M. Adélade and G. Sutre, Parametric analysis and abstraction of genetic regulatory networks, Proc. BioCONCUR 04, 2004.

R. Alur, T. Dang, and F. Ivan?i´ivan?i´c, Counter-Example Guided Predicate Abstraction of Hybrid Systems, Theor. Comput. Sci, vol.354, issue.2, pp.250-271, 2006.
DOI : 10.1007/3-540-36577-X_15

R. Alur, T. Henzinger, G. Lafferriere, and G. Pappas, Discrete abstractions of hybrid systems, Proceedings of the IEEE, vol.88, issue.7, pp.971-984, 2000.
DOI : 10.1109/5.871304

M. Antoniotti, A. Policriti, N. Ugel, and B. Mishra, Model Building and Model Checking for Biochemical Processes, Cell Biochemistry and Biophysics, vol.38, issue.3, pp.271-286, 2003.
DOI : 10.1385/CBB:38:3:271

R. Armoni, L. Fix, A. Flaisher, R. Gerth, B. Ginsburg et al., The ForSpec Temporal Logic: A New Temporal Property-Specification Language, Proceedings of the 8th International Conference on Tools and Algorithms for the Construction and Analysis of Systems TACAS'02 of Lecture Notes in Computer Science, pp.296-211, 2002.
DOI : 10.1007/3-540-46002-0_21

P. Ballarini, T. Mazza, A. Palmisano, and A. Csikasz-nagy, Studying Irreversible Transitions in a Model of Cell Cycle Regulation, Proceedings of the third International Workshop on Practical Applications of Stochastic Modelling, 2008.
DOI : 10.1016/j.entcs.2009.02.049

J. Barnat, L. Brim, I. Cerná, S. Drazan, and D. Safranek, Parallel Model Checking Large-Scale Genetic Regulatory Networks with DiVinE, From Biology to Concurrency and Back, 2007.
DOI : 10.1016/j.entcs.2007.12.001

H. Barringer, A. Goldberg, K. Havelund, and K. Sen, Rulebased runtime verification, Proceedings of the 5th International Conference on Verification, Model Checking, and Abstract Interpretation VMCAI'04 of Lecture Notes in Computer Science, pp.44-57, 2004.

G. Batt, D. Bergamini, H. De-jong, H. Gavarel, and R. Mateescu, Model Checking Genetic Regulatory Networks Using GNA and CADP, Eleventh International SPIN Workshop on Model Checking of Software, SPIN 2004, pp.158-163, 2004.
DOI : 10.1007/978-3-540-24732-6_12

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.114.7325

G. Batt, H. De-jong, M. Page, and J. Geiselmann, Symbolic reachability analysis of genetic regulatory networks using discrete abstractions, Automatica, vol.44, issue.4, pp.982-989, 2008.
DOI : 10.1016/j.automatica.2007.08.004

URL : https://hal.archives-ouvertes.fr/hal-00319485

G. Batt, D. Ropers, H. De-jong, J. Geiselmann, R. Mateescu et al., Validation of qualitative models of genetic regulatory networks by model checking: analysis of the nutritional stress response in Escherichia coli, Bioinformatics, vol.21, issue.Suppl 1, pp.19-28, 2005.
DOI : 10.1093/bioinformatics/bti1048

URL : https://hal.archives-ouvertes.fr/hal-00171939

G. Batt, B. Yordanov, C. Belta, and R. Weiss, Robustness analysis and tuning of synthetic gene networks, Bioinformatics, vol.23, issue.18, pp.2415-2422, 2007.
DOI : 10.1093/bioinformatics/btm362

I. Beer, S. Ben-david, C. Eisner, D. Fisman, A. Gringauze et al., The Temporal Logic Sugar, Proceedings of the 13th International Conference on Computer Aided Verification CAV of Lecture Notes in Computer Science, pp.363-367, 2001.
DOI : 10.1007/3-540-44585-4_33

I. Beer, S. Ben-david, and A. Landver, On-the-fly model checking of RCTL formulas, Proceedings of the 10th International Conference on Computer Aided Verification CAV'98 of Lecture Notes in Computer Science, pp.184-194, 1998.
DOI : 10.1007/BFb0028744

G. Bernot, J. Comet, A. Richard, and J. Guespin, Application of formal methods to biological regulatory networks: Extending Thomas, p.38, 2004.

K. Bettenbrock, S. Fischer, A. Kremling, K. Jahreis, T. Sauter et al., A Quantitative Approach to Catabolite Repression in Escherichia coli, Journal of Biological Chemistry, vol.281, issue.5, pp.2578-84, 2006.
DOI : 10.1074/jbc.M508090200

D. Bosnacki, H. Ten-eikelder, M. Steijaert, and E. De-vink, Stochastic Analysis of Amino Acid Substitution in Protein Synthesis, Computational Methods in Systems Biology, CMSB-08, pp.367-386, 2008.
DOI : 10.1007/978-3-540-88562-7_25

J. A. Brzozowski, Derivatives of Regular Expressions, Journal of the ACM, vol.11, issue.4, pp.481-494, 1964.
DOI : 10.1145/321239.321249

M. Calder, V. Vyshemirsky, D. Gilbert, and R. Orton, Analysis of signalling pathways using the PRISM model checker, Computational Methods in Systems Biology, CMSB-05, pp.79-90, 2005.

N. Chabrier-rivier, M. Chiaverini, V. Danos, F. Fages, and V. Schächter, Modeling and querying biomolecular interaction networks, Theoretical Computer Science, vol.325, issue.1, pp.25-44, 2004.
DOI : 10.1016/j.tcs.2004.03.063

URL : http://doi.org/10.1016/j.tcs.2004.03.063

C. Chaouiya, Petri net modelling of biological networks, Briefings in Bioinformatics, vol.8, issue.4, pp.210-219, 2007.
DOI : 10.1093/bib/bbm029

C. Chassagnole, N. Noisommit-rizzi, J. Schmid, K. Mauch, and M. Reuss, Dynamic modeling of the central carbon metabolism ofEscherichia coli, Biotechnology and Bioengineering, vol.81, issue.1, pp.53-73, 2002.
DOI : 10.1002/bit.10288

K. Chen, L. Calzone, A. Csikasz-nagy, F. Cross, B. Novak et al., Integrative Analysis of Cell Cycle Control in Budding Yeast, Molecular Biology of the Cell, vol.15, issue.8, pp.3841-3862, 2004.
DOI : 10.1091/mbc.E03-11-0794

A. Chutinan and B. Krogh, Verification of infinite-state dynamic systems using approximate quotient transition systems, IEEE Transactions on Automatic Control, vol.46, issue.9, 2001.
DOI : 10.1109/9.948467

A. Cimatti, E. Clarke, F. Giunchiglia, and M. Roveri, NUSMV: a new symbolic model checker, International Journal on Software Tools for Technology Transfer (STTT), pp.410-425, 2000.
DOI : 10.1007/s100090050046

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.4520

R. Cleaveland and B. Steffen, A linear-time model-checking algorithm for the alternation-free modal mu-calculus, Formal Methods in System Design, vol.27, issue.2, pp.121-147, 1993.
DOI : 10.1007/BF01383878

M. Dam, CTL??? and ECTL??? as fragments of the modal ??-calculus, Theoretical Computer Science, vol.126, issue.1, pp.77-96, 1994.
DOI : 10.1016/0304-3975(94)90269-0

W. Damm and D. Harel, Lscs: Breathing life into message sequence charts, Formal Methods in System Design, vol.19, issue.1, pp.45-80, 2001.
DOI : 10.1023/A:1011227529550

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.5472

H. De-jong, Modeling and Simulation of Genetic Regulatory Systems: A Literature Review, Journal of Computational Biology, vol.9, issue.1, pp.67-103, 2002.
DOI : 10.1089/10665270252833208

H. De-jong, J. Geiselmann, G. Batt, C. Hernandez, and M. Page, Qualitative simulation of the initiation of sporulation in, Bulletin of Mathematical Biology, vol.66, issue.2, pp.261-299, 2004.
DOI : 10.1016/j.bulm.2003.08.009

URL : https://hal.archives-ouvertes.fr/inria-00072061

H. De-jong, J. Gouzé, C. Hernandez, M. Page, T. Sari et al., Qualitative simulation of genetic regulatory networks using piecewise-linear models, Bulletin of Mathematical Biology, vol.66, issue.2, pp.301-340, 2004.
DOI : 10.1016/j.bulm.2003.08.010

URL : https://hal.archives-ouvertes.fr/hal-00173849

D. Dubnau and R. Losick, Bistability in bacteria, Molecular Microbiology, vol.196, issue.3, pp.564-572, 2006.
DOI : 10.1111/j.1365-2958.2005.04659.x

E. A. Emerson and J. Y. Halpern, "Sometimes" and "not never" revisited, Proceedings of the 10th ACM SIGACT-SIGPLAN symposium on Principles of programming languages , POPL '83, pp.127-140151, 1983.
DOI : 10.1145/567067.567081

E. A. Emerson and C. Lei, Efficient model checking in fragments of the propositional mu-calculus, Proceedings of the 1st International Symposium on Logic in Computer Science LICS'86, pp.267-278, 1986.

E. A. Emerson and C. Lei, Modalities for model checking: branching time logic strikes back, Science of Computer Programming, vol.8, issue.3, 1987.
DOI : 10.1016/0167-6423(87)90036-0

M. J. Fischer and R. E. Ladner, Propositional dynamic logic of regular programs, Journal of Computer and System Sciences, vol.18, issue.2, pp.194-211, 1979.
DOI : 10.1016/0022-0000(79)90046-1

J. Fisher and T. Henzinger, Executable cell biology, Nature Biotechnology, vol.2034, issue.11, pp.1239-1250, 2007.
DOI : 10.1038/nbt1356

J. Fisher, N. Piterman, A. Hajnal, and T. Henzinger, Predictive Modeling of Signaling Crosstalk during C. elegans Vulval Development, PLoS Computational Biology, vol.77, issue.5, p.92, 2007.
DOI : 0016-6731(1974)077[0071:TGOCE]2.0.CO;2

J. Fisher, N. Piterman, E. J. Hubbard, M. J. Stern, and D. Harel, Computational insights into Caenorhabditis elegans vulval development, Proceedings of the National Academy of Sciences of the USA, pp.1951-1956, 2005.
DOI : 10.1073/pnas.0409433102

URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC548551

H. Garavel, OPEN/C??SAR: An open software architecture for verification, simulation, and testing, Proceedings of the First International Conference on Tools and Algorithms for the Construction and Analysis of Systems TACAS'98, pp.68-84, 1998.
DOI : 10.1007/BFb0054165

H. Garavel, F. Lang, R. Mateescu, and W. Serwe, CADP??2006: A Toolbox for the Construction and Analysis of Distributed Processes, Proceedings of the 19th International Conference on Computer Aided Verification CAV of Lecture Notes in Computer Science, pp.158-163, 2007.
DOI : 10.1007/978-3-540-73368-3_18

URL : https://hal.archives-ouvertes.fr/inria-00189021

R. Ghosh and C. Tomlin, Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling, Systems Biology, vol.1, issue.1, pp.170-183, 2004.
DOI : 10.1049/sb:20045019

L. Glass and S. Kauffman, The logical analysis of continuous, non-linear biochemical control networks, Journal of Theoretical Biology, vol.39, issue.1, pp.103-129, 1973.
DOI : 10.1016/0022-5193(73)90208-7

F. Grognard, H. De-jong, and J. Gouzé, Piecewise-Linear Models of Genetic Regulatory Networks: Theory and Example, Biology and Control Theory: Current Challenges of Lecture Notes in Control and Information Science, pp.137-159, 2007.
DOI : 10.1007/978-3-540-71988-5_7

URL : https://hal.archives-ouvertes.fr/hal-01091662

R. Gutierrez-ríos, J. Freyre-gonzalez, O. Resendis, J. Collado-vides, M. Saier et al., Identification of regulatory network topological units coordinating the genome-wide transcriptional response to glucose in Escherichia coli, BMC Microbiology, vol.7, issue.1, p.53, 2007.
DOI : 10.1186/1471-2180-7-53

K. Hamaguchi, H. Hiraishi, and S. Yajima, Branching time regular temporal logic for model checking with linear time complexity, Proceedings of the 2nd International Conference on Computer Aided Verification CAV'90, pp.253-262, 1990.
DOI : 10.1007/BFb0023739

R. Hengge-aronis, The general stress response in Escherichia coli, Bacterial Stress Responses, pp.161-77, 2000.

G. Holzmann, The SPIN Model Checker ? Primer and Reference Manual, 2003.

G. Huisman, D. Siegele, M. Zambrano, and R. Kolter, Morphological and physiological changes during stationary phase, Escherichia coli and Salmonella: Cellular and Molecular Biology, pp.1672-82, 1996.

I. Iec, LOTOS ? a formal description technique based on the temporal ordering of observational behaviour, International Organization for Standardization ? Information Processing Systems ? Open Systems Interconnection, 1989.

S. Jha, B. Krogh, J. Weimer, and E. Clarke, Reachability for Linear Hybrid Automata Using Iterative Relaxation Abstraction, Proc. HSCC 2007, pp.287-300, 2007.
DOI : 10.1007/978-3-540-71493-4_24

C. Joubert and R. Mateescu, Distributed Local Resolution of Boolean Equation Systems, 13th Euromicro Conference on Parallel, Distributed and Network-Based Processing, pp.264-271, 2005.
DOI : 10.1109/EMPDP.2005.19

URL : https://hal.archives-ouvertes.fr/hal-00683892

C. Joubert and R. Mateescu, Distributed On-the-Fly Model Checking and Test Case Generation, Proceedings of the 13th International SPIN Workshop on Model Checking of Software SPIN, pp.126-145, 2006.
DOI : 10.1007/BFb0054166

URL : https://hal.archives-ouvertes.fr/inria-00071388

S. C. Kleene, Introduction to Metamathematics, 1952.

E. Klipp, B. Nordlander, R. Krüger, P. Gennemark, and S. Hohmann, Integrative model of the response of yeast to osmotic shock, Nature Biotechnology, vol.53, issue.8, pp.975-982, 2005.
DOI : 10.1016/0006-3002(58)90330-5

D. Kozen, Results on the Propositional ??-Calculus, DAIMI Report Series, vol.11, issue.146, pp.333-354, 1983.
DOI : 10.7146/dpb.v11i146.7420

H. Kugler, D. Harel, A. Pnueli, Y. Lu, and Y. Bontemps, Temporal Logic for Scenario-Based Specifications, Proceedings of the 11th International Conference on Tools and Algorithms for the Construction and Analysis of Systems TACAS'05 of Lecture Notes in Computer Science, pp.445-460, 2005.
DOI : 10.1007/978-3-540-31980-1_29

F. Lang, Exp.Open 2.0: A Flexible Tool Integrating Partial Order, Compositional, and On-The-Fly Verification Methods, Proceedings of the 5th International Conference on Integrated Formal Methods IFM'2005, 2005.
DOI : 10.1007/11589976_6

URL : https://hal.archives-ouvertes.fr/inria-00070339

K. G. Larsen, Proof systems for Hennessy-Milner Logic with recursion, Proceedings of the 13th Colloquium on Trees in Algebra and Programming CAAP'88, pp.215-230, 1988.
DOI : 10.1007/BFb0026106

J. Leloup and A. Goldbeter, Toward a detailed computational model for the mammalian circadian clock, Proceedings of the National Academy of Sciences of the USA, pp.7051-7056, 2003.
DOI : 10.1073/pnas.1132112100

Z. Manna and A. Pnueli, The Temporal Logic of Reactive and Concurrent Systems, volume I: Specification, 1992.

A. J. Martin, Compiling communicating processes into delay-insensitive VLSI circuits, Distributed Computing, vol.20, issue.8, pp.226-234, 1986.
DOI : 10.1007/BF01660034

URL : http://authors.library.caltech.edu/26661/2/postscript.pdf

R. Mateescu, CAESAR_SOLVE: A generic library for on-the-fly resolution of alternation-free Boolean equation systems, International Journal on Software Tools for Technology Transfer, vol.8, issue.1, pp.37-56, 2006.
DOI : 10.1007/s10009-005-0194-9

URL : https://hal.archives-ouvertes.fr/inria-00084628

R. Mateescu, P. Monteiro, E. Dumas, H. De-jong, S. D. Cha et al., Computation Tree Regular Logic for Genetic Regulatory Networks, Proceedings of the 6th International Symposium on Automated Technology for Verification and Analysis ATVA'08, pp.48-63, 2008.
DOI : 10.1007/11691617_8

URL : https://hal.archives-ouvertes.fr/inria-00277995

R. Mateescu and M. Sighireanu, Efficient on-the-fly model-checking for regular alternation-free mu-calculus, Science of Computer Programming, vol.46, issue.3, pp.255-281, 2003.
DOI : 10.1016/S0167-6423(02)00094-1

URL : https://hal.archives-ouvertes.fr/inria-00072755

R. Mateescu and D. Thivolle, A Model Checking Language for Concurrent Value-Passing Systems, Proceedings of the 15th International Symposium on Formal Methods FM'08, 2008.
DOI : 10.1007/978-3-540-68237-0_12

URL : https://hal.archives-ouvertes.fr/inria-00315312

P. Monteiro, D. Ropers, R. Mateescu, A. Freitas, and H. De-jong, Temporal logic patterns for querying dynamic models of cellular interaction networks, Bioinformatics, vol.24, issue.16, pp.227-233, 2008.
DOI : 10.1093/bioinformatics/btn275

URL : https://hal.archives-ouvertes.fr/inria-00357805

A. Regev and E. Shapiro, Cells as Computation, Nature, vol.419, issue.6905, p.343, 2002.
DOI : 10.1007/3-540-36481-1_1

D. Ropers, H. De-jong, M. Page, D. Schneider, and J. Geiselmann, Qualitative simulation of the carbon starvation response in Escherichia coli, Biosystems, vol.84, issue.2, pp.124-152, 2006.
DOI : 10.1016/j.biosystems.2005.10.005

URL : https://hal.archives-ouvertes.fr/hal-00171698

B. Schoeberl, C. Eichler-jonsson, E. Gilles, and G. Mller, Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors, Nature Biotechnology, vol.20, issue.4, pp.370-375, 2002.
DOI : 10.1038/nbt0402-370

J. Sepulchre, S. Reverchon, and W. Nasser, Modeling the onset of virulence in a pectinolytic bacterium, Journal of Theoretical Biology, vol.244, issue.2, pp.239-257, 2007.
DOI : 10.1016/j.jtbi.2006.08.010

URL : https://hal.archives-ouvertes.fr/hal-00138078

X. Shen, J. Collier, D. Dill, L. Shapiro, M. Horowitz et al., Architecture and inherent robustness of a bacterial cell-cycle control system, Proceedings of the National Academy of Sciences of the USA 105, pp.11340-11345, 2008.
DOI : 10.1073/pnas.0805258105

R. Streett, Propositional Dynamic Logic of looping and converse, Proceedings of the thirteenth annual ACM symposium on Theory of computing , STOC '81, pp.121-141, 1982.
DOI : 10.1145/800076.802492

A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, vol.5, issue.2, pp.285-309, 1955.
DOI : 10.2140/pjm.1955.5.285

URL : http://projecteuclid.org/download/pdf_1/euclid.pjm/1103044538

R. Thomas and M. Kaufman, Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.11, issue.1, pp.170-179, 2001.
DOI : 10.1063/1.1350439

R. Thomas, D. Thieffry, and M. Kaufman, Dynamical behaviour of biological regulatory networks???I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state, Bulletin of Mathematical Biology, vol.29, issue.2, pp.247-276, 1995.
DOI : 10.1007/BF02460618

W. Thomas, Computation tree logic and regular ??-languages, 1989.
DOI : 10.1007/BFb0013041

A. Tiwari and G. Khanna, Series of Abstractions for Hybrid Automata, Proc. HSCC 2002, pp.465-478, 2002.
DOI : 10.1007/3-540-45873-5_36

U. Viretta, A. Fussenegger, and M. , Modeling the Quorum Sensing Regulatory Network of Human-Pathogenic Pseudomonas aeruginosa, Biotechnology Progress, vol.20, issue.3, pp.670-678, 2004.
DOI : 10.1021/bp034323l

B. Vergauwen and J. Lewi, Efficient local correctness checking for single and alternating boolean equation systems, Proceedings of the 21st ICALP, pp.304-315, 1994.
DOI : 10.1007/3-540-58201-0_77

P. Wolper, A translation from full branching time temporal logic to one letter propositional dynamic logic with looping, 1982.

P. Wolper, Temporal logic can be more expressive, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981), pp.72-99, 1983.
DOI : 10.1109/SFCS.1981.44

URL : http://doi.org/10.1016/s0019-9958(83)80051-5

P. Let and B. {x-i-?-=-?-i, } 1?i?n be an equation block and ? 1 , ? 2 two propositional environments as stated in the hypothesis. The semantics of B in the context of an environment ? is determined by the associated functional ? ? : (2 S ) n ? (2 S ) n defined as follows, p.1