Functorial Question Answering

We study the relational variant of the categorical compositional distributional (DisCoCat) models of Coecke et al, where we replace vector spaces and linear maps by sets and relations. We show that RelCoCat models factorise through Cartesian bicategories, as a corollary we get logspace reductions from semantics and entailment to evaluation and containment of conjunctive queries respectively. Finally, we define question answering as an NP-complete problem.

[1]  Anne Preller,et al.  Natural language semantics in biproduct dagger categories , 2014, J. Appl. Log..

[2]  Anne Preller,et al.  Semantic Vector Models and Functional Models for Pregroup Grammars , 2011, J. Log. Lang. Inf..

[3]  Joachim Lambek,et al.  Type Grammar Revisited , 1997, LACL.

[4]  Jean-François Baget,et al.  Extensions of Simple Conceptual Graphs: the Complexity of Rules and Constraints , 2011, J. Artif. Intell. Res..

[5]  Anand Rajaraman,et al.  Conjunctive query containment revisited , 1997, Theor. Comput. Sci..

[6]  Anne Preller.  From Logical to Distributional Models , 2013, QPL.

[7]  Bob Coecke,et al.  Towards Compositional Distributional Discourse Analysis , 2018, CAPNS@QI.

[8]  Stephen Clark,et al.  The Frobenius anatomy of word meanings I: subject and object relative pronouns , 2013, J. Log. Comput..

[9]  Bob Coecke,et al.  Quantum Algorithms for Compositional Natural Language Processing , 2016, SLPCS@QPL.

[10]  Anne Preller.  Linear Processing with Pregroups , 2007, Stud Logica.

[11]  Boris Motik,et al.  Estimating the Cardinality of Conjunctive Queries over RDF Data Using Graph Summarisation , 2018, WWW.

[12]  Filippo Bonchi,et al.  Graphical Conjunctive Queries , 2018, CSL.

[13]  Stephen Clark,et al.  Mathematical Foundations for a Compositional Distributional Model of Meaning , 2010, ArXiv.

[14]  Michaël Thomazo,et al.  Conjunctive Query Answering Under Existential Rules - Decidability, Complexity, and Algorithms , 2013 .

[15]  Evan Patterson,et al.  Knowledge Representation in Bicategories of Relations , 2017, ArXiv.

[16]  Martha Lewis,et al.  Generalized relations in linguistics & cognition , 2018, Theor. Comput. Sci..

[17]  P. Selinger.  A Survey of Graphical Languages for Monoidal Categories , 2009, 0908.3347.

[18]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[19]  A. Carboni,et al.  Cartesian bicategories I , 1987 .

[20]  Anne Preller,et al.  Free compact 2-categories , 2007, Mathematical Structures in Computer Science.

[21]  Katarzyna Moroz.  A Savateev-Style Parsing Algorithm for Pregroup Grammars , 2009, FG.

[22]  Ashok K. Chandra,et al.  Optimal implementation of conjunctive queries in relational data bases , 1977, STOC '77.

[23]  S. Clark,et al.  A Compositional Distributional Model of Meaning , 2008 .

[24]  David I. Spivak,et al.  Graphical Regular Logic , 2018, ArXiv.

[25]  Dimitri Kartsaklis,et al.  Sentence entailment in compositional distributional semantics , 2015, Annals of Mathematics and Artificial Intelligence.

[26]  Samson Abramsky,et al.  Relating Structure and Power: Comonadic Semantics for Computational Resources , 2018, CSL.

[27]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[28]  Mati Pentus.  Models for the Lambek Calculus , 1995, Ann. Pure Appl. Log..

[29]  Wojciech Buszkowski,et al.  Pregroup Grammars and Context-free Grammars , 2007 .