The elementary theory of the recursively enumerable degrees is not 0-categorical (original) (raw)
1983
We completely characterize the finite ideals of <2) in this chapter as the set of all finite lattices. It is not known whether all finite lattices have finite homogeneous lattice tables, so we replace these tables with weakly homogeneous sequential lattice tables which are possessed by all finite lattices. We extend the methods of Chap. VI, using such tables to embed finite lattices as ideals of 9>. This embedding theorem is used to locate decidable fragments of Th(^); the V 2-theory of Q) is decidable, but the V 3theory of 3) is undecidable. Results from Appendices A.2 and B.2 are used in this chapter.
Some highly undecidable lattices
Annals of Pure and Applied Logic, 1990
Theorem 2.13. Let S be a nontrivial, Steinitz exchange system of infinite dimension over 0. Then the first-order theory of the lattice of closed substructures of S is of complexity at least that of second-order .logic on K,, (or equivalently, second-order number theory). Section 1 Let K be an algebraically closed field of characteristic 0 and infinite transcendence degree over Q, the rationals. Let k = cl(Q). Let 3 be the lattice of algebraically closed subfields of K. Let 3* consist of 6p together with several parameters, that is, constant symbols for several elements of 3 to be introduced shortly. We first show that _Y* has the logical complexity of second-order logic on K, and then we show how to eliminate the use of these parameters. Let B = {bi : i E Z} be a transcendence basis of K over k. As K is of infinite transcendence degree, K and Z have the same cardinality. Thus, it suffices to show how to translate all sentences of second-order logic on Z into sentences of the first-order theory of 9* (and later on 3). Furthermore, by folklore it suffices to show how to translate into sentences of the first-order theory of 3* only those sentences of second-order logic with quantification over elements of Z and quantification over functions from Z to 1. Notation. For any subset { wi :j E Z} of K, we let (w, : j E J) denote Cl({Wj :j E Z}). Similarly for any element w of K, we let (w) denote cl({ w}). Say x1,. .. , x, are algebraically independent elements of K. Say x E (x1,. .. , x,). We say x depends on xi,. .. , x, if xi E (x1,. .. , x, with Xi replaced by x) for i = 1,. .. , n. B can be split into two disjoint subsets Bx = {xi : i E Z} and BY = { yi : i E Z}. The parameters are K>=(B,), K;=(B,), Kg=(xt+yt: i EZ), and Kg=(Xiyt:i EZ). Let Id(u, v) be the formula in the language of lattices that says: (ia) u is one dimensional contained in Kg & (ib) v is one dimensional contained in Kc & (ii) (u join v) meet KS is one dimensional & (iii) (u join v) meet K; is one dimensional.
Embedding Finite Lattices Into the Computably Enumerable Degrees — a Status Survey
Logic Colloquium, 2000
We survey the current status of an old open question in classical computability theory: Which finite lattices can be em- bedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the
Theoretical Computer Science, 1992
Shore, R.A. and T.A. Slaman, The p-T degrees of the recursive sets: lattice embeddings, extensions of embeddings and the two-quantifier theory, Theoretical Computer Science 97 (1992) 2633284. Ambos-Spies (1984a) showed that the two basic nondistributive lattices can be embedded in R,.,, the polynomial-time Turing degrees of the recursive sets. We introduce more general techniques to extend his results to show that every recursive lattice can be embedded in R,,. In addition to lattice-theoretic representation theorems, we use the scheme of priority style arguments coupled with "looking back" techniques presented in Shinoda and Slaman (1988, 1990). We also generalize the density type results of Ladner (1975) and many others to settle the full extension of the embedding problem for R,.,. Combined with the logical analysis of sentences with one alternation of quantifiers (Shore 1978, Lerman 1983), these results suffice to decide the full U-theory of R,,. They also give a strong nonhomogeneity result: the p-time degrees of the sets recursive in (and, if desired, p-time above) two distinct sets A and E are almost never isomorphic. The situation for the p-time many-one degrees is quite different. We decide the extension of the embedding problem (differently than for R,,) but not the t/3-theory. A notion of reducibility 6, between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is reducible to a set B, A <, B, if *Research partially supported by NSF Grants DMS-8601048 and DMS-8902797.
Proceedings of the American Mathematical Society, 1979
It is shown that the generalized word problem for lattices is solvable. Moreover, one can recursively decide if two finitely presented lattices are isomorphic. It is also shown that the automorphism group of a finitely presented lattice is finite.