, For each R s.t. h(R) = Rev, for each R ? ls(R) and P ?{R }\{R} is consistent with E (This consistency can be verified by the condition (I, J) ? E s

?. If and P. \{r}?{r-}, reach(?, ?) = False for P and P satisfies Re, return P Bibliography

M. Emna-ben-abdallah, O. Folschette, M. Roux, and . Magnin, Exhaustive analysis of dynamical properties of biological regulatory networks with answer set programming, 2015 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), pp.281-285, 2015.

P. Parosh-aziz-abdulla, N. Bjesse, and . Eén, Symbolic reachability analysis based on SAT-solvers, International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp.411-425, 2000.

T. Akutsu, M. Hayashida, W. Ching, and M. Ng, Control of boolean networks: Hardness results and algorithms for tree structured networks, Journal of theoretical biology, vol.244, issue.4, pp.670-679, 2007.

K. Baba, R. Shibata, and M. Sibuya, Partial correlation and conditional correlation as measures of conditional independence, Australian & New Zealand Journal of Statistics, vol.46, issue.4, pp.657-664, 2004.

C. Baral, Knowledge representation, reasoning and declarative problem solving, 2003.

G. Bernot and F. Tahi, Behaviour preservation of a biological regulatory network when embedded into a larger network, Fundamenta Informaticae, vol.91, issue.3-4, pp.463-485, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00646001

R. Bonneau, J. David, P. Reiss, M. Shannon, L. Facciotti et al., The Inferelator: an algorithm for learning parsimonious regulatory networks from systems-biology data sets de novo, Genome biology, vol.7, issue.5, p.36, 2006.

M. Connie and . Borror, Practical nonparametric statistics, Journal of Quality Technology, vol.33, issue.2, p.260, 2001.

. Robert-k-brayton, D. Gary, A. Hachtel, F. Sangiovanni-vincentelli, A. Somenzi et al., VIS: A system for verification and synthesis, International conference on computer aided verification, pp.428-432, 1996.

R. Jerry, E. M. Burch, . Clarke, L. Kenneth, D. L. Mcmillan et al., Symbolic model checking: 10 20 states and beyond. Information and computation, vol.98, pp.142-170, 1992.

X. Chai, M. Magnin, and O. Roux, A heuristic for reachability problem in asynchronous binary automata networks, 2018.

X. Chai, T. Ribeiro, M. Magnin, O. Roux, and K. Inoue, Static analysis and stochastic search for reachability problem, 9th Static Analysis in Systems Biology, affiliated with Static Analysis Symposium, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01951251

X. Chai, T. Ribeiro, M. Magnin, O. Roux, and K. Inoue, Using reachability properties of logic program for revising biological models, Progress of International Conference on Inductive Logic Programming, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01951247

T. Chatain, S. Haar, and L. Paulevé, Boolean networks: Beyond generalized asynchronicity, International Workshop on Cellular Automata and Discrete Complex Systems, pp.29-42
URL : https://hal.archives-ouvertes.fr/hal-01768359

. Springer, , 2018.

A. Cheng, J. Esparza, and J. Palsberg, Complexity results for 1-safe nets, Theoretical Computer Science, vol.147, issue.1-2, pp.117-136, 1995.

A. Cimatti, E. Clarke, F. Giunchiglia, and M. Roveri, NuSMV: a new symbolic model checker, International Journal on Software Tools for Technology Transfer, vol.2, issue.4, pp.410-425, 2000.

E. Clarke, A. Biere, R. Raimi, and Y. Zhu, Bounded model checking using satisfiability solving. Formal methods in system design, vol.19, pp.7-34, 2001.

M. Edmund and . Clarke, The birth of model checking, 25 Years of Model Checking, pp.1-26, 2008.

M. Edmund, E. Clarke, and . Emerson, Design and synthesis of synchronization skeletons using branching time temporal logic, Workshop on Logic of Programs, pp.52-71, 1981.

M. Edmund, Q. Clarke, and . Wang, 2 5 years of model checking, International Andrei Ershov Memorial Conference on Perspectives of System Informatics, pp.26-40, 2014.

C. Daws and S. Tripakis, Model checking of real-time reachability properties using abstractions. Tools and Algorithms for the Construction and Analysis of Systems, pp.313-329, 1998.

A. De, L. Fuente, N. Bing, I. Hoeschele, and P. Mendes, Discovery of meaningful associations in genomic data using partial correlation coefficients, Bioinformatics, vol.20, issue.18, pp.3565-3574, 2004.

G. Didier, E. Remy, and C. Chaouiya, Mapping multivalued onto boolean dynamics, Journal of theoretical biology, vol.270, issue.1, pp.177-184, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00656334

E. Stanimirova-dimitrova, Polynomial models for systems biology: Data discretization and term order effect on dynamics, 2006.

E. Dubrova and M. Teslenko, A SAT-based algorithm for finding attractors in synchronous Boolean networks, IEEE/ACM transactions on computational biology and bioinformatics, vol.8, issue.5, pp.1393-1399, 2011.

J. Esparza, Reachability in live and safe free-choice Petri nets is NP-complete, Theoretical Computer Science, vol.198, issue.1-2, pp.211-224, 1998.

A. Fehnker, R. Huuck, P. Jayet, M. Lussenburg, and F. Rauch, Goanna-a static model checker, International Workshop on Formal Methods for Industrial Critical Systems, pp.297-300, 2006.

J. Foerster, Learning to communicate with deep multi-agent reinforcement learning, Advances in Neural Information Processing Systems, pp.2137-2145, 2016.

M. Folschette, L. Paulevé, M. Magnin, and O. Roux, Sufficient conditions for reachability in automata networks with priorities, Theoretical Computer Science, vol.608, pp.66-83, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01202671

N. Friedman and M. Linial, Iftach Nachman, and Dana Pe'er. Using Bayesian networks to analyze expression data, Journal of computational biology, vol.7, issue.3-4, pp.601-620, 2000.

M. Gebser, R. Kaminski, B. Kaufmann, M. Ostrowski, T. Schaub et al., Theory solving made easy with clingo 5, OASIcs-OpenAccess Series in Informatics, vol.52, 2016.

L. Glass, Classification of biological networks by their qualitative dynamics, Journal of Theoretical Biology, vol.54, issue.1, pp.85-107, 1975.

L. Glass and . Stuart-a-kauffman, The logical analysis of continuous, non-linear biochemical control networks, Journal of theoretical Biology, vol.39, issue.1, pp.103-129, 1973.

. Rh-hardin, . Rp-kurshan, M. Y. Sk-shukla, and . Vardi, A new heuristic for bad cycle detection using BDDs, International Conference on Computer Aided Verification, pp.268-278, 1997.

D. Harel, O. Kupferman, and M. Y. Vardi, On the complexity of verifying concurrent transition systems, Information and Computation, vol.173, issue.2, pp.143-161, 2002.

I. Harvey and T. Bossomaier, Time out of joint: Attractors in asynchronous random Boolean networks, Proceedings of the Fourth European Conference on Artificial Life, pp.67-75, 1997.

J. Hauke and T. Kossowski, Comparison of values of Pearson's and Spearman's correlation coefficients on the same sets of data. Quaestiones geographicae, vol.30, pp.87-93, 2011.

G. J. Holzmann, The model checker SPIN, IEEE Transactions on software engineering, vol.23, issue.5, pp.279-295, 1997.

P. John, F. Huelsenbeck, and . Ronquist, MRBAYES: Bayesian inference of phylogenetic trees, Bioinformatics, vol.17, issue.8, pp.754-755, 2001.

L. Clark and . Hull, The correlation coefficient and its prognostic significance, The Journal of Educational Research, vol.15, issue.5, pp.327-338, 1927.

K. Inoue, Logic programming for boolean networks, IJCAI proceedings-international joint conference on artificial intelligence, vol.22, p.924, 2011.

S. Kauffman, Homeostasis and differentiation in random genetic control networks, Nature, vol.224, pp.177-178, 1969.

Z. Khalis, J. Comet, A. Richard, and G. Bernot, The SMBioNet method for discovering models of gene regulatory networks, Genes, genomes and genomics, vol.3, issue.1, pp.15-22, 2009.

J. Kol?ák, S. David?afránek, L. Haar, and . Paulevé, Parameter Space Abstraction and Unfolding Semantics of Discrete Regulatory Networks, Theoretical Computer Science, 2018.

B. Larget, L. Donald, and . Simon, Markov chain monte carlo algorithms for the bayesian analysis of phylogenetic trees, Molecular biology and evolution, vol.16, issue.6, pp.750-759, 1999.

H. Li and Y. Wang, On reachability and controllability of switched boolean control networks, Automatica, vol.48, issue.11, pp.2917-2922, 2012.

H. Li, Y. Wang, and Z. Liu, Stability analysis for switched boolean networks under arbitrary switching signals, IEEE Transactions on Automatic Control, vol.59, issue.7, pp.1978-1982, 2014.

P. Lincoln and A. Tiwari, Symbolic systems biology: Hybrid modeling and analysis of biological networks, International Workshop on Hybrid Systems: Computation and Control, pp.660-672, 2004.

T. David-martínez-martínez, K. Ribeiro, and . Inoue, Learning probabilistic action models from interpretation transitions, Proceedings of the Technical Communications of the 31st International Conference on Logic Programming (ICLP 2015), pp.1-14, 2015.

W. Ernst and . Mayr, An algorithm for the general Petri net reachability problem, SIAM Journal on computing, vol.13, issue.3, pp.441-460, 1984.

. Kenneth-l-mcmillan, Symbolic model checking, Symbolic Model Checking, pp.25-60, 1993.

J. Peter-bro-miltersen, I. Radhakrishnan, and . Wegener, On converting CNF to DNF, Theoretical computer science, vol.347, issue.1-2, pp.325-335, 2005.

D. Nguyen-tuong and J. Peters, Model learning for robot control: a survey, Cognitive processing, vol.12, issue.4, pp.319-340, 2011.

R. Opgen-rhein and K. Strimmer, From correlation to causation networks: a simple approximate learning algorithm and its application to high-dimensional plant gene expression data, BMC systems biology, vol.1, issue.1, p.37, 2007.

L. Paulevé, Pint: a static analyzer for transient dynamics of qualitative networks with IPython interface, CMSB 2017 -15th conference on Computational Methods for Systems Biology, vol.10545, pp.309-316, 2017.

L. Paulevé, Reduction of qualitative models of biological networks for transient dynamics analysis, IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.15, issue.4, pp.1167-1179, 2018.

L. Paulevé, G. Andrieux, and H. Koeppl, Under-approximating cut sets for reachability in large scale automata networks, International Conference on Computer Aided Verification, pp.69-84, 2013.

L. Paulevé, C. Chancellor, M. Folschette, M. Magnin, and O. Roux, Logical Modeling of Biological Systems, chapter Analyzing Large Network Dynamics with Process Hitting, pp.125-166, 2014.

L. Paulevé, M. Magnin, and O. Roux, Refining dynamics of gene regulatory networks in a stochastic ?-calculus framework, Transactions on computational systems biology xiii, pp.171-191, 2011.

L. Paulevé, M. Magnin, and O. Roux, From the Process Hitting to Petri Nets and Back, 2012.

L. Paulevé, M. Magnin, and O. Roux, Static analysis of biological regulatory networks dynamics using abstract interpretation, Mathematical Structures in Computer Science, vol.22, issue.04, pp.651-685, 2012.

. James-l-peterson, Petri nets, ACM Computing Surveys (CSUR), vol.9, issue.3, pp.223-252, 1977.

W. John, . Pinney, G. David-r-westhead, and . Mcconkey, Petri net representations in systems biology, Biochemical Society Transactions, vol.31, issue.6, pp.1513-1515, 2003.

J. , P. Queille, and J. Sifakis, Specification and verification of concurrent systems in CESAR, International Symposium on programming, pp.337-351, 1982.

T. Ribeiro, M. Folschette, M. Magnin, O. Roux, and K. Inoue, Learning dynamics with synchronous, asynchronous and general semantics, the 28th International Conference on Inductive Logic Programming, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01826564

T. Ribeiro and K. Inoue, Learning prime implicant conditions from interpretation transition, Inductive Logic Programming, pp.108-125, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01710486

T. Ribeiro, S. Tourret, M. Folschette, M. Magnin, D. Borzacchiello et al., Inductive learning from state transitions over continuous domains, International Conference on Inductive Logic Programming, pp.124-139, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01655644

A. Rocca, N. Mobilia, É. Fanchon, T. Ribeiro, L. Trilling et al., ASP for construction and validation of regulatory biological networks, Logical Modeling of Biological Systems, pp.167-206, 2014.
URL : https://hal.archives-ouvertes.fr/hal-02272463

C. Rodrigues, P. Gérard, C. Rouveirol, and H. Soldano, Active learning of relational action models, International Conference on Inductive Logic Programming, pp.302-316

. Springer, , 2011.

A. Saadatpour, I. Albert, and R. Albert, Attractor analysis of asynchronous boolean models of signal transduction networks, Journal of theoretical biology, vol.266, issue.4, pp.641-656, 2010.

J. Saez-rodriguez, L. Simeoni, J. A. Lindquist, R. Hemenway, U. Bommhardt et al., A logical model provides insights into T cell receptor signaling, PLoS computational biology, vol.3, issue.8, p.163, 2007.

R. Samaga, J. Saez-rodriguez, L. G. Alexopoulos, K. Peter, S. Sorger et al., The logic of EGFR/ErbB signaling: theoretical properties and analysis of high-throughput data, PLoS computational biology, vol.5, issue.8, p.1000438, 2009.

T. Schaffter, D. Marbach, and D. Floreano, GeneNetWeaver: in silico benchmark generation and performance profiling of network inference methods, Bioinformatics, vol.27, issue.16, pp.2263-2270, 2011.

K. Shinozaki, K. Yamaguchi-shinozaki, and M. Seki, Regulatory network of gene expression in the drought and cold stress responses, Current opinion in plant biology, vol.6, issue.5, pp.410-417, 2003.

R. Singhania, M. Sramkoski, J. W. Jacobberger, and J. Tyson, A hybrid model of mammalian cell cycle regulation, PLoS computational biology, vol.7, issue.2, p.1001077, 2011.

S. El-houssine, Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach, Dynamics and stability of Systems, vol.4, issue.3-4, pp.565-583, 1989.

R. Tarjan, Depth-first search and linear graph algorithms, SIAM journal on computing, vol.1, issue.2, pp.146-160, 1972.

D. Thieffry and R. Thomas, Dynamical behaviour of biological regulatory networks-ii. immunity control in bacteriophage lambda, Bulletin of mathematical biology, vol.57, issue.2, pp.277-297, 1995.

R. Thomas, Logical analysis of systems comprising feedback loops, Journal of Theoretical Biology, vol.73, issue.4, pp.631-656, 1978.

R. Thomas and R. , Biological feedback, 1990.
URL : https://hal.archives-ouvertes.fr/hal-00087681

E. George-von-dassow, . Meir, M. Edwin, G. Munro, and . Odell, The segment polarity network is a robust developmental module, Nature, vol.406, issue.6792, p.188, 2000.

K. Vorobyov and P. Krishnan, Comparing model checking and static program analysis: A case study in error detection approaches, proceedings of 5th International Conference on Systems Software Verification, pp.1-7, 2010.

Q. Wang, P. Zuliani, S. Kong, S. Gao, and E. M. Clarke, SReach: A probabilistic bounded delta-reachability analyzer for stochastic hybrid systems, International Conference on Computational Methods in Systems Biology, pp.15-27, 2015.

B. Wo?na, A. Zbrzezny, and W. Penczek, Checking reachability properties for timed automata via SAT, Fundamenta Informaticae, vol.55, issue.2, pp.223-241, 2003.

Y. Yamamoto, A. Rougny, H. Nabeshima, K. Inoue, H. Moriya et al., Completing SBGN-AF networks by logic-based hypothesis finding, International Conference on Formal Methods in Macro-Biology, vol.109, p.110, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01110133