Showing papers on "Recursively enumerable language published in 1987"

Recursively enumerable sets and degrees

Abstract: TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.

Theory of Recursive Functions and Effective Computability

Abstract: Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular. A second group of topics has to do with generalizations of recursion theory. The third topics group mentioned is subrecursive computability and subrecursive hierarchies

Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets

Abstract: ..."The book, written by one of the main researchers on the field, gives a complete account of the theory of r.e. degrees...The definitions, results and proofs are always clearly motivated and explained before the formal presentation; the proofs are described with remarkable clarity and conciseness. The book is highly recommended to everyone interested in logic. It also provides a useful background to computer scientists, in particular to theoretical computer scientists." Acta Scientiarum Mathematicarum, Ungarn 1988 ..."The main purpose of this book is to introduce the reader to the main results and to the intricacies of the current theory for the recurseively enumerable sets and degrees. The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be." Zentralblatt fur Mathematik, 623.1988

Computation theory and logic

Abstract: Minimal pairs for polynomial time reducibilities.- Primitive recursive word-functions of one variable.- Existential fixed-point logic.- Unsolvable decision problems for PROLOG programs.- You have not understood a sentence, unless you can prove it.- On the minimality of K, F, and D or: Why loten is non-trivial.- A 5-color-extension-theorem.- Closure relations, Buchberger's algorithm, and polynomials in infinitely many variables.- The benefit of microworlds in learning computer programming.- Skolem normal forms concerning the least fixpoint.- Spectral representation of recursively enumerable and coenumerable predicates.- Aggregating inductive expertise on partial recursive functions.- Domino threads and complexity.- Modelling of cooperative processes.- A setting for generalized computability.- First-order spectra with one variable.- On the early history of register machines.- Randomness, provability, and the separation of Monte Carlo Time and space.- Representation independent query and update operations on propositional definite Horn formulas.- Direct construction of mutually orthogonal latin squares.- Negative results about the length problem.- Some results on the complexity of powers.- The Turing complexity of AF C*-algebras with lattice-ordered KO.- Remarks on SASL and the verification of functional programming languages.- Numerical stability of simple geometric algorithms in the plane.- Communication with concurrent systems via I/0-procedures.- A class of exp-time machines which can be simulated by polytape machines.- ???-Automata realizing preferences.- Ein einfaches Verfahren zur Normalisierung unendlicher Herleitungen.- Grammars for terms and automata.- Relative konsistenz.- Segment translation systems.- First steps towards a theory of complexity over more general data structures.- On the power of single-valued nondeterministic polynomial time computations.- A concatenation game and the dot-depth hierarchy.- Do there exist languages with an arbitrarily small amount of context-sensitivity?.- The complexity of symmetric boolean functions.

Hyperarithmetical index sets in recursion theory

Abstract: We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a II1-complete index set. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set. 0. Introduction. The present paper deals with index sets, i.e., sets of indices of partial recursive (p.r.) functions and recursively enumerable (r.e.) sets that are defined through the p.r. functions or r.e. sets they code. The early results in index sets used geometric arguments in oneor two-dimensional arrays: Rogers showed the S3 and Il3-completeness of the index sets of lecursive and simple sets, respectively, in a finite injury argument. Lachlan, D. A. Martin, R. W. Robinson, and Yates (1968, unpublished, later appearing in Tulloss [Tu71]) showed the Il4-completeness of the index set of maximal sets in an infinite injury argument. Tulloss [ibid.] also mentions for the first time the question whether the index set of quasimaximal sets is S5-complete. However, the geometric method was too complex at higher levels of the arithmetical hierarchy. During the 1970's, progress in index sets was mainly made in otter areas by several Russian mathematicians as well as L. Hay. Schwarz [Schta] was the first to introduce induction into index set proofs (in the r.e. degrees) and was able to show that the index sets of lown and highn r.e. sets are En+3 and En+4-complete, respectively. Solovay [JLSSta] then extended Schwarz's methods to show the S<,+l-completeness of the index sets of low<<, (lown for some n) and of high<<, (highn for some n) r.e. sets as well as the fI<,+l-completeness of the index set of intermediate degrees (degrees neither low<<, nor high<<,). In this paper, we exhibit a family of algebraically invariant properties L<,1,<,definable in 6, that yields index sets at any level of the hyperarithmetical hierarchy. The proof is based on induction and Lachlan's theorem [La68] that any S3-Boolean algebra is isomoi^phic to the lattice of r.e. supersets of some r.e. set (modulo finite sets). It uses tree arguments and the fact that the Cantor-Bendixson rank of a tree corresponds to certain properties of the lattice of r.e. supersets of the set constructed. An extension yields a Il1-complete index set. A corollary shows the S5-completeness of the index set of quasimaximal sets, thereby settling this longopen question. Further results classify the index sets of atomic sets and of r.e. sets major in a fixed nonrecursive r.e. set. Our notation is fairly standard and generally follows Soare's forthcoming book Recursively Enumerable Sets and Degrees [Sota]. Received by the editors July 7, 1986 and, in revised form, November 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03D25. This paper is an extended version of part of the author's thesis. He wishes to thank his thesis advisor, R. I. Soare, as well as T. A. Slaman and J. Steel for helpful suggestions and comments. (r)1987 American Mathematical Society 0002-9947/87 $1 00 + $.25 per page

Degrees of splittings and bases of recursively enumerable subspaces

Abstract: This paper analyzes the interrelationships between the (Turing) of r.e. bases and of r.e. splittings of r.e. vector spaces together with the relationship of the degrees of bases and the degrees of the vector spaces they generate. For an r.e. subspace V of Voo , we show that O! is the degree of an r.e. basis of V iff O! is the degree of an r.e. summand of V iff O! is the degree and dependence degree of an r.e. summand of V. This result naturally leads to explore several questions regarding the degree theoretic properties of pairs of summands and the ways in which bases may arise.

Lattice embeddings in the recursively enumerable truth table degrees

Abstract: It is shown that every finite lattice, and in fact every recursively presentable lattice, can be embedded in the r.e. tt-degrees by a map preserving least and greatest elements. The decidability of the I-quantifier theory of the Le. ttdegrees in the language with ~, v, /\\, 0, and 1 is obtained as a corollary. Introduction. A set A ~ w is truth table (tt) reducible to B ~ w (A ~ tt B) if answers to questions of the form \"n E A?\" are given by a finite Boolean combination, effectively determined from n, of answers to questions of the form \"k E B?\". Sets A and B are of the same tt-degree if A ~ It Band B ~ It A. We consider the structure consisting of the tt-degrees of recursively enumerable sets of natural numbers. Odifreddi [4] and Rogers [6] contain background information on the tt-degrees. In particular, Fejer and Shore [1] contains information about the r.e. tt-degrees and about questions relating to the decidability of the theory of the r.e. tt-degrees. They show there that every recursively presentable lattice can be embedded in the r.e. tt-degrees preserving least element. Using this, they show that the 3 theory of the r.e. tt-degrees in the language with ~ , V, 1\\,0 is decidable, and ask whether the 3 theory is still decidable when 1 is added to the language. This decidability question can be answered by determining which finite lattices can be embedded in the r.e. tt-degrees preserving least and greatest elements. Jockusch and Mohrherr [3] have shown that the diamond lattice, the pentagon lattice, and the I-n-llattices can be embedded preserving least and greatest elements, but leave open the general question, and even such special cases as the three generator Boolean algebra. The embedding used in their proof requires that the lattice in question have the property that no element which is the inf of a pair of incomparable elements of the lattice can be joined up to the 1 of the lattice (except by 1 itself). We show here that all finite lattices, and in fact all recursively presentable lattices, can be embedded in the r.e. tt-degrees preserving least and greatest elements (provided the lattice has distinct least and greatest elements). Our proof for the general lattices combines a generalization of the coding method used by Jockusch and Mohrherr with the strategy for preserving nonzero infs used by Fejer and Shore. We prove first, in §l, that every lattice with a finite representation (and distinct least and greatest elements) can be embedded, and then in §2 outline the modifications needed to embed a recursively presented (possibly infinite) lattice. As a Received by the editors December 26, 1985 and. in revised form. May 15. 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03D30; Secondary 03D25. 515 ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

An Elementary Proof of the Peters-Ritchie Theorem

Abstract: The mathematical results about various classes of transformational grammars continue to play a role in linguistic discussions. Peters and Ritchie (1973a) proved that transformational grammars of the “standard” sort with a context-sensitive base were equivalent to unrestricted rewriting systems (equivalently, Turing machines) in their weak generative capacity, that is, that there was such a grammar for every recursively enumerable language. The proof can be presented informally and is easy to grasp (see Bach, 1974, for an informal presentation of the proof).

Descriptive power of synchronized shuffle expressions

Abstract: As a model to describe the concurrent system with a synchronization mechanism, Holenderski has proposed the synchronized shuffle expression (SSE). SSE is a shuffle expression obtained by adding a synchronized shuffle operator with its closure and projection operator. This paper considers various subclasses of SSE. It is shown that the class of languages represented by the subclass excluding the projection operator is the same as the class of recursively enumerable languages. The hierarchical relations among the subclasses are described.