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Linear sampling and the ∀∃∀ case of the decision problem1

  • Stål O. Aanderaa (a1) and Harry R. Lewis (a2)


Let Q be the class of closed quantificational formulas ∀xuyM without identity such that M is a quantifier-free matrix containing only monadic and dyadic predicate letters and containing no atomic subformula of the form Pyx or Puy for any predicate letter P. In [DKW] Dreben, Kahr, and Wang conjectured that Q is a solvable class for satisfiability and indeed contains no formula having only infinite models. As evidence for this conjecture they noted the solvability of the subclass of Q consisting of those formulas whose atomic subformulas are of only the two forms Pxy, Pyu and the fact that each such formula that has a model has a finite model. Furthermore, it seemed likely that the techniques used to show this subclass solvable could be extended to show the solvability of the full class Q, while the syntax of Q is so restricted that it seemed impossible to express in formulas of Q any unsolvable problem known at that time.

In 1966 Aanderaa refuted this conjecture. He first constructed a very complex formula in Q having an infinite model but no finite model, and then, by an extremely intricate argument, showed that Q (in fact, the subclass Q2 defined below) is unsolvable ([Aa1], [Aa2]). In this paper we develop stronger tools in order to simplify and extend the results of [Aa2]. Specifically, we show the unsolvability of an apparently new combinatorial problem, which we shall call the linear sampling problem (defined in §1.2 and §2.3). From the unsolvability of this problem there follows the unsolvability of two proper subclasses of Q, which we now define. For each i ≥ 0, let Pi be a dyadic predicate letter and let Ri be a monadic predicate letter.



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The second author was supported in part by a fellowship from the International Business Machines Corporation and by the Advanced Research Projects Agency under contract F19628-71-C-0174. We are grateful to Burton Dreben and Warren Goldfarb for their helpful suggestions.



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[Aa1]Aanderaa, S. O., Some recursively undecidable problems in automata theory and quantification theory, Notices of the American Mathematical Society, vol. 13 (1966), p. 509.
[Aa2]Aanderaa, S. O., A new undecidable problem with applications in logic, Doctoral Dissertation, Harvard University, 1966 (unpublished).
[Be]Berger, R., The undecidability of the domino problem, Memoirs of the American Mathematical Society, No. 66, 1966.
[DG]Dreben, B. and Goldfarb, W. D., A systematic treatment of the decision problem (forthcoming).
[DKW]Dreben, B., Kahr, A. S. and Wang, H., Classification of AEA formulas by letter atoms, Bulletin of the American Mathematical Society, vol. 68 (1962), pp. 528–532.
[KMW]Kahr, A. S., Moore, E. F. and Wang, H., Entscheidungsproblem reduced to the AEA case, Proceedings of the National Academy of Sciences of the U.S.A., vol. 48 (1962), pp. 365–377.
[Po]Post, E., Formal reductions of the general combinatorial decision problem, American Journal of Mathematics, vol. 65 (1943), pp. 197–215.
[Wa]Wang, H., Dominoes and the AEA case of the decision problem, Proceedings of a Symposium on the Mathematical Theory of Automata, Polytechnic Institute of Brooklyn, 1962, pp. 23–55.

Linear sampling and the ∀∃∀ case of the decision problem1

  • Stål O. Aanderaa (a1) and Harry R. Lewis (a2)


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