arXiv subjects

Kyle A. Miller

Publications and source records attributed to Kyle A. Miller.

6 recordsLinked to original sources

arXiv2025

ScenicProver: A Framework for Compositional Probabilistic Verification of Learning-Enabled Systems

Full verification of learning-enabled cyber-physical systems (CPS) has long been intractable due to challenges including black-box components and complex real-world environments. Existing tools either provide formal guarantees for limited types of systems or test the system as a monolith, but no general framework exists for compositional analysis of learning-enabled CPS using varied verification techniques over complex real-world environments. This paper introduces ScenicProver, a verification framework that aims to fill this gap. Built upon the Scenic probabilistic programming language, the framework supports: (1) compositional system description with clear component interfaces, ranging from interpretable code to black boxes; (2) assume-guarantee contracts over those components using an extension of Linear Temporal Logic containing arbitrary Scenic expressions; (3) evidence generation through testing, formal proofs via Lean 4 integration, and importing external assumptions; (4) systematic combination of generated evidence using contract operators; and (5) automatic generation of assurance cases tracking the provenance of system-level guarantees. We demonstrate the framework's effectiveness through a case study on an autonomous vehicle's automatic emergency braking system with sensor fusion. By leveraging manufacturer guarantees for radar and laser sensors and focusing testing efforts on uncertain conditions, our approach enables stronger probabilistic guarantees than monolithic testing with the same computational budget.

cs.LO↗

arXiv2025

Fast Collection Operations from Indexed Stream Fusion

We present a system of efficient methods for traversing and combining associative collection data structures. A distinguishing feature of the system is that, like traditional sequential iterator libraries, it does not require specialized compiler infrastructure or staged compilation for efficiency and composability. By using a representation based on indexed streams, the library can express complex joins over input collections while using no intermediate allocations. We implement the library for the Lean, Morphic, and Rust programming languages and provide a mechanized proof of functional correctness in Lean.

cs.PL↗

arXiv2025

LeanLTL: A unifying framework for linear temporal logics in Lean

We propose LeanLTL, a unifying framework for linear temporal logics in Lean 4. LeanLTL supports reasoning about traces that represent either infinite or finite linear time. The library allows traditional LTL syntax to be combined with arbitrary Lean expressions, making it straightforward to define properties involving numerical or other types. We prove that standard flavors of LTL can be embedded in our framework. The library also provides automation for reasoning about LeanLTL formulas in a way that facilitates using Lean's existing tactics. Finally, we provide examples illustrating the utility of the library in reasoning about systems that come from applications.

cs.LO↗

arXiv2023

The homological arrow polynomial for virtual links

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface's skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled "whiskers" that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study $(\mathbb{Z}/n\mathbb{Z})$-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu's characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman-Murasugi-Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an "h-reduced" diagram $D$ is $4(c(D)-g(D)+1)$.

math.GT↗

arXiv2021

Formalizing Hall's Marriage Theorem in Lean

We formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, which is a community-driven effort to build a unified mathematics library for Lean. One goal of the mathlib project is to contain all of the topics of a complete undergraduate mathematics education. We provide three presentations of the main theorem statement: in terms of indexed families of finite sets, of relations on types, and of matchings in bipartite graphs. We also formalize a version of Kőnig's lemma (in terms of inverse limits) to boost the theorem to the case of countably infinite index sets. We give a description of the design of the recent mathlib library for simple graphs, and we also give a necessary and sufficient condition for a simple graph to carry a function.

math.CO↗

arXiv2018

Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $\mathfrak{sl}(N)$ Penrose polynomial for non-cubic graphs, giving contraction-deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.

math.GT↗