scholarly journals On Some Properties of Positive and Strongly Positive Arithmetical Sets

2018 ◽  
pp. 52-55
Author(s):  
Seda Manukian
Keyword(s):  
Transitive Closure ◽  
Proved Theorem ◽  
Recursively Enumerable Set ◽  
One Dimensional ◽  
Logical Description ◽  
Recursively Enumerable ◽  
Logical Operations ◽  
Arithmetical Formula

The notions of positive and strongly positive arithmetical sets are defined as in [1]-[4] (see, for example, [2], p. 33). It is proved (Theorem 1) that any arithmetical set is positive if and only if it can be defined by an arithmetical formula containing only logical operations ∃, &,∨ and the elementary subformulas having the forms 𝑥𝑥=0 or 𝑥𝑥=𝑦𝑦+1, where 𝑥𝑥and 𝑦𝑦 are variables.Corollary:the logical description of the class of positive sets is obtained from the logical description of the class of strongly positive sets replacing the list of operations &,∨ by the list ∃, &,∨. It is proved (Theorem 2) that for any one-dimensional recursively enumerable set 𝑀𝑀 there exists 6- dimensional strongly positive set 𝐻𝐻 such that 𝑥𝑥 ∈𝑀𝑀 holds if and only if (1, 2𝑥𝑥, 0, 0, 1, 0)∈𝐻𝐻+, where 𝐻𝐻+ is the transitive closure of 𝐻𝐻.

scholarly journals Monotone and 1–1 sets

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

AbstractAn infinite subset of ω 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.

Dynamic Programming and Hybrid Computation

SIMULATION ◽  
1964 ◽  
Vol 3 (5) ◽  
pp. 34-42
Author(s):  
R. Vichnevetsky ◽  
J.P. Waha
Keyword(s):  
Dynamic Programming ◽  
Control Systems ◽  
Operations Research ◽  
Applied Mathematics ◽  
Systems Of Equations ◽  
One Dimensional ◽  
Logical Operations ◽  
Engineering Control ◽  
Hybrid Computation ◽  
Ideal Tool

Dynamic programming finds its use in operations research, chem ical engineering, control systems, and other fields of applied mathematics; and is primarily concerned with the optimisation of multistage decision processes. Dynamic programming will split up the problem into a succession of N one-dimensional optimisation problems. The optimising action takes place at each discrete stage of the process. The technique is based on the so-called principle, of opti mality, as expressed by Bellmann. The paper describes the various techniques involved. A hybrid analogue/digital computer will serve as an ideal tool, having the ability to integrate differential systems of equations, and also to perform the logical operations required in connec tion with the optimisation of the return function.

Degrees of formal systems

1958 ◽  
Vol 23 (4) ◽  
pp. 389-392 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Sufficient Conditions ◽  
Formal System ◽  
Independent Interest ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
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 Gödel 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 ◽  
Vol 27 (2) ◽  
pp. 209-221
Author(s):  
Karol Pąk
Keyword(s):  
Normal Form ◽  
Linear Equations ◽  
Negative Solution ◽  
Recursively Enumerable Set ◽  
Pell’S Equation ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Finite Products ◽  
Special Case ◽  
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 − (a2 − 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 ◽  
Vol 04 (01) ◽  
pp. 95-112 ◽  
Author(s):  
NALINAKSHI NIRMAL ◽  
R. RAMA
Keyword(s):  
Regular Matrix ◽  
Recursively Enumerable Set ◽  
Recursively Enumerable ◽  
The Family ◽  
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.

Characterization of recursively enumerable sets

1972 ◽  
Vol 37 (3) ◽  
pp. 507-511 ◽  
Author(s):  
Jesse B. Wright
Keyword(s):  
Transitive Closure ◽  
Successor Function ◽  
Function Composition ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Abstract Setting ◽  
Recursively Enumerable Sets ◽  
Bracket Operation

AbstractLet N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N2 operations of transposition, composition, and bracketing are defined as follows: P∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃n ∈ M(Pn Q Rn).Theorem. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P∪, A∪M, R].The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.

Σ2-collection and the infinite injury priority method

1988 ◽  
Vol 53 (1) ◽  
pp. 212-221 ◽  
Author(s):  
Michael E. Mytilinaios ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degree ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Recursively Enumerable ◽  
Standard Application ◽  
Priority Method ◽  
Order Arithmetic ◽  
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 (P−) together with Σ2-collection (BΣ2). 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 P− + BΣn+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.

On Some Properties of 𝑟-Maximal Sets and 𝑄1–𝑁-Reducibility

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

Abstract It is shown that if 𝑀1, 𝑀2 are 𝑟-maximal sets and 𝑀1 ≡ 𝑄1–𝑁𝑀2, then 𝑀1 ≡ 𝑚𝑀2. In addition, we prove that there exists a simultaneously 𝑄1–𝑁- and 𝑊-complete recursively enumerable set which is not 𝑠𝑄-complete.

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