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Nonlinear Loop Invariant Generation using Gröbner Bases
, 2004
"... We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generati ..."
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Cited by 41 (4 self)
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We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generation has been focussed on the construction of linear invariants for linear programs. Consequently, there has been little progress toward nonlinear invariant generation. In this paper, we demonstrate a technique that encodes the conditions for a given template assertion being an invariant into a set of constraints, such that all the solutions to these constraints correspond to nonlinear (algebraic) loop invariants of the program. We discuss some tradeoffs between the completeness of the technique and the tractability of the constraintsolving problem generated. The application of the technique is demonstrated on a few examples.
Constructing Invariants for Hybrid Systems
 in Hybrid Systems: Computation and Control, LNCS 2993
, 2004
"... Abstract. An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideal ..."
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Cited by 36 (7 self)
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Abstract. An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideals over polynomial rings. We extend our previous work on the generation of algebraic invariants for discrete transition systems in order to generate algebraic invariants for hybrid systems. In doing so, we present a new technique to handle consecution across continuous differential equations. The techniques we present allow a tradeoff between the complexity of the invariant generation process and the strength of the resulting invariants. 1
Abstract interpretation based formal methods and future challenges, invited paper
 Informatics — 10 Years Back, 10 Years Ahead, volume 2000 of Lecture Notes in Computer Science
, 2001
"... Abstract. In order to contribute to the solution of the software reliability problem, tools have been designed to analyze statically the runtime behavior of programs. Because the correctness problem is undecidable, some form of approximation is needed. The purpose of abstract interpretation is to f ..."
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Cited by 28 (6 self)
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Abstract. In order to contribute to the solution of the software reliability problem, tools have been designed to analyze statically the runtime behavior of programs. Because the correctness problem is undecidable, some form of approximation is needed. The purpose of abstract interpretation is to formalize this idea of approximation. We illustrate informally the application of abstraction to the semantics of programming languages as well as to static program analysis. The main point is that in order to reason or compute about a complex system, some information must be lost, that is the observation of executions must be either partial or at a high level of abstraction. In the second part of the paper, we compare static program analysis with deductive methods, modelchecking and type inference. Their foundational ideas are briefly reviewed, and the shortcomings of these four methods are discussed, including when they should be combined. Alternatively, since program debugging is still the main program verification
Analysis of a Biphase Mark Protocol with Uppaal and PVS
"... The biphase mark protocol is a convention for representing both a string of bits and clock edges in a square wave. The protocol is frequently used for communication at the physical level of the ISO/OSI hierarchy, and is implemented on microcontrollers such as the Intel 82530 Serial Communications ..."
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Cited by 14 (1 self)
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The biphase mark protocol is a convention for representing both a string of bits and clock edges in a square wave. The protocol is frequently used for communication at the physical level of the ISO/OSI hierarchy, and is implemented on microcontrollers such as the Intel 82530 Serial Communications Controller. An important property of the protocol is that bit strings of arbitrary length can be transmitted reliably, despite differences in the clock rates of sender and receiver (drift), variations of the clock rates (jitter), and distortion of the signal after generation of an edge. In this article, we show how the protocol can be modelled naturally in terms of timed automata. We use the model checker Uppaal to derive the maximal tolerances on the clock rates, for different instances of the protocol, and to support the general parametric verification that we formalized using the proof assistant PVS. Based on the derived parameter constraints we propose instances of BMP that are correct (at least in our model) but have a faster bit rate than the instances that are commonly implemented in hardware.
Generation of basic semialgebraic invariants using convex polyhedra
 Static Analysis: Proceedings of the 12th International Symposium, volume 3672 of Lecture Notes in Computer Science
"... Abstract. A technique for generating invariant polynomial inequalities of bounded degree is presented using the abstract interpretation framework. It is based on overapproximating basic semialgebraic sets, i.e., sets defined by conjunctions of polynomial inequalities, by means of convex polyhedra. ..."
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Cited by 9 (0 self)
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Abstract. A technique for generating invariant polynomial inequalities of bounded degree is presented using the abstract interpretation framework. It is based on overapproximating basic semialgebraic sets, i.e., sets defined by conjunctions of polynomial inequalities, by means of convex polyhedra. While improving on the existing methods for generating invariant polynomial equalities, since polynomial inequalities are allowed in the guards of the transition system, the approach does not suffer from the prohibitive complexity of the methods based on quantifierelimination. The application of our implementation to benchmark programs shows that the method produces nontrivial invariants in reasonable time. In some cases the generated invariants are essential to verify safety properties that cannot be proved with classical linear invariants. 1