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Two-dimensional partial orderings: Undecidability

Published online by Cambridge University Press:  12 March 2014

Alfred B. Manaster  and
Joseph G. Rosenstein
Alfred B. Manaster
Affiliation:
University of California, San Diego, La Jolla, CA 92037
Joseph G. Rosenstein
Affiliation:
Rutgers University, New Brunswick, NJ 08903 Institute of Advanced Study, Princeton, NJ 08540
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Extract

In this paper we examine the class of two-dimensional partial orderings from the perspective of undecidability. We shall see that from this perspective the class of 2dpo's is more similar to the class of all partial orderings than to its one-dimensional subclass, the class of all linear orderings. More specifically, we shall describe an argument which lends itself to proofs of the following four results:

(A) the theory of 2dpo's is undecidable:

(B) the theory of 2dpo's is recursively inseparable from the set of sentences refutable in some finite 2dpo;

(C) there is a sentence which is true in some 2dpo but which has no recursive model;

(D) the theory of planar lattices is undecidable.

It is known that the theory of linear orderings is decidable (Lauchli and Leonard [4]). On the other hand, the theories of partial orderings and lattices were shown to be undecidable by Tarski [14], and that each of these theories is recursively inseparable from its finitely refutable statements was shown by Taitslin [13]. Thus, the complexity of the theories of partial orderings and lattices is, by (A), (B) and (D), already reflected in the 2dpo's and planar lattices.

As pointed out by J. Schmerl, bipartite graphs can be coded into 2dpo's, so that (A) and (B) could also be obtained by applying a Rabin-Scott style argument [9] to Rogers' result [11] that the theory of bipartite graphs is undecidable and to Lavrov's result [5] that the theory of bipartite graphs is recursively inseparable from the set of sentences refutable in some finite bipartite graph. (However, (C) and (D) do not seem to follow from this type of argument.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 1 , March 1980 , pp. 133 - 143
Copyright
Copyright © Association for Symbolic Logic 1980

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References

BIBLIOGRAPHY

[1]Feferman, S. and Vaught, R. L., The first-order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[2]Hanf, W., Nonrecursive tilings of the plane. I, this Journal, vol. 39 (1974), pp. 283–285.Google Scholar
[3]Kelly, D. and Rival, I., Planar lattices, Canadian Journal of Mathematics, vol. 27 (1975), pp. 636–665.CrossRefGoogle Scholar
[4]Läuchli, H. and Leonard, J., On the elementary theory of linear order, Fundamenta Mathematicae, vol. 59 (1966), pp. 109–116.CrossRefGoogle Scholar
[5]Lavrov, I. A., Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain elementary theories, Algebra i Logika, Seminar 2, vol. 1 (1963), pp. 5–19 (Russian).Google Scholar
[6]Manaster, A.B. and Rosenstein, J.G., Two-dimensional partial orderings: Recursive model theory, this Journal, vol. 45 (1980), pp. 121–132.Google Scholar
[7]Mostowski, A., A formula with no recursively enumerable model, Fundamenta Mathematicae, vol. 43 (1955), pp. 125–140.CrossRefGoogle Scholar
[8]Platt, C. R., Planar lattices and planar graphs, Journal of Combinatorial Theory, Series B, vol. 21 (1976), pp. 30–39.CrossRefGoogle Scholar
[9]Rabin, M. O., A simple method for undecidability proofs and some applications, Logic, Methodology and Philosophy of Science, Proceedings of the 1964 International Congress, North-Holland Amsterdam, 1965, pp. 58–68.Google Scholar
[10]Robinson, R. M., Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, vol. 12 (1971), pp. 177–209.CrossRefGoogle Scholar
[11]Rogers, H. Jr.,, Certain logical reduction and decision problems, Annals of Mathematics, vol. 64 (1956), pp. 264–284.CrossRefGoogle Scholar
[12]Schmerl, J., ℵ0-categoricity of partially ordered sets of width 2, Proceedings of the American Mathematical Society, vol. 63 (1977), pp. 299–305.Google Scholar
[13]Taitslin, M. A., Effective inseparability of the sets of identically true and finitely refutable formulae of elementary lattice theory, Algebra i Logika, vol. 3 (1962), pp. 24–38, MR 8 #1131.Google Scholar
[14]Tarski, A., Undecidability of the theory of lattices and projective geometry, this Journal, vol. 15 (1949), pp. 77–78.Google Scholar