Jiyu Zhu | Searcharxiv

arXiv2026

Quantifier Elimination Meets Treewidth

In this paper, we address the complexity barrier inherent in Fourier-Motzkin elimination (FME) and cylindrical algebraic decomposition (CAD) when eliminating a block of (existential) quantifiers. To mitigate this, we propose exploiting structural sparsity in the variable dependency graph of quantified formulas. Utilizing tools from parameterized algorithms, we investigate the role of treewidth, a parameter that measures the graph's tree-likeness, in the process of quantifier elimination. A novel dynamic programming framework, structured over a tree decomposition of the dependency graph, is developed for applying FME and CAD, and is also extensible to general quantifier elimination procedures. Crucially, we prove that when the treewidth is a constant, the framework achieves a significant exponential complexity improvement for both FME and CAD, reducing the worst-case complexity bound from doubly exponential to single exponential. Preliminary experiments on sparse linear real arithmetic (LRA) and nonlinear real arithmetic (NRA) benchmarks confirm that our algorithm outperforms the existing popular heuristic-based approaches on instances exhibiting low treewidth.

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arXiv2024

Reset Controller Synthesis by Reach-avoid Analysis for Delay Hybrid Systems

A reset controller plays a crucial role in designing hybrid systems. It restricts the initial set and redefines the reset map associated with discrete transitions, in order to guarantee the system to achieve its objective. Reset controller synthesis, together with feedback controller synthesis and switching logic controller synthesis, provides a correct-by-construction approach to designing hybrid systems. However, time-delay is an inevitable factor in hybrid systems, which can degrade control performance and render verification certificates obtained by abstracting away time-delay invalid in practice. In this paper, we investigate this issue in a practical manner by taking time-delay into account. We propose an approach that reduces the synthesis of reset controllers to the generation of reach-avoid sets for the hybrid system under consideration, which can be efficiently solved using off-the-shell convex optimization solvers.

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