Counterexample- and Simulation-Guided Floating-Point Loop Invariant Synthesis

We present an automated procedure for synthesizing sound inductive invariants for floating-point numerical loops. Our procedure generates invariants of the form of a convex polynomial inequality that tightly bounds the values of loop variables. Such invariants are a prerequisite for reasoning about the safety and roundoff errors of floating-point programs. Unlike previous approaches that rely on policy iteration, linear algebra or semi-definite programming, we propose a heuristic procedure based on simulation and counterexample-guided refinement. We observe that this combination is remarkably effective and general and can handle both linear and nonlinear loop bodies, nondeterministic values as well as conditional statements. Our evaluation shows that our approach can efficiently synthesize loop invariants for existing benchmarks from literature, but that it is also able to find invariants for nonlinear loops that today’s tools cannot handle.

Entire functions of exponential type not vanishing in the half-plane \(\Im z > k\), where \(k > 0\)

Let \(P (z)\) be a polynomial of degree \(n\) having no zeros in \(|z| &lt; k\), \(k \leq 1\), and let \(Q (z) := z^n \overline{P (1/{\overline {z}})}\). It was shown by Govil that if \(\max_{|z| = 1} |P^\prime (z)|\) and \(\max_{|z| = 1} |Q^\prime (z)|\) are attained at the same point of the unit circle \(|z| = 1\), then \[\max_{|z| = 1} |P'(z)| \leq \frac{n}{1 + k^n} \max_{|z| = 1} |P(z)|.\]<br />The main result of the present article is a generalization of Govil's polynomial inequality to a class of entire functions of exponential type.<br /><br />

Method of Detecting Unary Polynomial Inequality Likely Invariant

Many Studies has carried on the technology of detecting invariants, for example, Daikon can detect most of invariant by presetting some types of them. But unary polynomial inequality likely invariants are rarely discovered by traditional tools. An effective method to detect unary polynomial inequality likely invariant was proposed in this paper. Through analyzing the property of value ranges of unary polynomial inequality likely invariants, the algorithm set the threshold value and calculates neighbor distances to determine the form of invariants. Finally, experimental results are given to demonstrate the effectiveness of this method.

Computing Invariants for Hybrid Systems

This paper address the problem of generating invariants of hybrid systems. We present a new approach, for generating polynomial inequality invariants of hybrid systems through solving semi-algebraic systems and quantifier elimination. From the preliminary experiment results, we demonstrate the feasibility of our approach.

Markov type polynomial inequality for some generalized Hermite weight

ABSTRACT In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.