The elementary theory of the recursively enumerable degrees is not 0-categorical (original) (raw)
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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.
Reducibility orderings: Theories, definability and automorphisms
Annals of Mathematical Logic, 1980
introduced the notion of degree of unsolvability and the partial ordering ~ i on ~T, the set of such Tt, ring degrees, induced by Turing reducibility (Turing 13711, His paper with Kleene [ 1,4] contains the first serious analysis of this structure (~'r, ~x). They prove, for example, that all coantable partial orderings can be embedded in (ar, ~<~-). These embeddings show that the existential (it;st order~ theory of (~-r, ~r) is decidable, Next Spector [35], in a paper arising from Kleene's 1953 seminar, made an important inroad on the two quantifie, (i.e., VzI~ theory by showing that there is a minimal (Turing) degree. Sacks [31] extended these results and set forth some important conjectures on embeddings a~nd initial segments of ~'r. In particular he points out that one can prove the undecidability of the theory of (£ar, ~<v) by such results. This work inspired many papers by others eszabli,,hing better and better initial segment results. One milesto;~e was kachlan which showed that every countable distributive lattice can bc embedded as an initial segment of the Turing degrees. As the theory of distributive lattices was known to be undecidable, this sufficed to verify Sacks' conjezture that so is the theory of (@r, <~a-). (In fact it would have sufficed to embed all f-~nitc distributive lattices as was pointed out by Thomason [36] for hyperdegrees.) Two directions in which such results can be sharpened immediately come to mind, One is, where does the undecidability first arise in terms of quantifier complexity. The second is just how complicated is the full theory of (~, ~). (qhe results of Kleene and Post [ 14] showed only that the 3-theory was decidable while the coding of distributive lattices only showed that the full theory ha', degree at least 0'.) Further progress required further structural results. For the first question Lerman [20] supplied an essential ingredient by settling the full conjecture from Sacks , He showed that every finite lattice is embeddable ~, an initial segment of fib-. This can be combined with Kleene and Pc, st [14] to decide the 'q::l theory of
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
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.
Relational lattices: From databases to universal algebra
Journal of Logical and Algebraic Methods in Programming, 2016
Relational lattices are obtained by interpreting lattice connectives as natural join and inner union between database relations. Our study of their equational theory reveals that the variety generated by relational lattices has not been discussed in the existing literature. Furthermore, we show that addition of just the header constant to the lattice signature leads to undecidability of the quasiequational theory. Nevertheless, we also demonstrate that relational lattices are not as intangible as one may fear: for example, they do form a pseudoelementary class. We also apply the tools of Formal Concept Analysis and investigate the structure of relational lattices via their standard contexts. Furthermore, we show that the addition of typing rules and singleton constants allows a direct comparison with monotonic relational expressions of Sagiv and Yannakakis.
Substructure lattices of models of arithmetic
Annals of Mathematical Logic, 1979
We completely characterize those distributive lattices which can be obtained as elementary substructure lattices of models of Peano arithmetic. Stated concisely: every plausible distributive Mice occurs abundantly. Our proof employs the notion of a strongly definable type in many variables. With slight modifications the method also yields a characterization of those distributive lattices which can be obtained uniformly hy Gaifman's methods oi definable and end extensional l-types. As :J special case this gives another proof of two conjectures involving finite distributive lattices and models of arithmetic posed by Gaifman and initially proved by Schmurl. We also show that every minimal type (in the sense of Gaifman) satisfies a strong partitton property which we will call being "uniformly Ramsey". (2) VMbPA 3N> M Lt (N/M)= D. (3) D is complete, compactly generated, and eacf~ compact element of D hots C X,, compact predecessors.
The Journal of Symbolic Logic, 1996
We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theo...
Lecture Notes in Computer Science, 2014
Relational lattices are obtained by interpreting lattice connectives as natural join and inner union between database relations. Our study of their equational theory reveals that the variety generated by relational lattices has not been discussed in the existing literature. Furthermore, we show that addition of just the header constant to the lattice signature leads to undecidability of the quasiequational theory. Nevertheless, we also demonstrate that relational lattices are not as intangible as one may fear: for example, they do form a pseudoelementary class. We also apply the tools of Formal Concept Analysis and investigate the structure of relational lattices via their standard contexts. Furthermore, we show that the addition of typing rules and singleton constants allows a direct comparison with monotonic relational expressions of Sagiv and Yannakakis.