A computable and compositional semantics for hybrid automata

Hybrid Systems are systems having a mixed discrete and continuous behaviour that cannot be characterized faithfully using either only discrete or only continuous models. A good framework for hybrid systems should support their compositional description and analysis, since commonly systems are specified by a composition of smaller subsystems, to cope with the complexity of their monolithic representation. Moreover, since the reachability problem for hybrid systems is undecidable, one should investigate the conditions that guarantee approximate computability of composition, when only approximations to the exact problem data are available. In this paper, we propose an automata-based formalism (HIOA) for hybrid systems that is compositional and for which the evolution can be computed approximately. The main results are that the composition of compatible HIOA yields a pre-HIOA; a dominance result on the composition of HIOA by which we can replace any component in a composition by another one that exhibits the same external behaviour without affecting the behaviour of the composition; finally, the key result that the composition of two compatible upper(lower)-semicontinuous HIOA is a computable upper(lower)-semicontinuous pre-HIOA, which entails that the evolution of the composition is upper(lower)-semicomputable. A discussion on how compositionality/computability are handled in state-of-art libraries for reachability analysis closes the paper.

[1]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[2]  Alberto L. Sangiovanni-Vincentelli,et al.  Interchange Format for Hybrid Systems: Abstract Semantics , 2006, HSCC.

[3]  Davide Bresolin,et al.  A Platform-Based Design Methodology With Contracts and Related Tools for the Design of Cyber-Physical Systems , 2015, Proceedings of the IEEE.

[4]  Davide Bresolin,et al.  Computing the Evolution of Hybrid Systems using Rigorous Function Calculus , 2012, ADHS.

[5]  Goran Frehse,et al.  Modular, hierarchical models of control systems in SpaceEx , 2013, 2013 European Control Conference (ECC).

[6]  J. E. Rooda,et al.  Syntax and semantics of the compositional interchange format for hybrid systems , 2013, J. Log. Algebraic Methods Program..

[7]  Thomas A. Henzinger,et al.  Robust Undecidability of Timed and Hybrid Systems , 2000, HSCC.

[8]  Thomas A. Henzinger,et al.  Assume-Guarantee Reasoning for Hierarchical Hybrid Systems , 2001, HSCC.

[9]  Olivier Bournez,et al.  Approximate Reachability Analysis of Piecewise-Linear Dynamical Systems , 2000, HSCC.

[10]  Pieter Collins.  Semantics and Computability of the Evolution of Hybrid Systems , 2011, SIAM J. Control. Optim..

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

[12]  Xin Chen,et al.  Compositional Relational Abstraction for Nonlinear Hybrid Systems , 2017, ACM Trans. Embed. Comput. Syst..

[13]  Xin Chen,et al.  Decomposed Reachability Analysis for Nonlinear Systems , 2016, 2016 IEEE Real-Time Systems Symposium (RTSS).

[14]  Insup Lee,et al.  Compositional modeling and refinement for hierarchical hybrid systems , 2006, J. Log. Algebraic Methods Program..

[15]  George J. Pappas,et al.  Robustness of temporal logic specifications for continuous-time signals , 2009, Theor. Comput. Sci..

[16]  Thomas A. Henzinger,et al.  Modularity for Timed and Hybrid Systems , 1997, CONCUR.

[17]  Stefan Kowalewski,et al.  HyPro: A C++ Library of State Set Representations for Hybrid Systems Reachability Analysis , 2017, NFM.

[18]  Ramon R. H. Schiffelers,et al.  New Concepts in the Abstract Format of the Compositional Interchange Format , 2009, ADHS.

[19]  Davide Bresolin,et al.  Rigorous Continuous Evolution of Uncertain Systems , 2019, NSV@CAV.

[20]  Edward A. Lee,et al.  A modular formal semantics for Ptolemy† , 2013, Mathematical Structures in Computer Science.

[21]  Keijo Ruohonen.  An Effective Cauchy-Peano Existence Theorem for Unique Solutions , 1996, Int. J. Found. Comput. Sci..

[22]  Oded Maler,et al.  Reachability Analysis via Face Lifting , 1998, HSCC.

[23]  J. Lygeros,et al.  Computability of finite-time reachable sets for hybrid systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[24]  Olaf Stursberg,et al.  An assessment of the current status of algorithmic approaches to the verification of hybrid systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[25]  Matthias Althoff,et al.  Automated generation of hybrid system models for reachability analysis of nonlinear analog circuits , 2015, The 20th Asia and South Pacific Design Automation Conference.

[26]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[27]  Nicolas Halbwachs,et al.  Verification of Linear Hybrid Systems by Means of Convex Approximations , 1994, SAS.

[28]  Pieter Collins,et al.  Continuity and computability of reachable sets , 2005, Theor. Comput. Sci..

[29]  Alessandro Cimatti,et al.  HyDI: A Language for Symbolic Hybrid Systems with Discrete Interaction , 2011, 2011 37th EUROMICRO Conference on Software Engineering and Advanced Applications.

[30]  Oded Maler,et al.  Robust Satisfaction of Temporal Logic over Real-Valued Signals , 2010, FORMATS.

[31]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[32]  Davide Bresolin,et al.  Assume–guarantee verification of nonlinear hybrid systems with Ariadne , 2014 .

[33]  Marco Roveri,et al.  HRELTL: A temporal logic for hybrid systems , 2015, Inf. Comput..

[34]  Nancy A. Lynch,et al.  Hybrid I/O automata , 1995, Inf. Comput..