Nonlinear Polynomials, Interpolants and Invariant Generation for System Analysis

System Invariant properties at various locations play a critical role in enhancing confidence in the reliability of system behavior, be it software, hardware and hybrid systems. While there has recently been considerable interest in researching heuristics for generating loops invariants, almost all developments have focused on generating invariants typically handled using SMT solvers including propositional formulas, difference and octagonal formulas and linear formulas. While we have been investigating methods based on symbolic computation algorithms including Gröbner basis and approximate quantifier elimination for over a decade (see [14, 36, 42, 45, 26, 27, 43, 44, 28, 50, 49, 16, 18] for some of our papers), the SMT and CAV community have only recently started considering nonlinear polynomial invariants since many programs, especially linear filters, hybrid systems, and other applications, need nonlinear invariants for analysis of their behavior. We present an overview of our research with a focus on our work on nonlinear invariant generation [16] as well as interpolant generation [18] from the perspective of their role in software and hybrid system analysis. Our approach is in sharp contrast to some recent approaches in which nonlinear polynomials are approximated using linear inequalities and symbolic-numeric techniques. We also present new research on quantifier elimination heuristics for invariant generation and interpolant generation. Particularly, we give efficient algorithms for interpolant generation for quantifier-free theories of equality on uninterpreted symbols and octagonal formulas. We outline problems and challenges for future research.