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https://github.com/dmxlarchey/karp-miller
A Coq mechanization of the Karp-Miller algorithm based on Kruskal-AlmostFull
https://github.com/dmxlarchey/karp-miller
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A Coq mechanization of the Karp-Miller algorithm based on Kruskal-AlmostFull
- Host: GitHub
- URL: https://github.com/dmxlarchey/karp-miller
- Owner: DmxLarchey
- License: mpl-2.0
- Created: 2024-03-04T16:24:17.000Z (11 months ago)
- Default Branch: main
- Last Pushed: 2024-11-22T10:05:42.000Z (2 months ago)
- Last Synced: 2024-11-22T10:33:40.303Z (2 months ago)
- Language: Coq
- Size: 51.8 KB
- Stars: 1
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
```
(**************************************************************)
(* Copyright Dominique Larchey-Wendling [*] *)
(* *)
(* [*] Affiliation LORIA -- CNRS *)
(**************************************************************)
(* This file is distributed under the terms of the *)
(* Mozilla Public License Version 2.0, MPL-2.0 *)
(**************************************************************)
```
# What is this project
The project contains a correct by construction algorithm for deciding _coverability for Petri nets_, based on the construction of the
Karp-Miller tree. It crucially exploits Dickson's lemma from the [`Kruskal-AlmostFull`](https://github.com/DmxLarchey/Kruskal-AlmostFull) library
```coq
af_vec_fall2 n X R : af R → af (λ u v : vec X n, ∀ i, R u⦃i⦄ v⦃i⦄).
```
The project is loosely inspired from an [Mathematical Components based version](https://bitbucket.org/mituharu/karpmiller/src/master/)
based on the paper [_Formalization of Karp-Miller tree construction on Petri nets (CPP 2017)_](https://dl.acm.org/doi/10.1145/3018610.3018626).
It was started as a basis for further discussions with the team of [Jérôme Leroux of LaBRI](https://www.labri.fr/perso/leroux/).
The main statement that we prove here is the following:
```coq
(** with imports from Relations, KruskalTrees, KruskalFinite
and KruskalAFProp *)
Variables (NbPlaces : nat) (* number of places *)
(TrIdx : Type) (* type of indices of transitions *)
(TrIdx_fin : finite TrIdx). (* finitely many transitions *)
Notation place := (idx NbPlaces).
Notation marking := (vec nat NbPlaces).
(* Infix notations for the component wise sum and comparison of vectors *)
Infix "+ₘ" := (vec_scal plus) (at level 50, left associativity).
Infix "≦⁺" := (vec_fall2 le) (at level 70).
(* Description of a Petri net via its pre/post transitions *)
Variables (pre post : TrIdx → marking).
(* One Petri net transition *)
Inductive pn_trans : X → X → Prop :=
| pnt_intro t u : pn_trans (u +ₘ pre t) (u +ₘ post t).
(* Reachability and coverability *)
Definition pn_reachable a b := clos_refl_trans pn_trans a b.
Definition pn_coverable s a := ∃b, pn_reacheable s b ∧ a ≦⁺ b.
(* One of the main results *)
Theorem pn_coverable_dec s a : { pn_coverable s a } + { ~ pn_coverable s a }.
```
but notice that we also build the whole Karp-Miller tree as a variant. However,
its statement requires many more definitions.
# How to compile
First you need to install the dependencies via `opam`:
```console
opam update
opam install . --deps-only
make all
```
and then you can review the code, starting with the main file [`karp_miller.v`](theories/karp_miller.v).