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The integration of ICT (information and communications technology) in different applications is rapidly increasing in e.g. Embedded and Cyber physical systems, Communication protocols and Transportation systems. Hence, their reliability and dependability increasingly depends on software. Defects can be fatal and extremely costly (with regards to mass-production of products and safety-critical systems).

First, a model of the real system has to be built. In the simplest case, the model reflects all possible states that the system can reach and all possible transitions between states in a (labelled) State Transition System. When adding probabilities and discrete time to the model, we are dealing with so-called Discrete-time Markov chains which in turn can be extended with continuous timing to Continuous-time Markov chains. Both formalisms have been used widely for modeling and performance and dependability evaluation of computer and communication systems in a wide variety of domains. These formalisms are well understood, mathematically attractive while at the same time flexible enough to model complex systems.

Model checking focuses on the qualitative evaluation of the model. As formal verification method, model checking analyzes the functionality of the system model. A property that needs to be analyzed has to be specified in a logic with consistent syntax and semantics. For every state of the model, it is then checked whether the property is valid or not.

The main focus of this course is on quantitative model checking for Markov chains, for which we will discuss efficient computational algorithms. The learning objectives of this course are as follows:

– Express dependability properties for different kinds of transition systems .

– Compute the evolution over time for Markov chains.

– Check whether single states satisfy a certain formula and compute the satisfaction set for properties.

Q1. Which of the following CTL formulas expresses the mutual exclusion of two processes in all states of the system?

- Option 1

- Option 2

- Option 3

- Option 4

- Option 5

- Option 6

Q2. Which of the following CTL formulas expresses that every job will be answered eventually?

- Option 1

- Option 2

- Option 3

- Option 4

- Option 5

- Option 6

Q3. Which of the following CTL properties formulates a reset possibility for all states?

- Option 1

- Option 2

- Option 3

- Option 4

- Option 5

- Option 6

Q1. Given this TMR model:

Does the following property hold ?

- Yes
- No

Q2. Given this TMR model:

Does the following property hold for all states of the LTS?

- Yes
- No

Q3. Given this TMR model:

Does the following property hold for state up_3 of the LTS?

- Yes
- No

Q4. Given this TMR model:

Does the following property hold for state up_3 of the LTS?

- Yes
- No

Q5. Given this TMR model:

Does the following property hold for state up_1 of the LTS?

- Yes
- No

Q6. Given this TMR model:

Does the following property hold for state up_2 of the LTS?

- Yes
- No

Q7. Given this TMR model:

Does the following property hold for state up_2 of the LTS?

- Yes
- No

Q1. How is the collection of states called for which a certain CTL formula holds?

`Answer: `

Q2. The satisfaction set of the following formula is defined as:

- Option 1

- Option 2

Q3. Is the following a proper CTL formula?

- Yes
- No

Q4. Which operator stands at the root of the parse tree?

- highest binding operator
- lowest binding operator
- first operator of the formula

Q5. Which type of CTL formulas do you find in the leafs of the parse tree?

- atomic properties
- state formulas
- path formulas

Q6. Which operator is used to compute the satisfaction set of two disjunct CTL formulas?

- union
- intersection
- negation

Q7.

The above CTL formula can be reformulated as:

- Option 1

- Option 2

Q8.

The above CTL formula can be reformulated as:

- Option 1

- Option 2

Q9. What does CTL stand for?

- Computational Tree Logic
- Computer Tree Logic
- Computation Test Logic

Q10. What is the notion of time in CTL?

- continuous time
- branching time
- linear time

Q1. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q2. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q3. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q4. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q5. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q6. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q7. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q8. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q9. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q10. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q11. Consider the following transition system over AP = {black,green,red,yellow}

Which of the states are part of the satisfaction set of the following CTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q1. The columns in a transition probability matrix sum up to one.

- True
- False

Q2. The one-step probabilities can be found in the transition probability matrix.

- True
- False

Q3. The state probability distribution at time n

equals:

- Option 1

- Option 2

Q1. Consider the following DTMC:

Compute the transient-state probability of being in state S2 after one iterations given the vector p(0)= (0,0,0,0,1). (Please answer in real numbers not fractions and separate decimal digits by point not by comma)

`Answer: `

Q2. Consider the following DTMC:

Compute the transient-state probability of being in state S1 after two iterations given the vector p(0)= (0,0,0,0,1). (Please answer in real numbers not fractions and separate decimal digits by point not by comma)

`Answer: `

Q3. Consider the following DTMC:

Compute the transient-state probability of being in state S4 after one iterations given the vector p(0)= (0,0,0,0,1). (Please answer in real numbers not fractions and separate decimal digits by point not by comma)

`Answer: `

Q4. Consider the following DTMC:

Compute the transient-state probability of being in state S5 after two iterations given the vector p(0)= (0,0,0,0,1). (Please answer in real numbers not fractions and separate decimal digits by point not by comma)

`Answer: `

Q1. The entries per column in a transition probability matrix should sum up to the scalar one.

- True
- False

Q2. A state can be transient and recurrent.

- True
- False

Q3. A state is recurrent if we eventually return to it with probability one.

- True
- False

Q4. The period of a state is the greatest common divisor of the n for which pi,i (n) < 1.

- True
- False

Q5. An absorbing state is always recurrent.

- True
- False

Q6. In an irreducible and finite DTMC it holds that if one state is aperiodic, all states are aperiodic.

- True
- False

Q7. In a finite Markov chain a state is always positive recurrent if it is recurrent.

- True
- False

Q1. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 1 has the following properties (Multiple options may be correct):

- aperiodic
- periodic
- positive recurrent
- null-recurrent
- transient
- absorbing

Q2. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 2 has the following properties (Multiple options may be correct):

- aperiodic
- periodic
- positive recurrent
- null-recurrent
- transient
- absorbing

Q3. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 3 has the following properties (Multiple options may be correct):

- aperiodic
- periodic
- positive recurrent
- null-recurrent
- transient
- absorbing

Q4. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 1 has the following properties (Multiple options may be correct):

- aperiodic
- periodic
- positive recurrent
- null-recurrent
- transient
- absorbing

Q5. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 2 has the following properties (Multiple options may be correct):

- aperiodic
- periodic
- positive recurrent
- transient
- absorbing

Q6. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 3 has the following properties (Multiple answers may be correct):

- aperiodic
- periodic
- positive recurrent
- null-recurrent
- transient
- absorbing

Q7. Let the probability transition matrix of the following DTMC be given.

Draw the corresponding state-transition graph and classify the states of the DTMC. State 4 has the following properties (Multiple options may be correct):

- aperiodic
- periodic
- positive recurrent
- transient
- absorbing

Q8. Is the above DTMC irreducible?

- Yes
- No

Q1. Does the steady-state distribution exist for this Markov chain?

- Yes
- No

Q2. Consider the following DTMC:

Compute the steady-state probability of being in state S1 (rounded to the third decimal point)

`Answer: `

Q3. Consider the following DTMC:

Compute the steady-state probability of being in state S2 (rounded to the third decimal point)

`Answer: `

Q4. Consider the following DTMC:

Compute the steady-state probability of being in state S3 (rounded to the third decimal point)

`Answer: `

Q5. Consider the following DTMC:

Compute the steady-state probability of being in state S4 (rounded to the third decimal point)

`Answer: `

Q6. Consider the following DTMC:

Compute the steady-state probability of being in state S5 (rounded to the third decimal point)

`Answer: `

Q1. Imagine a signal processor in an embedded system. To process an input signal it must be stored in a buffer, and before it can be dropped two independent jobs that are carried out in sequence need to be completed. When an input signal arrives it may be stored in the buffer or be dropped without processing depending on how many signals are already in the buffer.

Which of the following expressions states that the probability of not being in a two_jobs state after one step is larger than 0.8.

- Option 1

- Option 2

- Option 3

Q2. In the same system, which of the following expressions states that the probability of having four signals in the buffer after four steps or less, without the buffer emptying, is smaller than 0.6.

- Option 1

- Option 2

- Option 3

Q3. In the same system, which of the following expressions states that the probability of having 4 signals in the buffer before hitting a one_job state is smaller than 0.01.

- Option 1

- Option 2

- Option 3

Q4. In the same system, which of the following expressions states that the probability of hitting a state, in which the probability of not being in a two_job state after one step is greater than 0.8, in 4 steps or less is smaller than 0.1?

- Option 1

- Option 2

- Option 3

Q1. Consider the following DTMC:

Which of the states are part of the satisfaction set of the following PCTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q2. Consider the following DTMC:

What is the exact probability of state S1 for the following PCTL formula: (Please answer in real numbers not fractions and separate decimal digits by point not by comma)

`Answer: `

Q1. All path starting in state s, fulfil the path formula

- with probability 1,

- with probability 1,

- with probability 0,

Q2. One can reduce model checking the time bounded until operator to a transient analysis in the reduced Markov chain, the following states are made absorbing:

- Option 1

- Option 2

- Option 3

Q3. In order to compute:

We need to compute

- for each state of the state-space.

- for each state in the the reduced Markov chain.

Q1. Consider the following DTMC:

Which of the states are part of the satisfaction set of the following PCTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q2. Consider the following DTMC:

What is the exact probability of state S2 for the following PCTL formula: (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q3. Consider the following DTMC:

Which of the states are part of the satisfaction set of the following PCTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q4. Consider the following DTMC:

What is the exact probability of state S2 for the following PCTL formula: (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q1. Consider the following DTMC:

Which of the states are part of the satisfaction set of the following PCTL formula:

- State 1
- State 2
- State 3
- State 4
- None of the States

Q2. Consider the following DTMC:

What is the exact probability of state S1 for the following PCTL formula:

`Answer: `

Q1. When model checking the unbounded Until operator, we need to solve the following system of equations:

- Option 1

- Option 2

- Option 3

Q2.

- True
- False

Q3. Worst time complexity of PCTL model checking is

- polynomial in the size of the DTMC.
- polynomial in the number of operators that are in the formula.
- polynomial in the size of the largest time-bound k.

Q1. Consider the following CTMC:

Which of the following matrices is the rate matrix for the CTMC ?

- Option 1

- Option 2

- Option 3

- Option 4

Q2. Consider the following CTMC:

Which of the following matrices is the generator matrix for the CTMC ?

- Option 1

- Option 2

- Option 3

- Option 4

Q3. Consider the following CTMC:

Which of the following matrices is the probability matrix of the embedded DTMC of this CTMC ?

- Option 1

- Option 2

- Option 3

- Option 4

Q1. The state-residence times in a CTMC are distributed according to a

- Poission distribution
- negative exponential distribution
- geometric distribution

Q2. The generator matrix is a stochastic matrix

- True
- False

Q3. The rows in the generator matrix sum up to

- zero
- one

Q1. Given the following CTMC:

Compute the steady state of being in state S1. (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q2. Given the following CTMC:

Compute the steady state of being in state S2. (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q3. Given the following CTMC:

Compute the steady state of being in state S3. (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q4. The CTMC in questions 3 consists of how many components?

(just enter the correct number)

`Answer: `

Q1. Given the following generator matrix:

Draw the state diagram and derive the number of bottom strongly connected component in this CTMC.

`Answer: `

Q2. Given the following generator matrix:

What is the probability of reaching the bscc that includes state 4 starting in state 2 ? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q3. Given the following generator matrix:

What is the steady state probability of state 4 starting in state 2 ? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q1. The uniformisation constant needs to be

- the mininum of the diagonal entries of the Q-matrix
- at least as large as the minimum absolute value of the entries of the Q-matrix
- at most as large as the minimum absolute value of the entries of the Q-matrix

Q2. The uniformisation constant determines the speed of the embedded DTMC.

- True
- False

Q3. Using uniformisation the transient probabilities in the CTMC are computed as the weighted average of the transient probabilities in the embedded DTMC for all possible step numbers.

- True
- False

Q1. Given the following generator matrix:

What is the smallest possible uniformization rate that can be used ?

`Answer: `

Q2. Given the following generator matrix:

Derive probability matrix of the uniformized DTMC with the smallest possible uniformization rate. Let the matrix be denoted as follows:

What is the numeric value of a1,1? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q3. Given the following generator matrix:

Derive probability matrix of the uniformized DTMC with the smallest possible uniformization rate. Let the matrix be denoted as follows:

What is the numeric value of a1,2? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q4. Given the following generator matrix:

Derive probability matrix of the uniformized DTMC with the smallest possible uniformization rate. Let the matrix be denoted as follows:

What is the numeric value of a1,3? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer:`

Q5. Given the following generator matrix:

What is the numeric value of a2,1? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q6. Given the following generator matrix:

What is the numeric value of a2,2? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q7. Given the following generator matrix:

What is the numeric value of a2,3? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q8. Given the following generator matrix:

What is the numeric value of a3,1? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q9. Given the following generator matrix:

What is the numeric value of a3,2? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q10. Given the following generator matrix:

What is the numeric value of a3,3? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q11. Given the following generator matrix:

What is the minimum number of steps that is necessary to compute the transient probabilities for an error at most epsilon = 10^-4 for time t = 0.1 ?

`Answer: `

Q12. Given the following generator matrix:

What is the minimum number of steps that is necessary to compute the transient probabilities for an error at most epsilon = 10^-4 for time t = 0.2 ?

`Answer: `

Q13. Given the following generator matrix:

What is the transient probability of being in state 1 given the initial distribution [1,0,0] for time t = 0.1 ? (Please answer in real numbers not fractions and separate decimal digits by point not by comma. Rounded to the third decimal point.)

`Answer: `

Q1. Let the following simple model of an assembly line be given:

After a startup phase, the assembly line is either waiting for the first piece to be processed or it performs an integrity check, after which it repeats the startup phase. Each piece is assembled in three steps, During each step, a failure can occur, which enforces a time-consuming recovery and the removal of the work piece. Afterwards an integrity check with following restart is performed. After its successful assembly a piece has to be removed, then the assembly line moves again to the waiting state.

Which CSL formula describes the following requirement:

The probability that after system start, a system check is done, is below 9 %.

- Option 1

- Option 2

- Option 3

- Option 4

Q2. Let the following simple model of an assembly line be given:

After a startup phase, the assembly line is either waiting for the first piece to be processed or it performs an integrity check, after which it repeats the startup phase. Each piece is assembled in three steps, During each step, a failure can occur, which enforces a time-consuming recovery and the removal of the work piece. Afterwards an integrity check with following restart is performed. After its successful assembly a piece has to be removed, then the assembly line moves again to the waiting state.

Which CSL formula describes the following requirement:

In at least 98 % of all cases, a job can be completed without failures.

- Option 1

- Option 2

- Option 3

- Option 4

Q3. Let the following simple model of an assembly line be given:

After a startup phase, the assembly line is either waiting for the first piece to be processed or it performs an integrity check, after which it repeats the startup phase. Each piece is assembled in three steps, During each step, a failure can occur, which enforces a time-consuming recovery and the removal of the work piece. Afterwards an integrity check with following restart is performed. After its successful assembly a piece has to be removed, then the assembly line moves again to the waiting state.

Which CSL formula describes the following requirement:

In at most 90 % of all cases, a job can be completed within at most 9 time units.

- Option 1

- Option 2

- Option 3

- Option 4

Q4. Let the following simple model of an assembly line be given:

Which CSL formula describes the following requirement:

With probability more than 0.95, a job is completed within less than 7 time units, without either a failure or a system check.

- Option 1

- Option 2

- Option 3

- Option 4

Q5. Let the following simple model of an assembly line be given:

Which CSL formula describes the following requirement:

The system is available 99 % of the time (available means that the system is either working or ready to start working).

- Option 1

- Option 2

- Option 3

- Option 4

Q1. Continuous Stochastic Logic follows a branching notion of time

- True
- False

Q2. In CSL one can reason about the validity of a path formula on individual states.

- True
- False

Q3. The probability that a next formula holds for a given state is independent of the time indicated in the superscript of the next.

- True
- False

Q4. In case a goal state (Psi) is reachable in one step, the probability that the until formula

holds, can be simply computed by model checking the next operator for the same time-bound.

- True
- False

Q1. Consider the following labelled CTMC.

Check if the white state 4 satisfies the following formula:

- Yes
- No

Q2. Consider the following labelled CTMC.

What is the exact result for the following formula for the red state 1?

`Answer: `

Q3. Consider the following labelled CTMC.

Check if the white state 4 satisfies the following formula:

- Yes
- No

Q4. Consider the following labelled CTMC.

What is the exact result for the following formula for the red state 1?

`Answer: `

Q1. In an irreducible CTMC the satisfaction set of a steady-state formula is either the full state-space or the empty set.

- True
- False

Q2. In a reducible CTMC the satisfaction set of a steady-state operator cannot be a non-empty subset of the state-space.

- True
- False

Q3. A bottom-strongly-connected-component is a component of the CTMC that

- is absorbing
- is positive recurrent
- is irreducible
- is periodic
- is transient

Q1. Consider the following labelled CTMC.

What is the smallest possible uniformization rate that can be used ?

`Answer: `

Q2. Consider the following labelled CTMC.

Check if the red state 1 satisfies the following formula using uniformisation.

How many steps do you need to take before you can say whether this property is satisfied or not?

`Answer: `

Q3. Consider the following labelled CTMC.

Check if the red state 1 satisfies the following formula using uniformisation. Using the previously calculated amount of steps, what is the maximum absolute error between the value you calculate for

`Answer: `

Q4. Consider the following labelled CTMC.

Check if the red state 1 satisfies the following formula using uniformisation. Using the previously calculated amount of steps, what is the true value for

`Answer: `

Q1. To compute the satisfaction set of a time-bounded until operator, we must

- first uniformize the CTMC and then make the goal states and the violating states absorbing.
- first make the goal states and the violating states absorbing, then uniformise the resulting CTMC.
- uniformize and make the goal states and the violating states absorbing. However, it can be done in both orders.

Q2. To perform uniformisation up to some predefined error-bound does guarantee that one can decide whether a state fulfils the time-bounded until formula under investigation.

- True
- False

Q3. The number of steps that is performed in the uniformised DTMC up to some time t is distributed according to a

- geometric distribution
- negative exponential distribution
- Poisson distribution

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