Long Qian

Jonathan Hellwig, Long Qian, André Platzer

This paper presents differential-algebraic refinement logic (dARL) with which one can deductively verify both properties and relations of differential-algebraic programs (DAPs) that extend hybrid dynamical systems with differential-algebraic equations (DAEs). A refinement calculus is introduced that enables the sound comparison of trajectories of differential-algebraic equations, crucially utilizing a novel trace-based semantics. This enables the incremental verification/simplification of complicated DAEs, while ensuring correctness at each step by the soundness of the calculus. The calculus is shown to be complete for certifying index reductions of DAEs, providing trustworthy syntactic proofs of correctness at each step of the reduction.

André Platzer, Long Qian

This article establishes the completeness of an axiomatization for the robust safety of dynamical systems with polynomial differential equations on bounded time horizons. Safety properties of robust systems are uniformly reduced to a sound axiomatization of polynomial invariants, resulting in reliable logical proofs of correctness. Approximate decidability results are also established: there is a computable algorithm such that, given any perturbation parameter $δ$, it either produces a symbolic proof of robust safety (hence correctly decides the dynamical system to be robustly safe), or correctly decides that the system is not robustly safe under a perturbation of level $δ$. In contrast to earlier works, this article crucially leverages results from subanalytic geometry to retain a level of exactness, thereby establishing positive results of provability/decidability allowing for arbitrary bounded (semialgebraic) initial/post conditions even without positive separation at their (topological) boundaries. This enables the generation of proofs of inductive safety beyond finite time horizons for general hybrid dynamical systems.

Long Qian, Eric Wang, Bernardo Subercaseaux et al.

We consider the problem of finding and enumerating polyominos that can be folded into multiple non-isomorphic boxes. While several computational approaches have been proposed, including SAT, randomized algorithms, and decision diagrams, none has been able to perform at scale. We argue that existing SAT encodings are hindered by the presence of global constraints (e.g., graph connectivity or acyclicity), which are generally hard to encode effectively and hard for solvers to reason about. In this work, we propose a new SAT-based approach that replaces these global constraints with simple local constraints that have substantially better propagation properties. Our approach dramatically improves the scalability of both computing and enumerating common box unfoldings: (i) while previous approaches could only find common unfoldings of two boxes up to area 88, ours easily scales beyond 150, and (ii) while previous approaches were only able to enumerate common unfoldings up to area 30, ours scales up to 60. This allows us to rule out 46, 54, and 58 as the smallest areas allowing a common unfolding of three boxes, thereby refuting a conjecture of Xu et al. (2017).