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== References == {{Reflist}} * [[Foto Afrati|Foto N. Afrati]], Vinayak Borkar, [[Michael J. Carey (computer scientist)|Michael Carey]], Neoklis Polyzotis, [[Jeffrey Ullman|Jeffrey D. Ullman]], [https://web.archive.org/web/20140810063150/http://www.edbt.org/Proceedings/2011-Uppsala/papers/edbt/a1-afrati.pdf Map-Reduce Extensions and Recursive Queries], EDBT 2011, March 22β24, 2011, Uppsala, Sweden, {{isbn|978-1-4503-0528-0}} * {{Cite book | last1 = Aho | first1 = A. V. | author1-link = Alfred Aho | last2 = Ullman | first2 = J. D. | author2-link = Jeffrey Ullman | doi = 10.1145/567752.567763 | chapter = Universality of data retrieval languages | title = Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of programming languages - POPL '79 | pages = 110β119 | year = 1979 }} * {{Cite book | last1 = Benedikt | first1 = M. | last2 = Senellart | first2 = P. | doi = 10.1007/978-1-4614-1168-0_10 | chapter = Databases | title = Computer Science. The Hardware, Software and Heart of It| pages = 169β229| year = 2011 | isbn = 978-1-4614-1167-3 | editor1-last = Blum| editor1-first = Edward K.| editor2-last = Aho| editor2-first = Alfred V. }} * {{cite book|author1=Heinz-Dieter Ebbinghaus|author2=JΓΆrg Flum|title=Finite Model Theory|url=https://archive.org/details/finitemodeltheor0000ebbi|url-access=registration|year=1999|publisher=Springer|isbn=978-3-540-28787-2|edition=2nd|pages=[https://archive.org/details/finitemodeltheor0000ebbi/page/123 123]β124, 151β161, 220β235}} * {{cite conference | doi=10.1109/SWAT.1971.4 | contribution-url=http://mercury.pr.erau.edu/~siewerts/cs332/documents/Papers/Transitive-Closure/Transitive-Closure-with-Boolean-Matrices.pdf | first1=M.J. |last1=Fischer |first2=A.R. |last2=Meyer | contribution=Boolean matrix multiplication and transitive closure | editor=Raymond E. Miller and John E. Hopcroft | title=Proc. 12th Ann. Symp. on Switching and Automata Theory (SWAT) | publisher=IEEE Computer Society | pages=129–131 | date=Oct 1971 }} * {{cite book|author1=Erich GrΓ€del|author2=Phokion G. Kolaitis|author3=Leonid Libkin |author4=Maarten Marx |author5=Joel Spencer |author6=Moshe Y. Vardi |author7=Yde Venema |author8=Scott Weinstein|title=Finite Model Theory and Its Applications|year=2007|publisher=Springer|isbn=978-3-540-68804-4|pages=151β152}} * Keller, U., 2004, ''[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.8266 Some Remarks on the Definability of Transitive Closure in First-order Logic and Datalog]'' (unpublished manuscript)* {{citation|first=Leonid|last=Libkin|author-link=Leonid Libkin|title=Elements of Finite Model Theory|year=2004|publisher=Springer|isbn=978-3-540-21202-7|url-access=registration|url=https://archive.org/details/elementsoffinite00libk}} * {{citation|last1=Lidl|first1= R.|last2=Pilz|first2=G.|year=1998|title=''Applied abstract algebra''|edition=2nd|series= [[Undergraduate Texts in Mathematics]]|publisher= Springer|isbn=0-387-98290-6}} * {{cite journal|last=Munro| first=Ian | doi=10.1016/0020-0190(71)90006-8 | url= | title=Efficient determination of the transitive closure of a directed graph | journal=Information Processing Letters | volume=1 | number=2 | pages=56–58 | date=Jan 1971 }} * {{Cite book|last=Nuutila|first=Esko|url=http://worldcat.org/oclc/912471702|title=Efficient transitive closure computation in large digraphs|date=1995|publisher=Finnish Academy of Technology|isbn=951-666-451-2|oclc=912471702}} * {{cite book|author1=Abraham Silberschatz|author2=Henry Korth|author3=S. Sudarshan|title=Database System Concepts|year=2010|edition=6th|publisher=McGraw-Hill|isbn=978-0-07-352332-3|title-link=Database System Concepts}} [http://codex.cs.yale.edu/avi/db-book/db6/appendices-dir/c.pdf Appendix C] (online only)
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