Bican Xia

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.

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.

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.

Symbolic decision procedure for termination of linear programs

Tiwari proved that the termination of a class of linear programs is decidable in Tiwari (Proceedings of CAV’04. Lecture notes in computer science, vol 3114, pp 70–82, 2004). The decision procedure proposed therein depends on the computation of Jordan forms. Thus, people may draw a wrong conclusion from this procedure, if they simply apply floating-point computation to compute Jordan forms. In this paper, we first use an example to explain this problem, and then present a symbolic implementation of the decision procedure. Thus, the rounding error problem is therefore avoided. Moreover, we also show that the symbolic decision procedure is as efficient as the numerical one given in Tiwari (Proceedings of CAV’04. Lecture notes in computer science, vol 3114, pp 70–82, 2004). The complexity of former is max{O(n6), O(nm+3)}, while that of the latter is O(nm+3), where n is the number of variables of the program and m is the number of its Boolean conditions. In addition, for the case when the characteristic polynomial of the assignment matrix is irreducible, we design a more efficient symbolic algorithm whose complexity is max(O(n6), O(mn3)).

Generating non-linear interpolants by semidefinite programming

Interpolation-based techniques have been widely and successfully applied in the verification of hardware and software, e.g., in bounded-model checking, CEGAR, SMT, etc., in which the hardest part is how to synthesize interpolants. Various work for discovering interpolants for propositional logic, quantifier-free fragments of first-order theories and their combinations have been proposed. However, little work focuses on discovering polynomial interpolants in the literature. In this paper, we provide an approach for constructing non-linear interpolants based on semidefinite programming, and show how to apply such results to the verification of programs by examples.

Program Verification by Reduction to Semi-algebraic Systems Solving

The discovery of invariants and ranking functions plays a central role in program verification. In our previous work, we investigated invariant generation and non-linear ranking function discovering of polynomial programs by reduction to semi-algebraic systems solving. In this paper we will first summarize our results on the two topics and then show how to generalize the approach to discovering more expressive invariants and ranking functions, and applying to more general programs.

Computing reachable sets of linear vector fields revisited

The reachability problem is one of the most important issues in the verification of hybrid systems. But unfortunately the reachable sets for most of hybrid systems are not computable except for some special families. In our previous work, we identified a family of vector fields, whose state parts are linear with real eigenvalues, while input parts are exponential functions, and proved its reachability problem is decidable. In this paper, we investigate another family of vector fields, whose state parts are linear, but with pure imagine eigenvalues, while input parts are trigonometric functions, and prove its reachability problem is decidable also. To the best of our knowledge, the two families are the largest families of linear vector fields with a decidable reachability problem. In addition, we present an approach on how to abstract general linear dynamical systems to the first family. Comparing with existing abstractions for linear dynamical systems, experimental results indicate that our abstraction is more precise.

Decidability of the Reachability for a Family of Linear Vector Fields

The reachability problem is one of the most important issues in the verification of hybrid systems. Computing the reachable sets of differential equations is difficult, although computing the reachable sets of finite state machines is well developed. Hence, it is not surprising that the reachability of most of hybrid systems is undecidable. In this paper, we identify a family of vector fields and show its reachability problem is decidable. The family consists of all vector fields whose state parts are linear, while input parts are non-linear, possibly with exponential expressions. Such vector fields are commonly used in practice.To the best of our knowledge, the family is one of the most expressive families of vector fields with a decidable reachability problem.The decidability is achieved by proving the decidability of the extension of Tarski’s algebra with some specific exponential functions, which has been proved in [21, 22] due to Strzebonski. In this paper, we propose another decision procedure, which is more efficient when all constraints are open sets. The experimental results indicate the efficiency of our approach, even better than existing approaches based on approximation and numeric computation in general.

Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF

An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities both strict and nonstrict is proposed. The algorithm is based on a key observation that quadratic polynomial inequalities can be linearized if they are concave. A generalization of Motzkins transposition theorem is proved, which is used to generate an interpolant between two mutually contradictory conjunctions of polynomial inequalities, using semi-definite programming in time complexity