research-article

Algorithmic Compression of Finite Tree Languages by Rigid Acyclic Grammars

Authors:

Sebastian Eberhard,

Stefan HetzlAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 18, Issue 4

Article No.: 26, Pages 1 - 20

https://doi.org/10.1145/3127401

Published: 21 September 2017 Publication History

Abstract

We present an algorithm to optimally compress a finite set of terms using a vectorial totally rigid acyclic tree grammar. This class of grammars has a tight connection to proof theory, and the grammar compression problem considered in this article has applications in automated deduction. The algorithm is based on a polynomial-time reduction to the MaxSAT optimization problem. The crucial step necessary to justify this reduction consists of applying a term rewriting relation to vectorial totally rigid acyclic tree grammars. Our implementation of this algorithm performs well on a large real-world dataset.

References

[1]

Brian Alspach, Peter Eades, and Gordon Rose. 1983. A lower-bound for the number of productions required for a certain class of languages. Discrete Applied Mathematics 6, 2 (1983), 109--115.

[2]

Josep Argelich, Chu Min Li, Felip Manya, and Jordi Planes. 2008. The first and second max-SAT evaluations.Journal on Satisfiability, Boolean Modeling and Computation 4, 2--4 (2008), 251--278.

[3]

Walter Bucher. 1981. A note on a problem in the theory of grammatical complexity. Theoretical Computer Science 14 (1981), 337--344.

[4]

Walter Bucher, Hermann A. Maurer, and Karel Culik II. 1984. Context-free complexity of finite languages. Theoretical Computer Science 28 (1984), 277--285.

[5]

Walter Bucher, Hermann A. Maurer, Karel Culik II, and Detlef Wotschke. 1981. Concise description of finite languages. Theoretical Computer Science 14 (1981), 227--246.

[6]

Giorgio Busatto, Markus Lohrey, and Sebastian Maneth. 2008. Efficient memory representation of XML document trees. Information Systems 33, 45 (2008), 456--474. Selected Papers from the 10th International Symposium on Database Programming Languages (DBPL’05).

Digital Library

[7]

Samuel R. Buss. 1995. On Herbrand’s theorem. In Logic and Computational Complexity. Lecture Notes in Computer Science, Vol. 960. Springer, 195--209.

[8]

Cezar Câmpeanu, Nicolae Santean, and Sheng Yu. 1998. Minimal cover-automata for finite languages. In Proceedings of the 3rd International Workshop on Implementing Automata (WIA’98) (Lecture Notes in Computer Science), Jean-Marc Champarnaud, Denis Maurel, and Djelloul Ziadi (Eds.), Vol. 1660. Springer, 43--56.

[9]

Cezar Câmpeanu, Nicolae Santean, and Sheng Yu. 2001. Minimal cover-automata for finite languages. Theoretical Computer Science 267, 1--2 (2001), 3--16.

Digital Library

[10]

Hubert Comon, Max Dauchet, Rémi Gilleron, Florent Jacquemard, Denis Lugiez, Christof Löding, Sophie Tison, and Marc Tommasi. 2007. Tree Automata Techniques and Applications. (2007). Available on: http://www.grappa.univ-lille3.fr/tata.

[11]

Nachum Dershowitz and David A. Plaisted. 2001. Rewriting. In Handbook of Automated Reasoning. Vol. 1. 535--610.

[12]

Sebastian Eberhard and Stefan Hetzl. 2015a. Compressibility of finite languages by grammars. In Proceedings of the 17th International Workshop on Descriptional Complexity of Formal Systems, (DCFS’15), J. Shallit and A. Okhotin (Eds.). 93--104.

[13]

Sebastian Eberhard and Stefan Hetzl. 2015b. Inductive theorem proving based on tree grammars. Annals of Pure and Applied Logic 166, 6 (2015), 665--700.

[14]

Sebastian Eberhard and Stefan Hetzl. 2017. On the compressibility of finite languages and formal proofs. http://www.dmg.tuwien.ac.at/hetzl/research/compressibility_proofs.pdf. To appear.

[15]

Gabriel Ebner, Stefan Hetzl, Alexander Leitsch, Giselle Reis, and Daniel Weller. 2017. On the generation of quantified lemmas. (2017). Submitted.

[16]

Gabriel Ebner, Stefan Hetzl, Giselle Reis, Martin Riener, Simon Wolfsteiner, and Sebastian Zivota. 2016. System description: GAPT 2.0. In Proceedings of hte 8th International Joint Conference on Automated Reasoning (IJCAR’16).

Digital Library

[17]

Adriá Gascón, Guillem Godoy, and Florent Jacquemard. 2009. Closure of tree automata languages under innermost rewriting. Electronic Notes in Theoretical Computer Science 237 (2009), 23--38. Proceedings of the 8th International Workshop on Reduction Strategies in Rewriting and Programming (WRS’08).

Digital Library

[18]

Jacques Herbrand. 1930. Recherches sur la théorie de la démonstration. Ph.D. Dissertation. Université de Paris.

[19]

Stefan Hetzl. 2012. Applying tree languages in proof theory. In Language and Automata Theory and Applications (LATA’12) (Lecture Notes in Computer Science), Adrian-Horia Dediu and Carlos Martín-Vide (Eds.), Vol. 7183. Springer, 301--312.

Digital Library

[20]

Stefan Hetzl, Alexander Leitsch, Giselle Reis, Janos Tapolczai, and Daniel Weller. 2014b. Introducing quantified cuts in logic with equality. In Proceedings of the 7th International Joint Conference on Automated Reasoning (IJCAR’14) (Lecture Notes in Computer Science), Stéphane Demri, Deepak Kapur, and Christoph Weidenbach (Eds.), Vol. 8562. Springer, 240--254.

[21]

Stefan Hetzl, Alexander Leitsch, Giselle Reis, and Daniel Weller. 2014a. Algorithmic introduction of quantified cuts. Theoretical Computer Science 549 (2014), 1--16.

[22]

Stefan Hetzl, Alexander Leitsch, and Daniel Weller. 2012. Towards algorithmic cut-introduction. In Logic for Programming, Artificial Intelligence and Reasoning (LPAR’18) (Lecture Notes in Computer Science), Vol. 7180. Springer, 228--242.

Digital Library

[23]

Florent Jacquemard, Francis Klay, and Camille Vacher. 2009. Rigid tree automata. In Language and Automata Theory and Applications (LATA’09) 2009 (Lecture Notes in Computer Science), Adrian Horia Dediu, Armand-Mihai Ionescu, and Carlos Martín-Vide (Eds.), Vol. 5457. Springer, 446--457.

Digital Library

[24]

Florent Jacquemard, Francis Klay, and Camille Vacher. 2011. Rigid tree automata and applications. Information and Computation 209, 3 (2011), 486--512.

Digital Library

[25]

John C. Kieffer and En-hui Yang. 2000. Grammar based codes: A new class of universal lossless source codes. IEEE Transactions on Information Theory 46 (2000), 737--754.

Digital Library

[26]

N. Jesper Larsson and Alistair Moffat. 1999. Offline dictionary-based compression. In Proceedings of the Data Compression Conference, 1999 (DCC’99). 296--305.

[27]

Markus Lohrey. 2012. Algorithmics on SLP-compressed strings: A survey. Groups Complexity Cryptology 4, 2 (2012), 241--299.

[28]

Markus Lohrey, Sebastian Maneth, and Roy Mennicke. 2013. XML tree structure compression using RePair. Information Systems 38, 8 (2013), 1150--1167.

Digital Library

[29]

Ruben Martins, Vasco M. Manquinho, and Inês Lynce. 2014. Open-WBO: A modular MaxSAT solver. In Theory and Applications of Satisfiability Testing (SAT’14). 438--445.

[30]

Craig G. Nevill-Manning and Ian H. Witten. 1997. Identifying hierarchical strcture in sequences: A linear-time algorithm. Journal of Artificial Intelligence Research 7 (1997), 67--82.

Digital Library

[31]

Gordon D. Plotkin. 1970. A note on inductive generalization. Machine Intelligence 5, 1 (1970), 153--163.

[32]

Gordon D. Plotkin. 1971. A further note on inductive generalization. Machine Intelligence 6 (1971), 101--124.

[33]

John C. Reynolds. 1970. Transformational systems and the algebraic structure of atomic formulas. Machine Intelligence 5, 1 (1970), 135--151.

[34]

Sherif Sakr. 2009. XML compression techniques: A survey and comparison. Journal of Computer and System Sciences 75, 5 (2009), 303--322.

Digital Library

[35]

James A. Storer and Thomas G. Szymanski. 1982. Data compression via textual substitution. Journal of the ACM 29, 4 (Oct. 1982), 928--951.

Digital Library

[36]

Geoff Sutcliffe. 2010. The TPTP world - Infrastructure for automated reasoning. In Proceedings of the 16th International Conference on Logic for Programming Artificial Intelligence and Reasoning (Lecture Notes in Artificial Intelligence), E. Clarke and A. Voronkov (Eds.). Springer-Verlag, 1--12.

[37]

Ole Tange. 2011. GNU parallel - The command-line power tool. ;login: The USENIX Magazine 36, 1 (Feb. 2011), 42--47.

[38]

Grigori S. Tseitin. 1983. On the complexity of derivation in propositional calculus. In Automation of Reasoning: Classical Papers in Computational Logic. Vol. 2. Springer, 466--483.

[39]

Zsolt Tuza. 1987. On the context-free production complexity of finite languages. Discrete Applied Mathematics 18, 3 (1987), 293--304.

Digital Library

[40]

Kazunori Yamagata, Tomoyuki Uchida, Takayoshi Shoudai, and Yasuaki Nakamura. 2003. An effective grammar-based compression algorithm for tree structured data. In Inductive Logic Programming, Tamás Horváth and Akihiro Yamamoto (Eds.). Lecture Notes in Computer Science, Vol. 2835. Springer, Berlin, 383--400.

Cited By

Cerna DKutsia T(2020)Higher-order pattern generalization modulo equational theoriesMathematical Structures in Computer Science10.1017/S096012952000011030:6(627-663)Online publication date: 20-May-2020
https://doi.org/10.1017/S0960129520000110
Cerna D(2020)Anti-unification and the theory of semiringsTheoretical Computer Science10.1016/j.tcs.2020.10.020848(133-139)Online publication date: Dec-2020
https://doi.org/10.1016/j.tcs.2020.10.020
Ebner GHetzl SLeitsch AReis GWeller D(2019)On the Generation of Quantified LemmasJournal of Automated Reasoning10.1007/s10817-018-9462-863:1(95-126)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s10817-018-9462-8
Show More Cited By

Index Terms

Algorithmic Compression of Finite Tree Languages by Rigid Acyclic Grammars
1. Theory of computation
  1. Formal languages and automata theory

Recommendations

Tree Compression Using String Grammars

We study the compressed representation of a ranked tree by a (string) straight-line program (SLP) for its preorder traversal, and compare it with the well-studied representation by straight-line context free tree grammars (which are also known as tree ...
Tree Structure Compression with RePair
DCC '11: Proceedings of the 2011 Data Compression Conference

Larsson and Moffat's RePair algorithm is generalized from strings to trees. The new algorithm (TreeRePair) produces straight-line linear context-free tree (SLT) grammars which are smaller than those produced by previous grammar-based compressors such as ...
Categorial Grammars and Tree Languages

We consider a representation of the structure of derivations in Ajdukiewicz and Lambek categorial grammars called derived trees. We define derived tree languages of those grammars based on syntactic calculi formalized in Gentzen style and in natural ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 18, Issue 4

October 2017

251 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/3143777

Editor:
Orna Kupferman
School of Computer Science and Engineering, Hebrew University, Israel

Issue’s Table of Contents

Copyright © 2017 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 the author(s) 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].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 21 September 2017

Accepted: 01 June 2017

Revised: 01 June 2017

Received: 01 April 2016

Published in TOCL Volume 18, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Vienna Science Fund (WWTF)
FWF project

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
130
Total Downloads

Downloads (Last 12 months)2
Downloads (Last 6 weeks)0

Reflects downloads up to 06 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Cerna DKutsia T(2020)Higher-order pattern generalization modulo equational theoriesMathematical Structures in Computer Science10.1017/S096012952000011030:6(627-663)Online publication date: 20-May-2020
https://doi.org/10.1017/S0960129520000110
Cerna D(2020)Anti-unification and the theory of semiringsTheoretical Computer Science10.1016/j.tcs.2020.10.020848(133-139)Online publication date: Dec-2020
https://doi.org/10.1016/j.tcs.2020.10.020
Ebner GHetzl SLeitsch AReis GWeller D(2019)On the Generation of Quantified LemmasJournal of Automated Reasoning10.1007/s10817-018-9462-863:1(95-126)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s10817-018-9462-8
Eberhard SEbner GHetzl S(2018)Complexity of Decision Problems on Totally Rigid Acyclic Tree GrammarsDevelopments in Language Theory10.1007/978-3-319-98654-8_24(291-303)Online publication date: 5-Aug-2018
https://doi.org/10.1007/978-3-319-98654-8_24

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents