Decidability of the two-quantifier theory of the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices (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.
A Decidable Fragment of the Elementary Theory of the Lattice of Recursively Enumerable Sets
Transactions of the American Mathematical Society, 1980
A natural class of sentences about the lattice of recursively enumerable sets modulo finite sets is shown to be decidable. This class properly contains the class of sentences previously shown to be decidable by Lachlan. New structure results about the lattice of recursively enumerable sets are proved which play an important role in the decision procedure.
Degree theoretic definitions of the low2 recursively enumerable sets
Journal of Symbolic Logic, 1995
The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤TB, where arbitrary (Turing) machines, φe, can be used; access to information about (the oracle) B is unlimited and the lengths of computations are potentially unbounded. Many other interesting reducibilities result from restricitng one or more of these facets of the procedure. Thus, for example, the strongest notion considered is one-one reducibility on sets: A ≤1B iff there is a one-one recursive (= effective) function f such that x Є A ⇔ f(x) Є B. Many-one (≤m) reducibility simply allows f to be many-one. Other intermediate reducibilities include truth-table (≤tt) and weak truth-table (≤wtt). The latter imposes a recursive bound f(x) on the information about B that can be used to compute A(x). The former also bounds the length of comp...
Contiguity and distributivity in the enumerable Turing degrees
Journal of Symbolic Logic, 1997
We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no m-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.
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
Lecture Notes in Computer Science, 1999
In this paper we establish a link between satisfiability of universal sentences with respect to varieties of distributive lattices with operators and satisfiability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of universal sentences in such varieties, and then use results from automated theorem proving to obtain decidability and complexity results for the universal theory of some such varieties.
Undecidability results for low complexity degree structures
Proceedings of Computational Complexity. Twelfth Annual IEEE Conference, 1997
We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and many-one reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras which w as introduced in [7]. The method can be applied in fact to all hyper-polynomial time classes.