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引用次数: 0
摘要
本文提出了线性系统 \(\xi '=A\xi +u/)的可达性决策程序,其中矩阵 \(A's/)特征值可以是任意代数数,输入 u 是三角-指数多项式的向量。如果初始集只包含一个点,那么所考虑的可达性问题就简化为三角-指数多项式符号的可判定性,然后通过相关指数函数和三角函数的泰勒展开,简化为一系列单变量多项式不等式的验证。如果初始集合是开放的半代数,我们将提出一种基于 OpenCAD 的判定程序,以及一种从三角-指数多项式的符号判定程序中衍生出来的实根隔离算法。实验结果表明我们的方法非常有效。在Schanuel猜想的假设下,除了几种情况外,上述程序都能在有界时间内完成。
In this paper, we propose a decision procedure of reachability for a linear system \(\xi '=A\xi +u\), where the matrix \(A's\) eigenvalues can be arbitrary algebraic number and the input u is a vector of trigonometric-exponential polynomials. If the initial set contains only one point, the reachability problem under consideration is reduced to the decidability of the sign of trigonometric-exponential polynomial and then achieved by being reduced to verification of a series of univariate polynomial inequalities through Taylor expansions of the related exponential functions and trigonometric functions. If the initial set is open semi-algebraic, we will propose a decision procedure based on OpenCAD and an algorithm of real root isolation derived from the sign-deciding procedure for the trigonometric-exponential polynomials. The experimental results indicate the efficiency of our approach. Under the assumption of Schanuel’s Conjecture, the above procedures are complete for bounded time except for several cases.
期刊介绍:
Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics.
Topics of interest include:
• semantics of programming languages
• models and modeling languages for concurrent, distributed, reactive and mobile systems
• models and modeling languages for timed, hybrid and probabilistic systems
• specification, program analysis and verification
• model checking and theorem proving
• modal, temporal, first- and higher-order logics, and their variants
• constraint logic, SAT/SMT-solving techniques
• theoretical aspects of databases, semi-structured data and finite model theory
• theoretical aspects of artificial intelligence, knowledge representation, description logic
• automata theory, formal languages, term and graph rewriting
• game-based models, synthesis
• type theory, typed calculi
• algebraic, coalgebraic and categorical methods
• formal aspects of performance, dependability and reliability analysis
• foundations of information and network security
• parallel, distributed and randomized algorithms
• design and analysis of algorithms
• foundations of network and communication protocols.