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Code related to Keller's conjecture

Quick Start Guide

The wrapper Python script requires the python-igraph package, you can install it with

$ pip3 install --user python-igraph

To compile all the necessary tools, run

$ make all

To generate the files and proofs for the cases s = {3,4,6}, special targets can be invoked, e.g.:

$ make s3

Compilation Details

One can specify the location of the Boost installation in order to compile the pprsearch tool:

$ make tools/pprsearch/pprsearch BOOST_ROOT=/path/to/boost

Toolchain Details

The verification targets s<n> for n∊{3,4,6} run a number of steps verifying the addition of symmetry breaking clauses to the encoding of the clique problem for the corresponding Keller graphs. These steps are also make targets which can be invoked separately:

  • s<n>-python: run the script and generate the files for the verification step. This step generates several CNF and DRAT files numbered in sequence from the initial file s<n>.0.cnf, going through intermediate CNF formulas with some symmetry breaking clauses added, up to s<n>.cnf which is the final version of the CNF. This last version contains all the symmetry breaking clauses and is the formula passed to the SAT solver. Along with each formula, a DRAT file justifying the addition of the next batch of symmetry breaking clauses is generated. This target also creates a file s<n>.dnf which is a DNFtautology containing all the assignments that need to be tried by a SAT solver in order to prove the formula unsatisfiable. The output CNF and DNF files of this call are recorded in the outputs directory.
  • s<n>-drat-trim: This target verifies the DRAT files generated by the s<n>-python target. The verification is sequential.
  • s<n>-tautology: This target verifies that the DNF file output by s<n>-python is indeed a tautology.

Of these 3 steps of the verification process, the first two can take a significant amount of time (days or even weeks). In order to speed up the verification process, we provide a set of scripts that allow for the generation and checking of the proof files to happen in parallel. We strongly recommend running the verification in parallel as indicated in the next section.

SLURM Scripts to Run the DRAT Verification

The scripts in the slurm folder automate the parallel verification of the transformations done on the CNF formulation of Keller's conjecture.


All shell commands are meant to be run in the top directory of this repository. The first batch of files with the *.ippr extension are generated by the script. For instance, for s=3, running

$ python3 3 s3 ./Keller-encode

will generate these files.

If needed, this can be run in the cluster environment with

$ sbatch <sbatch-arguments> slurm/ 3

After this step is complete, parallel verification can be done with

$ slurm/ 3

Once these jobs are done, the files with the *.log extension will contain the output of the verification by drat-trim. Each file should have the line

c VERIFIED derivation: all lemmas preserve satisfiability

You can edit the variables at the top of the verification script according to the queue names and time limits of your cluster environment.

ACL2 Verified Checker

The proofs of unsatisfiability can be verified using drat-trim. Alternatively, you can use the ACL2 verified checker for these proofs. To build the ACL2 checker you will need a Common Lisp implementation. This build has been tested with Steel Bank Common Lisp (sbcl) and Clozure (ccl).

Targets are provided for getting the sources, certifying the checker and building a binary for it:

$ make acl2-checker

The default Common Lisp implementation is sbcl. You can specify your own implementation:

$ make acl2-checker LISP=ccl

You can test this installation with the provided examples:

$ acl2-8.2/books/projects/sat/lrat/cube/ examples/s3-12345.cnf examples/s3-12345.clrat examples/s3-12345-out.cnf


code related to Keller's conjecture



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