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On the complexity of equivalence of specifications of infinite objects

Published: 09 September 2012 Publication History

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.
[2]
M. Bidoit, R. Hennicker, and A. Kurz. Observational Logic, Constructor-based Logic, and Their Duality. Theor. Comput. Sci., 298:471--510, 2003.
[3]
S. R. Buss and G. Rosu. Incompleteness of Behavioral Logics. ENTCS, 33:61--79, 2000.
[4]
J. Castro and F. Cucker. Nondeterministic ω-Computations and the Analytical Hierarchy. Logik u. Grundlagen d. Math, 35:333--342, 1989.
[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.
[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.
[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.
[8]
J. Endrullis, H. Geuvers, J. G. Simonsen, and H. Zantema. Levels of Undecidability in Rewriting. Information and Computation, 209(2):227--245, 2011.
[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.
[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.
[11]
D. P. Friedman and D. S. Wise. CONS Should Not Evaluate its Arguments. In ICALP, pages 257--284, 1976.
[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.
[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.
[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.
[15]
G. Malcolm. Hidden Algebra and Systems of Abstract Machines. In Proc. Symp. on New Models for Software Architecture (IMSA), 1997.
[16]
G. Roşu. Hidden Logic. PhD thesis, University of California, 2000.
[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.
[18]
H. Rogers, Jr. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.
[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.
[20]
J. J. M. M. Rutten. A Tutorial on Coinductive Stream Calculus and Signal Flow Graphs. Theor. Comput. Sci., 343:443--481, 2005.
[21]
D. Sangiorgi and J. J. M. M. Rutten. Advanced Topics in Bisimulation and Coinduction. Cambridge University Press, 2012.
[22]
J. R. Shoenfield. Degrees of Unsolvability. North-Holland, 1971.
[23]
B. A. Sijtsma. On the Productivity of Recursive List Definitions. ACM Transactions on Programming Languages and Systems, 11(4):633--649, 1989.
[24]
Terese. Term Rewriting Systems. Cambridge University Press, 2003.
[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.

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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
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]

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Association for Computing Machinery

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Publication History

Published: 09 September 2012
Published in SIGPLAN Volume 47, Issue 9

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Author Tags

  1. complexity
  2. equality
  3. equational specifications
  4. infinite objects
  5. lambda terms
  6. semantics

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