Automata Learning -- Expect Delays!

Gabriel Dengler; Holger Hermanns; Sven Apel

REVIEW 4 major objections 5 minor 83 references

Automata Learning -- Expect Delays!

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Active learning of automata with stochastic delays works best split into two stages — learn the input/output logic first, then route a postman tour that samples every transition's delay enough times — beating sampling inside classic L* by a

desk verdict A genuinely new two-phase method for learning stochastic transition delays, with the central caveat that its state-separation case rests on an uncertified d-step confluence parameter the paper itself admits can fail. read the letter →

arxiv 2508.16384 v1 pith:64KYD4XJ submitted 2025-08-22 cs.FL cs.SE

Gabriel Dengler , Sven Apel , Holger Hermanns This is my paper

classification cs.FLcs.SE MSC 68Q4568Q32

keywords activeautomatalearningMealydelaymachinesstochastictransitiondelaysRuralPostmanProblemL*algorithmd-stepconfluencedistributionsamplingblack-boxtiminganalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is the first, by its own account, to build a formal framework for actively learning automata whose transitions carry stochastic delays. Its central claim is that delay sampling should be separated from logical behavior learning: first learn the minimal Mealy machine with classic L* querying, then use the learned structure to compute a minimum-cost postman route that visits every transition at least k times, so each delay distribution can be estimated from k samples. When identical input/output behaviors can stem from different delay characteristics, the paper expands the learned machine into a history-indexed machine Md and merges back only states whose sampled delays are statistically indistinguishable. The payoff the paper demonstrates is practical: across a wide benchmark range, the two-stage method outperforms the naive sampling-based L* by a large margin, and a relational-database join-order case study shows the method can discover that the order of operations changes completion time from tens to hundreds of milliseconds.

What carries the argument

Mealy delay machine (MDM): a Mealy machine with a per-transition CDF of delay, the object to be learned. The d-step confluence criterion — two I/O-equivalent states must reconverge within d steps — justifies the expanded machine Md, whose states are (minimal-state, last-d-steps history) pairs, injectively bisimilar to the true machine (Theorem 1) and of size O(|S'|·|I|^d) (Theorem 2). The Rural Postman Problem over the learned graph, solved by min-cost flow, yields the cheapest tour whose edge multiplicities give k samples per transition, with resets modeled as edges to the root. A final merge collapses states with statistically equal sampled delays.

What would settle it

Take a system whose delays depend on history but whose logical states never reconverge (the paper's non-confluent example), run the learner with d=1, and inspect the delay samples on one expanded state: if the empirical distribution is bimodal, or its deviation from the true branch CDFs does not shrink as k grows, the d-step confluence premise is what keeps the reported delays trustworthy. Complementary check: on a d-step confluent system, estimated CDFs should converge to the true ones as k grows at the predicted rate.

Watch

Extended reading notes

Core claim

Mealy delay machines (MDMs) are Mealy machines whose transitions carry continuous probability distributions of delay, observable only through samples. The paper's central proposal: learn the minimal underlying Mealy machine with classic L* query learning, then solve a directed Rural Postman Problem over that machine to build the cheapest input sequence visiting every transition k times. Since identical input/output histories can hide different delays, the learner expands the learned machine into Md, whose states carry the last d steps of history, samples Md, then merges states with statistically indistinguishable delays. The paper claims this is the first formal framework for active learning

Load-bearing premise

The load-bearing premise is that the true system is d-step confluent for the learner's chosen d — any two input/output-identical states must reconverge within d steps — because if d is too small, delay samples on a state of the expanded machine blend several genuinely different distributions and the reported delay curves are a mixture, not a faithful estimate.

Editorial extensions

If this is right

Sampling effort decouples from the L* query tree: machines with large alphabets that push sampling-based L* past 10^8 actions (OpenSSH, TCP servers) stay feasible when the expanded machine Md is used for sample collection.
Raising d never loses distinctions already made, since Md is injectively bisimilar to Md+1 (Lemma 2); a larger d can only improve the faithfulness of the learned delay structure, at exponential state cost.
The merge step, together with the stutter-free and unique-root assumptions, recovers a minimal MDM that matches the true delay structure when the assumptions hold, so the final model is both minimal and delay-accurate.
The database case study shows the method works against real measured delays: it learns that join-completion time depends on the last one or two joins in the order, information invisible to logic-only learning.
The k-sample budget can be set before learning from accuracy targets: the deviation bound for empirical CDFs (or, for exponential delays, a closed-form relative-error formula) gives the k needed for a chosen error and confidence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is an adaptive sampling budget: instead of a fixed global k, allocate extra samples to transitions whose estimated delay distributions are still far apart or high-variance, reusing the same postman-route machinery with per-edge weights.
The paper's admitted mixture problem is itself diagnosable: when the true machine is not d-step confluent at the chosen d, samples on a state of the expanded machine should show excess dispersion or multi-modality, so a variance or modality check on collected samples could offer a data-driven way to raise d.
The database join-order study is the seed of a general black-box performance profiler: the same pipeline could attach measured timing distributions to logical states of compilers, network stacks, or query planners, and flag histories whose delay differs despite identical logical results.
The exponential dependence of Md on d and alphabet size sets the practical ceiling; the empirical tables show the unique-root assumption is the lever that keeps real benchmarks tractable, and a statistical stopping criterion for d would make the method's main tuning parameter self-tuning.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a careful read. The core idea is real: separate I/O learning from delay sampling, then plan the delay collection as a directed Rural Postman Problem over the learned machine. That separation is a clear improvement over the naive L*-with-repeated-queries baseline, and the MDM model with continuous CDFs per transition is cleanly defined. The Md expansion and the injective-bisimilarity theorems are new and mostly solid, and the empirical sweep over the public benchmark suite is broad. Credit also goes to the artifact: a reproduction package and code are provided, which is more than many papers in this area do.

The soft spots are in proportion. The biggest one is exactly what the stress test flags: the Sec. 5 construction is only sound when the true MDM is stutter-free and d-step confluent, and d is an input with no selection or certification procedure. The paper says so itself in Sec. 5.2: d is generally unknown, there are MDMs with no suitable d, and too small a d gives "an uncontrolled mixture of multiple CDFs." Lemma 2 only says larger d is no worse; it does not tell you when d is large enough. Since the later state-merging step compares already-mixed samples, it cannot split a bad state and recover the true delays. So for the state-separation cases the advertised faithful estimate is conditional on an assumption the method cannot verify. The Sec. 4 minimal-MM case does not have this problem.

Two smaller issues. First, the RPP reduction is described as min-cost, but the general RPP is NP-complete and the implementation falls back to an arborescence heuristic when mandatory edges are not weakly connected. That is fine as engineering, but the optimality claim should be scoped to the cases where min-cost flow actually applies. Second, the empirical section reports single runs without error bars or statistical tests, and the database study also appears unrepeated. The consistent margin over the baseline is suggestive, not conclusive. The 20% distribution-difference threshold derived from the exponential sample-size calculation is not circular; it is a reasonable heuristic, though its sensitivity is unexplored.

For someone working in active automata learning, especially with timing or performance: yes, send it to serious peer review. It deserves referee time. The revisions that matter are a procedure or certificate for d, honest scoping of the RPP optimality, and repeated experiments with variance reported.

Referee Report

4 major / 5 minor

Summary. The paper proposes an active automata learning framework for Mealy machines with stochastic transition delays (Mealy delay machines, MDMs). It separates the learning of I/O behavior from the collection of delay samples: first an ordinary L*-style algorithm learns the underlying minimal Mealy machine, then the learner plans a low-cost circuit of inputs—via a reduction to the directed Rural Postman Problem—that visits every transition at least k times to estimate delay CDFs. In the more general case where I/O-equivalent states may have distinct delay behavior, the paper constructs an expanded machine Md under assumptions of stutter-freeness and d-step confluence, samples delays on Md, and then merges statistically equivalent delay states. Theoretical results include injective bisimilarity of the true machine to Md (Theorem 1), a bound on |Md| (Theorem 2), monotonicity in d (Lemma 2), and a general product construction (Theorem 3). The empirical evaluation compares the sampling-aware L* baseline with the Md-based method on a public benchmark suite and a PostgreSQL join-order case study, reporting substantially fewer actions for the new method.

Significance. If the results hold, this is a useful first step toward active learning of automata with stochastic delays, and the separation of I/O learning from delay sampling is a natural and potentially impactful idea. The paper ships an artifact, contains a formal semantics for MDMs in Appendix A, and provides a reproducible benchmark setup, which are strengths. However, the central correctness guarantee for the state-separation part depends on a parameter d that is neither known in advance nor certified, and the empirical evaluation uses a mean-based equality criterion and single runs. These issues do not invalidate the algorithmic idea but limit the strength of the claims as currently stated.

major comments (4)

[Sec. 5.2, 'Selection of d' and Def. 6] The correctness of the Md construction and the subsequent merging step requires the SUL to be stutter-free and d-step confluent for the chosen d. The paper admits that d is generally unknown and that some MDMs have no suitable d (Fig. 2b). If d is too small, a state of Md corresponds to multiple true states and sampling yields 'an uncontrolled mixture of multiple CDFs,' as the authors state. The later merging step compares those already-mixed samples and cannot split them, so D' may not be a faithful delay model. Lemma 2 only gives monotonicity, not a stopping rule or certificate. The benchmarks choose d with knowledge of the true system (M1 requires d≥2; the database SUL is 3-step confluent), so the failure mode is not exercised. Please either provide a data-driven method to choose/certify d, or explicitly restrict the formal claims to 'given a d that is known to satisfy the confluence
[Sec. 6, 'Experimental setup' and Appendix C] The distribution-equality test used for state merging is a heuristic based on the sample mean: two distributions are considered different only if the relative deviation of the means exceeds 20% (and the absolute deviation is at least 0.01). This does not test equality of CDFs as required by Def. 4 and the problem statement. Two distributions with the same mean but different shapes (e.g., exponential vs. Erlang with the same mean) will be merged, yielding an incorrect D'. Please either use a proper CDF-level test (e.g., Kolmogorov–Smirnov) or explicitly narrow the target class to distributions whose differences are captured by the mean.
[Sec. 4.2 and Appendix D] The text says the sampling sequence will be constructed 'with minimum total cost,' but then notes the general RPP is NP-complete and, when mandatory edges are not weakly connected, the implementation resorts to a minimum spanning arborescence heuristic. Appendix D further states that 'we used a heuristic approach to find a solution to the RPP.' Therefore the evaluated procedure is not the optimal one in general. Please state which experiments use the exact min-cost flow formulation and which use the heuristic, and report the heuristic's suboptimality gap where possible.
[Fig. 4 and Appendix D] The empirical evaluation reports single runs without repetitions, confidence intervals, or statistical tests. More importantly, it reports only the number of steps and resets, not the quality of the learned delay model (e.g., whether the merged D' correctly recovers the true delay states and whether the estimated CDFs are close to the true ones). Since the equality tests are stochastic, a single run may silently produce an incorrect D'. Please report repeated runs and accuracy metrics for the learned delays.

minor comments (5)

[Sec. 2, Def. 5 vs. Def. 8] The definition of a sink in Sec. 2 uses δ(s,i)=s for all i, while Def. 8 uses δ(s,i)∼IO s for all i. Align the terminology.
[Sec. 4.2, last paragraph] 'minimum spanning arborescence of minimum weight' is redundant; use 'minimum weight spanning arborescence.'
[Fig. 4] The y-axis is labeled 'Total number of steps and resets,' but the plot shows a single total. Clarify whether resets are included in the total or counted separately.
[Appendix D tables] The repeated column header 'Using expansion Md' is confusing; the left block appears to be the adapted L* baseline. Reformat the tables to make the columns unambiguous.
[Sec. 7, d=0 result] The text says for d=0 the algorithm 'reports that ... the commit will be delayed with a mean duration of 191 ms,' then immediately says this is not a meaningful average. Rephrase to avoid appearing contradictory.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained, and the selection-of-d limitation is a correctness risk, not a circular reduction.

full rationale

The derivation chain is self-contained. The MDM model (Def. 3) and the learning problem are defined independently of the proposed algorithm; the algorithm separates I/O learning from delay sampling and uses an RPP tour to cover transitions (Sec. 4.2), which is an optimization over the learned MM, not a prediction fitted to the delay data. The expansion construction Md (Sec. 5.2) is defined purely from the learned minimal MM and the parameters d and the loop-free/stutter-free/confluence assumptions (Defs. 5-7); Theorem 1 is proved from these definitions and does not import the conclusion. The state-merging step compares fresh delay samples and is explicitly optional. The self-citations (e.g., [9,27,31,45]) provide background semantics and model-family context; none carries a load-bearing uniqueness theorem or ansatz. The limitation the paper itself states in Sec. 5.2 -- that a too-small d yields 'an uncontrolled mixture of multiple CDFs' -- is a correctness risk for the parameter choice, not a circular step: the output D' is not defined in terms of the samples in a way that forces the claimed performance. The empirical threshold of 20% is derived from the exponential sample-size calculation in Appendix C and the synthetic benchmarks use exponential delays; this is a transparent calibration match, but it does not make the method's performance claim reduce by construction, and the same threshold is applied to both compared methods. Overall, no prediction or first-principles result in the paper is equivalent to its inputs by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method is parameterized by k, d, and statistical thresholds, and rests on structural assumptions (stutter-free, d-step confluence, sink delay-ignorant) about the system under learning. These assumptions are clearly stated, but the most fragile one, d-step confluence, cannot be checked a priori and its failure can invalidate the sampling scheme.

free parameters (4)

k = 1000
Number of delay samples per transition. Derived in Appendix C from a 0.081 relative error bound at 99% confidence for exponential distributions, but chosen globally as a constant for the method.
d = varied 0..5 in experiments
Confluence depth used for state separation. It is a user-selected parameter that must be large enough to distinguish delay-relevant histories; too small a d can mix CDFs (Sec. 5.2).
distribution equality thresholds = 20% relative deviation, 0.01 absolute deviation
Thresholds used to decide whether two sets of delay samples come from the same distribution. Calibrated in Appendix C for exponential samples; not validated for other delay families.
RPP edge cost = 1 (constant)
Each input action and each reset is assumed to cost 1 in the RPP formulation. Context-specific costs are stated as supported but only the constant choice is evaluated.

assumptions (6)

domain assumption The teacher provides a perfect equivalence oracle for the underlying time-abstract Mealy machine.
Stated in Sec. 1 and used throughout: 'the teacher has full knowledge of the time-abstract behavior' and the problem statement assumes 'an EO for the underlying MM'. This is standard in AAL but strong for real systems.
domain assumption Stutter-free condition (Def. 5): self-loops in the minimal MM indicate no state change in the MDM.
Required for the expansion construction in Sec. 5.2; if violated, transitions between distinct states with the same I/O behavior would be missed.
domain assumption d-step confluence (Def. 6): I/O-equivalent states have identical delay behavior after at most d steps.
Load-bearing for the expanded machine Md and for state separation. The authors note that non-confluent cases (Fig. 2b) lead to uncontrolled CDF mixtures.
domain assumption Sink delay-ignorant (Def. 8): sink nodes do not split by delay behavior.
Assumed across all empirical studies (Sec. 6) to reduce the expansion size; not theoretically justified beyond convenience.
domain assumption The learner obtains independent samples from each transition's delay CDF on every traversal.
Formalized in the MDM semantics (Appendix A); independence underlies the DKW/Chebyshev sample-size arguments and the statistical tests.
domain assumption Visiting each transition at least k times yields statistically sufficient estimates of the delay distributions.
The problem statement in Sec. 3 fixes k as the decisive accuracy parameter; the paper does not address how to choose k when delays are not exponential.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Automata Learning -- Expect Delays!." pith.science (2026). https://pith.science/paper/64KYD4XJ

@misc{pith2026250816384,
  author       = {Pith},
  title        = {Pith review of: Automata Learning -- Expect Delays!},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64KYD4XJ}},
  note         = {Machine review of arXiv:2508.16384}
}

abstract

This paper studies active automata learning (AAL) in the presence of stochastic delays. We consider Mealy machines that have stochastic delays associated with each transition and explore how the learner can efficiently arrive at faithful estimates of those machines, the precision of which crucially relies on repetitive sampling of transition delays. While it is possible to na\"ively integrate the delay sampling into AAL algorithms such as $L^*$, this leads to considerable oversampling near the root of the state space. We address this problem by separating conceptually the learning of behavior and delays such that the learner uses the information gained while learning the logical behavior to arrive at efficient input sequences for collecting the needed delay samples. We put emphasis on treating cases in which identical input/output behaviors might stem from distinct delay characteristics. Finally, we provide empirical evidence that our method outperforms the na\"ive baseline across a wide range of benchmarks and investigate its applicability in a realistic setting by studying the join order in a relational database.

Figures

Figures reproduced from arXiv: 2508.16384 by the authors.

**Figure 1.** Minimal MM M′ The useful case. To make the analysis approachable, we propose restrictions on what the delay behavior induces compared to the I/O behavior alone. The first one states that self-loops in the underlying minimal MM indicate that no changes to the system are made. This prevents transitions between distinct states of D that have the same I/O behavior, which is why we use the term stutter-free: Definition … view at source ↗

**Figure 2.** Confluent and non-confluent expansions of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Expanded Md for minimal MM M′ from [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Number of total actions for different models and selections o [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: Impact of k (d = 0) Evaluation. RQ1: Across all cases, the samplingbased L ∗ is outperformed by the approach using the expanded MM Md by a large margin [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

83 extracted references · 56 canonical work pages

[1]

Alur, R., Dill, D.L.: A theory of timed automata. Theor. Co mput. Sci. 126(2), 183–235 (1994). https://doi.org/10.1016/0304-3975 (94)90010-8, https://doi.org/10.1016/0304-3975(94)90010-8 18 G. Dengler et al

doi:10.1016/0304-3975 1994
[2]

In: Biere, A., Parker, D

An, J., Chen, M., Zhan, B., Zhan, N., Zhang, M.: Learning On e-Clock Timed Au- tomata. In: Biere, A., Parker, D. (eds.) Tools and Algorithms for the Construction and Analysis of Systems. pp. 444–462. Springer Internation al Publishing, Cham (2020)

2020
[3]

Science China Information Sciences 64(9), 192103 (Aug 2021)

An, J., Wang, L., Zhan, B., Zhan, N., Zhang, M.: Learning real-time automata. Science China Information Sciences 64(9), 192103 (Aug 2021). https://doi.org/10.1007/s11432-019- 2767-4, https://doi.org/10.1007/s11432-019-2767-4

doi:10.1007/s11432-019- 2021
[4]

ACM Trans

An, J., Zhan, B., Zhan, N., Zhang, M.: Learning nondetermi nistic real- time automata. ACM Trans. Embed. Comput. Syst. 20(5s) (Sep 2021). https://doi.org/10.1145/3477030, https://doi.org/10.1145/3477030

doi:10.1145/3477030 2021
[5]

Information and Computation 75(2), 87–106 (1987)

Angluin, D.: Learning regular sets from queries and coun- terexamples. Information and Computation 75(2), 87–106 (1987). https://doi.org/https://doi.org/10.1016/0890 -5401(87)90052-6, https://www.sciencedirect.com/science/article/pii/0890540187900526

work page arXiv 1987
[7]

Scandinavian Journal of Statistics 23(4), 419–441 (1996), http://www.jstor.org/stable/4616418

Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics 23(4), 419–441 (1996), http://www.jstor.org/stable/4616418

work page arXiv 1996
[8]

In: Wani, M.A., Sethi, I.K., Shi, W., Qu, G., Raicu, D.S., Jin, R

Bacci, G., Ing´ olfsd´ ottir, A., Larsen, K.G., Reynouard , R.: Active Learn- ing of Markov Decision Processes using Baum-Welch algorith m. In: Wani, M.A., Sethi, I.K., Shi, W., Qu, G., Raicu, D.S., Jin, R. (eds.) 20th IEEE International Conference on Machine Learning and Appli- cations, ICMLA 2021, Pasadena, CA, USA, December 13-16, 202 1. pp. 1203–1208....

arXiv 2021
[9]

In: Flum, J., Gr¨ adel, E., Wilke, T

Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.: Rea chability in continuous- time markov reward decision processes. In: Flum, J., Gr¨ adel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives [in Honor of Wo lfgang Thomas]. Texts in Logic and Games, vol. 2, pp. 53–72. Amsterdam Univer sity Press (2008)

2008

Show all 83 references

[10]

Bertrand, N., Bouyer, P., Brihaye, T., Menet, Q., Baier, C., Gr¨ oßer, M., Jurdzinski, M.: Stochastic Timed Automata. Log. Method s Com- put. Sci. 10(4) (2014). https://doi.org/10.2168/LMCS-10(4:6)2014, https://doi.org/10.2168/LMCS-10(4:6)2014

2014 doi
[11]

Oxford University Press (1986)

Biggs, N., Lloyd, E.K., Wilson, R.J.: Graph Theory, 1736 -1936. Oxford University Press (1986)

1936
[12]

Stochastic Models 21(2-3), 303–326 (2005)

Bobbio, A., Andr´ as Horv´ ath and Mikl´ os Telek: Matching three moments with min- imal acyclic phase type distributions. Stochastic Models 21(2-3), 303–326 (2005). https://doi.org/10.1081/STM-200056210

2005 doi
[13]

CoRR abs/2403.02019 (2024)

Bruy` ere, V., Garhewal, B., P´ erez, G.A., Staquet, G., V aandrager, F.W.: Active Learning of Mealy Machines with Timers. CoRR abs/2403.02019 (2024). https://doi.org/10.48550/ARXIV.2403.02019, https://doi.org/10.48550/arXiv.2403.02019 Automata Learning – Expect Delays! 19

2024 doi arXiv
[14]

In: 2019 49th Annual IEEE/IFI P International Con- ference on Dependable Systems and Networks (DSN)

Buchholz, P., Dohndorf, I., Kriege, J.: An online approa ch to estimate parameters of phase-type distributions. In: 2019 49th Annual IEEE/IFI P International Con- ference on Dependable Systems and Networks (DSN). pp. 100–1 11 (June 2019). https://doi.org/10.1109/DSN.2019.00024

2019
[15]

IEEE Trans A utom

Caldwell, B., Cardell-Oliver, R., French, T.: Learning Time Delay Mealy Machines From Programmable Logic Controllers. IEEE Trans A utom. Sci. Eng. 13(2), 1155–1164 (2016). https://doi.org/10.1109/TASE.20 15.2496242, https://doi.org/10.1109/TASE.2015.2496242

2016
[16]

In: Carrasco, R.C., Oncina, J

Carrasco, R.C., Oncina, J.: Learning Stochastic Regula r Grammars by Means of a State Merging Method. In: Carrasco, R.C., Oncina, J. (ed s.) Grammatical Inference and Applications, Second International Colloqu ium, ICGI-94, Alicante, Spain, September 21-23, 1994, Proceedings. ...

1994 doi
[17]

IEEE Transactions on Software Engineering SE-4(3), 178–187 (1978)

Chow, T.: Testing Software Design Modeled by Finite-Sta te Machines. IEEE Transactions on Software Engineering SE-4(3), 178–187 (1978). https://doi.org/10.1109/TSE.1978.231496

1978
[18]

Scientia Sinica 14, 1396–1400 (1965)

Chu, Y.J.: On the shortest arborescence of a directed gra ph. Scientia Sinica 14, 1396–1400 (1965)

1965
[19]

Activity analysis and production and allocation (1951)

Dantzig, G.B.: Application of the simplex method to a tra nsportation problem. Activity analysis and production and allocation (1951)

1951
[20]

D’Argenio, P.R., Katoen, J.: A theory of stochas- tic systems part I: Stochastic automata. Inf. Comput. 203(1), 1–38 (2005). https://doi.org/10.1016/J.IC.2005.07 .001, https://doi.org/10.1016/j.ic.2005.07.001

2005 doi
[21]

In: Johansson, K.H., Yi, W

David, A., Larsen, K.G., Legay, A., Nyman, U., Wasowski, A.: Timed I/O au- tomata: a complete speciﬁcation theory for real-time systems. In: Johansson, K.H., Yi, W. (eds.) Proceedings of the 13th ACM International Conf erence on Hy- brid Systems: Computation and Control, HSCC ...

2010
[22]

Deng, Y., Hennessy, M.: On the semantics of Markov automa ta. Inf. Comput. 222, 139–168 (2013). https://doi.org/10.1016/J.IC.2012.10 .010, https://doi.org/10.1016/j.ic.2012.10.010

2013 doi
[23]

The Annals of Mathematical Stat istics 27(3), 642 – 669 (1956)

Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic Mi nimax Char- acter of the Sample Distribution Function and of the Classi- cal Multinomial Estimator. The Annals of Mathematical Stat istics 27(3), 642 – 669 (1956). https://doi.org/10.1214/aoms/1177 728174, https://doi.org...

1956
[24]

Mathematical Programming 5(1), 88–124 (Dec 1973)

Edmonds, J., Johnson, E.L.: Matching, Euler tours and th e Chi- nese postman. Mathematical Programming 5(1), 88–124 (Dec 1973). https://doi.org/10.1007/BF01580113, https://doi.org/10.1007/BF01580113

1973 doi
[25]

Journal of Rese arch of the national Bureau of Standards B 71(4), 233–240 (1967)

Edmonds, J., et al.: Optimum branchings. Journal of Rese arch of the national Bureau of Standards B 71(4), 233–240 (1967)

1967
[26]

Operations research 43(3), 399–414 (1995)

Eiselt, H.A., Gendreau, M., Laporte, G.: Arc routing pro blems, part ii: The rural postman problem. Operations research 43(3), 399–414 (1995)

1995
[27]

In: Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, LICS 2010, 11-14 July 2010, Edinburgh, Un ited Kingdom

Eisentraut, C., Hermanns, H., Zhang, L.: On Probabilist ic Automata in Contin- uous Time. In: Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, LICS 2010, 11-14 July 2010, Edinburgh, Un ited Kingdom. pp. 342–351. IEEE Computer Society (2010). https://...

2010 doi
[28]

Feldmann, A., Whitt, W.: Fitting mixtures of exponentia ls to long-tail dis- tributions to analyze network. Perform. Evaluation 31(3-4), 245–279 (1998). https://doi.org/10.1016/S0166-5316(97)00003-5

1998 doi
[29]

Feller, W.: An introduction to probability theory and it s applications, Volume 2, vol. 81. John Wiley & Sons (1991)

1991
[30]

In: Proceedings o f the 2021 ACM SIG- COMM 2021 Conference

Ferreira, T., Brewton, H., D’Antoni, L., Silva, A.: Prog nosis: closed-box analy- sis of network protocol implementations. In: Proceedings o f the 2021 ACM SIG- COMM 2021 Conference. p. 762–774. SIGCOMM ’21, Association for Computing Machinery, New York, NY, USA (2021). https:...

2021
[31]

Hartmanns, A., Hermanns, H.: In the quantitative automa ta zoo. Sci. Com- put. Program. 112, 3–23 (2015). https://doi.org/10.1016/J.SCICO.2015.08 .009, https://doi.org/10.1016/j.scico.2015.08.009

2015 doi
[32]

In: Wainwright, R.L., Corchado, J.M., Bechini, A., Hong, J

Henderson, K., Gallagher, B., Eliassi-Rad, T.: EP-MEAN S: an eﬃcient nonpara- metric clustering of empirical probability distributions . In: Wainwright, R.L., Corchado, J.M., Bechini, A., Hong, J. (eds.) Proceedings of the 30th An- nual ACM Symposium on Applied Computing, Sala...

2015
[33]

In: Bertrand, N., Jansen, N

Henry, L., J´ eron, T., Markey, N.: Active Learning of Timed Automata with Unob- servable Resets. In: Bertrand, N., Jansen, N. (eds.) Formal Modeling and Analysis of Timed Systems - 18th International Conference, FORMATS 2020, Vienna, Aus- tria, September 1-3, 2020, Proceeding...

2020 doi
[34]

Journal of the American Statistical Associa tion 58(301), 13–30 (1963)

Hoeﬀding, W.: Probability inequalities for sums of boun ded ran- dom variables. Journal of the American Statistical Associa tion 58(301), 13–30 (1963). https://doi.org/10.1080/01621459. 1963.10500830, https://www.tandfonline.com/doi/abs/10.1080/01621459.1963.10500830

1963
[35]

In: Field, T., Harrison, P.G., Bradley, J.T., Harder, U

Horv´ ath, A., Telek, M.: Phﬁt: A general phase-type ﬁtti ng tool. In: Field, T., Harrison, P.G., Bradley, J.T., Harder, U. (eds.) Computer P erformance Evalua- tion, Modelling Techniques and Tools 12th International Conference, TOOLS 2002, London, UK, April 14-17, 2002, Proce...

2002
[36]

Stochastic Models 23(2), 167–194 (2007)

Horv´ ath, A., Telek, M.: Matching more than three moment s with acyclic phase type distributions. Stochastic Models 23(2), 167–194 (2007). https://doi.org/10.1080/15326340701300712

2007 doi
[37]

In: Dudin, A.N., Turck, K.D

Horv´ ath, G.: Moment matching-based distribution ﬁtti ng with generalized hyper-erlang distributions. In: Dudin, A.N., Turck, K.D. ( eds.) Analytical and Stochastic Modelling Techniques and Applications - 20th In ternational Con- ference, ASMTA 2013, Ghent, Belgium, July 8-10...

2013 doi
[39]

Communications in Statistics

Johnson, M.A., Taaﬀe, M.R.: Matching moments to phase di stributions: Mixtures of erlang distributions of common order. Communications in Statistics. Stochastic Models 5(4), 711–743 (1989). https://doi.org/10.1080/15326348908807131

1989 doi
[40]

Jonsson, B., Vaandrager, F.: Learning Mealy machines wi th timers. Tech. rep., Tech. rep (2018)

2018
[41]

Khayari, R.E.A., Sadre, R., Haverkort, B.R.: Fitting wo rld-wide web request traces with the em-algorithm. Perform. Evaluation 52(2-3), 175–191 (2003). https://doi.org/10.1016/S0166-5316(02)00179-7

2003 doi
[42]

arXiv preprint arXiv:1207.6381 (2012)

Kir´ aly, Z., Kov´ acs, P.: Eﬃcient implementations of minimum-cost ﬂow algorithms. arXiv preprint arXiv:1207.6381 (2012)

2012 arXiv
[43]

In: Li, Y., Tahar, S

Kogel, P., Kl¨ os, V., Glesner, S.: Learning Mealy Machines with Local Timers. In: Li, Y., Tahar, S. (eds.) Formal Methods and Software Engineering - 24th International Conference on Formal Engineering Methods, ICFEM 2023, Bris bane, QLD, Aus- tralia, November 21-24, 2023, Pro...

2023 doi
[44]

In: Finkbeiner, B., Kov´ acs, L

Kruger, L., Junges, S., Rot, J.: Small Test Suites for Act ive Automata Learning. In: Finkbeiner, B., Kov´ acs, L. (eds.) Tools and Algorithms for the Construction and Analysis of Systems. pp. 109–129. Springer Nature Switz erland, Cham (2024)

2024
[45]

In: de Alfaro, L., Gilmo re, S

L´ opez, G.G.I., Hermanns, H., Katoen, J.: Beyond memory less distributions: Model checking semi-markov chains. In: de Alfaro, L., Gilmo re, S. (eds.) Pro- cess Algebra and Probabilistic Methods, Performance Model ing and Veriﬁ- cation: Joint International Workshop, PAPM-PROBM...

2001 doi
[46]

In: Proceedings of 5-th Berkeley Symposium on Math ematical Statistics and Probability/University of California Press (1967)

MacQueen, J.: Some methods for classiﬁcation and analys is of multivariate obser- vations. In: Proceedings of 5-th Berkeley Symposium on Math ematical Statistics and Probability/University of California Press (1967)

1967
[48]

In: Fahrenberg, U., Legay, A., Thrane, C.R

Mao, H., Chen, Y., Jaeger, M., Nielsen, T.D., Larsen, K.G ., Nielsen, B.: Learning Markov Decision Processes for Model Checking. In: Fahrenberg, U., Legay, A., Thrane, C.R. (eds.) Proceedings Quantities in Fo rmal Methods, QFM 2012, Paris, France, 28 August 2012. EPTCS, vol. 1...

2012 doi
[49]

The Annals of Probability 18(3), 1269 – 1283 (1990)

Massart, P.: The Tight Constant in the Dvoretzky-Kiefer - Wolfowitz Inequality. The Annals of Probability 18(3), 1269 – 1283 (1990). https://doi.org/10.1214/aop/1176990 746, https://doi.org/10.1214/aop/1176990746

1990
[50]

In: Margaria, T

de Matos Pedro, A., Crocker, P.A., de Sousa, S.M.: Learni ng Stochas- tic Timed Automata from Sample Executions. In: Margaria, T. , Stef- fen, B. (eds.) Leveraging Applications of Formal Methods, V eriﬁcation and Validation. Technologies for Mastering Change - 5th Int ernation...

2012 doi
[51]

In: Bar- rett, C.W., Davies, M.D., Kahsai, T

Mediouni, B.L., Nouri, A., Bozga, M., Bensalem, S.: Impr oved Learn- ing for Stochastic Timed Models by State-Merging Algorithm s. In: Bar- rett, C.W., Davies, M.D., Kahsai, T. (eds.) NASA Formal Meth ods - 9th International Symposium, NFM 2017, Moﬀett Field, CA, U SA, May 16-...

2017 doi
[52]

Meijer, J., van de Pol, J.: Sound black-box checking in th e LearnLib. Innov. Syst. Softw. Eng. 15(3-4), 267–287 (2019). https://doi.org/10.1007/S11334-019-00342-6, https://doi.org/10.1007/s11334-019-00342-6

2019 doi
[53]

Mohri, M.: Foundations of machine learning (2018)

2018
[54]

Innovations in Systems and Software Engineering 18(3), 417–426 (Sep 2022)

Muˇ skardin, E., Aichernig, B.K., Pill, I., Pferscher, A ., Tappler, M.: AALpy: an active automata learning library. Innovations in Systems and Software Engineering 18(3), 417–426 (Sep 2022). https://doi.org/10.1007/s11334 -022-00449-3

2022 doi
[55]

Software and Systems Modeling 23(2), 503–524 (Apr 2024)

Muˇ skardin, E., Tappler, M., Aichernig, B.K., Pill, I.: Active model learning of stochastic reactive systems (extended version). Software and Systems Modeling 23(2), 503–524 (Apr 2024). https://doi.org/10.1007/s10270 -024-01158-0

2024 doi
[56]

Neider, D., Smetsers, R., Vaandrager, F., Kuppens, H.: B enchmarks for Automata Learning and Conformance Testing, pp. 390–416. Springer In ternational Publish- ing, Cham (2019). https://doi.org/10.1007/978-3-030-22348-9 23

2019 doi
[57]

In: Larsen, K.G., Skou, A

Nicollin, X., Sifakis, J.: An overview and synthesis on t imed process algebras. In: Larsen, K.G., Skou, A. (eds.) Computer Aided V eriﬁca- tion, 3rd International Workshop, CAV ’91, Aalborg, Denmar k, July, 1-4, 1991, Proceedings. Lecture Notes in Computer Science, vol. 575, ...

1991 doi
[58]

CoRR abs/1009.4004 (2010), http://arxiv.org/abs/1009.4004

Nielsen, F.: A family of statistical symmetric divergen ces based on jensen’s in- equality. CoRR abs/1009.4004 (2010), http://arxiv.org/abs/1009.4004

2010 arXiv
[59]

Panchenko, A., Th¨ ummler, A.: Eﬃcient phase-type ﬁttin g with aggregated traﬃc traces. Perform. Evaluation 64(7- 8), 629–645 (2007). https://doi.org/10.1016/J.PEVA.200 6.09.002, https://doi.org/10.1016/j.peva.2006.09.002

2007 doi
[60]

In: Deussen, P

Park, D.M.R.: Concurrency and automata on inﬁnite seque nces. In: Deussen, P. (ed.) Theoretical Computer Science, 5th GI-Conference, Karlsruhe, Ger- many, March 23-25, 1981, Proceedings. Lecture Notes in Comp uter Science, vol. 104, pp. 167–183. Springer (1981). https://doi.or...

1981 doi
[61]

Computers & Operations Research 22(8), 819–828 (1995)

Pearn, W., Wu, T.: Algorithms for the rural postman prob- lem. Computers & Operations Research 22(8), 819–828 (1995). https://doi.org/https://doi.org/10.1016/0305-0548(94)00070-O, https://www.sciencedirect.com/science/article/pii/030505489400070O

1995
[62]

Peled, D., Vardi, M.Y., Yannakakis, M.: Black Box Checki ng, pp. 225–240. Springer US, Boston, MA (1999). https://doi.org/10.1007/ 978-0-387-35578-8 13, https://doi.org/10.1007/978-0-387-35578-8_13

1999 doi
[63]

In: Proceedings of the Sixth Int ernational Confer- ence on Computer Vision (ICCV-98), Bombay, India, January 4 -7, 1998

Rubner, Y., Tomasi, C., Guibas, L.J.: A metric for distri butions with appli- cations to image databases. In: Proceedings of the Sixth Int ernational Confer- ence on Computer Vision (ICCV-98), Bombay, India, January 4 -7, 1998. pp. 59–

1998
[64]

Segala, R., Lynch, N.A.: Probabilistic simulations for probabilistic processes. Nord. J. Comput. 2(2), 250–273 (1995) Automata Learning – Expect Delays! 23

1995
[65]

In: 1st Internationa l Con- ference on Quantitative Evaluation of Systems (QEST 2004), 27- 30 September 2004, Enschede, The Netherlands

Sen, K., Viswanathan, M., Agha, G.: Learning Continuous Time Markov Chains from Sample Executions. In: 1st Internationa l Con- ference on Quantitative Evaluation of Systems (QEST 2004), 27- 30 September 2004, Enschede, The Netherlands. pp. 146–155. IEEE Computer Society (2004)...

2004 arXiv
[66]

https://doi.org/10.110 9/ICCV.1998.710701, https://doi.org/10.1109/ICCV.1998.710701

IEEE Computer Society (1998). https://doi.org/10.110 9/ICCV.1998.710701, https://doi.org/10.1109/ICCV.1998.710701

1998
[67]

ACM Trans

Shijubo, J., Waga, M., Suenaga, K.: Probabilistic black -box checking via active MDP learning. ACM Trans. Embed. Comput. Syst. 22(5s), 148:1–148:26 (2023). https://doi.org/10.1145/3609127, https://doi.org/10.1145/3609127

2023 doi
[68]

In: Bertolino, A ., Canfora, G., El- baum, S.G

Siegmund, J., Siegmund, N., Apel, S.: Views on internal a nd external va- lidity in empirical software engineering. In: Bertolino, A ., Canfora, G., El- baum, S.G. (eds.) 37th IEEE/ACM International Conference o n Software Engineering, ICSE 2015, Florence, Italy, May 16-24, 20...

2015 doi
[69]

The Annals of Mathematical Statist ics 19(2), 279 – 281 (1948)

Smirnov, N.: Table for Estimating the Goodness of Fit of E m- pirical Distributions. The Annals of Mathematical Statist ics 19(2), 279 – 281 (1948). https://doi.org/10.1214/aoms/1177 730256, https://doi.org/10.1214/aoms/1177730256

1948
[70]

In: Heinz, J., de la Higuer a, C., Oates, T

Steﬀen, B., Howar, F., Isberner, M.: Active Automata Lea rning: From DFAs to Interface Programs and Beyond. In: Heinz, J., de la Higuer a, C., Oates, T. (eds.) Proceedings of the Eleventh International Confer ence on Grammati- cal Inference, ICGI 2012, University of Maryland, ...

2012
[71]

Steﬀen, B., Howar, F., Merten, M.: Introduction to Activ e Automata Learn- ing from a Practical Perspective, pp. 256–296. Springer Ber lin Heidel- berg, Berlin, Heidelberg (2011). https://doi.org/10.100 7/978-3-642-21455-4 8, https://doi.org/10.1007/978-3-642-21455-4_8

2011 doi
[72]

Stoelinga, M.: An Introduction to Probabilistic Automa ta. Bull. EATCS 78, 176– 198 (2002)

2002
[73]

In: Proceedings of the 36th Internatio nal Conference on Neural Information Processing Systems

Suilen, M., Sim˜ ao, T.D., Parker, D., Jansen, N.: Robustanytime learning of Markov decision processes. In: Proceedings of the 36th Internatio nal Conference on Neural Information Processing Systems. NIPS ’22, Curran Associates Inc., Red Hook, NY, USA (2024)

2024
[74]

, Larsen, K.G.: L∗-based learning of Markov decision processes (extended version)

Tappler, M., Aichernig, B.K., Bacci, G., Eichlseder, M. , Larsen, K.G.: L∗-based learning of Markov decision processes (extended version). Formal Aspects of Com- puting 33(4), 575–615 (Aug 2021). https://doi.org/10.1007/s00165 -021-00536-5

2021 doi
[75]

Telek, M., Horv´ ath, G.: A minimal representation of markov arrival processes and a moments matching method. Perform. Evaluation 64(9-12), 1153–1168 (2007). https://doi.org/10.1016/j.peva.2007.06.001

2007 doi
[76]

In: ´Abrah´ am, E., Jr., M.M

Teng, Y., Zhang, M., An, J.: Learning Deterministic Mult i-Clock Timed Automata. In: ´Abrah´ am, E., Jr., M.M. (eds.) Proceedings of the 27th ACM International Conference on Hybrid Systems: Co mpu- tation and Control, HSCC 2024, Hong Kong SAR, China, May 14-1 6,

2024
[77]

In: Fisman, D., Rosu, G

Vaandrager, F., Garhewal, B., Rot, J., Wißmann, T.: A New Approach for Active Automata Learning Based on Apartness. In: Fisman, D., Rosu, G. (eds.) Tools and Algorithms for the Construction and Analysis of Systems. pp . 223–243. Springer International Publishing, Cham (2022)

2022
[78]

IEEE Trans

Th¨ ummler, A., Buchholz, P., Telek, M.: A novel approach for phase-type ﬁtting with the EM algorithm. IEEE Trans. Dependable Secur. Comput . 3(3), 245–258 (2006). https://doi.org/10.1109/TDSC.2006.27 24 G. Dengler et al

2006 doi
[79]

Cybe rnetics 9(4), 653–665 (Jul 1973)

Vasilevskii, M.P.: Failure diagnosis of automata. Cybe rnetics 9(4), 653–665 (Jul 1973). https://doi.org/10.1007/BF0106 8590, https://doi.org/10.1007/BF01068590

1973 doi
[80]

In: Bouyer, P., van de Pol, J

Vaandrager, F., Melse, I.: New Fault Domains for Conform ance Test- ing of Finite State Machines. In: Bouyer, P., van de Pol, J. (e ds.) 36th International Conference on Concurrency Theory (CONC UR 2025). Leibniz International Proceedings in Informatics (LIPIcs ), vol. 348, pp...

2025 doi
[81]

In: Ames, A.D., Seshia, S.A., Deshmukh , J

Waga, M.: Falsiﬁcation of cyber-physical systems with r obustness-guided black-box checking. In: Ames, A.D., Seshia, S.A., Deshmukh , J. (eds.) HSCC ’20: 23rd ACM International Conference on Hybrid Syste ms: Com- putation and Control, Sydney, New South Wales, Australia, A pril 21-24,
[82]

Verwer, S., de Weerdt, M., Witteveen, C.: Eﬃciently iden tifying deterministic real-time automata from labeled data. Mach. Learn. 86(3), 295–333 (2012). https://doi.org/10.1007/S10994-01 1-5265-4, https://doi.org/10.1007/s10994-011-5265-4

2012 doi
[85]

In: Enea, C., Lal, A

Waga, M.: Active Learning of Deterministic Timed Automa ta with Myhill-Nerode Style Characterization. In: Enea, C., Lal, A. (eds.) Comput er Aided Veriﬁcation. pp. 3–26. Springer Nature Switzerland, Cham (2023) A Semantics of Mealy Delay Machines This section spells out the fo...

2023
[86]

We will show that R fulﬁlls all requirements for being a bisimulation relation between M and M′ according to Deﬁnition 2

We deﬁne relation R = {(s, s′) | s ∼IO s′} (for this use of ∼IO to match Deﬁnition 1 we work in the disjoint union of the two MMs with arbitrary initial state but deﬁne the relation on S × S′). We will show that R fulﬁlls all requirements for being a bisimulation relation betw...

2001
[2020]

11:1–11:13

pp. 11:1–11:13. ACM (2020). https://doi.org/10.114 5/3365365.3382193, https://doi.org/10.1145/3365365.3382193

2020
[2024]

6:1–6:11

pp. 6:1–6:11. ACM (2024). https://doi.org/10.1145/ 3641513.3650124, https://doi.org/10.1145/3641513.3650124

2024

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Alur, R., Dill, D.L.: A theory of timed automata. Theor. Co mput. Sci. 126(2), 183–235 (1994). https://doi.org/10.1016/0304-3975 (94)90010-8, https://doi.org/10.1016/0304-3975(94)90010-8 18 G. Dengler et al

doi:10.1016/0304-3975 1994

[2] [2]

In: Biere, A., Parker, D

An, J., Chen, M., Zhan, B., Zhan, N., Zhang, M.: Learning On e-Clock Timed Au- tomata. In: Biere, A., Parker, D. (eds.) Tools and Algorithms for the Construction and Analysis of Systems. pp. 444–462. Springer Internation al Publishing, Cham (2020)

2020

[3] [3]

Science China Information Sciences 64(9), 192103 (Aug 2021)

An, J., Wang, L., Zhan, B., Zhan, N., Zhang, M.: Learning real-time automata. Science China Information Sciences 64(9), 192103 (Aug 2021). https://doi.org/10.1007/s11432-019- 2767-4, https://doi.org/10.1007/s11432-019-2767-4

doi:10.1007/s11432-019- 2021

[4] [4]

ACM Trans

An, J., Zhan, B., Zhan, N., Zhang, M.: Learning nondetermi nistic real- time automata. ACM Trans. Embed. Comput. Syst. 20(5s) (Sep 2021). https://doi.org/10.1145/3477030, https://doi.org/10.1145/3477030

doi:10.1145/3477030 2021

[5] [5]

Information and Computation 75(2), 87–106 (1987)

Angluin, D.: Learning regular sets from queries and coun- terexamples. Information and Computation 75(2), 87–106 (1987). https://doi.org/https://doi.org/10.1016/0890 -5401(87)90052-6, https://www.sciencedirect.com/science/article/pii/0890540187900526

work page arXiv 1987

[6] [7]

Scandinavian Journal of Statistics 23(4), 419–441 (1996), http://www.jstor.org/stable/4616418

Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics 23(4), 419–441 (1996), http://www.jstor.org/stable/4616418

work page arXiv 1996

[7] [8]

In: Wani, M.A., Sethi, I.K., Shi, W., Qu, G., Raicu, D.S., Jin, R

Bacci, G., Ing´ olfsd´ ottir, A., Larsen, K.G., Reynouard , R.: Active Learn- ing of Markov Decision Processes using Baum-Welch algorith m. In: Wani, M.A., Sethi, I.K., Shi, W., Qu, G., Raicu, D.S., Jin, R. (eds.) 20th IEEE International Conference on Machine Learning and Appli- cations, ICMLA 2021, Pasadena, CA, USA, December 13-16, 202 1. pp. 1203–1208....

arXiv 2021

[8] [9]

In: Flum, J., Gr¨ adel, E., Wilke, T

Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.: Rea chability in continuous- time markov reward decision processes. In: Flum, J., Gr¨ adel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives [in Honor of Wo lfgang Thomas]. Texts in Logic and Games, vol. 2, pp. 53–72. Amsterdam Univer sity Press (2008)

2008