Recursively enumerable generic sets

1982 ◽  
Vol 47 (4) ◽  
pp. 809-823 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Essential Feature ◽  
Turing Degree ◽  
New Class ◽  
Recursively Enumerable ◽  
Simple Set ◽  
Recursively Enumerable Sets

AbstractWe show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0′. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice ℰ of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions.

1-genericity in the enumeration degrees

1988 ◽  
Vol 53 (3) ◽  
pp. 878-887 ◽  
Author(s):  
Kate Copestake
Keyword(s):  
Partial Function ◽  
Partial Ordering ◽  
Turing Degree ◽  
Turing Degrees ◽  
Original Formulation ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Generic Set

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

Nonisomorphism of lattices of recursively enumerable sets

1993 ◽  
Vol 58 (4) ◽  
pp. 1177-1188 ◽  
Author(s):  
John Todd Hammond
Keyword(s):  
Boolean Algebra ◽  
Isomorphism Class ◽  
Boolean Algebras ◽  
Turing Degree ◽  
Symmetric Difference ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Definition Of

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

On degrees of unsolvability and complexity properties

1975 ◽  
Vol 40 (4) ◽  
pp. 529-540 ◽  
Author(s):  
Ivan Marques
Keyword(s):  
Turing Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Complete Sets ◽  
Turing Complete ◽  
Degrees Of Unsolvability

In this paper we present two theorems concerning relationships between degrees of unsolvability of recursively enumerable sets and their complexity properties.The first theorem asserts that there are nonspeedable recursively enumerable sets in every recursively enumerable Turing degree. This theorem disproves the conjecture that all Turing complete sets are speedable, which arose from the fact that a rather inclusive subclass of the Turing complete sets, namely, the subcreative sets, consists solely of effectively speedable sets [2]. Furthermore, the natural construction to produce a nonspeedable set seems to lower the degree of the resulting set.The second theorem says that every speedable set has jump strictly above the jump of the recursive sets. This theorem is an expected one in view of the fact that all sets which are known to be speedable jump to the double jump of the recursive sets [4].After this paper was written, R. Soare [8] found a very useful characterization of the speedable sets which greatly facilitated the proofs of the results presented here. In addition his characterization implies that an r.e. degree a contains a speed-able set iff a′ > 0′.

scholarly journals When catalytic P systems with one catalyst can be computationally complete

2021 ◽  
Author(s):  
Artiom Alhazov ◽  
Rudolf Freund ◽  
Sergiu Ivanov
Keyword(s):  
Computational Complexity ◽  
P Systems ◽  
Membrane Systems ◽  
Computational Completeness ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractCatalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

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