H10-LMCS-v1.1

Specific Instructions regarding the “Hilbert Tenth problem in Coq” Paper

The code accompanying this paper is part of the Coq Library of Undecidability Proofs in the branch coq-8.13, library which depends on some external packages (Equations and MetaCoq). Therefore it is necessary to clone and checkout this release and install the whole library following the Manual installation instructions below.

However, to save some time, the final build with make all can be replaced by the command

followed by either:

Any of the above two make commands should take considerably less time than the >30min necessary for the full build of the library.

Notice that the files H10/summary.v and H10/standalone.v are specific to this release and are not included in the coq-8.13 branch of the Coq Library of Undecidability Proofs.

Coq Library of Undecidability Proofs

CI

The Coq Library of Undecidability Proofs contains mechanised reductions to establish undecidability results in Coq. The undecidability proofs are based on a synthetic approach to undecidability, where a problem P is considered undecidable if its decidability in Coq would imply the decidability of the halting problem of single-tape Turing machines in Coq. As in the traditional literature, undecidability of a problem P in the library is often established by constructing a many-one reduction from an undecidable problem to P.

For more information on the structure of the library, the synthetic approach, and included problems see Publications below, our Wiki, look at the slides or the recording of the talk on the Coq Library of Undecidability proofs at CoqPL ‘20.

The library is a collaborative effort, growing constantly and we invite everybody to contribute undecidability proofs!

Problems in the Library

The problems in the library can mostly be categorized into seed problems, advanced problems, and target problems.

Seed problems are simple to state and thus make for good starting points of undecidability proofs, often leading to easier reductions to other problems.

Advanced problems do not work well as seeds, but they highlight the potential of our library as a framework for mechanically checking pen&paper proofs of potentially hard undecidability results.

Target problems are very expressive and thus work well as targets for reduction, with the aim of closing loops in the reduction graph to establish the inter-reducibility of problems.

Seed Problems

Advanced Problems

Halting Problems for Traditional Models of Computation

Problems from Logic

Other Problems

Target Problems

Installation Instructions

If you can use opam 2 on your system, you can follow the instructions here. If you cannot use opam 2, you can use the noopam branch of this repository, which has no dependencies, but less available problems.

Install from opam

We recommend creating a fresh opam switch:

opam switch create coq-library-undecidability 4.07.1+flambda
eval $(opam env)

Then the following commands install the library:

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-library-undecidability.1.0.0+8.13

Manual installation

You need Coq 8.13 built on OCAML >= 4.07.1, the Smpl package, the Equations package, and the MetaCoq package for Coq. If you are using opam 2 you can use the following commands to install the dependencies on a new switch:

opam switch create coq-library-undecidability 4.07.1+flambda
eval $(opam env)
opam repo add coq-released https://coq.inria.fr/opam/released
opam install . --deps-only

Building the undecidability library

Compiled interfaces

The library is compatible with Coq’s compiled interfaces (vos) for quick infrastructural access.

Troubleshooting

Windows

If you use Visual Studio Code on Windows 10 with Windows Subsystem for Linux (WSL), then local opam switches may cause issues. To avoid this, you can use a non-local opam switch, i.e. opam switch create 4.07.1+flambda.

Coq version

Be careful that this branch only compiles under Coq 8.13. If you want to use a different Coq version you have to change to a different branch. Due to compatibility issues, not every branch contains exactly the same problems. We recommend to use the newest branch if possible.

Publications

Papers and abstracts on the library

A Coq Library of Undecidable Problems. Yannick Forster, Dominique Larchey-Wendling, Andrej Dudenhefner, Edith Heiter, Dominik Kirst, Fabian Kunze, Gert Smolka, Simon Spies, Dominik Wehr, Maximilian Wuttke. CoqPL ‘20. https://popl20.sigplan.org/details/CoqPL-2020-papers/5/A-Coq-Library-of-Undecidable-Problems

Papers and abstracts on problems and proofs included in the library

How to contribute

Fork the project on GitHub, add a new subdirectory for your project and your files, then file a pull request. We have guidelines for the directory structure of projects.

Contributors

Parts of the Coq Library of Undecidability Proofs reuse generic code initially developed as a library for the lecture “Introduction to Computational Logics” at Saarland University, which was written by a subset of the above contributors, Sigurd Schneider, and Jan Christian Menz.

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