Boosting isomorphic model filtering with invariants

The enumeration of finite models is very important to the working discrete mathematician (algebra, graph theory, etc) and hence the search for effective methods to do this task is a critical goal in discrete computational mathematics. However, it is hindered by the possible existence of many isomorphic models, which usually only add noise. Typically, they are filtered out a posteriori, a step that might take a long time just to discard redundant models. This paper proposes a novel approach to split the generated models into mutually non-isomorphic blocks. To do that we use well-designed hand-crafted invariants as well as randomly generated invariants. The blocks are then tackled separately and possibly in parallel. This approach is integrated into Mace4 (the most popular tool among mathematicians) where it shows tremendous speed-ups for a large variety of algebraic structures.

[1]  M. Elhamdadi,et al.  Automorphism Groups of Quandles , 2010, 1012.5291.

[2]  William McCune,et al.  Mace4 Reference Manual and Guide , 2003, ArXiv.

[3]  D. Marker.  Model theory : an introduction , 2002 .

[4]  K. Claessen,et al.  New Techniques that Improve MACE-style Finite Model Finding , 2007 .

[5]  Jian Zhang.  Constructing finite algebras with FALCON , 2004, Journal of Automated Reasoning.

[6]  N. Sloane.  The on-line encyclopedia of integer sequences , 2018, Notices of the American Mathematical Society.

[7]  Bart Selman,et al.  Boosting Combinatorial Search Through Randomization , 1998, AAAI/IAAI.

[8]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[9]  Gilles Audemard,et al.  Predicting and Detecting Symmetries in FOL Finite Model Search , 2006, Journal of Automated Reasoning.

[10]  J. Araújo,et al.  Filtering Isomorphic Models by Invariants , 2021 .

[11]  P. Cameron.  Permutation groups , 1996 .

[12]  Gilles Audemard,et al.  The eXtended Least Number Heuristic , 2001, IJCAR.

[13]  Hantao Zhang,et al.  SEM: a System for Enumerating Models , 1995, IJCAI.

[14]  G. Nagy.  Computing with quasigroups and loops in GAP , 2005 .

[15]  Pierre Hansen,et al.  NP-hardness of Euclidean sum-of-squares clustering , 2008, Machine Learning.

[16]  James M. Crawford,et al.  Symmetry-Breaking Predicates for Search Problems , 1996, KR.

[17]  Laurent Henocque,et al.  A Hybrid Method for Finite Model Search in Equational Theories , 1999, Fundam. Informaticae.

[18]  Joao Marques-Silva,et al.  Complete unrestricted backtracking algorithms for Satisfiability , 2002 .

[19]  Joao Marques-Silva,et al.  Using Randomization and Learning to Solve Hard Real-World Instances of Satisfiability , 2000, CP.

[20]  Ian P. Gent,et al.  Minion: A Fast Scalable Constraint Solver , 2006, ECAI.

[21]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[22]  Brendan D. McKay,et al.  Isomorph-Free Exhaustive Generation , 1998, J. Algorithms.

[23]  Mikolás Janota,et al.  Towards Smarter MACE-style Model Finders , 2018, LPAR.

[24]  Jian Zhang,et al.  A Powerful Technique to Eliminate Isomorphism in Finite Model Search , 2006, IJCAR.

[25]  Martin E. Malandro.  Enumeration of finite inverse semigroups , 2013, Semigroup Forum.

[26]  Thierry Boy de la Tour,et al.  An Isomorph-Free SEM-Like Enumeration of Models , 2005, Electron. Notes Theor. Comput. Sci..

[27]  Giles Reger,et al.  Symmetry Avoidance in MACE-Style Finite Model Finding , 2019, FroCos.