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41
Computing differential invariants of hybrid systems as fixedpoints
, 2008
"... Abstract. We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, ..."
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Cited by 58 (21 self)
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Abstract. We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixedpoint algorithm works with a compositional verification logic for hybrid systems. To improve the verification power, we further introduce a saturation procedure that refines the system dynamics successively with differential invariants until safety becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout maneuvers in air traffic management and collision avoidance in train control.
Dimensions in program synthesis
"... Program Synthesis, which is the task of discovering programs that realize user intent, can be useful in several scenarios: enabling people with no programming background to develop utility programs, helping regular programmers automatically discover tricky/mundane details, program understanding, dis ..."
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Cited by 54 (20 self)
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Program Synthesis, which is the task of discovering programs that realize user intent, can be useful in several scenarios: enabling people with no programming background to develop utility programs, helping regular programmers automatically discover tricky/mundane details, program understanding, discovery of new algorithms, and even teaching. This paper describes three key dimensions in program synthesis: expression of user intent, space of programs over which to search, and the search technique. These concepts are illustrated by brief description of various program synthesis projects that target synthesis of a wide variety of programs such as standard undergraduate textbook algorithms (e.g., sorting, dynamic programming), program inverses (e.g., decoders, deserializers), bitvector manipulation routines, deobfuscated programs, graph algorithms, text-manipulating routines, mutual exclusion algorithms, etc. Categories and Subject Descriptors D.1.2 [Programming Techniques]:
Differential-Algebraic Dynamic Logic for Differential-Algebraic Programs
"... Abstract. We generalise dynamic logic to a logic for differential-algebraic programs, i.e., discrete programs augmented with first-order differentialalgebraic formulas as continuous evolution constraints in addition to first-order discrete jump formulas. These programs characterise interacting discr ..."
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Cited by 41 (28 self)
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Abstract. We generalise dynamic logic to a logic for differential-algebraic programs, i.e., discrete programs augmented with first-order differentialalgebraic formulas as continuous evolution constraints in addition to first-order discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differential-algebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.
Formal verification of hybrid systems
, 2011
"... In formal verification, a designer first constructs a model, with mathematically precise semantics, of the system under design, and performs extensive analysis with respect to correctness requirements. The appropriate mathematical model for embedded control systems is hybrid systems that combines th ..."
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Cited by 34 (0 self)
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In formal verification, a designer first constructs a model, with mathematically precise semantics, of the system under design, and performs extensive analysis with respect to correctness requirements. The appropriate mathematical model for embedded control systems is hybrid systems that combines the traditional state-machine based models for discrete control with classical differential-equations based models for continuously evolving physical activities. In this article, we briefly review selected existing approaches to formal verification of hybrid systems, along with directions for future research.
Synthesizing Switching Logic using Constraint Solving
"... A new approach based on constraint solving techniques was recently proposed for verification of hybrid systems. This approach works by searching for inductive invariants of a given form. In this paper, we extend that work to automatic synthesis of safe hybrid systems. Starting with a multi-modal d ..."
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Cited by 19 (11 self)
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A new approach based on constraint solving techniques was recently proposed for verification of hybrid systems. This approach works by searching for inductive invariants of a given form. In this paper, we extend that work to automatic synthesis of safe hybrid systems. Starting with a multi-modal dynamical system and a safety property, we present a sound technique for synthesizing a switching logic for changing modes so as to preserve the safety property. By construction, the synthesized hybrid system is well-formed and is guaranteed safe. Our approach is based on synthesizing a controlled invariant that is sufficient to prove safety. The generation of the controlled invariant is cast as a constraint solving problem. When the system, the safety property, and the controlled invariant are all expressed only using polynomials, the generated constraint is an ∃ ∀ formula in the theory of reals, which we solve using SMT solvers. The generated controlled invariant is then used to arrive at the maximally liberal switching logic.
Proving termination of integer term rewriting
- In Proc. RTA ’09, LNCS 5595
, 2009
"... Abstract. When using rewrite techniques for termination analysis of programs, a main problem are pre-defined data types like integers. We extend term rewriting by built-in integers and adapt the dependency pair framework to prove termination of integer term rewriting automatically. 1 ..."
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Cited by 18 (11 self)
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Abstract. When using rewrite techniques for termination analysis of programs, a main problem are pre-defined data types like integers. We extend term rewriting by built-in integers and adapt the dependency pair framework to prove termination of integer term rewriting automatically. 1
Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded ..."
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Cited by 18 (17 self)
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyber-physical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multi-agent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic
Solving non-linear polynomial arithmetic via sat modulo linear arithmetic.
- In CADE,
, 2009
"... Abstract. Polynomial constraint-solving plays a prominent role in several areas of engineering and software verification. In particular, polynomial constraint solving has a long and successful history in the development of tools for proving termination of programs. Well-known and very efficient tec ..."
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Cited by 13 (5 self)
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Abstract. Polynomial constraint-solving plays a prominent role in several areas of engineering and software verification. In particular, polynomial constraint solving has a long and successful history in the development of tools for proving termination of programs. Well-known and very efficient techniques, like SAT algorithms and tools, have been recently proposed and used for implementing polynomial constraint solving algorithms through appropriate encodings. However, powerful techniques like the ones provided by the SMT (SAT modulo theories) approach for linear arithmetic constraints (over the rationals) are underexplored to date. In this paper we show that the use of these techniques for developing polynomial constraint solvers outperforms the best existing solvers and provides a new and powerful approach for implementing better and more general solvers for termination provers.
Verification and synthesis using real quantifier elimination
, 2011
"... We present the application of real quantifier elimination to formal verification and synthesis of continuous and switched dynamical systems. Through a series of case studies, we show how first-order formulas over the reals arise when formally analyzing models of complex control systems. Existing off ..."
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Cited by 13 (3 self)
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We present the application of real quantifier elimination to formal verification and synthesis of continuous and switched dynamical systems. Through a series of case studies, we show how first-order formulas over the reals arise when formally analyzing models of complex control systems. Existing off-the-shelf quantifier elimination procedures are not successful in eliminating quantifiers from many of our benchmarks. We therefore automatically combine three established software components: virtual subtitution based quantifier elimination in Reduce/Redlog, cylindrical algebraic decomposition implemented in Qepcad, and the simplifier Slfq implemented on top of Qepcad. We use this combination to successfully analyze various models of systems including adaptive cruise control in automobiles, adaptive flight control system, and the classical inverted pendulum problem studied in control theory.
Switching Logic Synthesis for Reachability
, 2010
"... We consider the problem of driving a system from some initial configuration to a desired configuration while avoiding some unsafe configurations. The system to be controlled is a dynamical system that can operate in different modes. The goal is to synthesize the logic for switching between the modes ..."
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Cited by 13 (6 self)
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We consider the problem of driving a system from some initial configuration to a desired configuration while avoiding some unsafe configurations. The system to be controlled is a dynamical system that can operate in different modes. The goal is to synthesize the logic for switching between the modes so that the desired reachability property holds. In this paper, we first present a sound and complete inference rule for proving reachability properties of single mode continuous dynamical systems. Next, we present an inference rule for proving controlled reachability in multi-modal continuous dynamical systems. From a constructive proof of controlled reachability, we show how to synthesize the desired switching logic. We show that our synthesis procedure is sound and produces only non-zeno hybrid systems. In practice, we perform a constructive proof of controlled reachability by solving an Exists-Forall formula in the theory of reals. We present an approach for solving such formulas that combines symbolic and numeric solvers. We demonstrate our approach on some examples. All results extend naturally to the case when, instead of reachability, interest is in until properties.
