Showing papers on "Recursively enumerable language published in 1985"

Cuts, consistency statements and interpretations

Abstract: Interpretability in reflexive theories, especially in PA, has been studied in many papers; see e.g. [3], [6], [7], [10], [11], [15], [26]. It has been shown that reflexive theories exhibit many nice properties, e.g. (1) if T, S are recursively enumerable reflexive, then T is interpretable in S iff every Π1 sentence provable in T is provable in S; and (2) if S is reflexive, T is recursively enumerable and locally interpretable in S (i.e. every finite part of T is interpretable in S), then T is globally interpretable in S (Orey's theorem, cf. [3]).In this paper we want to study such statements for nonreflexive theories, especially for finitely axiomatizable theories (which are never reflexive). These theories behave differently, although they may be quite close to reflexive theories, as e.g. GB to ZF. An important fact is that in such theories one can define proper cuts. By a cut we mean a formula with one free variable which defines a nonempty initial segment of natural numbers closed under the successor function. The importance of cuts for interpretations in GB was realized already by Vopěnka and Hajek in [30]. Pioneering work was done by Solovay in [24]. There he developed the method of “shortening of cuts”. Using this method it is possible to replace any cut by a cut which is contained in it and has some desirable additional properties; in particular it can be closed under + and ·. This introduces ambiguity in the concept of arithmetic in theories which admit proper cuts, namely, which cut (closed under + and ·) should be called the arithmetic of the theory? Cuts played the crucial role also in [20].

On minimal pairs of enumeration degrees

Abstract: For sets of natural numbers A and B, A is enumeration reducible to B if there is some effective algorithm which when given any enumeration of B will produce an enumeration of A. Gutteridge [5] has shown that in the upper semilattice of the enumeration degrees there are no minimal degrees (see Cooper [3]), and in this paper we study those pairs of degrees with gib 0. Case [1] constructed a minimal pair. This minimal pair construction can be relativised to any gib, and following a suggestion of Jockusch we can also fix one of the degrees and still construct the pair. These methods yield an easier proof of Case's exact pair theorem for countable ideals. 0″ is an upper bound for the minimal pair constructed in §1, and in §2 we improve this bound to any Σ2-high Δ2 degree. In contrast to this we show that every low degree c bounds a degree a which is not in any minimal pair bounded by c. The structure of the co-r.e. e-degrees is isomorphic to that of the r.e. Turing degrees, and Gutteridge has constructed co-r.e. degrees which form a minimal pair in the e-degrees. In §3 we show that if a, b is any minimal pair of co-r.e. degrees such that a is low then a, b is a minimal pair in the e-degrees (and so Gutteridge's result follows). As a corollary of this we can embed any countable distributive lattice and the two nondistributive five-element lattices in the e-degrees below 0′. However the lowness assumption is necessary, as we also prove that there is a minimal pair of (high) r.e. degrees which is not a minimal pair in the e-degrees (under the isomorphism). In §4 we present more concise proofs of some unpublished work of Lagemann on bounding incomparable pairs and embedding partial orderings.As usual, {Wi}i ∈ ω is the standard listing of the recursively enumerable sets, Du is the finite set with canonical index u and {‹ m, n ›}m, n ∈ ω is a recursive, one-to-one coding of the pairs of numbers onto the numbers. Capital italic letters will be variables over sets of natural numbers, and lower case boldface letters from the beginning of the alphabet will vary over degrees.

A Completeness Theorem for Recursively Defined Types

Abstract: In this paper the notion of recursively defined type for a functional language is studied. The semantics of types (which are interpreted as subsets of a type-free domain following /MIL/) is built by successive approximations. An alternative approach, using metic spaces, has been given in /MPS/.

Index sets in the hyperarithmetical hierarchy

Abstract: In [I] the problem about the possibility of "structural" discription of the families of recursively enumerable sets, whose index sets belong to a given class of the arithmetical hierarchy, is posed. In this article, we will study the possibility of this description in terms of completely enumerable families, and that too in the following more general situation: for abstract classes of numbered sets and for various hierarchies.

The family of one-counter languages is closed under quotient

Abstract: We study, first, the operation of quotient in connection with rational transductions. We show, afterwards, that Rocl, the family of one counter languages is closed under quotient by a context-free language. On the contrary, every recursively enumerable language is the quotient of two linear languages.

Variations on promptly simple sets

Abstract: In this paper we answer the question of whether all low sets with the splitting property are promptly simple. Further we try to make the role of lowness properties and prompt simplicity in the construction of automorphisms of the lattice of r.e. (recursively enumerable) sets more perspicuous. It turns out that two new properties of r.e. sets, which are dual to each other, are essential in this context: the prompt and the low shrinking property. In an earlier paper [4] we had shown (using Soare's automorphism construction [10] and [12]) that all r.e. generic sets are automorphic in the lattice ℰ of r.e. sets under inclusion. We called a set A promptly simple if Ā is infinite and there is a recursive enumeration of A and the r.e. sets ( W e ) e ∈ N such that if W e is infinite then there is some element (or equivalently: infinitely many elements) x of W e such that x gets into A “promptly” after its appearance in W e (i.e. for some fixed total recursive function f we have x ∈ A f ( s ) , where s is the stage at which x entered W e ). Prompt simplicity in combination with lowness turned out to capture those properties of r.e. generic sets that were used in the mentioned automorphism result. In a following paper with Shore and Stob [7] we studied an ℰ-definable consequence of prompt simplicity: the splitting property.

Terminal weighted grammars and picture description

Abstract: Motivated by the idea of describing parquet deformations using grammars and also of describing an infinite number of terminals starting with only a finite set, this paper defines a terminal weighted grammar, where the terminal generated at any step of a derivation is defined as a function of time It is seen that terminal weighted regular grammars generate exactly the class of recursively enumerable sets Terminal weighted matrix grammars are used to describe parquet deformations The extension of terminal weights to array grammars is also discussed

Embedding the diamond lattice in the recursively enumerable truth-table degrees

Abstract: It is shown that the four element Boolean algebra can be embedded in the recursively enumerable truth-table degrees with least and greatest elements preserved. Corresponding results for other lattices and other reducibilites are also discussed. For sets A, B c w, we say that A is a truth-table (tt) reducible to B if there exists an effective procedure for reducing any question of the form "n E A?" to an equivalent finite Boolean combination of questions of the form "k E B?" Then, A, B are said to have the same tt-degree if each is tt-reducible to the other, and tt-degrees have a natural ordering induced by tt-reducibility. (See [1, 6 and 8] for information on tt-degrees.) We show the existence of two incomparable recursively enumerable (r.e.) tt-degrees with supremum 0' (the highest r.e. tt-degree) and infimum 0 (the lowest). In other words, the four-element Boolean algebra (known also as the diamond lattice) can be embedded as a lattice in the r.e. tt-degrees with least and greatest elements preserved. We also obtain analogous results with the diamond lattice replaced by each of the two five-element nondistributive lattices (pentagon and 1-3-1) and with tt-reducibility replaced by many of its restricted forms, such as bounded truth-table and positive reducibility [2]. The history of this problem is as follows. A. H. Lachlan proved in his well-known "nondiamond theorem" [5, Theorem 5] that the diamond lattice cannot be embedded in the r.e. Turing degrees with 0 and 1 preserved. His proof simultaneously establishes the corresponding result for r.e. weak truth-table (wtt) degrees [6]. Lachlan also showed in [4] that no two incomparable r.e. many-one (m) degrees can have supremum 0', so the diamond lattice cannot be embedded in the r.e. m-degrees with 1 preserved. The trend of these results makes it reasonable to conjecture that the diamond lattice cannot be embedded in the r.e. tt-degrees with 0 and 1 preserved, although in the other direction D. Posner [7] proved that the Turing degrees below 0' are complemented. In [6, Theorem 6.6] P. G. Odifreddi announced that in fact the diamond lattice can be embedded in the r.e. tt-degrees with 0 and 1 preserved. His construction involved splitting a creative set K into two disjoint r.e. Received by the editors April 9, 1984. 1980 Mathematics Subject Classification. Primary 03D30; Secondary 03D25.

Minimal polynomial degrees of nonrecursive sets

Abstract: This is the second paper exploring a new link between recursion theory and computational complexity theory. From its inception part of the work in complexity theory has been guided by an analogy with recursion theory. However, here we explore a more concrete and direct connection. This connection stems from the notion of honest polynomial-time reducibi l i ty, ~ , defined in Homer [3]. ~ is a sl ight strengthening of the usual polynomial-time Turing reducibi l i ty. (Precise definitions and the main theorems from [3] are presented in the next section.) The particular problem studied is the existence of minimal sets for this reducibi l i ty. A set A is -minimal i f A ~ P and for any B, i f B ~ A then B~ P or A ~ B. In [3] two opposing results were proved. First i t was shown that no recursive set is of minimal ~ -degree. Second, that i f a set recursive in O" has minimal degree then P i NP. The method used to prove this theorem is recursion-theoretic in nature and is an adaption of the proof of the existence of minimal Turing degrees. The result gives a concrete connection between recursion theory and a fundamental problem of complexity theory. The results presented in this paper carry these methods and ideas further, There is clearly a gap between the f i r s t result, dealing with recursive sets and the second, dealing with sets recursive in 0". The intention here is to narrow that gap by looking at r.e. sets and nonrecursive sets with particular properties, but with respect to the same question of ~ -minimality. in section 3 we take another look at the second theorem mentioned above. A partial converse of this result is proved, l inking this problem to the existence of certain one-way functions. Sections 4 and 5 are devoted to trying to strengthen the f i r s t theorem mentioned above. Recursively enumerable sets are considered f i r s t . I t is shown that no recursively enumerable set which is P-immune can be ~ -minimal. (A set is P-immune i f i t contains no in f in i te polynomial time subset.) Arbitrary sets are next considered and i t is shown that i f a set and i ts complement are both P-immune then i t is not ~ -minimal. Topics for future work are discussed in the last section.

Unification in a Many-sorted Calculus with Declarations

Abstract: The many-sorted first order calculus ∑RP* is extended to a many-sorted calculus, which allows declarations, i.e. a term t of sort S can be declared to be of some lesser sort S’. The heart of such a calculus is the unification algorithm for terms, which respects the declarations. In this paper it is shown, that the set of most general DS-unifiers is recursively enumerable and that such a set may be infinite. Furthermore, it is shown, that it is undecidable, whether two terms are DS-unifiable.

The Incompleteness Theorems and Related Results for Nonconstructive Theories

Abstract: After the discovery of the arithmetical hierarchy Mostowski realized that there are further general recursion-theoretic facts which lie in the foundation of the incompleteness theorems and related results. In his articles (see Mostowski, 1979) he has made attempts to investigate formal systems from the recursion-theoretic point of view. Particularly, he has shown that the restriction to recursively enumerable systems is irrelevant for the 1st incompleteness theorem.

The decidable normal modal logics are not recursively enumerable

Abstract: LEMMA 2. S, is decidable if X is recursive. Proof Any wff A will be a theorem of S, iff A is valid on (i) all frames G, (see [ 1, p. 358]), and (ii) all frames & for (Y $ X. Let A be of modal degree n. Test A on Fn+z and Gn+Z. Gn+s is already a frame for Sx, and if A fails on F,,+z, then it will fail on F,, which is a frame for Sx. Otherwise A will be valid on every frame F, and G, , for (Y > n + 1. So test A on all F,,, and G,, where m < n + 1. If A fails on G, then, since G, is a frame for S,, A 4 S,. If A is valid on every G, (m < n + l), then test A on every F, (m < n + 1) such that m $Z X. Since X is recursive and m is bounded, this process is finite. If A fails on some such F, , then A f$ S,. Otherwise A E S,.

Complete Symmetry in D2L Systems and Cellular Automata

Abstract: We introduce completely symmetric D2L systems and cellular automata by means of an additional restriction on the corresponding symmetric devices. Then we show that completely symmetric D2L systems and cellular automata are still able to simulate Turing machine computations. As corollaries we obtain new characterizations of the recursively enumerable languages and of some space-bounded complexity classes.