On Some Properties of π‘Ÿ-Maximal Sets and 𝑄1–𝑁-Reducibility

2002 β—½  
Vol 9 (1) β—½  
pp. 161-166
Author(s):  
R. Omanadze
Keyword(s):  
Recursively Enumerable Set β—½  
Recursively Enumerable

Abstract It is shown that if 𝑀1, 𝑀2 are π‘Ÿ-maximal sets and 𝑀1 ≑ 𝑄1–𝑁𝑀2, then 𝑀1 ≑ π‘šπ‘€2. In addition, we prove that there exists a simultaneously 𝑄1–𝑁- and π‘Š-complete recursively enumerable set which is not 𝑠𝑄-complete.

scholarly journals Monotone and 1–1 sets

1982 β—½  
Vol 33 (1) β—½  
pp. 62-75 β—½  
Author(s):  
D. B. Madan β—½  
R. W. Robinson
Keyword(s):  
Recursive Function β—½  
Infinite Subset β—½  
Recursively Enumerable Set β—½  
Recursively Enumerable

AbstractAn infinite subset of Ο‰ is monotone (1–1) if every recursive function is eventually monotone on it (eventually constant on it or eventually 1–1 on it). A recursively enumerable set is co-monotone (co-1–1) just if its complement is monotone (1–1). It is shown that no implications hold among the properties of being cohesive, monotone, or 1–1, though each implies r-cohesiveness and dense immunity. However it is also shown that co-monotone and co-1–1 are equivalent, that they are properly stronger than the conjunction of r-maximality and dense simplicity, and that they do not imply maximality.

Degrees of formal systems

10.2307/2964013 β—½  
1958 β—½  
Vol 23 (4) β—½  
pp. 389-392 β—½  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Sufficient Conditions β—½  
Formal System β—½  
Independent Interest β—½  
Recursively Enumerable Set β—½  
First Order β—½  
Formal Systems β—½  
Recursively Enumerable β—½  
Recursively Enumerable Sets β—½  
Axiomatizable Theory

In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of GΓΆdel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable.In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A. (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.

scholarly journals Formalization of the MRDP Theorem in the Mizar System

2019 β—½  
Vol 27 (2) β—½  
pp. 209-221
Author(s):  
Karol Pąk
Keyword(s):  
Normal Form β—½  
Linear Equations β—½  
Negative Solution β—½  
Recursively Enumerable Set β—½  
Pell’S Equation β—½  
Recursively Enumerable β—½  
Recursively Enumerable Sets β—½  
Finite Products β—½  
Special Case β—½  
Martin Davis

Summary This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem. In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 βˆ’ (a2 βˆ’ 1)y2 = 1 [8] and its solutions considered as two sequences $\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$ . We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9]. In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form. The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

TERMINAL WEIGHTED L-SYSTEMS

1990 β—½  
Vol 04 (01) β—½  
pp. 95-112 β—½  
Author(s):  
NALINAKSHI NIRMAL β—½  
R. RAMA
Keyword(s):  
Regular Matrix β—½  
Recursively Enumerable Set β—½  
Recursively Enumerable β—½  
The Family β—½  
L Systems

Terminal weights are attached to L-systems by replacing each terminal generated by an OL-system by fa(i) in the ith step of a derivation. The family of terminal weighted OL languages will be equal to the recursively enumerable set. Terminal weights are attached to EOL-regular matrix languages and also to OL array languages. Parquet deformations are generated by TWEOL-RMS.

Ξ£2-collection and the infinite injury priority method

1988 β—½  
Vol 53 (1) β—½  
pp. 212-221 β—½  
Author(s):  
Michael E. Mytilinaios β—½  
Theodore A. Slaman
Keyword(s):  
Turing Degree β—½  
Recursively Enumerable Set β—½  
First Order β—½  
Recursively Enumerable β—½  
Standard Application β—½  
Priority Method β—½  
Order Arithmetic β—½  
Bounded Fragment

AbstractWe show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (Pβˆ’) together with Ξ£2-collection (BΞ£2). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither lown nor highn-1, while true, cannot be established in Pβˆ’ + BΞ£n+1. Consequently, no bounded fragment of first order arithmetic establishes the facts that the highn and lown jump hierarchies are proper on the recursively enumerable degrees.

Countable lattice-ordered groups

1983 β—½  
Vol 94 (1) β—½  
pp. 29-33 β—½  
Author(s):  
A. M. W. Glass
Keyword(s):  
Free Lattice β—½  
Lattice Ordered Group β—½  
Ordered Group β—½  
Recursively Enumerable Set β—½  
Ordered Groups β—½  
Recursively Enumerable β—½  
Lattice Ordered Groups β—½  
Natural Analogues β—½  
Group Words β—½  
Combinatorial Group

A lattice-ordered group is a group and a lattice such that the group operation distributes through the lattice operations (i.e. f(g ∨ h)k = fgk ∨ fhk and dually). Lattice-ordered groups are torsion-free groups and distributive lattices. They further satisfy f ∧ g = (fβˆ’1 ∨ gβˆ’1)βˆ’1 and f ∨ g = (fβˆ’1 ∧ gβˆ’1)βˆ’1. Since the lattice is distributive, each lattice-ordered group word can be written in the form ∨A ∧B ωαβ where A and B are finite and each ωαβ is a group word in {xi: i ∈ I}. Unfortunately, even for free lattice-ordered groups, this form is not unique. We will use the prefix l- for maps between lattice-ordered groups that preserve both the group and lattice operations, and e for the identity element. A presentation (xi;rj(x) = e)i∈I, j∈J is the quotient of the free lattice-ordered group F on {xi: i∈I} by the l-ideal (convex normal sublattice subgroup) generated by its subset {rj(x): j ∈ J}. {xi: i ∈ I} is called a generating set and {ri(x):j∈J} a defining set of relations. If I and J are finite we have a finitely presented lattice-ordered group. If we can effectively enumerate all lattice-ordered group words r1(x), r2(x),… in xi; i∈I}. If I is finite and J (for this enumeration) is a recursively enumerable set, we say that we have a recursively presented lattice-ordered group. Throughout Z denotes the group of integers and ℝ the real line.Our purpose in this paper is to prove the natural analogues of three theorems from combinatorial group theory (5), chapter IV, theorems 4Β·9, 3Β·1 and 3Β·5-in particular, theorem C is a natural analogue of an unpublished theorem of Philip Hall (4).

Maximal and Cohesive vector spaces

10.2307/2272868 β—½  
1977 β—½  
Vol 42 (3) β—½  
pp. 400-418 β—½  
Author(s):  
J. B. Remmel
Keyword(s):  
Vector Space β—½  
Quotient Space β—½  
Effective Procedure β—½  
Vector Spaces β—½  
Recursively Enumerable Set β—½  
Infinite Dimensional β—½  
Recursively Enumerable β—½  
Vector Addition β—½  
Zero Vector β—½  
Recursive Set

Let N denote the natural numbers. If A βŠ† N, we write Δ€ for the complement of A in N. A set A βŠ† N is cohesive if (i) A is infinite and (ii) for any recursively enumerable set W either W ∩ A or ∩ A is finite. A r.e. set M βŠ† N is maximal if is cohesive.A recursively presented vector space (r.p.v.s.) U over a recursive field F consists of a recursive set U βŠ† N and operations of vector addition and scalar multiplication which are partial recursive and under which U becomes a vector space. A r.p.v.s. U has a dependence algorithm if there is a uniform effective procedure which applied to any n-tuple Ξ½0, Ξ½1, …, Ξ½nβˆ’1 of elements of U determines whether or not Ξ½0, Ξ½1 …, Ξ½nβˆ’1 are linearly dependent. Throughout this paper we assume that if U is a r.p.v.s. over a recursive field F then U is infinite dimensional and U = N. If W βŠ† U, then we say W is recursive (r.e., etc.) iff W is a recursive (r.e., etc.) subset of N. If S βŠ† U, we write (S)* for the subspace generated by S. If V1 and V2 are subspaces of U such that V1 ∩ V2 ={} (where is the zero vector of U), then we write V1 βŠ• V2 for (V1 βˆͺ V2)*. If V1 βŠ† V2βŠ†U are subspaces, we write V2/V1 for the quotient space.

Sign in / Sign up

Export Citation Format

Share Document