Naijun Zhan

Computing semi-algebraic invariants for polynomial dynamical systems

In this paper, we consider an extended concept of invariant for polynomial dynamical systems (PDSs) with domain and initial condition, and establish a sound and complete criterion for checking semi-algebraic invariants (SAIs) for such PDSs. The main idea is encoding relevant dynamical properties as conditions on the high order Lie derivatives of polynomials occurring in the SAI. A direct consequence of this criterion is a relatively complete method of SAI generation based on template assumption and semi-algebraic constraint solving. Relative completeness means if there is an SAI in the form of a predefined template, then our method can indeed find one.

Refinement and verification in component-based model-driven design

Modern software development is complex as it has to deal with many different and yet related aspects of applications. In practical software engineering this is now handled by a UML-like modelling approach in which different aspects are modelled by different notations. Component-based and object-oriented design techniques are found effective in the support of separation of correctness concerns of different aspects. These techniques are practised in a model-driven development process in which models are constructed in each phase of the development. To ensure the correctness of the software system developed, all models constructed in each phase are verifiable. This requires that the modelling notations are formally defined and related in order to have tool support developed for the integration of sophisticated checkers, generators and transformations. This paper summarises our research on the method of Refinement of Component and Object Systems (rCOS) and illustrates it with experiences from the work on the Common Component Modelling Example (CoCoME). This gives evidence that the formal techniques developed in rCOS can be integrated into a model-driven development process and shows where it may be integrated in computer-aided software engineering (CASE) tools for adding formally supported checking, transformation and generation facilities.

Discovering non-linear ranking functions by solving semi-algebraic systems

Differing from [6] this paper reduces non-linear ranking function discovering for polynomial programs to semi-algebraic system solving, and demonstrates how to apply the symbolic computation tools, DISCOVERER and QEPCAD, to some interesting examples.

A calculus for hybrid CSP

Hybrid Communicating Sequential Processes (HCSP) is an extension of CSP allowing continuous dynamics. We are interested in applying HCSP to model and verify hybrid systems. This paper is to present a calculus for a subset of HCSP as a part of our efforts in modelling and verifying hybrid systems. The calculus consists of two parts. To deal with continuous dynamics, the calculus adopts differential invariants. A brief introduction to a complete algorithm for generating polynomial differential invariants is presented, which applies DISCOVERER, a symbolic computation tool for semi-algebraic systems. The other part of the calculus is a logic to reason about HCSP process, which involves communication, parallelism, real-time as well as continuous dynamics. This logic is named as Hybrid Hoare Logic. Its assertions consist of traditional pre- and post-conditions, and also Duration Calculus formulas to record execution history of HCSP process.

A model of component-based programming

Component-based programming is about how to create application programs from prefabricated components with new software that provides both glue between the components, and new functionality. Models of components are required to support black-box compositionality and substitutability by a third party as well as interoperability. However, the glue codes and programs designed by users of the components for new applications in general do not require these features, and they can be even designed in programming paradigms different from those of the components. In this paper, we extend the rCOS calculus of components with a model for glue programs and application programs that is different from that of components. We study the composition of a glue program with components and prove that the components glued by the glue program yield a new component.

Program Verification by Using DISCOVERER

Recent advances in program verification indicate that various verification problems can be reduced to semi-algebraic system (SAS for short) solving. An SAS consists of polynomial equations and polynomial inequalities. Algorithms for quantifier elimination of real closed fields are the general method for those problems. But the general method usually has low efficiency for specific problems. To overcome the bottleneck of program verification with a symbolic approach, one has to combine special techniques with the general method. Based on the work of complete discrimination systems of polynomials [33,31],, we invented new theories and algorithms [32,30,35] for SAS solving and partly implemented them as a real symbolic computation tool in Maple named DISCOVERER. In this paper, we first summarize the results that we have done so far both on SAS-solving and program verification with DISCOVERER, and then discuss the future work in this direction, including SAS-solving itself, termination analysis and invariant generation of programs, and reachability computation of hybrid systems etc.

Generating polynomial invariants with DISCOVERER and QEPCAD

This paper investigates howto apply the techniques on solving semi-algebraic systems to invariant generation of polynomial programs. By our approach, the generated invariants represented as a semi-algebraic system are more expressive than those generated with the well-established approaches in the literature, which are normally represented as a conjunction of polynomial equations. We implement this approach with the computer algebra tools DISCOVERER and QEPCAD1.We also explain, through the complexity analysis, why our approach is more efficient and practical than the one of [17] which directly applies first-order quantifier elimination.

Verifying Chinese Train Control System under a Combined Scenario by Theorem Proving

In this paper, we investigate how to formalize and verify the System Requirements Specification SRS of Chinese Train Control System Level 3 CTCS-3, which includes a set of basic operational scenarios that cooperate with each other to achieve the desired behavior of trains. It is absolutely necessary to prove that the cooperation of basic scenarios indeed completes the required behavior. As a case study, a combined scenario with several basic scenarios integrated is studied in this paper. We model each scenario as a Hybrid CSP HCSP process, and specify its properties using Hybrid Hoare Logic HHL. Given such an annotated HCSP model, the deductive verification of conformance of the model to the properties is then carried out. For the purpose, we implement a theorem prover of HHL in Isabelle/HOL, with which the process including modelling and verification of annotated HCSP models can be mechanized. In particular, we provide a machine-checked proof for the combined scenario, with the result indicating a design error in SRS of CTCS-3.

Recent advances in program verification through computer algebra

In this paper, we summarize the results on program verification through semi-algebraic systems (SASs) solving that we have obtained, including automatic discovery of invariants and ranking functions, symbolic decision procedure for the termination of a class of linear loops, termination analysis of nonlinear systems, and so on.

Formal Modelling, Analysis and Verification of Hybrid Systems

Hybrid systems is a mathematical model of embedded systems, and has been widely used in the design of complex embedded systems. In this chapter, we will introduce our systematic approach to formal modelling, analysis and verification of hybrid systems. In our framework, a hybrid system is modelled using Hybird CSP HCSP, and specified and reasoned about by Hybrid Hoare Logic HHL, which is an extension of Hoare logic to hybrid systems. For deductive verification of hybrid systems, a complete approach to generating polynomial invariants for polynomial hybrid systems is proposed; meanwhile, a theorem prover for HHL that can provide tool support for the verification has been implemented. We give some case studies from real world, for instance, Chinese High-Speed Train Control System at Level 3 CTCS-3. In addition, based on our invariant generation approach, we consider how to synthesize a switching logic for a considered hybrid system by reduction to constraint solving, to meet a given safety, liveness, optimality requirement, or any of their combinations. We also discuss other issues of hybrid systems, e.g., stability analysis.