On the T-degrees of partial functions

1985 ◽  
Vol 50 (3) ◽  
pp. 580-588 ◽  
Author(s):  
Paolo Casalegno
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Order Theory ◽  
Partial Functions ◽  
First Order ◽  
First Order Theory ◽  
Degrees Of Unsolvability

AbstractLet 〈, ≤ 〉 be the usual structure of the degrees of unsolvability and 〈, ≤ 〉 the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of 〈, ≤ 〉: as a corollary, the first order theory of 〈, ≤ 〉 is recursively isomorphic to that of 〈, ≤ 〉. We also show that 〈, ≤ 〉 and 〈, ≤ 〉 are not elementarily equivalent.

The model-theoretic significance of complemented existential formulas

1981 ◽  
Vol 46 (4) ◽  
pp. 843-850 ◽  
Author(s):  
Volker Weispfenning
Keyword(s):  
Distributive Lattice ◽  
Model Theory ◽  
Order Theory ◽  
Equivalence Classes ◽  
First Order ◽  
First Order Theory

Let T be an inductive, first-order theory in a language L, let E(L) denote the set of existential L-formulas, and let E(T) denote the distributive lattice of equivalence-classes φT of formulas φ ∈ E(L) with respect to equivalence in T. We consider three types of ‘complements’ in E(T): Let φT, ψT ∈ E(T) and suppose φT ∏ ψT = 0. Then ψT is a complement of φT, if φT ∐ ψT = 1; ψT is a pseudo-complement of φT, if for all μT ∈ E(T), (φT ∐ ψT) = 0 implies μT ≤ ψT; ψT is a “weak complement of φT, if for all μT ∈ E(T), (φT ⋰ ψT) ∐ μT = 0 implies μT = 0. The following facts are obvious: A complement of φT is also a pseudo-complement of φT and a pseudo-complement of φT is also a weak complement of φT. Any φT has at most one pseudo-complement; it is denoted by φT*. The relations ‘ψT is the complement of φT’ and ‘ψT is a weak complement of φT’ are symmetrical. We call φT (weakly, pseudo-) complemented if φT has a (weak, pseudo-) complement, and we call E(T) (weakly, pseudo-) complemented if every φT is (weakly, pseudo-) complemented.The object of this note is to characterize (weakly, pseudo-) complemented existential formulas in model-theoretic terms, and conversely to characterize some classical notions of Robinson style model theory in terms of these formulas. The following theorems illustrate the second approach.

Noninitial segments of the α-degrees

1973 ◽  
Vol 38 (3) ◽  
pp. 368-388 ◽  
Author(s):  
John M. Macintyre
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Recursion Theory ◽  
Partial Ordering ◽  
First Order ◽  
Order Language ◽  
Suitable Notion ◽  
The Universe ◽  
Degrees Of Unsolvability

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.

Computational Fortmalism: Abstract Combinatory View-Point and Related First Order Logical Framework

1981 ◽  
Vol 4 (1) ◽  
pp. 3-18
Author(s):  
Vincenzo Manca
Keyword(s):  
Algebraic Approach ◽  
Order Theory ◽  
Logical Framework ◽  
View Point ◽  
Partial Functions ◽  
First Order ◽  
First Order Theory ◽  
Natural Notion

A natural notion of computational formalism is here characterized by elaborating on the algebraic approach with Uniformly Reflexive Structures. Namely by dealing with the undefined within a first-order theory admitting partial functions. Here, quite generally, we obtain several limitative theorems about the computational formalisms.

First-order-theory-based thermoelastic stability of functionally graded material circular plates

AIAA Journal ◽  
2002 ◽  
Vol 40 ◽  
pp. 1444-1450 ◽  
Author(s):  
M. Najafizadeh ◽  
M. R. Eslami
Keyword(s):  
Functionally Graded Material ◽  
Order Theory ◽  
Functionally Graded ◽  
Circular Plates ◽  
First Order ◽  
First Order Theory ◽  
Graded Material

scholarly journals On equations and first-order theory of one-relator monoids

2021 ◽  
pp. 104745
Author(s):  
Albert Garreta ◽  
Robert D. Gray
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory

A reciprocal theorem for a first order theory of electrostriction

1963 ◽  
Vol 14 (2) ◽  
pp. 148-155 ◽  
Author(s):  
Robin J. Knops
Keyword(s):  
Order Theory ◽  
Reciprocal Theorem ◽  
First Order ◽  
First Order Theory
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