Showing papers on "Recursively enumerable language published in 1980"

Fixed Point Languages, Equality Languages, and Representation of Recursively Enumerable Languages

Abstract: Fixed point languages and equality languages of homomorphisms and dgsm mappings are consid- ered. Some basic properties of these classes of languages are proved, and it is shown how to use them to represent recursively enumerable sets. In particular, very simple languages are introduced which play the same role for the class of recursively enumerable languages that the Dyck languages play for the class of context-free languages. Finally, a new type of acceptor for defining equality languages is introduced. KEY WOADS AND PHRASES: equality language, fLxed point language, recursively enumerable language, determin- istic sequential machine, Turing machine, Post correspondence problem, shuffle, AFL generator, representation of languages

Decomposition of recursively enumerable degrees

Abstract: It is shown that any nonzero recursively enumerable degree can be expressed as the join of two distinct such degrees having a greatest lower bound. Let a nonrecursive r.e. set A be given. We shall show how to enumerate r.e. sets Bo, B1 and C such that C is recursive inA, B?U B1=A, B0nB1=0, Bl-i iTBi EC (i =0O,1) and deg C = deg(B? E C) n deg(Bl ED C). One corollary of this result is that the so-called "nondiamond" theorem [1, Theorem 5 ] cannot be improved to read: if bog b, are r.e. degrees such that bo U b, = 0' and bolb1 then bo and b, have no greatest lower bound in the upper semilattice of r.e. degrees. The same conclusion has been reached by J. R. Shoenfield and R. I. Soare [4] who independently and at about the same time as the present author constructed r.e. degrees bog b, such that bolbl, bo U b, = 0', and bo n b, exists. Our construction combines the technique for constructing minimal pairs from Lachlan [1, Theorem 1] and Yates [6] with the method of Sacks's splitting theorem [3, Theorem 1]. Our interest in this topic was awakened by Soare's article [5] where he asks whether improvement of the nondiamond theorem is possible. We first establish some notation. Let and K('?, 'I'): i be standard enumerations of all partial recursive (p.r.) functionals and of all ordered pairs of p.r. functionals. We assume given an enumeration of A and simultaneous uniformly effective enumerations of the p.r. functionals (Di, gI?, and 'I'. In describing the enumeration of Bo, B1, and C we often use our notations for sets and functionals to denote current approximations to them. If we wish to specify the approximation from a stage t other than the current one we append "[t]" to an expression. Thus (D1(B&)[t] denotes the finite function obtained by applying the finite functional (D1[t], defined by the axioms of (Di enumerated before stage t, to B?[t] the set of numbers enumerated in Bo before stage t. We use the same notation for a set and its characteristic function. In the construction below we shall be satisfying the following requirements: Nj: If 4'?(Bo (D C), 4JV(B1 E C) are the same total function then that total function is recursive in C, Received by the editors November 28, 1978 and, in revised form, September 14, 1979. AMS (MOS) subject classifications (1970). Primary 02F30; Secondary 02F25. O 1980 American Mathematical Society 0002-9939/80/0000-0377/$02.50

A new proof of the theorem on exponential diophantine representation of enumerable sets

Abstract: A new proof is given for the well-known theorem of Putnam, Davis, and Robinson on exponential diophantine representation of recursively enumerable sets. Starting from the usual definition of r.e. sets via Turing machines, a new method of arithmetization is given. This new method leads directly to a purely existential exponential formula. The new proof may be more suitable for a course on the theory of algorithms because it requires less knowledge of number theory.

A decidable fragment of the elementary theory of the lattice of recursively enumerable sets

Abstract: 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. 0. Introduction. Much of the recent work dealing with S, the lattice of recursively enumerable sets, has dealt with global properties of S such as automorphisms and decidability, rather than local properties of S, i.e., properties of definable classes of recursively enumerable sets. Two of the major results are Lachlan's [3] decision procedure for a natural fragment of the elementary theory of S *, the quotient lattice of S by the ideal of finite sets, Soare's result [15] on the existence of automorphisms carrying any maximal set into any other maximal set. More recently, Shore [13] has determined the definable automorphism bases for £. These global results have inspired new local results, in that they have naturally led to the discovery of important new S-definable classes of recursively enumerable sets whose properties have been investigated. Lachlan's result led to the discovery of small sets, Soare's result led to the discovery of ¿/-simple sets, and Shore's result led to the discovery of nowhere simple sets; the first two of these classes have been studied by Lerman and Soare [8], and the third class by Shore [14]. The class of ¿-simple sets proved to be of particular importance, in that it led to the refutation of conjectures of Martin and Shoenfield which imply that the degrees of elements of any S-definable class can be characterized by a finite set of equalities and inequalities involving the jumps of those degrees. Evidence for these conjectures included Martin's result [9] that a is the degree of a maximal set if and only if a' = 0\", and results of Lachlan [2] and Shoenfield [12] that a is the degree of an atomless set if and only if a\" > 0\". In this paper, we give a decision procedure for a larger fragment of the Received by the editors April 14, 1978 and, in revised form, November 3, 1978. AMS (MOS) subject classifications (1970). Primary 02F25; Secondary 02G05.

A homomorphic characterization of time and space complexity classes of languages

Abstract: Recently, it has been shown that for each recursively enumerable language there exists an erasing homomorphism h 0 and homomorphisms h 1,h 2 such that L= h 0(e(h 1,h 2)) where (e(h 1,h 2)) is the set of minimal words on which h 1 and h 2 agree. Here we show that by restrictions on the erasing h 0 we obtain most time-complexity language classes, and by restrictions on the pair (h 1 h 2) we characterize all space complexity language classes.

Very special languages and representations of recursively enumerable languages via computation histories

Abstract: A method of encoding the computation histories of a wide class of machines is introduced and used to derive several representation theorems for the class of recursively enumerable languages. In particular it is demonstrated that any recursively enumerable language K ⊂ Σ* can be represented as K = Φ Σ ( R ∩ D 1 ⋮ D 2 ), where D 1 and D 2 are fixed semi-Dyck languages, 〈 is the shuffle operation, R is a regular language depending on K and Φ Σ is a weak identity homomorphism. This result is the natural analog for the recursively enumerable languages of the Chomsky-Shutzenberger representation of the context-free languages.

Post's Problem, Absoluteness and Recursion in Finite Types

Abstract: The unresolved character of the power set operation stymies the solution of elementary problems arising in Kleene's theory of recursion in objects of finite type. E.g. Post's problem for 3 E has a positive solution if V=L (NORMANN, 1975), and a negative if AD holds. Let δ be the class of all sets R ⊆ 2ω such that R is recursive in 3 E, b for some real b . A forcing construction shows δ is not recursively enumerable in 3 E when there is a recursively regular well-ordering of 2″ recursive in 3 E. It follows that the concepts of Σ * , and weak Σ * , definability differ. With the aid of the notions of indexicality and ordinal recursiveness, an absolute version of Post's problem for 3 E is devised, and studied when certain recursively enumerable projecta are equal.

Embedded Implicational Dependencies and their Inference Problem.

Abstract: It is shown that the general inference problem for embedded implicational dependencies (EIDs) is undecidable. For the more important case of finite inference (i.e., inference for finite data bases), the problem is not even recursively enumerable (r.e.); rather, it is complete in co-r.e. These results hold even for typed EIDs without equality, as well as for (untyped) template dependencies. The case for typed template dependencies remains open. The complexity of the inference problem for full dependencies has also been characterized - it is complete in exponential time for full implicational dependencies, and even for full typed template dependencies.

The unaxiomatizability of a quantified intensional logic

Abstract: In this paper we show that an intensional logic TP of [4], with quantification over "individual concepts", is not recursively enumerable, and that it has, in fact, the degree of unsolvability of second order logic. TP has predicate letters, individual variables ranging over functions on a domain of contexts, propositional variables, V, ~, 3, =, and a T-operator T, which takes an individual variable x, and a wff A into a new wff TxA. TxA may be read 'A is the case at (or as of) x'. The quantifier V binds individual variables introduced with the T-operator, as well as those that appear with predicate letters and identity. A more formal account of the syntax of TP can be reconstructed from the truth clauses for the semantics of TP given in Section 1.

Rekursionstheorie und Komplexitätstheorie auf effektiven CPO-S

Abstract: Effective cpo-s are a useful tool for introducing computability on a large class of sets. In this report Blum's theory of computational complexity is generalized to certain effective cpo-s with effective weight. At first, as a generalization of Rogers's Isomorphism Theorem for Godel numberings of the partial recursive functions it is proved that two "admissible numberings" of the recursively enumerable elements of an effective cpo are recursively isomorphic. The concept of effective weight on an effective cpo is introduced and recursive cpo-elements are defined. In analogy to Blum's approach two axioms for cpo-complexity are introduced. For cpo-elements with zero weight a hierarchy theorem, the speedup theorem and the gap theorem are proved. Using "extended" cpo-s the results can be applied to the recursive elements of a cpo.

A Note on Natural Creative Sets and Goedel Numberings

Abstract: Creative sets (or the complete recursively enumerable sets) play an important role in logic and mathematics and they are known to be recursively isomorphic. Therefore, on the one hand, all the creative sets can be viewed as equivalent, on the other hand, we intuitively perceive some creative sets as more ``natural and simpler'''' than others. In this note, we try to capture this intuitive concept precisely by defining a creative set to be natural if all other recursively enumerable sets can be reduced to it by computationally simple reductions and show that these natural creative sets are all isomorphic under the same type of computationally simple mappings. The same ideas are also applied to define natural Goedel numberings.

On the distribution of independent formulae of number theory

Abstract: It follows by Godel's incompleteness Theorem [6] that any effective sound system of logic for elementary arithmetic must be incomplete. We show that in any effective sound system of logic for elementary arithmetic, there exist valid unprovable formulae that are quite small relative to the complexity of the logical system. Also, such formulae are quite dense. In fact, the situation is about as bad as it could possibly be. That is , no infinite axiom system for elementary arithmetic can be much more compact than a listing of all the valid formulae. The unprovable formulae we construct express predicates in the classes &Sgr2 and π 2 of the Kleene arithmetic hierarchy [12]. The construction yields a set of short formulae, at least one of which must be valid and unprovable, but the construction does not tell us which one is valid and unprovable. We also construct small valid unprovable formulae expressing a relation in the class π 1 of the Kleene arithmetic hierarchy. These latter formulae are not as small. We do not know how small the independent formulae corresponding to the class π 1 are. The constructions are based on the concept of a restricted oracle, first introduced in [10] and further developed in [11]. The proofs make use of the recent result of Matijasevic [9] concerning the relationship between recursively enumerable sets and Diophantine equations.