research-article

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Authors:

Philippe Schnoebelen,

Georg ZetzscheAuthors Info & Claims

LICS '17: Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science

Article No.: 81, Pages 1 - 12

Published: 20 June 2017 Publication History

Abstract

We consider first-order logic over the subword ordering on finite words where each word is available as a constant. Our first result is that the Σ₁ theory is undecidable (already over two letters).

We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ₁ fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ₂ fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

References

[1]

P. A. Abdulla, M. F. Atig, Y. Chen, L. Holík, A. Rezine, P. Rümmer, and J. Stenman, "String constraints for verification," in Proc. CAV 2014, LNCS 8559. Springer, 2014, pp. 150--166.

Digital Library

[2]

P. A. Abdulla, A. Collomb-Annichini, A. Bouajjani, and B. Jonsson, "Using forward reachability analysis for verification of lossy channel systems," Form. Methods Sys. Des., vol. 25, no. 1, pp. 39--65, 2004.

Digital Library

[3]

J. Berstel, Transductions and Context-Free Languages. Stuttgart: B. G. Teubner, 1979.

[4]

P. Bouyer, N. Markey, J. Ouaknine, Ph. Schnoebelen, and J. Worrell, "On termination and invariance for faulty channel machines," Form. Asp. Comp., vol. 24, no. 4--6, pp. 595--607, 2012.

[5]

J. R. Büchi and S. Senger, "Definability in the existential theory of concatenation and undecidable extensions of this theory," Z. Math. Logik Grundlag. Math., vol. 34, no. 4, pp. 337--342, 1988.

[6]

M. Cadilhac, A. Finkel, and P. McKenzie, "Affine Parikh automata," RAIRO Theor. Inf. Appl., vol. 46, no. 4, pp. 511--545, 2012.

[7]

H. Comon and R. Treinen, "Ordering constraints on trees," in Proc. CAAP '94, LNCS 787. Springer, 1994, pp. 1--14.

Digital Library

[8]

V. Diekert, P. Gastin, and M. Kufleitner, "A survey on small fragments of first-order logic over finite words," Int. J. Found. Comput. Sci., vol. 19, no. 3, pp. 513--548, 2008.

[9]

V. G. Durnev, "Undecidability of the positive ∀∃<sup>3</sup>-theory of a free semigroup," Sib. Math. J., vol. 36, no. 5, pp. 917--929, 1995.

[10]

J. Ferrante and C. W. Rackoff, The computational complexity of logical theories, ser. Lecture Notes in Mathematics. Springer, 1979, vol. 718.

[11]

V. Ganesh, M. Minnes, A. Solar-Lezama, and M. C. Rinard, "Word equations with length constraints: What's decidable?" in Proc. HVC 2012, LNCS 7857. Springer, 2013, pp. 209--226.

Digital Library

[12]

S. Ginsburg and E. H. Spanier, "Bounded regular sets," Proc. Amer. Math. Soc., vol. 17, no. 5, pp. 1043--1049, 1966.

[13]

G. Gottlob, N. Leone, and H. Veith, "Second order logic and the weak exponential hierarchies," in Proc. MFCS '95, LNCS 969. Springer, 1995, pp. 66--81.

Digital Library

[14]

Ch. Haase, S. Schmitz, and Ph. Schnoebelen, "The power of priority channel systems," Log. Methods Comput. Sci., vol. 10, no. 4:4, 2014.

[15]

C. Haase, "Subclasses of Presburger arithmetic and the weak EXP hierarchy," in Proc. CSL-LICS 2014. ACM, 2014, pp. 47:1--47:10.

Digital Library

[16]

S. Halfon, Ph. Schnoebelen, and G. Zetzsche, "Decidability, complexity, and expressiveness of first-order logic over the subword ordering," full version of this paper; available at http://www.arxiv.org/abs/1701.07470.

[17]

L. Hemachandra, "The strong exponential hierarchy collapses," J. Comput. System Sci., vol. 39, no. 3, pp. 299--322, 1989.

[18]

P. Hooimeijer and W. Weimer, "StrSolve: solving string constraints lazily," Autom. Softw. Eng., vol. 19, no. 4, pp. 531--559, 2012.

[19]

O. H. Ibarra, "Reversal-bounded multicounter machines and their decision problems," J. ACM, vol. 25, no. 1, pp. 116--133, 1978.

Digital Library

[20]

A. Jeż, "Recompression: A simple and powerful technique for word equations," J. ACM, vol. 63, no. 1, pp. 4:1--4:51, 2016.

Digital Library

[21]

P. Karandikar, M. Niewerth, and Ph. Schnoebelen, "On the state complexity of closures and interiors of regular languages with subwords and superwords," Theoret. Comput. Sci., vol. 610, pp. 91--107, 2016.

Digital Library

[22]

P. Karandikar and Ph. Schnoebelen, "Decidability in the logic of subsequences and supersequences," in Proc. FST&TCS 2015, LIPIcs 45. Leibniz-Zentrum für Informatik, 2015, pp. 84--97.

[23]

P. Karandikar, "Generalized Post embedding problems," Theory of Computing Systems, vol. 56, no. 4, pp. 697--716, 2015.

Digital Library

[24]

P. Karandikar, "The height of piecewise-testable languages with applications in logical complexity," in Proc. CSL 2016, LIPIcs 62. Leibniz-Zentrum für Informatik, 2016, pp. 37:1--37:22.

[25]

F. Klaedtke and H. Rueß, "Monadic second-order logics with cardinalities," in Proc. ICALP 2003, LNCS 2719. Springer, 2003, pp. 681--696.

Digital Library

[26]

O. Klíma and L. Polák, "Alternative automata characterization of piecewise testable languages," in Proc. DLT 2013, LNCS 7907. Springer, 2013, pp. 289--300.

[27]

D. C. Kozen, "Lower bounds for natural proof systems," in Proc. FOCS'77. IEEE, 1977, pp. 254--266.

Digital Library

[28]

O. V. Kudinov, V. L. Selivanov, and L. V. Yartseva, "Definability in the subword order," in Proc. CiE 2010, LNCS 6158. Springer, 2010, pp. 246--255.

Digital Library

[29]

D. Kuske, "Theories of orders on the set of words," RAIRO Theor. Inf. Appl., vol. 40, no. 1, pp. 53--74, 2006.

[30]

Y. V. Matiyasevich, Hilbert's Tenth Problem. Cambridge, Massachusetts: MIT Press,1993.

Digital Library

[31]

D. C. Oppen, "A 2<sup>2<sup>2<sup>pn</sup></sup></sup> upper bound on the complexity of Presburger arithmetic," J. Comput. System Sci., vol. 16, no. 3, pp. 323--332, 1978.

[32]

W. Plandowski, "An efficient algorithm for solving word equations," in Proc. STOC 2006. ACM Press, 2006, pp. 467--476.

Digital Library

[33]

W. V. Quine, "Concatenation as a basis for arithmetic," J. Symb. Logic, vol. 11, no. 4, pp. 105--114, Dec. 1946.

[34]

L. J. Stockmeyer, "The complexity of decision problems in automata theory and logic," Ph.D. dissertation, Department of Electical Engineering, MIT, Jul. 1974, available as Report MAC-TR-133.

Cited By

Demri SQuaas K(2021)Concrete domains in logicsACM SIGLOG News10.1145/3477986.34779888:3(6-29)Online publication date: 28-Jul-2021
https://dl.acm.org/doi/10.1145/3477986.3477988
Burn TOng LRamsay SWagner DGorla D(2021)Initial limit datalogProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470527(1-13)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470527

Recommendations

Alternating complexity of counting first-order logic for the subword order
Abstract
This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word. We are interested in the counting extension of first-order logic by ...
Expressiveness and succinctness of first-order logic on finite words
First-order logic with two variables and unary temporal logic
Special issue: LICS'97

We investigate the power of first-order logic with only two valiables over ω-words and finite words, a logic denoted by FO². We prove that FO² can express precisely the same properties as linear temporal logic with only the unary temporal operators: "...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

LICS '17: Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science

June 2017

1068 pages

ISBN:9781509030187

Conference Chairs:
Luca Aceto
Reykjavik University
,
Anna Ingólfsdóttir
Reykjavik University

Sponsors

SIGLOG: ACM Special Interest Group on Logic and Computation

Publisher

IEEE Press

Publication History

Published: 20 June 2017

Check for updates

Qualifiers

Research-article

Conference

LICS '17

Sponsor:

SIGLOG

LICS '17: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science

June 20 - 23, 2017

Reykjavík, Iceland

Acceptance Rates

Overall Acceptance Rate 215 of 622 submissions, 35%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
26
Total Downloads

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

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Demri SQuaas K(2021)Concrete domains in logicsACM SIGLOG News10.1145/3477986.34779888:3(6-29)Online publication date: 28-Jul-2021
https://dl.acm.org/doi/10.1145/3477986.3477988
Burn TOng LRamsay SWagner DGorla D(2021)Initial limit datalogProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470527(1-13)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470527

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 Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents