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The Complexity of Equivariant Unification

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Automata, Languages and Programming (ICALP 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3142))

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Abstract

Nominal logic is a first-order theory of names and binding based on a primitive operation of swapping rather than substitution. Urban, Pitts, and Gabbay have developed a nominal unification algorithm that unifies terms up to nominal equality. However, because of nominal logic’s equivariance principle, atomic formulas can be provably equivalent without being provably equal as terms, so resolution using nominal unification is sound but incomplete. For complete resolution, a more general form of unification called equivariant unification, or “unification up to a permutation” is required. Similarly, for rewrite rules expressed in nominal logic, a more general form of matching called equivariant matching is necessary.

In this paper, we study the complexity of the decision problem for equivariant unification and matching. We show that these problems are NP-complete in general. However, when one of the terms is essentially first-order, equivariant and nominal unification coincide. This shows that equivariant unification can be performed efficiently in many interesting common cases: for example, any purely first-order logic program or rewrite system can be run efficiently on nominal terms.

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References

  1. Albert, M.H., Lawrence, J.: Unification in varieties of groups: nilpotent varieties. Canadian Journal of Mathematics 46(6), 1135–1149 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cheney, J., Urban, C.: Alpha-Prolog: A logic programming language with names, binding and alpha-equivalence. In: Demoen, B., Lifschitz, V. (eds.) ICLP 2004. LNCS, vol. 3132, pp. 269–283. Springer, Heidelberg (2004) (to appear)

    Chapter  Google Scholar 

  3. Engebretsen, L., Holmerin, J., Russell, A.: Inapproximability results for equations over finite groups. Theoretical Computer Science 312(1), 17–45 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fernández, M., Gabbay, M., Mackie, I.: Nominal rewriting (January 2004) (submitted)

    Google Scholar 

  5. Finkelstein, H.: Solving equations in groups: a survey of Frobenius’ Theorem. Periodica Mathematica Hungarica 9(3), 187–204 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Formal Aspects of Computing 13, 341–363 (2002)

    Article  MATH  Google Scholar 

  7. Goldmann, M., Russell, A.: The complexity of solving equations over finite groups. Information and Computation 178, 253–262 (2002)

    MATH  MathSciNet  Google Scholar 

  8. Hofmann, M.: Semantical analysis of higher-order abstract syntax. In: Proc. 14th Symp. on Logic in Computer Science, July 1999, pp. 204–213. IEEE, Los Alamitos (1999)

    Google Scholar 

  9. Huet, G.: A unification algorithm for typed λ-calculus. Theoretical Computer Science 1, 27–67 (1975)

    Article  MathSciNet  Google Scholar 

  10. Martelli, A., Montanari, U.: An efficient unification algorithm. Transactions on Programming Languages and Systems 4(2), 258–282 (1982)

    Article  MATH  Google Scholar 

  11. Miller, D.: A logic programming language with lambda-abstraction, function variables, and simple unification. J. Logic and Computation 1(4), 497–536 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Pfenning, F., Elliott, C.: Higher-order abstract syntax. In: Proc. ACM SIGPLAN Conf. on Programming Language Design and Implementation (PLDI 1989), pp. 199–208. ACM Press, New York (1989)

    Google Scholar 

  13. Pitts, A.M.: Nominal logic, a first order theory of names and binding. Information and Computation 183, 165–193 (2003)

    Article  MathSciNet  Google Scholar 

  14. Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (1965)

    Article  MATH  Google Scholar 

  15. Urban, C., Cheney, J.: Avoiding equivariance in Alpha-Prolog (2004) (submitted)

    Google Scholar 

  16. Urban, C., Pitts, A.M., Gabbay, M.J.: Nominal unification. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 513–527. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Cornell University,  

    James Cheney

Authors
  1. James Cheney
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord - Ed. Omega, 240 Jordi Girona Salgado, 1-3 E-08034, Barcelona

    Josep Díaz

  2. Department of Mathematics and Turku Centre for Computer Science TUCS, University of Turku, 20014, Turku, Finland

    Juhani Karhumäki

  3. Department of Mathematics, University of Turku, Turku, Finland

    Arto Lepistö

  4. Laboratory for Foundations of Computer Science, University of Edinburgh,  

    Donald Sannella

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Cheney, J. (2004). The Complexity of Equivariant Unification. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_30

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  • DOI: https://doi.org/10.1007/978-3-540-27836-8_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22849-3

  • Online ISBN: 978-3-540-27836-8

  • eBook Packages: Springer Book Archive

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