Polygraphs of finite derivation type

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This survey presents Squier's results in the contemporary language of polygraphs and higher-dimensional categories, with new proofs and relations between them.

[1]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[2]  Philippe Malbos,et al.  Higher-dimensional normalisation strategies for acyclicity , 2010, 1011.0558.

[3]  H. Brown,et al.  Computational Problems in Abstract Algebra , 1971 .

[4]  Yves Métivier.  About the Rewriting Systems Produced by the Knuth-Bendix Completion Algorithm , 1983, Inf. Process. Lett..

[5]  D. Anick,et al.  On the homology of associative algebras , 1986 .

[6]  Samuel Mimram,et al.  A Homotopical Completion Procedure with Applications to Coherence of Monoids , 2013, RTA.

[7]  Ross Street,et al.  The algebra of oriented simplexes , 1987 .

[8]  Yves Guiraud,et al.  Termination orders for 3-dimensional rewriting , 2006, ArXiv.

[9]  M. Newman.  On Theories with a Combinatorial Definition of "Equivalence" , 1942 .

[10]  Philippe Malbos,et al.  Higher-dimensional categories with finite derivation type , 2008, 0810.1442.

[11]  Craig C. Squier,et al.  Word problems and a homological niteness condition for monoids , 1987 .

[12]  Friedrich Otto,et al.  Finite Derivation Type Implies the Homological Finiteness Condition FP_3 , 1994, J. Symb. Comput..

[13]  Matthias Jantzen.  A Note on a Special One-Rule Semi-Thue System , 1985, Inf. Process. Lett..

[14]  Terese.  Term rewriting systems , 2003, Cambridge tracts in theoretical computer science.

[15]  Michael Barr,et al.  Category theory for computing science , 1995, Prentice Hall International Series in Computer Science.

[16]  Kenneth S. Brown,et al.  The Geometry of Rewriting Systems: A Proof of the Anick-Groves-Squier Theorem , 1992 .

[17]  J. R. J. Groves.  Rewriting systems and homology of groups , 1990 .

[18]  Yves Lafont,et al.  Church-Rooser property and homology of monoids , 1991, Mathematical Structures in Computer Science.

[19]  Yves Lafont,et al.  Algebra and Geometry of Rewriting , 2007, Appl. Categorical Struct..

[20]  Jesus Gonzalez,et al.  The BP-theory of 2-fold products of real projective spaces , 2008 .

[21]  Philippe Malbos,et al.  Coherence in monoidal track categories , 2010, Mathematical Structures in Computer Science.

[22]  Ross Street,et al.  Limits indexed by category-valued 2-functors , 1976 .

[23]  J. Huebschmann,et al.  Identities among relations , 1982 .

[24]  Yves Lafont.  A new finiteness condition for monoids presented by complete rewriting systems , 1995 .

[25]  Friedrich Otto,et al.  String-Rewriting Systems , 1993, Text and Monographs in Computer Science.

[26]  Daniel E. Cohen,et al.  A Monoid which is Right FP∞ but not Left FP1 , 1992 .

[27]  Yuji Kobayashi,et al.  Complete rewriting systems and homology of monoid algebras , 1990 .

[28]  R. Lyndon,et al.  Combinatorial Group Theory , 1977 .

[29]  FRANÇOIS MÉTAYER,et al.  Resolutions by Polygraphs , 2003 .

[30]  François Métayer,et al.  Cofibrant objects among higher-dimensional categories , 2008 .

[31]  Philippe Malbos,et al.  Coherent presentations of Artin monoids , 2012, Compositio Mathematica.

[32]  Friedrich Otto,et al.  A Finiteness Condition for Rewriting Systems , 1994, Theor. Comput. Sci..

[33]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[34]  Emil L. Post.  Recursive Unsolvability of a problem of Thue , 1947, Journal of Symbolic Logic.

[35]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

[36]  Friedrich Otto,et al.  Infinite Convergent String-Rewriting Systems and Cross-Sections for Finitely Presented Monoids , 1998, J. Symb. Comput..

[37]  Stephen J. Pride.  Low-Dimensional homotopy Theory for Monoids , 1995, Int. J. Algebra Comput..

[38]  Robert Cremanns.  Finiteness conditions for rewriting systems , 1995 .

[39]  Philippe Malbos,et al.  Identities among relations for higher-dimensional rewriting systems , 2009, 0910.4538.

[40]  Samuel Mimram,et al.  Computing Critical Pairs in 2-Dimensional Rewriting Systems , 2010, RTA.

[41]  Albert Burroni,et al.  Higher-Dimensional Word Problems with Applications to Equational Logic , 1993, Theor. Comput. Sci..

[42]  Friedrich Otto,et al.  For Groups the Property of Having Finite Derivation Type is Equivalent to the Homological Finiteness Condition FP_3 , 1996, J. Symb. Comput..

[43]  Paliath Narendran,et al.  A Finite Thue System with Decidable Word Problem and without Equivalent Finite Canonical System , 1985, Theor. Comput. Sci..