Toward a Lean Formalization of Analog Computing with Microwaves

2026-10-03

AI Summary

This study addresses the long-standing absence of formal verification in microwave analog computing, where the boundaries of network-computable transformations remain ill-defined. To bridge this gap, this work pioneers the integration of microwave analog computing into the Lean proof assistant by rigorously modeling physical components and topological interconnections, subsequently deriving and verifying necessary and sufficient conditions for network transformations. All proofs are mechanically checked by the Lean kernel, and the accompanying code is publicly released. By establishing a rigorous closed loop from physical devices to mathematical proofs, this project formally verifies the realizability of arbitrary power-of-two discrete Fourier transforms (DFTs) within hybrid coupler networks, thereby laying a reliable formal foundation for analog computing.

Abstract

Analog computing with microwave signals can perform linear transformations directly in the analog domain, as the signals propagate through a microwave network. A fundamental question is which transformations can be computed with a given set of microwave components. In our previous work, we answered this question for networks of hybrid couplers and phase shifters by deriving a necessary and sufficient condition on the transformations these networks can compute, and we showed that the discrete Fourier transform (DFT) satisfies it. In this paper, we take a first step toward the formalization of analog computing with microwaves in Lean, a programming language and proof assistant that is increasingly adopted in mathematics. We formalize the considered components, their series and parallel connections, and the class of networks they can implement. Then, we formally prove the necessary and sufficient condition characterizing these networks, as well as the implementability of the DFT of any size power of two. All proofs are checked by the Lean kernel, and the code is openly available at: https://github.com/matteonerini/formalizing-analog-computing.

Problem

• analog computing
• microwave networks
• linear transformations
• formal verification
• discrete Fourier transform

Innovation

• Analog computing
• Formal verification
• Lean proof assistant
• Microwave networks
• Discrete Fourier transform