Baier, C: On Algorithmic Verification methods for Probabilistic Systems. Habilitation thesis, Univ. Mannheim, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... n tends to 1, of the approximations hx s;n i n2IN , where x s;0 = 0 and for n = 1; 2; x s;n = max 8 : X t2S (t) Delta x t;n Gamma1 X t2S yes (t) j s a 9 = Alternatively, the values p max s (OE U ) can also be computed by solving linear optimization problems [25,2,30]. On the other hand, to establish whether s 2 S satisfies Pwq [OE U ] where w2 f; g, we calculate the minimum probability : p min s (OE U ) inffp A s (OE U ) j A 2 A(S)g and compare the result to the threshold q, i.e. establish the inequality p min s w q. We can calculate p min s ....

....probability 0 and maximal minimal probability 1 by means of graph based analysis. Further details on these precomputation algorithms can be found in [25,31,32] Furthermore, using such precomputation steps the model checking for j= fair can be reduced to that for ordinary satisfaction j= see [26,30,32] for further details. For DTMCs, which are a subset of CPSs, model checking of until reduces to solving the following linear equation system in jS j unknowns: x s = X t2S s (t) Delta x t X t2S yes s (t) 14 where s is the unique distribution such that s a s . This ....

C. Baier, On algorithmic verification methods for probabilistic systems, Habilitation thesis, Fakultat fur Mathematik & Informatik, Universitat Mannheim (1999).

....of (bi )simulation. At each step, the condition for (bi )simulation may in general involve checking for inclusion between convex polytopes, spanned by different possible outcomes of a nondeterministic choice. Decision procedures for checking probabilistic (bi )simulation can be found in e.g. [3, 4]. These relations may be used to reason about probabilistic processes by means of algebraic operators in a compositional manner. A process expression may be generated by process constants, process variables and algebraic operators described in section 4. We define a process context to be a ....

.... by Bianco and de Alfaro [8, 21] Symbolic modelchecking algorithms of these logics was presented by Baier et al. in [6] For more detailed information on the interesting topic of model checking probabilistic systems, we refer the reader to the excellent works by Alfaro [20] and Baier [4]. 8 CONCLUSION and TRENDS In this chapter, we have dealt with a number of classical process algebraic issues in a rather general setting allowing both discrete probabilistic choice as well as nondeterminism. In particular, we have shown how non probabilistic process algebraic operators, in a ....

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation, thesis, University of Mannheim, 1999.

....selects one of possibly several probability distributions available in each state. Model checking of until properties in this case is performed through a reduction to a linear optimization problem. The basic algorithm (without fairness) can be found in [11] with fairness in [8] improved in [18, 21, 4]) The above two classes of models (DTMCs and MDPs) cannot express real time. An extension of the logic pCTL to cater for long run average properties, based on the labelling of states of an MDP with expected time, together with the corresponding algorithm, appeared in [18, 19] The timed automata ....

....to the logic CTL, which has a much more expressive counter part called CTL , 2 one can formulate PBTL (pCTL ) see e.g. 11, 8] The model checking algorithms, however, are expensive. For a comprehensive overview of issues in probabilistic verification the interested reader should consult [31, 18, 4] and [6, 26] ....

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, submitted, 1998.

.... Somenzi [47] Based on [31] an MTBDD based symbolic model checking procedure for purely probabilistic processes (state labelled discrete Markov chains) for the logic PCTL of [31] a probabilistic variant of CTL) was first presented in [4] and since extended to concurrent probabilistic systems in [2] (without implementation) The algorithm for checking PCTL until properties reduces to solving a system of linear equations . In this paper we consider models for concurrent probabilistic systems based on Markov Decision Processes [9] similar to those of [50, 22, 43, 10, 6, 24] These are ....

....to store the full size matrix by storing the component matrices instead and reformulating steady state probability calculation in terms of the component matrices. Existing implementation work in this area includes Petri net tools such as [16, 39] In this paper we adapt and extend the ideas of [4, 2] in order to represent concurrent probabilistic systems in terms of MTBDDs. The differences with the corresponding work in numerical analysis of Markov chains are: we allow nondeterminism as well as probability; we work with probability matrices, not generator matrices of continuous time Markov ....

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, submitted, 1998.

....our methods. First, consider the PTCTL model checking method based on the region graph. The time complexity of PBTL model checking is polynomial in the size of the system (measured by the number of states and transitions) and linear in the size of the formula [6] see also the recent improvement [5]) Since the translation from PTCTL to the extended PBTL has no effect on the size of the formula, it follows that the model checking for PTCTL against probabilistic timed systems will be polynomial in the size of the region graph and linear in the size of the PTCTL formula. Note that the addition ....

C. Baier. On algorithmic verification methods for probabilistic systems, 1998. Habilitation thesis, University of Mannheim.

....finite state model. If non determinism is replaced by randomized, i.e. probabilistic decisions, the resulting model boils down to a finite state discrete time Markov chain (DTMC) For these models, qualitative and quantitative model checking algorithms have been investigated extensively, see e.g. [3, 5, 6, 10, 13, 17, 29]. In a qualitative setting it is checked whether a property holds with probability 0 or 1; in a quantitative setting it is verified whether the probability for a certain property meets a given lower or upper bound. Markov chains are memoryless. In the discrete time setting this is reflected by ....

....methods have successfully been applied for much larger systems (up to 10 7 states) 14] For these reasons, E MC 2 supports all of the above mentioned iterative methods to solve (1) Probabilistic path formulas. Calculating the probabilities Prob(s, #) proceeds as in the discrete time case [13, 17, 5], except for the time bounded until 3 In [8] the above linear system of equations is defined in a slightly di#erent way, by characterizing the steady state probabilities in terms of the embedded DTMC. that is particular to the continuous time case. More precisely: Next: Prob(s, X#) is obtained ....

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, Univ. of Mannheim, 1999.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, submitted, 1998.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation Thesis, Universitat Mannheim, 1998.

No context found.

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation Thesis, Universitat Mannheim, 1998.

....2 ) # # B2 (s 1 ) # # B2 (s 2 ) 1. Subsequently, calculation of #B2 (s 2 ) yields 2 3. Thus, # s2 ,s 3 (s 0 ) 1 2 2 3 1 2 = 5 6 which indeed exceeds 0.5. Computing Prob(s, #) The next step and until operator can be handled as in the discrete time probabilistic case, cf. [18, 25, 5]. This entails that model checking for these formulas can be carried out by well known methods. The values Prob(s, X#) can be calculated by multiplying the transition probability matrix P with the (boolean) vector i # = i # (s) s#S characterising Sat(#) i.e. i # (s) 1 if s = #, and 0 ....

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis (submitted), Univ. Mannheim, 1998.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, Univ. of Mannheim, 1999.

No context found.

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, Univ. of Mannheim, 1999.

....0 y s 1 y s ub(f 1 ; s) ub(f 2 ; s) y s ub(f 2 ; s) max ( X t2S (t) Delta y t : 2 Steps(s) where P s2S y s is maximal. The existence of unique solutions of the optimization problems given in the definition above follows from fixed point arguments for maps on state vectors [3]. Similarly, we can unwind the operators 3f = tt U f) and 2f = 3:f ) Consider, for example, 3f : ffl The vector (lb(3f; s) s2S is the unique solution of the following optimization problem: 0 x s 1 x s lb(f; s) x s min ( X t2S (t) Delta x t : 2 Steps(s) where P s2S x s is ....

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, submitted, 1998.

.... IR 0 [0; 1] where 3 Omega (F ) s; t) 8 : 1 if s j= Phi 2 P s 0 2S Q(s; s 0 ) Delta R t 0 e GammaE(s) Deltax Delta F (s 0 ; t Gammax) dx if s j= Phi 1 : Phi 2 0 otherwise: The first two results of Theorem 1 are identical to the discrete time probabilistic case, cf. [14, 19, 4]. This entails that model checking for these formulas can be carried out by well known methods: Prob(s; X Phi) s2S can be obtained by multiplying the transition probability matrix P with the (boolean) vector i Phi = i Phi (s) s2S characterising Sat( Phi) i.e. i Phi (s) 1 if s j= ....

C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis (submitted), Univ. Mannheim, 1998.

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Baier, C: On Algorithmic Verification methods for Probabilistic Systems. Habilitation thesis, Univ. Mannheim, 1998.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis (submitted), Univ. Mannheim, 1998.

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Baier, C: On Algorithmic Verification methods for Probabilistic Systems. Habilitation thesis, Univ. Mannheim, 1998.

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C. Baier. On Algorithmic Verification Methods for Probabilistic Systems. Habilitation thesis, Fakultat fur Mathematik & Informatik, Universitat Mannheim, 1998.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, Universit at Mannheim, 1998.

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C. Baier, On algorithmic verification methods for probabilistic systems, Habilitation thesis, Fakultat fur Mathematik & Informatik, Universitat Mannheim (1999).

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, Fakultat fur Mathematik & Informatik, Universitat Mannheim, 1998.

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C. Baier, On algorithmic verification methods for probabilistic systems, Habilitationsschrift, FMI, Universitaet Mannheim, 1998.

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C. Baier. On Algorithmic Verification Methods for Probabilistic Systems. Habilitationsschrift, Universitat Mannheim, 1998.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, submitted, 1998.

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C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis, submitted, 1998.

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