skip to main content
research-article

On the complexity of equivalence of specifications of infinite objects

Published:09 September 2012Publication History
Skip Abstract Section

Abstract

We study the complexity of deciding the equality of infinite objects specified by systems of equations, and of infinite objects specified by λ-terms. For equational specifications there are several natural notions of equality: equality in all models, equality of the sets of solutions, and equality of normal forms for productive specifications. For λ-terms we investigate Böhm-tree equality and various notions of observational equality. We pinpoint the complexity of each of these notions in the arithmetical or analytical hierarchy.

We show that the complexity of deciding equality in all models subsumes the entire analytical hierarchy. This holds already for the most simple infinite objects, viz. streams over {0,1}, and stands in sharp contrast to the low arithmetical ϖ02-completeness of equality of equationally specified streams derived in [17] employing a different notion of equality.

References

  1. H. P. Barendregt. The Lambda Calculus, its Syntax and Semantics. North-Holland, 1984.Google ScholarGoogle Scholar
  2. M. Bidoit, R. Hennicker, and A. Kurz. Observational Logic, Constructor-based Logic, and Their Duality. Theor. Comput. Sci., 298:471--510, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. S. R. Buss and G. Rosu. Incompleteness of Behavioral Logics. ENTCS, 33:61--79, 2000.Google ScholarGoogle Scholar
  4. J. Castro and F. Cucker. Nondeterministic ω-Computations and the Analytical Hierarchy. Logik u. Grundlagen d. Math, 35:333--342, 1989.Google ScholarGoogle Scholar
  5. T. Coquand. Infinite Objects in Type Theory. In Postproc. Conf. on Types for Proofs and Programs (TYPES 1993), volume 806 of LNCS, pages 62--78. Springer, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. M. Dezani-Ciancaglini and E. Giovannetti. From Böhm's Theorem to Observational Equivalences: an Informal Account. In BOTH'01, volume 50 of ENTCS, 2001.Google ScholarGoogle Scholar
  7. M. Dezani-Ciancaglini, P. Severi, and F.-J. de Vries. Böhm's theorem for Berarducci trees. In CATS 2000 Computing: the Australasian Theory Symposium, volume 31 of ENTCS, 2000.Google ScholarGoogle Scholar
  8. J. Endrullis, H. Geuvers, J. G. Simonsen, and H. Zantema. Levels of Undecidability in Rewriting. Information and Computation, 209(2):227--245, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. J. Endrullis, C. Grabmayer, and D. Hendriks. Data-Oblivious Stream Productivity. In Proc. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2008), number 5330 in LNCS, pages 79--96. Springer, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. J. Endrullis, C. Grabmayer, and D. Hendriks. Complexity of Fractran and Productivity. In Proc. Conf. on Automated Deduction (CADE 22), volume 5663 of LNCS, pages 371--387, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. D. P. Friedman and D. S. Wise. CONS Should Not Evaluate its Arguments. In ICALP, pages 257--284, 1976.Google ScholarGoogle Scholar
  12. H. Geuvers. Inductive and Coinductive Types with Iteration and Recursion. In Proc. Workshop on Types for Proofs and Programs (TYPES 1992), pages 193--217, 1992.Google ScholarGoogle Scholar
  13. C. Grabmayer, J. Endrullis, D. Hendriks, J. W. Klop, and L. S. Moss. Automatic Sequences and Zip-Specifications. In Proc. Symp. on Logic in Computer Science (LICS 2012). IEEE Computer Society, 2012. To appear. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. P. Henderson and J. H. Morris, Jr. A Lazy Evaluator. In Proc. ACM SIGACT-SIGPLAN Symp. on Principles on programming languages (POPL), pages 95--103. ACM, 1976. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. G. Malcolm. Hidden Algebra and Systems of Abstract Machines. In Proc. Symp. on New Models for Software Architecture (IMSA), 1997.Google ScholarGoogle Scholar
  16. G. Roşu. Hidden Logic. PhD thesis, University of California, 2000.Google ScholarGoogle Scholar
  17. G. Roşu. Equality of Streams is a Π02-complete Problem. In Proc. ACM SIGPLAN Conf. on Functional Programming (ICFP), pages 184--191. ACM, 2006. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. H. Rogers, Jr. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. J. J. M. M. Rutten. Behavioural Differential Equations: a Coinductive Calculus of Streams, Automata, and Power Series. Theor. Comput. Sci., 308(1-3):1--53, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. J. J. M. M. Rutten. A Tutorial on Coinductive Stream Calculus and Signal Flow Graphs. Theor. Comput. Sci., 343:443--481, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. D. Sangiorgi and J. J. M. M. Rutten. Advanced Topics in Bisimulation and Coinduction. Cambridge University Press, 2012. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. J. R. Shoenfield. Degrees of Unsolvability. North-Holland, 1971.Google ScholarGoogle Scholar
  23. B. A. Sijtsma. On the Productivity of Recursive List Definitions. ACM Transactions on Programming Languages and Systems, 11(4):633--649, 1989. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Terese. Term Rewriting Systems. Cambridge University Press, 2003.Google ScholarGoogle Scholar
  25. M. Walicki and S. Meldal. Nondeterminism vs. underspecification. In Proc. of the World Multiconference on Systemics, Cybernetics and Informatics, ISAS-SCI 2001, pages 551--555. IIIS, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. On the complexity of equivalence of specifications of infinite objects

          Recommendations

          Comments

          Login options

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

          Sign in

          Full Access

          • Published in

            cover image ACM SIGPLAN Notices
            ACM SIGPLAN Notices  Volume 47, Issue 9
            ICFP '12
            September 2012
            368 pages
            ISSN:0362-1340
            EISSN:1558-1160
            DOI:10.1145/2398856
            Issue’s Table of Contents
            • cover image ACM Conferences
              ICFP '12: Proceedings of the 17th ACM SIGPLAN international conference on Functional programming
              September 2012
              392 pages
              ISBN:9781450310543
              DOI:10.1145/2364527

            Copyright © 2012 ACM

            Publisher

            Association for Computing Machinery

            New York, NY, United States

            Publication History

            • Published: 9 September 2012

            Check for updates

            Qualifiers

            • research-article

          PDF Format

          View or Download as a PDF file.

          PDF

          eReader

          View online with eReader.

          eReader
          About Cookies On This Site

          We use cookies to ensure that we give you the best experience on our website.

          Learn more

          Got it!