SEKE 2012 Proceedings - Knowledge Systems Institute

Verification of Cyber-Physical Systems Based on Differential-Algebraic Temporal Dynamic Logic
Xiaoxiang Zhai, Bixin Li, Min Zhu, Jiakai Li, Qiaoqiao Chen, Shunhui Ji
School of Computer Science and Engineering, Southeast University, Nanjing, China
Email: {xxzhai, bx.li}@seu.edu.cn, kongs@139.com, jiakai li@seu.edu.cn,
joe 0701@126.com, shunhuiji@163.com
Abstract
Differential temporal dynamic logic (dTL) is an approach
for specifying and verifying properties of cyber-physical systems
(CPS) and it can handle with temporal behaviors for CPS.
The hybrid programs (HP), as operating model of dTL, only
contain differential equations that can be solved in polynomial
arithmetic, which results that dTL can only specify and verify
CPS of simple dynamics. However, differential-algebraic dynamic
logic (DAL) solves the problem through the introduction of differential
invariants, but lacks verification capabilities for properties
with temporality. This paper combines the advantages of dTL
and DAL, and proposes differential-algebraic temporal dynamic
logic (DATL). We have achieved the following results: a trace
semantics for differential-algebraic programs (DAP), four new
rules based on the rules of dTL and DAL, a proof of the soundness
of the new rules, and the specification and verification of safety
of aircraft collision avoidance system with DATL. Our theory
together with a case study demonstrates that DATL overcomes
the constraints that differential equations must be solvable in
polynomial arithmetic and can be used to specify and verify
temporal properties of CPS.
Keywords—Cyber-physical systems; property verification; differential
temporal dynamic logic; differential-algebraic dynamic logic;
differential-algebraic temporal dynamic logic; aircraft collision avoidance
system;
I. Introduction
Cyber-physical systems (CPS) are integrations of computation
and physical processes. Embedded computers and networks
monitor and control the physical processes, usually with feedback
loops where physical processes affect computations and
This work is supported partially by National Natural Science Foundation
of China under Grant No. 60973149, partially by the Open Funds
of State Key Laboratory of Computer Science of Chinese Academy
of Sciences under Grant No.SYSKF1110, partially by Doctoral Fund
of Ministry of Education of China under Grant No. 20100092110022,
and partially by the College Industrialization Project of Jiangsu Province
under Grant No.JHB2011-3.
Correspondence to: Bixin Li, School of Computer Science and Engineering,
Southeast University, Nanjing, China. E-mail: bx.li@seu.edu.cn
vice versa [1]. Many approaches have been proposed to verify
properties of CPS. They are primarily divided into two categories:
model checking and theorem proving. Because CPS do not admit
equivalent finite-state abstractions [2] and due to general limits
of numerical approximation, model checkers are still more successful
in falsification than in verification. Differential temporal
dynamic logic (dTL) and differential-algebraic dynamic logic
(DAL) are approaches falling the scope of theorem proving.
As operating model of dTL, HP have limited expressiveness
and cannot model complex CPS. For example, fluctuations and
errors in the physical processes cannot be expressed. As operating
model of DAL, DAP make up for the shortcomings of HP via the
introduction of quantifiers and differential invariants. However,
DAL cannot verify properties with temporality, and dTL has
the ability by introducing temporal operator and expanding the
relevant calculus rules. This paper combines the advantages
of dTL and DAL and proposes differential-algebraic temporal
dynamic logic (DATL) to verify properties of CPS with complex
dynamics and temporality.
This paper is organized as follows: next section first introduces
an aircraft collision avoidance system, then defines trace
semantics of DAP under temporal behavior. The aircraft collision
avoidance system is modeled using DAP and safety of the
system is specified as a DATL formula. Section 3 introduces
four new calculus rules for DATL by inheritance, expansion and
improvement of the rules of the DAL and dTL, and proves
the correctness of the new rules. The property of the aircraft
collision avoidance system is specified as a DATL formula and
we use DATL sequent calculus to formally verify the property.
Finally, a conclusion of research in this paper is drawn and some
expectations are brought forward for the future.
II. CPS Modeling and Property Specification
A. Aircraft Collision Avoidance System
There are a number of aircrafts flying in the air, their flight
dynamics can be described by a group of differential equations.
The differential equations of (∗) denote flight dynamics of
any two aircrafts X and Y, where point x(x 1 , x 2 , x 3 ) and point
y(y 1 , y 2 , y 3 ) denote three-dimensional coordinates of aircrafts X
and Y respectively, d(d 1 , d 2 , d 3 ) and e(e 1 , e 2 , e 3 ) denote speed of
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