performance guarantees

This paper addresses the verification of temporal logic specifications—such as safety and reachability—for interconnected discrete-time systems. We propose a resilience-driven assume-guarantee (A/G) contract computation framework. Our method models inter-subsystem couplings as structured disturbances and introduces a resilience metric to guide iterative contract refinement, maximizing the assumption set (e.g., within a spherical domain) while preserving correctness and monotonicity. The approach supports finite-horizon specifications for both linear systems and general finite-horizon specifications for nonlinear systems. We provide theoretical guarantees for two subsystems and extend the results to arbitrary multi-subsystem networks. Numerical experiments on linear systems and a case study on a nonlinear DC microgrid demonstrate satisfaction of temporal specifications and validate the framework’s effectiveness in compositional reasoning.

This work addresses the lack of machine-verifiable formalizations of line search methods in nonlinear optimization, which has hindered algorithmic reliability. Within the Lean 4 theorem prover, it presents the first systematic formalization of several classical line search criteria—including Armijo, Goldstein, Wolfe, and their nonmonotone variants—alongside rigorous definitions of gradient descent, descent directions, and backtracking step-size selection. The study fully verifies the Zoutendijk convergence theorem within this framework, thereby establishing a comprehensive formal foundation for line search theory. This contribution significantly enhances the verifiability and trustworthiness of nonlinear optimization algorithms through mechanized mathematical reasoning.

This work establishes the first polynomial sensitivity lower bounds for randomized approximation algorithms solving constraint satisfaction problems (CSPs), filling a key theoretical gap. To overcome the limitation of classical lower-bound techniques—which fail to preserve sensitivity—the authors innovatively adapt the PCP framework into a sensitivity-preserving variant, integrating Hamming distance metrics and analysis within the LOCAL model of distributed computing. The results yield tight polynomial sensitivity lower bounds for fundamental problems including Maximum Clique, Minimum Vertex Cover, and Maximum Cut. Concurrently, they imply tight round-complexity lower bounds for these problems in the LOCAL model. This is the first systematic demonstration of a deep connection between algorithmic sensitivity and distributed computational complexity, laying the foundation for a unified theory bridging the robustness of approximation algorithms and the scalability of distributed computation.

This work addresses the long-standing bottleneck in prophet inequality research—manual construction of worst-case instances for tight competitive ratio proofs—by proposing the first unified framework that directly models tightness analysis as a computationally tractable optimization problem. Methodologically, it formalizes worst-case instance search as a convex optimization and linear programming problem subject to extremal probability distribution constraints, integrating tools from random-order theory and extremal probability analysis, thereby replacing the traditional decoupled paradigm of “algorithm analysis + counterexample construction.” Contributions include: (i) the first automated computation of tight competitive ratios; (ii) a unified derivation of tight bounds for multiple prophet inequality variants, yielding several new results; and (iii) rigorous verification of the optimality of several classical bounds. The framework significantly enhances the systematicity, scalability, and reliability of tightness proofs in online stochastic optimization.

Formal theories of algorithms have long been confined to non-interactive settings, leaving interactive and nondeterministic algorithms without rigorous foundational treatment. Method: This work introduces a unified formal framework encompassing both non-interactive and interactive, deterministic and nondeterministic algorithms. It proposes the “prototype algorithm” as an abstract computational model and rigorously defines its behavioral semantics. Three equivalence relations—behavioral, implementation, and specification equivalence—are formally introduced; their relationships are established, and specification equivalence is proven to be the appropriate criterion for capturing essential algorithmic identity. Contribution: The framework breaks the traditional boundaries of algorithm definitions, providing the first formal foundation for interactive algorithms. It establishes a layered, extensible meta-theory of algorithms and delivers a rigorous logical basis for reasoning about algorithmic essence, correctness verification, and cross-model comparison—thereby unifying previously fragmented formal approaches under a coherent theoretical umbrella.

This work addresses the absence of a mechanized formal verification framework for primal-dual algorithm analysis. It presents the first systematic formalization in Isabelle/HOL that supports rigorous correctness and performance verification of such algorithms, unifying a diverse range of instances—from the classical Hungarian algorithm to modern Adwords algorithms—within a single coherent framework. By achieving machine-checked proofs for multiple primal-dual algorithms, this study not only establishes their formal correctness but also develops a reusable library of verified components. The resulting infrastructure offers a novel paradigm for trustworthy verification of combinatorial optimization algorithms, enhancing both reliability and reusability in formal methods applied to algorithmic analysis.

This work addresses the lack of machine-verifiable foundations in control theory for cyber-physical systems by developing an open-source formal library within the Lean interactive theorem prover. The library formalizes Lyapunov stability theory and the small-gain theorem, supporting continuous, discrete, and hybrid dynamical systems. A key contribution is a unified formulation of Lyapunov’s theorem applicable to both points and sets, alongside a relational definition of input–output systems that avoids well-posedness assumptions, enabling a fully formalized proof of the small-gain theorem. Leveraging mathematical tools such as neighborhood filters, the project establishes a scalable verification framework for control theory, laying the groundwork for trustworthy, machine-checked validation of cyber-physical systems.

Standard simulation techniques struggle to verify liveness properties, and existing notions of fair simulation are often too complex for interactive verification. This work proposes a family of “approximate fair simulation” relations tailored to transition systems equipped with Büchi fairness conditions. By simplifying the nested inductive–coinductive structures inherent in traditional approaches, our method introduces a stronger and more user-friendly reasoning mechanism. We formalize this framework within a fixed-point logic and develop a corresponding deductive system, which we mechanize and prove correct in the Rocq proof assistant. Case studies demonstrate the effectiveness and practicality of our approach for interactive verification of fairness properties.

Addressing the challenge of jointly enforcing safety and actuator constraints in cyber-physical systems—and the lack of end-to-end formal verification in existing zonotope-based reachability analysis—this paper proposes a scalable safety control envelope verification framework. Methodologically, it employs control-invariant sets as safety certificates, integrating efficient zonotope reachability analysis with KeYmaera X’s deductive logic reasoning; formal verification of hybrid systems is achieved via compact numerical evidence generation and quantifier elimination. Our key contribution is the first principled integration of zonotope-based analysis with high-assurance logical verification, enabling rapid and trustworthy certification of high-dimensional control envelopes. Experimental evaluation demonstrates substantial improvements in both verification efficiency and reliability, with successful validation across multiple safety-critical systems.

This work proposes a novel method for the automated discovery and verification of lower confidence bounds on the mean. By introducing a general relaxation framework parameterized by order statistics, the problem of finding optimal confidence bounds is formulated as a computationally tractable optimization problem, which unifies classical results such as Hoeffding’s inequality. The approach integrates mixed-integer linear programming with optimization relaxation theory to enable, for the first time, the automatic construction and formal verification of confidence bounds. In particular, when the order-statistic function is linear—as in the case of Hoeffding-type bounds—the method yields a mixed-integer linear program of linear size, allowing efficient approximation and rigorous validation of the target confidence bound.