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The computational complexity of the unconstrained limited domino problem (with implications for logical decision problems)

  • Martin Fürer
Section IV: Decision Problems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)

Abstract

The unconstrained domino or tiling problem is the following. Given a finite set T (of tiles), sets H,VT×T and a cardinal kω, does there exist a function :k×k->T such that (t(i,j),t(i+1,j))εH and (t(i,j),t(i,j+1))εV for all i,j<k ?

The unlimited domino problem (k infinite) has played an important role in the process of finding the undecidability proof for the ∀ε∀ prefix class of predicate calculus. The limited domino problem is similarly connected to some decidable prefix classes.

The limited domino problem is NP-complete, if k is given in unary. The same was conjectured for k given in binary, but we show an Θ (c n ) nondeterministic time lower bound (upper bound O(d n )). As a consequence, the non-deterministic time complexity of the decision problem for the Вε ВВ class of predicate calculus lies between Θ (c n/log n ) and O(d n/log n ).

Keywords

Filing System Turing Machine Tiling System Binary Number Origin Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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6. References

  1. [1]
    Berger, R. The Undecidability of the Domino Problem, Memoirs of the American Mathematical Society 66, 1966.Google Scholar
  2. [2]
    Büchi, J. R. Turing Machines and the Entscheidungsproblem. Mathematische Annalen 148 (1962), pp. 201–213.Google Scholar
  3. [3]
    Dreben, B., and W. D. Goldfarb.The Decision Problem: Solvable Classes of Quantificational Formulas. Reading, Mass. Addison-Wesley, 1979.Google Scholar
  4. [4]
    Fürer, M. Alternation and the Ackermann Case of the Decision Problem. L'Enseignement Mathématique 27 (1981), pp. 137–162.Google Scholar
  5. [5]
    Garey, M. R. and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.Google Scholar
  6. [6]
    Kahr, A. S., E. F. Moore and Hao Wang. Entscheidungsproblem Reduced to the ∀∃∀ Case. Proc. Nat. Acad. Sci. USA 48, (1962), pp. 365–377.Google Scholar
  7. [7]
    Lewis, Harry R. Complexity of Solvable Cases of the Decision Problem for Predicate Calculus. Proc. 19th Symp. on Foundations of Computer Science, 1978, pp. 35–47.Google Scholar
  8. [8]
    Lewis, Harry R. Complexity Results for Classes of Quantificational Formulas. Journal of Computer and System Sciences 21, Dec. 1980, pp. 317–353.Google Scholar
  9. [9]
    Lewis, Harry R.Unsolvable Classes of Quantificational Formulas. Addison-Wesley, Advanced Book Program, Reading, Massachusetts, 1979.Google Scholar
  10. [10]
    Lewis, Harry R. and Chr. H. Papadimitriou. Elements of the Theory of Computation. Prentice-Hall, 1981.Google Scholar
  11. [11]
    Robinson Raphael. M. Undecidability and Nonperiodicity for Tilings of the Plane. Inventiones Mathematicae 12, (1971), pp. 177–209.Google Scholar
  12. [12]
    Seiferas, J. I., M. J. Fischer and A. R. Meyer. Separating Nondeterministic Time Complexity Classes. Journal of the Association for Computing Machinery 25, (1978), pp. 146–167.Google Scholar
  13. [13]
    van Emde Boas, P. Dominoes are Forever. Report on the 1stGTI-workshop Paderborn, FRG, Reihe Theoretische Informatik 13, L. Priese Ed., Univ. of Paderborn 1982, pp. 75–95.Google Scholar
  14. [14]
    Wang, Hao. Proving Theorems by Pattern Recognition II. The Bell System Technical Journal 40, (1961), pp. 1–41.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Martin Fürer
    • 1
  1. 1.University of ZürichSwitzerland

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