Showing papers on "Recursively enumerable language published in 1968"

On the lattice of recursively enumerable sets

Abstract: This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper is to advance the search for a decision procedure for the elementary theory of r.e. sets. More precisely, an effective method is sought for deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c is true of the r.e. sets. The main achievement of the paper is the characterisation of the hh-simple sets as those coinfinite r.e. sets whose r.e. supersets form a Boolean algebra. The reader is referred to Davis's book [1] for basic information about the partial recursive (p.r.) functions and about r.e. sets. Other background material required for a proper understanding of the present paper consists of [8], [3, Theorem 2], [10, Introduction and ?4], and [5] where the contributions have been listed in their natural order. We take the formulation of the lower predicate calculus given in Abraham Robinson [9]. Natural numbers are denoted by lower case Roman letters and sets of them by lower case Greek letters. The empty set is denoted by 0 and the set of all natural numbers by v. The complement of any set a is denoted by a'; a is called cofinite or coinfinite just if a' is finite or infinite respectively. For sets a, / we write aoc3 just if the set (a -3 u (3 a) is finite; otherwise we write aZ 3. By function we mean a map of some subset of v x v x **. x v into v; functions will be denoted by upper case Roman letters as will relations on the natural numbers. The informal logical signs used are v , &, ->, , (x), (Ex), -,> which are to be read as " or ", " and ", " implies ", "not", "for all x", "there exists x", "is equivalent to" respectively. Let ,' be a finite class of propositions; then s otherwise sup a is to be oo. The plan of the paper is as follows. In the first section we give a brief discussion of the elementary theory of r.e. sets and prove that its decision problem is of the same degree as that of the elementary theory of the lattice obtained by taking the equivalence classes of r.e. sets with respect to -. In ?2 we prove the main theorem which states: if a is an r.e. subset of an r.e. set / then either there exists a recursive subset 8 of:/ such that a u 8 = 3 or there exists a recursive sequence {8il of disjoint finite subsets of / such that 8i a is nonempty for all i. This theorem was inspired by

On Recursively Unsolvable Problems in Topology and Their Classification

Abstract: Publisher Summary This chapter discusses recursively unsolvable problems in topology and their classification The decision problems that have been considered, pertain to diffeomorphism homeomorphism, combinatorial equivalence, and homotopy equivalence For each dimension n ≥4 and for each recursively enumerable degree of unsolvability D , there exists a recursive class C ( n , D ) of finite presentations of n -manifolds, endowed with a differentiable and a compatible combinatorial structure The chapter discusses a topological analogue of the Markov–Addison–Feeney–Adjan–Rabin theorem

Degrees of Recursively Enumerable Sets Which Have No Maximal Supersets

Abstract: The purpose of this paper is to present two new theorems concerning the degrees of coinfinite recursively enumerable (r.e.) sets which have no maximal supersets Let the class of all such degrees be denoted by A. Martin in [2] conjectured that there was some equality or inequality involving a′ or a″ characterizing the degrees a in A. Martin himself proved ([2, Corollary 4.1]) that a′ = 0″ is sufficient for ar r.e. degree a to be in A, and Robinson [3] announced that a′ ≥ 0″ is necessary. In this paper we improve both of these theorems by a factor of the jump, i.e., we shall show that a″ = 0″ is sufficient for an r.e. degree a to be in A, and that a″ ≥ 0″ is necessary.

Finite generation of recursively enumerable sets

Abstract: Suppose we wish to build up the class of recursively enumerable sets by starting with the set DI of natural numbers and constructing new sets from those already obtained using as little auxiliary machinery as possible. One way would be to start with a finite number of functions F1, * * *, Fk (of one variable, from and to 9t) such that every recursively enumerable set can be obtained from t by constructing new sets Fj [3 I where 5 is a previously obtained set. We can think of F1, * * *, Fk as unary operations on sets of natural numbers. Any set 8 obtained in this way is the range of a function F obtained by composition from F1, **, Fk. If we consider the values of F1, * * *, Fk as given, then the number of steps needed to compute Fn does not depend on n. Hence for all xe8, there exists a proof that xCS of bounded length in terms of F1, * * *, Fk (just as there is a one-step proof that a composite number is composite in terms of multiplication). We say a set of natural numbers is generated by F1, * * *, Fk if it is the range of a function obtained by composition from F1, * * * , Fk. Also a class e of sets is generated by F1, * * *, Fk if every nonempty set of e is generated by F1, l * , Fk and every set generated by Fl, @ , Fk is in e. EXAMPLE. Let Go, G1, * * * be the primitive recursive functions listed systematically so that the function G given by

Complete recursively enumerable sets

Abstract: The main purpose of this paper is to improve upon the main theorem of Martin [2 ]. Martin gave a sufficient condition for a recursively enumerable (r.e.) set to be complete. By a slight modification we weaken Martin's condition so that it becomes both necessary and sufficient. Next we indicate briefly why Martin's condition for completeness is not a necessary one. Finally, we discuss applications of our theorem and the problem of formulating a notion of effectively maximal set. I am grateful for the referee's suggestions, particularly with regard to the statement of the theorem. Let W0, W1, * * be a standard enumeration of all r.e. sets. Let B00 f91, * * be a standard enumeration of all partial recursive functions of one argument. The representing function KA of a set of natural numbers A is defined by: KA(X) =0 if x is in A, KA(X) = 1 if x is in AT (the complement of A). Let L(x) denote the set of natural numbers

Degrees of Unsolvability in Formal Grammars

Abstract: The following theorem is a refinement of an unsolvability result due to E. Post: For any recursively enumerable degree D of recursive unsolvability there is a recursive class of sequences (of the same length) of nonempty words on an alphabet A such that the Post correspondence decision problem for that class is of degree D. This theorem is proved and then applied to obtain degree analogues of the ambiguity problem and the common program problem for the class of context-free grammars.

Decision Problems About Algebraic and Logical Systems as a Whole and Recursively Enumerable Degrees of Unsolvability

Abstract: Publisher Summary This chapter discusses decision problems of arbitrarily preassigned recursively enumerable degree of unsolvability and decision problems concerned with recognizing that a system of a certain kind enjoys a specified property or with recognizing that two systems of a certain kind are in a specified relation The latter decision problems are also known as “global decision problems” The chapter also discusses several meta-decision problems about groups that are formed by compounding the kind of “first order” decision problems A recursive class of finite presentations of groups is always given by means of some generic presentation depending on a single parameter ranging over some recursive set of words Finally, the chapter presents several unsolvability results regarding the homomorphic images of a fixed finitely presented group

One equation to rule them all

Abstract: : The report describes an application of recursive function theory to Hilbert's tenth problem. It is proved that if a particular exhibited diophantine equation has no nontrivial solutions, then all recursively enumerable sets are diophantine. Hence, if the exhibited diophantine equation has no nontrivial solutions, then Hilbert's tenth problem is recursively unsolvable. The methods used can be readily adapted to obtained various other hypotheses about which demonstrations can be made similar to the one given in this study. It has not yet been proved that the 'one equation to rule them all' has no nontrivial solution, but so far the search for counterexamples has been fruitless.

On the completeness of some transfinite recursive progressions of axiomatic theories

Abstract: The well-known incompleteness results of Godel assert that there is no recursively enumerable set of sentences of formalized first order arithmetic which entails all true statements of that theory. It is equally well known that by introducing sufficiently nonconstructive rules, such as the ω-rule of induction, completeness can be re-established.Starting from the work of Turing [4] Feferman in [1] developed another method, viz. the study of transfinite recursive progressions of theories, for closing the gap between Godel (recursively enumerable sets of axioms yield incompleteness) and Tarski (number-theoretic truth is not arithmetically definable).