Abstract
Error coefficients are ubiquitous in systems. In particular, errors in reasoning verification must be considered regarding safety-critical systems. We present a reasoning method that can be applied to systems described by the polynomial error assertion (PEA). The implication relationship between PEAs can be converted to an inclusion relationship between zero sets of PEAs; the PEAs are then transformed into first-order polynomial logic. Combined with the quantifier elimination method, based on cylindrical algebraic decomposition, the judgment of the inclusion relationship between zero sets of PEAs is transformed into judgment error parameters and specific error coefficient constraints, which can be obtained by the quantifier elimination method. The proposed reasoning method is validated by proving the related theorems. An example of intercepting target objects is provided, and the correctness of our method is tested through large-scale random cases. Compared with reasoning methods without error semantics, our reasoning method has the advantage of being able to deal with error parameters.
Highlights
We introduce the definitions of polynomial error assertion (PEA) and the zero set of PEAs (Definitions 4 and 5) that are contributed by this study
The literature [19] is valuable for linear error assertions, but it is invalid for nonlinear PEAs
For non-linear assertions without polynomials, Taylor expansion can be used to expand them into polynomials for approximation
Summary
Formal verification includes two main approaches: model-checking and theoremproving. Model-checking, proposed by Clarke and Emerson, verifies safety properties by searching the state space [1,2]. Theorem-proving uses logic and mathematical reasoning to verify the safety properties of a piece of software [3] These two approaches have complementary advantages [4,5] and are widely used in industries [6,7,8]. The real polynomial algebraic transition system extends the domain space of multivalued logic to the real number domain, which can more effectively describe complex systems and verify their safety properties [11]. The authors previously proposed a reasoning method for linear error assertion (LEA) [19] In this method, combined with the convex properties of LEA, the conclusion that vertexes of the precursor assertion are contained in the zero set of the successor assertion can be used to determine whether there is an implication relationship between precursor assertion and successor assertion. Our reasoning method can be used to verify systems described by PEAs
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