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Isabelle

Our preprint on formally verified neurosymbolic trajectory learning is out on arXiv

    Formally Verified Neurosymbolic Trajectory Learning via Tensor-based Linear Temporal Logic on Finite Traces

    Astract:

    We present a novel formalisation of tensor semantics for linear temporal logic on finite traces (LTLf), with formal proofs of correctness carried out in the theorem prover Isabelle/HOL. We demonstrate that this formalisation can be integrated into a neurosymbolic learning process by defining and verifying a differentiable loss function for the LTLf constraints, and automatically generating an implementation that integrates with PyTorch. We show that, by using this loss, the process learns to satisfy pre-specified logical constraints. Our approach offers a fully rigorous framework for constrained training, eliminating many of the inherent risks of ad-hoc, manual implementations of logical aspects directly in an “unsafe” programming language such as Python, while retaining efficiency in implementation.

    Paper: https://arxiv.org/abs/2501.13712

    Our formalisation of Linear Resources and Process Compositions has been published in the Archive of Formal Proof

      Abstract

      In this entry we formalise a framework for process composition based on actions that are specified by their input and output resources. We verify their correctness by translating compositions of process into deductions of intuitionistic linear logic. As part of the verification we derive simple conditions on the compositions which ensure well-formedness of the corresponding deduction.

      We describe an earlier version of this formalisation in our article Linear Resources in Isabelle/HOL, which also includes a formalisation of manufacturing processes in the simulation game Factorio.

      Our formalisation of Lie Groups and Algebras has been published in the Archive of Formal Proof

        Abstract

        Lie Groups are formalised as locales, building on the AFP theory of Smooth Manifolds. We formalise the diffeomorphism group of a manifold, and the action of a Lie group on a manifold. The general linear group is shown to be a Lie group by proving properties of the determinant, and matrix inverses. We also develop a theory of smooth vector fields on a manifold , defined as smooth maps from the manifold to its tangent bundle . We employ a shortcut that avoids difficulties in defining the tangent bundle as a manifold, but which still leads to vector fields with the properties one would expect. We then construct the Lie algebra of a Lie group as an algebra of left-invariant smooth vector fields.

        Schmoetten R. and Fleuriot J. D. (2024). Lie Groups and Algebras. Archive of Formal Proofs. ISSN: 2150-914x.

        Our paper “Linear Resources in Isabelle/HOL” has just been published in the Journal of Automated Reasoning

          Abstract:

          We present a formal framework for process composition based on actions that are specified by their input and output resources. The correctness of these compositions is verified by translating them into deductions in intuitionistic linear logic. As part of the verification we derive simple conditions on the compositions which ensure well-formedness of the corresponding deduction when satisfied. We mechanise the whole framework, including a deep embedding of ILL, in the proof assistant Isabelle/HOL. Beyond the increased confidence in our proofs, this allows us to automatically generate executable code for our verified definitions. We demonstrate our approach by formalising part of the simulation game Factorio and modelling a manufacturing process in it. Our framework guarantees that this model is free of bottlenecks.

          Smola, F., Fleuriot, J.D. Linear Resources in Isabelle/HOL. J Autom Reasoning 68, 9 (2024). https://doi.org/10.1007/s10817-024-09698-2

          Paper accepted at CICM 2022

            Our paper, “Re-imagining the Isabelle Archive of Formal Proofs” (MacKenzie, Huch, Vaughan and Fleuriot), has been accepted at the 15th Conference on Intelligent Computer Mathematics (CICM 2022).

            New Archive of Formal Proof Website

              A project involving Carlin MacKenzie, James Vaughan and Jacques Fleuriot has resulted in a re-design and re-implementation of the Archive of Formal Proofs, which is a collection of proof libraries, examples, and larger scientific developments, mechanically checked in the theorem prover Isabelle. The revamped website (https://www.isa-afp.org) is now live and will serve the Isabelle community across the world. The Edinburgh team worked with Fabian Huch of TU Munich to integrate their work into the AFP infrastructure.