneider-Kamp, Stephan Falke:

Mechanizing Dependency Pairs

2004-01 ∗ Fachgruppe Informatik: Jahresbericht 2003

35

2004-02 Benedikt Bollig, Martin Leucker: Message-Passing Automata are expres-

sively equivalent to EMSO logic

2004-03 Delia Kesner, Femke van Raamsdonk, Joe Wells (eds.): HOR 2004 – 2nd

International Workshop on Higher-Order Rewriting

2004-04 Slim Abdennadher, Christophe Ringeissen (eds.): RULE 04 – Fifth In-

ternational Workshop on Rule-Based Programming

2004-05 Herbert Kuchen (ed.): WFLP 04 – 13th International Workshop on Func-

tional and (Constraint) Logic Programming

2004-06 Sergio Antoy, Yoshihito Toyama (eds.): WRS 04 – 4th International

Workshop on Reduction Strategies in Rewriting and Programming

2004-07 Michael Codish, Aart Middeldorp (eds.): WST 04 – 7th International

Workshop on Termination

2004-08 Klaus Indermark, Thomas Noll: Algebraic Correctness Proofs for Com-

piling Recursive Function Definitions with Strictness Information

2004-09 Joachim Kneis, Daniel Molle, Stefan Richter, Peter Rossmanith: Param-

eterized Power Domination Complexity

2004-10 Zinaida Benenson, Felix C. Gartner, Dogan Kesdogan: Secure Multi-

Party Computation with Security Modules

2005-01 ∗ Fachgruppe Informatik: Jahresbericht 2004

2005-02 Maximillian Dornseif, Felix C. Gartner, Thorsten Holz, Martin Mink: An

Offensive Approach to Teaching Information Security: “Aachen Summer

School Applied IT Security”

2005-03 Jurgen Giesl, Rene Thiemann, Peter Schneider-Kamp: Proving and Dis-

proving Termination of Higher-Order Functions

2005-04 Daniel Molle, Stefan Richter, Peter Rossmanith: A Faster Algorithm for

the Steiner Tree Problem

2005-05 Fabien Pouget, Thorsten Holz: A Pointillist Approach for Comparing

Honeypots

2005-06 Simon Fischer, Berthold Vocking: Adaptive Routing with Stale Informa-

tion

2005-07 Felix C. Freiling, Thorsten Holz, Georg Wicherski: Botnet Tracking: Ex-

ploring a Root-Cause Methodology to Prevent Distributed Denial-of-

Service Attacks

2005-08 Joachim Kneis, Peter Rossmanith: A New Satisfiability Algorithm With

Applications To Max-Cut

2005-09 Klaus Kursawe, Felix C. Freiling: Byzantine Fault Tolerance on General

Hybrid Adversary Structures

2005-10 Benedikt Bollig: Automata and Logics for Message Sequence Charts

2005-11 Simon Fischer, Berthold Vocking: A Counterexample to the Fully Mixed

Nash Equilibrium Conjecture

2005-12 Neeraj Mittal, Felix Freiling, S. Venkatesan, Lucia Draque Penso: Ef-

ficient Reductions for Wait-Free Termination Detection in Faulty Dis-

tributed Systems

2005-13 Carole Delporte-Gallet, Hugues Fauconnier, Felix C. Freiling: Revisiting

Failure Detection and Consensus in Omission Failure Environments

2005-14 Felix C. Freiling, Sukumar Ghosh: Code Stabilization

2005-15 Uwe Naumann: The Complexity of Derivative Computation

36

2005-16 Uwe Naumann: Syntax-Directed Derivative Code (Part I: Tangent-

Linear Code)

2005-17 Uwe Naumann: Syntax-directed Derivative Code (Part II: Intraprocedu-

ral Adjoint Code)

2005-18 Thomas von der Maßen, Klaus Muller, John MacGregor, Eva Geis-

berger, Jorg Dorr, Frank Houdek, Harbhajan Singh, Holger Wußmann,

Hans-Veit Bacher, Barbara Paech: Einsatz von Features im Software-

Entwicklungsprozess - Abschlußbericht des GI-Arbeitskreises “Features”

2005-19 Uwe Naumann, Andre Vehreschild: Tangent-Linear Code by Augmented

LL-Parsers

2005-20 Felix C. Freiling, Martin Mink: Bericht uber den Workshop zur Ausbil-

dung im Bereich IT-Sicherheit Hochschulausbildung, berufliche Weiter-

bildung, Zertifizierung von Ausbildungsangeboten am 11. und 12. Au-

gust 2005 in Koln organisiert von RWTH Aachen in Kooperation mit

BITKOM, BSI, DLR und Gesellschaft fuer Informatik (GI) e.V.

2005-21 Thomas Noll, Stefan Rieger: Optimization of Straight-Line Code Revis-

ited

2005-22 Felix Freiling, Maurice Herlihy, Lucia Draque Penso: Optimal Random-

ized Fair Exchange with Secret Shared Coins

2005-23 Heiner Ackermann, Alantha Newman, Heiko Roglin, Berthold Vocking:

Decision Making Based on Approximate and Smoothed Pareto Curves

2005-24 Alexander Becher, Zinaida Benenson, Maximillian Dornseif: Tampering

with Motes: Real-World Physical Attacks on Wireless Sensor Networks

2006-01 ∗ Fachgruppe Informatik: Jahresbericht 2005

2006-02 Michael Weber: Parallel Algorithms for Verification of Large Systems

2006-03 Michael Maier, Uwe Naumann: Intraprocedural Adjoint Code Generated

by the Differentiation-Enabled NAGWare Fortran Compiler

2006-04 Ebadollah Varnik, Uwe Naumann, Andrew Lyons: Toward Low Static

Memory Jacobian Accumulation

2006-05 Uwe Naumann, Jean Utke, Patrick Heimbach, Chris Hill, Derya Ozyurt,

Carl Wunsch, Mike Fagan, Nathan Tallent, Michelle Strout: Adjoint

Code by Source Transformation with OpenAD/F

2006-06 Joachim Kneis, Daniel Molle, Stefan Richter, Peter Rossmanith: Divide-

and-Color

2006-07 Thomas Colcombet, Christof Loding: Transforming structures by set in-

terpretations

2006-08 Uwe Naumann, Yuxiao Hu: Optimal Vertex Elimination in Single-

Expression-Use Graphs

2006-09 Tingting Han, Joost-Pieter Katoen: Counterexamples in Probabilistic

Model Checking

2006-10 Mesut Gunes, Alexander Zimmermann, Martin Wenig, Jan Ritzerfeld,

Ulrich Meis: From Simulations to Testbeds - Architecture of the Hybrid

MCG-Mesh Testbed

2006-11 Bastian Schlich, Michael Rohrbach, Michael Weber, Stefan Kowalewski:

Model Checking Software for Microcontrollers

2006-12 Benedikt Bollig, Joost-Pieter Katoen, Carsten Kern, Martin Leucker:

Replaying Play in and Play out: Synthesis of Design Models from Sce-

narios by Learning

37

2006-13 Wong Karianto, Christof Loding: Unranked Tree Automata with Sibling

Equalities and Disequalities

2006-14 Danilo Beuche, Andreas Birk, Heinrich Dreier, Andreas Fleischmann,

Heidi Galle, Gerald Heller, Dirk Janzen, Isabel John, Ramin Tavakoli

Kolagari, Thomas von der Maßen, Andreas Wolfram: Report of the GI

Work Group “Requirements Management Tools for Product Line Engi-

neering”

2006-15 Sebastian Ullrich, Jakob T. Valvoda, Torsten Kuhlen: Utilizing optical

sensors from mice for new input devices

2006-16 Rafael Ballagas, Jan Borchers: Selexels: a Conceptual Framework for

Pointing Devices with Low Expressiveness

2006-17 Eric Lee, Henning Kiel, Jan Borchers: Scrolling Through Time: Improv-

ing Interfaces for Searching and Navigating Continuous Audio Timelines

2007-01 ∗ Fachgruppe Informatik: Jahresbericht 2006

2007-02 Carsten Fuhs, Jurgen Giesl, Aart Middeldorp, Peter Schneider-Kamp,

Rene Thiemann, and Harald Zankl: SAT Solving for Termination Anal-

ysis with Polynomial Interpretations

2007-03 Jurgen Giesl, Rene Thiemann, Stephan Swiderski, and Peter Schneider-

Kamp: Proving Termination by Bounded Increase

2007-04 Jan Buchholz, Eric Lee, Jonathan Klein, and Jan Borchers: coJIVE: A

System to Support Collaborative Jazz Improvisation

2007-05 Uwe Naumann: On Optimal DAG Reversal

2007-06 Joost-Pieter Katoen, Thomas Noll, and Stefan Rieger: Verifying Con-

current List-Manipulating Programs by LTL Model Checking

2007-07 Alexander Nyßen, Horst Lichter: MeDUSA - MethoD for UML2-based

Design of Embedded Software Applications

2007-08 Falk Salewski and Stefan Kowalewski: Achieving Highly Reliable Em-

bedded Software: An empirical evaluation of different approaches

2007-09 Tina Kraußer, Heiko Mantel, and Henning Sudbrock: A Probabilistic

Justification of the Combining Calculus under the Uniform Scheduler

Assumption

2007-10 Martin Neuhaußer, Joost-Pieter Katoen: Bisimulation and Logical

Preservation for Continuous-Time Markov Decision Processes

2007-11 Klaus Wehrle: 6. Fachgesprach Sensornetzwerke

2007-12 Uwe Naumann: An L-Attributed Grammar for Adjoint Code

2007-13 Uwe Naumann, Michael Maier, Jan Riehme, and Bruce Christianson:

Second-Order Adjoints by Source Code Manipulation of Numerical Pro-

grams

2007-14 Jean Utke, Uwe Naumann, Mike Fagan, Nathan Tallent, Michelle Strout,

Patrick Heimbach, Chris Hill, and Carl Wunsch: OpenAD/F: A Modular,

Open-Source Tool for Automatic Differentiation of Fortran Codes

2007-15 Volker Stolz: Temporal assertions for sequential and current programs

2007-16 Sadeq Ali Makram, Mesut Gunec, Martin Wenig, Alexander Zimmer-

mann: Adaptive Channel Assignment to Support QoS and Load Balanc-

ing for Wireless Mesh Networks

2007-17 Rene Thiemann: The DP Framework for Proving Termination of Term

Rewriting

38

2007-20 Joost-Pieter Katoen, Daniel Klink, Martin Leucker, and Verena Wolf:

Three-Valued Abstraction for Probabilistic Systems

2007-22 Heiner Ackermann, Paul W. Goldberg, Vahab S. Mirrokni, Heiko Roglin,

and Berthold Vocking: Uncoordinated Two-Sided Markets

∗ These reports are only available as a printed version.

Please contact biblio@informatik.rwth-aachen.de to obtain copies.

39