Article

Free access

Presburger arithmetic with bounded quantifier alternation

Authors:

C. R. Reddy,

D. W. LovelandAuthors Info & Claims

STOC '78: Proceedings of the tenth annual ACM symposium on Theory of computing

Pages 320 - 325

https://doi.org/10.1145/800133.804361

Published: 01 May 1978 Publication History

PDF eReader

Abstract

This paper concerns both the complexity aspects of PA and the pragmatics of improving algorithms for dealing with restricted subcases of PA for uses such as program verification. We improve the Cooper-Presburger decision procedure and show that the improved version permits us to obtain time and space upper bounds for PA classes restricted to a bounded number of alternations of quantifiers. The improvement is one exponent less than the upper bounds for the decision problem for full PA. The pragmatists not interested in complexity bounds can read the results as substantiation of the intuitive feeling that the improvement to the Cooper-Presburger algorithm is a real, rather than ineffectual, improvement. (It can be easily shown that the bounds obtained here are not achievable using the Cooper-Presburger procedure).

References

[1]

Borosh I. and Treybig, L. B. Bounds on positive integral solutions of linear diophantine equations. Proc. AMS 55, March, 1976.

Crossref

Google Scholar

[2]

Cooper, D. C. Theorem-proving in arithmetic without multiplication. Machine Intell. 7, J. Wiley, 1972.

Google Scholar

[3]

Ferrante, J. and Rackoff, C. A decision procedure for the first order theory of real addition with order. SIAM J. Comp., March, 1975.

Google Scholar

[4]

Fischer, M. and Rabin, M. O. Super-exponential complexity of Presburger arithmetic. Project MAC. Tech. Memo 43, MIT, Cambridge, 1974.

Digital Library

Google Scholar

[5]

Oppen, D. C. Elementary bounds for Presburger arithmetic. |5th SIGACT,# May, 1973.

Digital Library

Google Scholar

[6]

Presburger, M. Uber die Vollstandigkeit eines gewissen Systems der Arithmetic ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Compte-Rendus dei Congres des Math. des pays slavs, Warsaw, 1929.

Google Scholar

[7]

Shostak, R. An efficient decision algorithm for arithmetic with function symbols. Talk at Workshop on Auto. Deduction, Aug. 1977.

Google Scholar

[8]

Suzuki, N. and Jefferson, D. Verification decidability of Presburger array programs. CMU Comp. Sci. Report, June, 1977.

Google Scholar

Cited By

View all

Bednarczyk BDemri SFervari RMansutti A(2023)On Composing Finite Forests with Modal LogicsACM Transactions on Computational Logic10.1145/356995424:2(1-46)Online publication date: 3-Apr-2023
https://dl.acm.org/doi/10.1145/3569954
Sankaran A(2022)Feferman–Vaught Decompositions for Prefix Classes of First Order LogicJournal of Logic, Language and Information10.1007/s10849-022-09384-932:1(147-174)Online publication date: 12-Nov-2022
https://doi.org/10.1007/s10849-022-09384-9
Ye ZPeng Y(2019)Sequential Cross-Modal Hashing Learning via Multi-scale Correlation MiningACM Transactions on Multimedia Computing, Communications, and Applications10.1145/335633815:4(1-20)Online publication date: 26-Dec-2019
https://dl.acm.org/doi/10.1145/3356338
Show More Cited By

Presburger arithmetic with bounded quantifier alternation
1. Theory of computation

Recommendations

Bounds on the automata size for Presburger arithmetic

Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by the automata-based approach for Presburger arithmetic. In this article, we give an ...
On the Automata Size for Presburger Arithmetic
LICS '04: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states ...
Alternation and Bounded Concurrency Are Reverse Equivalent

Numerous models of concurrency have been considered in the framework of automata. Among the more interesting concurrency models are classical nondeterminism and pure concurrency, the two facets of alternation, and the bounded concurrency model. Bounded ...

Comments

Information & Contributors

Information

Published In

STOC '78: Proceedings of the tenth annual ACM symposium on Theory of computing

May 1978

342 pages

ISBN:9781450374378

DOI:10.1145/800133

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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 May 1978

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Acceptance Rates

STOC '78 Paper Acceptance Rate 38 of 120 submissions, 32%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

67
Total Citations
View Citations
154
Total Downloads

Downloads (Last 12 months)61
Downloads (Last 6 weeks)12

Reflects downloads up to 16 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Bednarczyk BDemri SFervari RMansutti A(2023)On Composing Finite Forests with Modal LogicsACM Transactions on Computational Logic10.1145/356995424:2(1-46)Online publication date: 3-Apr-2023
https://dl.acm.org/doi/10.1145/3569954
Sankaran A(2022)Feferman–Vaught Decompositions for Prefix Classes of First Order LogicJournal of Logic, Language and Information10.1007/s10849-022-09384-932:1(147-174)Online publication date: 12-Nov-2022
https://doi.org/10.1007/s10849-022-09384-9
Ye ZPeng Y(2019)Sequential Cross-Modal Hashing Learning via Multi-scale Correlation MiningACM Transactions on Multimedia Computing, Communications, and Applications10.1145/335633815:4(1-20)Online publication date: 26-Dec-2019
https://dl.acm.org/doi/10.1145/3356338
Liu SHuang Z(2019)Efficient Image Hashing with Geometric Invariant Vector Distance for Copy DetectionACM Transactions on Multimedia Computing, Communications, and Applications10.1145/335539415:4(1-22)Online publication date: 16-Dec-2019
https://dl.acm.org/doi/10.1145/3355394
Kua JArmitage GBranch PBut J(2019)Adaptive Chunklets and AQM for Higher-Performance Content StreamingACM Transactions on Multimedia Computing, Communications, and Applications10.1145/334438115:4(1-24)Online publication date: 16-Dec-2019
https://dl.acm.org/doi/10.1145/3344381
Sturm TKauers MOvchinnikov ASchost É(2018)Thirty Years of Virtual SubstitutionProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209030(11-16)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209030
Nguyen DPak IHatami HMcKenzie PKing V(2017)Complexity of short Presburger arithmeticProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055435(812-820)Online publication date: 19-Jun-2017
https://dl.acm.org/doi/10.1145/3055399.3055435
Veanes MBjørner NNachmanson LBereg S(2017)Monadic DecompositionJournal of the ACM10.1145/304048864:2(1-28)Online publication date: 30-Apr-2017
https://dl.acm.org/doi/10.1145/3040488
Nguyen DPak I(2017)Short Presburger Arithmetic Is Hard2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.13(37-48)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.13
Reynolds AKing TKuncak V(2017)Solving quantified linear arithmetic by counterexample-guided instantiationFormal Methods in System Design10.1007/s10703-017-0290-y51:3(500-532)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s10703-017-0290-y
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Get Access

Login options

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

Cited By

Recommendations

Bounds on the automata size for Presburger arithmetic

On the Automata Size for Presburger Arithmetic

Alternation and Bounded Concurrency Are Reverse Equivalent

Comments

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Acceptance Rates

Other Metrics

Article Metrics

Other Metrics

Cited By

PDF

eReader

Login options

Full Access

Abstract

References

Cited By

Recommendations

Bounds on the automata size for Presburger arithmetic

On the Automata Size for Presburger Arithmetic

Alternation and Bounded Concurrency Are Reverse Equivalent

Comments

Information

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Acceptance Rates

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

PDF

eReader

Get Access

Login options

Full Access

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations