Skip to main content
SpringerLink
Log in
Book cover

Description Logic, Theory Combination, and All That pp 497–522Cite as

Asymmetric Unification and Disunification

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11560))

Abstract

We compare two kinds of unification problems: Asymmetric Unification and Disunification, which are variants of Equational Unification. Asymmetric Unification is a type of Equational Unification where the instances of the right-hand sides of the equations are in normal form with respect to the given term rewriting system. In Disunification we solve equations and disequations with respect to an equational theory for the case with free constants. We contrast the time complexities of both and show that the two problems are incomparable: there are theories where one can be solved in polynomial time while the other is NP-hard. This goes both ways. The time complexity also varies based on the termination ordering used in the term rewriting system.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Symmetric and asymmetric unification were contrasted in [16].

  2. 2.

    For the theory ACUN that we consider in Sect. 6, disunification can be reduced to asymmetric unification.

  3. 3.

    Gaussian elimination over \(\mathbb {Z}_2\), or GF(2), is discussed in several papers [9, 25].

  4. 4.

    We follow the notation and procedure similar to Greenwell and Kertzner [19].

References

  1. Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979)

    Article  MathSciNet  Google Scholar 

  2. Baader, F.: Unification in commutative theories. J. Symb. Comput. 8(5), 479–497 (1989)

    Article  MathSciNet  Google Scholar 

  3. Baader, F.: Unification in commutative theories, Hilbert’s basis theorem, and Gröbner bases. J. ACM 40(3), 477–503 (1993)

    Article  MathSciNet  Google Scholar 

  4. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  5. Baader, F., Nutt, W.: Combination problems for commutative/monoidal theories or how algebra can help in equational unification. Appl. Algebra Eng. Commun. Comput. 7(4), 309–337 (1996)

    Article  MathSciNet  Google Scholar 

  6. Baader, F., Schulz, K.U.: Combination techniques and decision problems for disunification. Theor. Comput. Sci. 142(2), 229–255 (1995)

    Article  MathSciNet  Google Scholar 

  7. Baader, F., Snyder, W.: Unification theory. In: Handbook of Automated Reasoning, vol. 1, pp. 445–532 (2001)

    Chapter  Google Scholar 

  8. Bachmair, L.: Canonical Equational Proofs. Birkhauser, Boston (1991)

    Book  Google Scholar 

  9. Bogdanov, A., Mertens, M.C., Paar, C., Pelzl, J., Rupp, A.: A parallel hardware architecture for fast Gaussian elimination over GF(2). In: 14th IEEE Symposium on Field-Programmable Custom Computing Machines (FCCM 2006), Napa, CA, USA, 24–26 April 2006, Proceedings, pp. 237–248. IEEE Computer Society (2006)

    Google Scholar 

  10. Brahmakshatriya, S., Danturi, S., Gero, K.A., Narendran, P.: Unification problems modulo a theory of until. In: Korovin, K., Morawska, B. (eds.) 27th International Workshop on Unification, UNIF 2013, Eindhoven, Netherlands, 26 June 2013. EPiC Series in Computing, vol. 19, pp. 22–29. EasyChair (2013). http://www.easychair.org/publications/?page=723757558

  11. Büchi, J.R.: Weak second-order arithmetic and finite automata. Math. Logic Q. 6(1–6), 66–92 (1960)

    Article  MathSciNet  Google Scholar 

  12. Buntine, W.L., Bürckert, H.-J.: On solving equations and disequations. J. ACM 41(4), 591–629 (1994)

    Article  MathSciNet  Google Scholar 

  13. Bürckert, H.-J., Herold, A., Schmidt-Schauss, M.: On equational theories, unification, and (un)decidability. J. Symb. Comput. 8(1–2), 3–49 (1989)

    Article  MathSciNet  Google Scholar 

  14. Comon, H.: Disunification: a survey. In: Lassez, J.-L., Plotkin, G.D. (eds.) Computational Logic - Essays in Honor of Alan Robinson, pp. 322–359. The MIT Press (1991)

    Google Scholar 

  15. Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Am. Math. Soc. 98(1), 21–51 (1961)

    Article  MathSciNet  Google Scholar 

  16. Erbatur, S., et al.: Asymmetric unification: a new unification paradigm for cryptographic protocol analysis. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 231–248. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_16

    Chapter  Google Scholar 

  17. Erbatur, S., Kapur, D., Marshall, A.M., Meadows, C., Narendran, P., Ringeissen, C.: On asymmetric unification and the combination problem in disjoint theories. In: Muscholl, A. (ed.) FoSSaCS 2014. LNCS, vol. 8412, pp. 274–288. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54830-7_18

    Chapter  MATH  Google Scholar 

  18. Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  19. Greenwell, R.N., Kertzner, S.: Solving linear diophantine matrix equations using the Smith normal form (more or less). Int. J. Pure Appl. Math. 55(1), 49–60 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Guo, Q., Narendran, P., Wolfram, D.A.: Complexity of nilpotent unification and matching problems. Inf. Comput. 162(1–2), 3–23 (2000)

    Article  MathSciNet  Google Scholar 

  21. Jouannaud, J.-P., Kirchner, C.: Solving equations in abstract algebras: a rule-based survey of unification. In: Computational Logic - Essays in Honor of Alan Robinson, pp. 257–321 (1991)

    Google Scholar 

  22. Kaltofen, E., Krishnamoorthy, M.S., Saunders, B.D.: Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Algebraic Discrete Methods 8(4), 683–690 (1987)

    Article  MathSciNet  Google Scholar 

  23. Kannan, R.: Solving systems of linear equations over polynomials. Theor. Comput. Sci. 39, 69–88 (1985)

    Article  MathSciNet  Google Scholar 

  24. Klaedtke, F., Ruess, H.: Parikh automata and monadic second-order logics with linear cardinality constraints (2002)

    Google Scholar 

  25. Koç, Ç.K., Arachchige, S.N.: A fast algorithm for Gaussian elimination over GF(2) and its implementation on the GAPP. J. Parallel Distrib. Comput. 13(1), 118–122 (1991)

    Article  Google Scholar 

  26. Liu, Z.: Dealing efficiently with exclusive-OR, abelian groups and homomorphism in cryptographic protocol analysis. Ph.D. thesis, Clarkson University (2012)

    Google Scholar 

  27. Lynch, C., Morawska, B.: Basic syntactic mutation. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 471–485. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45620-1_37

    Chapter  Google Scholar 

  28. MacDuffee, C.C.: The Theory of Matrices. Chelsea Publishing Company, New York (1961)

    Google Scholar 

  29. Nutt, W.: Unification in monoidal theories. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 618–632. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52885-7_118

    Chapter  Google Scholar 

  30. Rotman, J.J.: Advanced Modern Algebra. Prentice Hall. 1st edn (2002); 2nd printing (2003)

    Google Scholar 

  31. Vardi, M.Y., Wilke, T.: Automata: from logics to algorithms. Logic Automata 2, 629–736 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We wish to thank Franz Baader for all his remarkable contributions in this field. We also wish to thank the anonymous reviewers for their detailed comments and suggestions which helped us greatly in improving this paper.

Author information

Authors and Affiliations

  1. University of Mary Washington, Fredericksburg, USA

    Veena Ravishankar

  2. The College of Saint Rose, Albany, USA

    Kimberly A. Cornell

  3. University at Albany-SUNY, Albany, USA

    Paliath Narendran

Authors
  1. Veena Ravishankar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kimberly A. Cornell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paliath Narendran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Veena Ravishankar .

Editor information

Editors and Affiliations

  1. University of Bremen, Bremen, Germany

    Carsten Lutz

  2. University of Manchester, Manchester, UK

    Uli Sattler

  3. University of Iowa, Iowa City, IA, USA

    Cesare Tinelli

  4. TU Dresden, Dresden, Germany

    Anni-Yasmin Turhan

  5. University of Liverpool, Liverpool, UK

    Frank Wolter

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ravishankar, V., Cornell, K.A., Narendran, P. (2019). Asymmetric Unification and Disunification. In: Lutz, C., Sattler, U., Tinelli, C., Turhan, AY., Wolter, F. (eds) Description Logic, Theory Combination, and All That. Lecture Notes in Computer Science(), vol 11560. Springer, Cham. https://doi.org/10.1007/978-3-030-22102-7_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22102-7_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22101-0

  • Online ISBN: 978-3-030-22102-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation