Hereditarily finite representations of natural numbers and self-delimiting codes
Author: 
Paul TarauAuthors Info & Claims
Paul TarauAuthors Info & ClaimsPages 11 - 18
Abstract
Using a bijection between natural numbers and hereditarily finite functions we derive a new reversible variable length self-delimiting code through a bitstring representation in a balanced parenthesis language. The code features the ability to encode arbitrarily nested data types, can represent huge (low "complexity") numbers, and is decodable from its beginning or its end. Besides its possible practical applications to media stream encodings, a comparison with the well-known Elias omega code and a conjecture about its asymptotic behavior under the Kraft inequality suggest it as an interesting object of study for experimental mathematicians.
The paper is organized as a self-contained literate Haskell program inviting the reader to explore its content independently. Its code is available at http://logic.cse.unt.edu/tarau/research/2010/selfdelim.hs
References
[1]
}}Artem Alimarine, Sjaak Smetsers, Arjen van Weelden, Marko van Eekelen, and Rinus Plasmeijer. There and back again: arrows for invertible programming. In Haskell '05: Proceedings of the 2005 ACM SIGPLAN workshop on Haskell, pages 86--97, New York, NY, USA, 2005. ACM Press.
[2]
}}J. Berstel and L. Boasson. Balanced grammars and their languages. Lecture Notes In Computer Science, pages 3--25, 2002.
[3]
}}J. Berstel and L. Boasson. Formal properties of XML grammars and languages. Acta Informatica, 38 (9): 649--671, 2002.
[4]
}}Cristian Calude and Arto Salomaa. Algorithmically coding the universe. In Developments in Language Theory, World Scientific, pages 472--492, 1994.
[5]
}}G. J. Chaitin. Information-theoretic incompleteness. Applied Mathematics and Computation, 52 (1): 83--101, 1992.
[6]
}}Conal Elliott. Module: Data.Bijections. Haskell source code library at: http://haskell.org/haskellwiki/TypeCompose.
[7]
}}C. Deppe and H. Schnettler. On the 3/4-Conjecture for Fix-Free Codes. DMTCS Proceedings, (1), 2006.
[8]
}}P. Elias. Universal codeword sets and representations of the integers. IEEE Transactions on Information Theory, 21 (2): 194--203, 1975.
[9]
}}K. Gödel. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38: 173--198, 1931.
[10]
}}Juris Hartmanis and Theodore P. Baker. On Simple Goedel Numberings and Translations. In Jacques Loeckx, editor, ICALP, volume 14 of Lecture Notes in Computer Science, pages 301--316. Springer, 1974. ISBN 3-540-06841-4. URL http://dblp.uni-trier.de/db/conf/icalp/icalp74.html#HartmanisB74.
[11]
}}John Hughes. Generalizing Monads to Arrows. Science of Computer Programming 37, pp. 67--111, May 2000.
[12]
}}Laszlo Kalmar. On the Reduction of the Decision Problem. First Paper. Ackermann Prefix, A Single Binary Predicate. The Journal of Symbolic Logic, 4 (1): 1--9, mar 1939. ISSN 0022-4812.
[13]
}}Richard Kaye and Tin Lock Wong. On Interpretations of Arithmetic and Set Theory. Notre Dame J. Formal Logic Volume, 48 (4): 497--510, 2007.
[14]
}}Laurence Kirby. Addition and multiplication of sets. Math. Log. Q., 53 (1): 52--65, 2007.
[15]
}}Donald Knuth. The Art of Computer Programming, Volume 4, draft, 2006. http://www-cs-faculty.stanford.edu/~knuth/taocp.html.
[16]
}}L. G. Kraft. A device for quantizing, grouping, and coding amplitude-modulated pulses. Master's thesis, Massachusetts Institute of Technology. Dept. of Electrical Engineering, 1949.
[17]
}}Ming Li and Paul Vitányi. An introduction to Kolmogorov complexity and its applications. Springer-Verlag New York, Inc., New York, NY, USA, 1993. ISBN 0-387-94053-7.
[18]
}}J. Liebehenschel. Ranking and unranking of a generalized Dyck language and the application to the generation of random trees. Séminaire Lotharingien de Combinatoire, 43: 19, 2000.
[19]
}}Conrado Martinez and Xavier Molinero. Generic algorithms for the generation of combinatorial objects. In Branislav Rovan and Peter Vojtas, editors, MFCS, volume 2747 of Lecture Notes in Computer Science, pages 572--581. Springer, 2003.
[20]
}}John McCarthy. Recursive functions of symbolic expressions and their computation by machine, part i. Commun. ACM, 3 (4): 184--195, 1960. ISSN 0001-0782. http://doi.acm.org/10.1145/367177.367199.
[21]
}}Erik Meijer and Graham Hutton. Bananas in Space: Extending Fold and Unfold to Exponential Types. In FPCA, pages 324--333, 1995.
[22]
}}Jozef Pepis. Ein verfahren der mathematischen logik. The Journal of Symbolic Logic, 3 (2): 61--76, jun 1938. ISSN 0022-4812.
[23]
}}Julia Robinson. General recursive functions. Proceedings of the American Mathematical Society, 1 (6): 703--718, dec 1950. ISSN 0002-9939.
[24]
}}Frank Ruskey and Andrzej Proskurowski. Generating binary trees by transpositions. J. Algorithms, 11: 68--84, 1990.
[25]
}}S. A. Savari. On minimum-redundancy fix-free codes. In Proc. of the Data Compression Conference, Snowbird, UT, 2009.
[26]
}}Moto-o Takahashi. A Foundation of Finite Mathematics. Publ. Res. Inst. Math. Sci., 12 (3): 577--708, 1976.
[27]
}}Paul Tarau. Declarative Combinatorics: Isomorphisms, Hylomorphisms and Hereditarily Finite Data Types in Haskell, January 2009. http://arXiv.org/abs/0808.2953, unpublished draft, 104 pages.
[28]
}}Paul Tarau. A Groupoid of Isomorphic Data Transformations. In J. Carette, L. Dixon, C. S. Coen, and S. M. Watt, editors, Intelligent Computer Mathematics, 16th Symposium, Calculemus 2009, 8th International Conference MKM 2009, pages 170--185, Grand Bend, Canada, July 2009. Springer, LNAI 5625.
[29]
}}Paul Tarau. Isomorphisms, Hylomorphisms and Hereditarily Finite Data Types in Haskell. In Proceedings of ACM SAC'09, pages 1898--1903, Honolulu, Hawaii, March 2009. ACM.
[30]
}}Paul Tarau. "Everything Is Everything" Revisited: Shapeshifting Data Types with Isomorphisms and Hylomorphisms. Complex Systems, (18), 2010.
[31]
}}J. Wen and J. D. Villasenor. Reversible variable length codes for efficient and robust image and video coding. In Proceedings Data Compression Conference, pages 471--480, 1998.
Index Terms
- Hereditarily finite representations of natural numbers and self-delimiting codes
Recommendations
On Minimum-Redundancy Fix-Free Codes
DCC '09: Proceedings of the 2009 Data Compression ConferenceFix-free codes are variable length codes in which no codeword is the prefix or suffix of another codeword. They are used in video compression standards because their property of efficient decoding in both the forward and backward directions assists with ...
132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers
We describe the recursive structures of the set of two-stack sortable permutations which avoid 132 and the set of two-stack sortable permutations which contain 132 exactly once. Using these results and standard generating function techniques, we ...
Comments
Information & Contributors
Information
Published In
September 2010
62 pages
Copyright © 2010 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 25 September 2010
Check for updates
Author Tags
Qualifiers
- Research-article
Conference
ICFP '10
Sponsor:
ICFP '10: ACM SIGPLAN International Conference on Functional Programming
September 25, 2010
Maryland, Baltimore, USA
Acceptance Rates
MSFP '10 Paper Acceptance Rate 4 of 8 submissions, 50%;
Overall Acceptance Rate 4 of 8 submissions, 50%
Upcoming Conference
ICFP '25
- Sponsor:
- sigplan
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 90Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 25 Jan 2025
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in