Showing papers on "Recursively enumerable language published in 1983"

Initial and final algebra semantics for data type specifications, two characterization theorems

Abstract: We prove that those data types which may be defined by conditional equation specifications and final algebra semantics are exactly the cosemicomputable data types-those data types which are effectively computable, but whose inequality relations are recursively enumerable. And we characterize the computable data types as those data types which may be specified by conditional equation specifications using both initial algebra semantics and final algebra semantics. Numerical bounds for the number of auxiliary functions and conditional equations required are included in both theorems.

Pseudojump operators. I. The r.e. case

Abstract: Call an operator J on the power set of co a pseudo jump operator if J(A) is uniformly recursively enumerable in A and A is recursive in J(A) for all subsets A of c. Thus the (Turing) jump operator is a pseudo jump operator, and any existence proof in the theory of r.e. degrees yields, when relativized, one or more pseudo jump operators. Extending well-known results about the jump, we show that for any pseudo jump operator J, every degree > 0' has a representative in the range of J, and that there is a nonrecursive r.e. set A with J(A) of degree 0'. The latter result yields a finite injury proof in two steps that there is an incomplete high r.e. degree, and by iteration analogous results for other levels of the H", Ln hierarchy of r.e. degrees. We also establish a result on pairs of pseudo jump operators. This is combined with Lachlan's result on the impossibility of combining splitting and density for r.e. degrees to yield a new proof of Harrington's result that 0' does not split over all lower r.e. degrees. 1. We prove some generalizations of theorems about the Turing jump operator (denoted A H A') to theorems about operators of the form A * A ED WA, for an arbitrary fixed Godel number e. (Here A ED B is the recursive join of A and B and WA is the eth set r.e. in A in a fixed standard enumeration.) For instance, Friedberg (see (25, Theorem 4.1)) showed that there is a nonrecursive r.e. set A with A' TK (where K is a complete r.e. set). We prove by a finite injury priority argument similar to Friedberg's that for every e there is a nonrecursive r.e. set A with A ED WA-T K. Now Friedberg's argument relative to an arbitrary oracle B yields a fixed Godel

Characterization of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets

Abstract: We show that the lattice of supersets of a recursively enumerable (r.e.) set A is effectively isomorphic to the lattice of all r.e. sets if and only if the complement A of A is infinite and _{e WIe n A finite)

On the generative power of regular pattern grammars

Abstract: It is shown that for every recursively enumerable language L $$ \subseteq $$ ?* there exists a selective substitution grammar with a regular selector over a binary alphabet that generates L¢5, where ¢??. By requiring additional structural properties of the (already simple) selectors the language generating power is reduced in such a way that the resulting class lies strictly in between the family of EOL languages and the family of context-sensitive languages. For this class of languages some decision problems and normal forms are considered.

Abstract dependence, recursion theory, and the lattice of recursively enumerable filters

Abstract: DEPENDENCE, RECURSION THEORY, AND THE LATTICE OF RECURSIVELY ENUMERABLE FILTERS

Generalized Heterogeneous Algebras and Partial Interpretations

Abstract: A notion of heterogeneous algebras generalizing the concepts of total, partial and regular algebras is introduced consisting of a family of carrier sets, a family of total functions and a family of definedness predicates Partial interpretations are families of partial functions mapping such generalized algebras homomorphically onto partial heterogeneous algebras Classes of generalized algebras can be specified equationally by generalized abstract types This notion of abstract type is particularly well suited for the description of programming languages since the notion of homomorphism between generalized algebras allows to obtain fully abstract models in a uniform way — as weakly terminal models Sufficient conditions for the existence of initial and terminal models and interpretations are given, the model classes of generalized abstract types is analysed using lattice-theoretic methods and the relationship to the classical concept of partial functions is explained The main advantage of this approach — a uniform treatment of "strict" and "nonstrict" functions — is shown by an extended example where all recursively enumerable (finite and infinite) sequences of natural numbers are specified as a generalized abstract type with nonstrict basic functions

Degrees of Recursively Enumerable Topological Spaces

Abstract: In [5], Metakides and Nerode introduced the study of recursively enumerable (r.e.) substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff considered X , a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets of ω give rise to r.e. subsets of X . The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset of X ? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above. In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that given A any r.e. set and any r.e. open subset of X , there exists an r.e. open set ℋ which is a subset of and is dense in (in a topological sense) and in which A is coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree of A . We then go on and establish various results (both existential and universal) on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2.

Recursive properties of isomorphism types

Abstract: For Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.

Recursively enumerable extensions of R1 by finite functions

Abstract: Let R1 be the total recursive functions from IN to IN, and Fin the set of all partial functions from IN to IN having a finite initial segment of IN as domain Motivated by earlier studies on "simulation-universal automata" (BUCHBERGER&MENZEL 77/ MAIER, MENZEL&SPERSCHNEIDER 82/ MENZEL &SPERSCHNEIDER 82) we ask what it means that R1UF is recursively enumerable (re), for a subfamily F of Fin We show that each such family F is, in a certain sense, very rich A (simple) corollary is that it must be dense in Fin wrt the usual product topology on Fin As a consequence one obtains simple but useful necessary conditions on F to make R1UF re We also consider the class Ext(R1) :={F⊑Fin | R1UF re} as a whole It is also quite rich in structure (eg if viewed as an upper semi-lattice wrt union of families of functions) A sufficient criterion for F to be in Ext(R1) provides examples of families "naturally in Ext(R1)", thus demonstrating that richness On the other hand, there are families which belong to Ext(R1) in a more nonstandard way Main results: There is an F in Ext(R1) which is itself not re For F in Ext(R1), F is re iff F is "effectively dense in Fin", in some appropriate sense

Decision problem for separated distributive lattices

Abstract: It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1, Y2 such that Y1 c X1, Y2 c X2 and Y1 U Y2 = X1 U X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable. Introduction. A distributive lattice with 0 is separated if it satisfies the following separation property: for every x1, x2 there are Yi < x1 and Y2 < x2 such that Yil Y2 are disjoint (i.e. Yi A Y2 = 0) and Yi V Y2 x1 V x2. Alistair Lachlan introduced separated distributive lattices in [La] in connection with his study of the first-order theory of the lattice of recursively enumerable sets. He mentioned to me a question whether the first-order theory of separated distributive lattices is decidable. The answer is negative: in ?2 a known undecidable theory is interpreted in the firstorder theory of separated distributive lattices. The known undecidable theory is the first-order theory of the following structures: a Boolean algebra with a distinguished subalgebra. About undecidability of it see [Ru]. Actually the first version of the undecidability proof used the closure algebra CACD of Cantor Discontinuum, i.e. the Boolean algebra of subsets of Cantor Discontinuum with the closure operation. CACD is easily interpretable in the separated distributive lattice of functions f from Cantor Discontinuum into {0, 1, 2} such thatf1(2) is clopen. By [GS1] a finitely axiomatizable essentially undecidable arithmetic reduces to the first-order theory of CACD, hence to the first-order theory of the mentioned separated distributive lattice of functions, hence to the first-order theory of separated distributive lattices. The last step is somewhat complicated by the fact that [GS1] does not interpret the standard model N of arithmetic in CACD. (Even though [GS2] reduces the second-order theory of N to the first-order theory of CACD, [GS3] proves that N cannot be interpreted in CACD.) However the Boolean algebra of subsets of Cantor Discontinuum with a distinguished subalgebra of clopen (closed and open) sets is easily interpretable in CACD. This way I came to use Rubin's result which made the undecidability proof very simple. From the other side the cited result of [GS1] can be used to reprove Rubin's theorem and Received October 12 1980; revised August 30, 1981. 'The results were obtained and the paper was written during the Logic Year in the Institute for Advanced Studies of Hebrew University. ? 1983, Association for Symbolic Logic 0022-4812/83/4801-0020/$01.40

The basic theory of partial α-recursive operators

Abstract: In this paper, we investigate the theory of partial α-recursive operators and functionals, α an admissible ordinal, which are defined in terms of α-enumeration reducibility. The theory bifurcates into the study of weak operators and functionate, and of operators and functionate proper. The status of the representative theorems of the classical theory (when α=ω) is examined relative to both kinds of operators and functionals. Especial attention is given to the difficulties, when such exist, encountered in generalizing a classical result, whether simple or profound, to level α. In the course of the investigation we are led to consider briefly topics such as the structure theory of completely recursively enumerable classes of α-recursively enumerable sets. This is natural since this theory bears on the properties of effective operations at level α. The paper provides the framework for the further investigation of this and allied topics.

Logic of Reduced Power Structures

Abstract: ?0. Introduction and notation. We start with the framework upon which this paper is based. The most useful reference for these notions is [2]. For any nonempty index set I and any proper filter D on S(I) (the power set of I), we denote by d'I/D the reduced power of a? modulo D as defined in [2, pp. 167-169]. The first-order language associated with d'I/D will always be the same language as associated with aq/. We denote the 2-element Boolean algebra by 2 and 2I/D denotes the reduced power of it modulo D. We point out the intimate connection between the structures sdI/D and 2I/D given in [2, pp. 341-345]. Moreover, we assume as known the definition of Horn formula and Horn sentence as given in [2, p. 328] along with the fundamental theorem that q is a reduced product sentence iff q is provably equivalent to a Horn sentence [2, Theorem 6.2.5] (iff q is a 2-direct product sentence and a reduced power sentence [2, Proposition 6.2.6(ii)]). For a theory T (any set of sentences), a? k T denotes that a'? is a model of T. In addition to the above we assume as known the elementary characteristics (due to Tarski) associated with a complete theory of a Boolean algebra, and we adopt the notation of [3], [10], or [6] to denote such an elementary characteristic or the corresponding complete theory. We frequently will use Ershov's theorem which asserts that for each there exist an index set I and filter D such that 2I/D k [3] or [2, Lemma 6.3.21]. We call d'I/D a reduced power of type , if 2I/D k . We never allow in this paper the trivial Boolean algebra or its elementary characteristic to occur. In ?1 of this paper, we study for a given first-order language L and elementary characteristic the set V( ) of all sentences q of L such that for all structures sq/ of L, sdI/D k q whenever 2I/D k . An element q of V( ) is called a valid sentence of the power structures of type (note that V( ) is the usual set of valid sentences of L). It is shown that for a recursive language L, V( ) is recursively enumerable. Moreover, any model of V( ) is elementarily equivalent to sdI/D for some s'Z and where 2I/D k . A syntactic characterization is given for those sentences of L preserved by reduced powers of type . In ?2, for each k < w a characterization is given for when a 11k-sentence q of L is preserved by all reduced powers. An improvement upon the theorem of Keisler

In conclusion, the author expresses his gratitude to A. A. Makhnev for advice and remarks which helped in proving the theorem. LITERATURE CITED

Abstract: In this paper we describe a general method for interpreting how Turing machines perform computations in finitely axiomatizable theories. The method can be used to construct finitely axiomatizable theories whose properties are determined by Turing machine computations. This method is used to prove that the Lindenbaum boolean algebra for any recursively enumerable theory T is recursively isomorphic to the Lindenbaum algebra of a suitable finitely axiomatizable theory F In addition, the axiom system of the theory T and the recursive isomorphism of the boolean algebras T and ~ can be found uniformly with respect to a recursively enumerable index for the axiom system of T. This solves a problem due to Hanf [13], given as No. 22 in Friedman's list [12]. Hanf [14] announced the solution of this problem in 1975, but no proof was ever published. We then use the above method of interpretation to s~udy the properties of simple models. We construct a finitely axiomatizable complete theory possessing a nonconstructivizable simple model, thereby solving a problem posed by Harrington [15]. In Sec. 7 we estimate the complexity of some classes of formulas. The interpretations are based on the author's construction of a complete finitely axiomatizable superstable theory. The models for this theory

An algorithmically unsolvable problem in analysis

Abstract: The decision problem of distinguishing between the cases when the Laplace-Beltrami operator on the covering space of a compact manifold has 0 in its spectrum or is bounded away from 0 is algorithmically unsolvable in any class of manifolds that includes all 4-dimensional ones. The proof depends on a result of Brooks connecting the spectrum with the amenability of the fundamental group. Algorithmically unsolvable problems abound in logic, algebra, combinatorial topology, and diophantine equations (for a review and references see, for instance, [1]). In this note we point out an algorithmically unsolvable decision problem whose natural context is analysis on Riemannian manifolds. Let M be a compact differentiable manifold, 7T1(M) its fundamental group, M its universal cover. If g is a Riemannian metric on M (and so also on M), let Xo(M, g) denote the infimum of the spectrum of the Laplace-Beltrami operator -/ acting in L2(M, g). Whether XO(M, g) = 0 or XO(M, g) > 0 holds is independent of g and is, therefore, a diffeomorphism invariant of M. The following question then arises: Is there an algorithm for deciding which of the two cases holds? To make this question meaningful one considers a recursively enumerable class C of compact differentiable manifolds, i.e. one that can be put into one-to-one correspondence with a recursively enumerable set of positive integers which serve as code names for the manifolds (for this notion see [2, ?1.1]). An algorithm is then understood to be a Turing machine which, when presented with the code number of any M in C, computes in a finite number of steps which of the above alternatives holds. THEOREM. If C is such that for every finitely presented group G there is an M in C with g1(M) = G, then there is no algorithm for the above decision problem. The proof depends on a recent theorem of Brooks [3] which establishes a connection between the condition XA(M) = 0 and g1(M), namely, the condition holds if and only if gl1(M) is an amenable group. Thus, to prove our theorem, it is only necessary to show that amenability is an algorithmically undecidable property in the class of finitely presented groups. Amenability may be defined in various ways (see [4]), but from our point of view the essential fact is that amenability is a so-called Markov property. A property P of finitely presented groups is called a Markov property whenever (A) there is at least one finitely presented group G1 with the Received by the editors June 9, 1982. 1980 Mathematics Subject Classification. Primary 03D35; Secondary 31 C 12, 58C40.

The Structures of Recursion Theory

Abstract: Publisher Summary This chapter discusses the structures of the recursion theory The notion of relative computability is central to the formation of algebraic structures based on the concept of algorithm This notion is used to compare sets based on the amount of information they contain An algorithm can be viewed as a program for a digital computer with unlimited space and time to perform computations A set S is recursively enumerable if it is the range of a recursive function Thus the class of recursively enumerable sets is the class of sets that constitute the output of a computer for a particular algorithm A degree is recursively enumerable if it contains a recursively enumerable set