Automated Symbolic Reachability Analysis; with Application to Delta-Notch Signaling Automata

This paper describes the implementation of predicate abstraction techniques to automatically compute symbolic backward reachable sets of high dimensional piecewise affine hybrid automata, used to model Delta-Notch biological cell signaling networks. These automata are analyzed by creating an abstraction of the hybrid model, which is a finite state discrete transition system, and then performing the computation on the abstracted system. All the steps, from model generation to the simplification of the reachable set, have been automated using a variety of decision procedure and theorem-proving tools. The concluding example computes the reach set for a four cell network with 8 continuous and 256 discrete states. This demonstrates the feasibility of using these tools to compute on high dimensional hybrid automata, to provide deeper insight into realistic biological systems.

[1]  H. Hong.  An improvement of the projection operator in cylindrical algebraic decomposition , 1990, ISSAC '90.

[2]  Thao Dang,et al.  d/dt: a verification tool for hybrid systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[3]  Alberto Bemporad,et al.  Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems , 2000, HSCC.

[4]  Hassen Saïdi,et al.  Construction of Abstract State Graphs with PVS , 1997, CAV.

[5]  Thomas A. Henzinger,et al.  HYTECH: a model checker for hybrid systems , 1997, International Journal on Software Tools for Technology Transfer.

[6]  Claire J. Tomlin,et al.  Lateral Inhibition through Delta-Notch Signaling: A Piecewise Affine Hybrid Model , 2001, HSCC.

[7]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[8]  César Muñoz,et al.  An Overview of SAL , 2000 .

[9]  E Mjolsness,et al.  Delta-Notch lateral inhibitory patterning in the emergence of ciliated cells in Xenopus: experimental observations and a gene network model. , 2000, Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing.

[10]  H. Wong-Toi,et al.  A procedure for reachability analysis of rectangular automata , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[11]  Scott McCallum,et al.  An Improved Projection Operation for Cylindrical Algebraic Decomposition of Three-Dimensional Space , 1988, J. Symb. Comput..

[12]  Hyoung Seok Hong,et al.  Qualitative Modeling of Hybrid Systems , 2001 .

[13]  Bruce H. Krogh,et al.  Verification of infinite-state dynamic systems using approximate quotient transition systems , 2001, IEEE Trans. Autom. Control..

[14]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[15]  Ashish Tiwari,et al.  Series of Abstractions for Hybrid Automata , 2002, HSCC.

[16]  Rajeev Alur,et al.  Reachability Analysis of Hybrid Systems via Predicate Abstraction , 2002, HSCC.

[17]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[18]  Benjamin Kuipers,et al.  Qualitative Modeling and Heterogeneous Control of Global System Behavior , 2002, HSCC.

[19]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[20]  Benjamin Kuipers,et al.  Proving Properties of Continuous Systems: Qualitative Simulation and Temporal Logic , 1997, Artif. Intell..

[21]  Ian M. Mitchell,et al.  Overapproximating Reachable Sets by Hamilton-Jacobi Projections , 2003, J. Sci. Comput..

[22]  Stavros Tripakis,et al.  Verification of Hybrid Systems with Linear Differential Inclusions Using Ellipsoidal Approximations , 2000, HSCC.

[23]  Claire J. Tomlin,et al.  A hybrid systems approach to modeling and analyzing planar cell polarity , 2002 .

[24]  Jill P. Mesirov,et al.  Computational Biology , 2018, Encyclopedia of Parallel Computing.