A selection theorem

1983 ◽  
Vol 48 (3) ◽  
pp. 585-594
Author(s):  
Lefteris Miltiades Kirousis
Keyword(s):  
Partial Function ◽  
Existential Quantifier ◽  
Selection Theorem ◽  
Unbounded Subset ◽  
Image Position ◽  
Existential Quantification

In [1978] Harrington and MacQueen proved that if B is an (A, E)-semirecursive subset of A, such that the functions in BA can be coded as elements of A in an (A, E)-recursive way, then ENV(A, E) is closed under the existential quantifier ∃T ∈ B.Later Moschovakis showed that if ENV(Vκ, ∈, E) is closed under the quantifier ∃t ∈ λ, where λ is the p-cofinality of κ, thenthe p-cofinality of κ is the least ordinal λ for which there exists a (κ, <, E)-recursive partial function ƒ into κ, such that ƒ∣λ is total from λ onto an unbounded subset of κ.In this paper we prove that for any infinite ordinal κ if p-card(κ) = κ, then ENV(κ, <, E) is closed under ∃t ∈ μ, for μ < p-cf(κ); p-cf(κ) is the “boldface” analog of p-cf((κ) and p-card(κ) is defined similarly.From this follows that for any infinite ordinal κ the following two statements are equivalent.(i) ENV(κ, <, E) is closed under bounded existential quantification.(ii) ENV(κ, <, E) = ENV(κ, <, E#) or p-cf(κ) = κ.We also show that we cannot omit any of the hypotheses in the above theorem.We follow mainly the notation of Kechris and Moschovakis [1977].

Relativised quantification: Some canonical varieties of sequence-set algebras

1998 ◽  
Vol 63 (1) ◽  
pp. 163-184 ◽  
Author(s):  
Hajnal Andréka ◽  
Robert Goldblatt ◽  
István Németi
Keyword(s):  
Existential Quantifier ◽  
Cylindric Algebras ◽  
Order Model ◽  
Assigned Value ◽  
First Order ◽  
Cartesian Space ◽  
A Value ◽  
Quantificational Logic ◽  
Image Position ◽  
Existential Quantification

This paper explores algebraic aspects of two modifications of the usual account of first-order quantifiers.Standard first-order quantificational logic is modelled algebraically by cylindric algebras. Prime examples of these are algebras whose members are sets of sequences: given a first-order model U for a language that is based on the set {υκ: κ < α} of variables, each formula φ is represented by the setof all those α-length sequences x = 〈xκ: κ < α〉 that satisfy φ in U. Such a sequence provides a value-assignment to the variables (υκ is assigned value xκ), but it may also be viewed geometrically as a point in the α-dimensional Cartesian spaceαU of all α-length sequences whose terms come from the underlying set U of U. Then existential quantification is represented by the operation of cylindrification. To explain this, define a binary relation Tκ on sequences by putting xTκy if and only if x and y differ at most at their κth coordinate, i.e.,Then for any set X ⊆ αU, the setis the “cylinder” generated by translation of X parallel to the κth coordinate axis in αU. Given the standard semantics for the existential quantifier ∃υκ asit is evident that

On closure under direct product

1958 ◽  
Vol 23 (2) ◽  
pp. 149-154 ◽  
Author(s):  
C. C. Chang ◽  
Anne C. Morel
Keyword(s):  
Normal Form ◽  
Direct Product ◽  
Disjunctive Normal Form ◽  
Negative Answer ◽  
Sufficient Condition ◽  
Natural Question ◽  
Existential Quantifier ◽  
Relational Systems ◽  
Image Position

In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question.We shall discuss relational systems of the formwhere A and R are non-empty sets; each element of R is an ordered triple 〈a, b, c〉, with a, b, c ∈ A.1 If the triple 〈a, b, c〉 belongs to the relation R, we write R(a, b, c); if 〈a, b, c〉 ∉ R, we write (a, b, c). If x0, x1 and x2 are variables, then R(x0, x1, x2) and x0 = x1 are predicates. The expressions (x0, x1, x2) and x0 ≠ x1 will be referred to as negations of predicates.We speak of α1, …, αn as terms of the disjunction α1 ∨ … ∨ αn and as factors of the conjunction α1 ∧ … ∧ αn. A sentence (open, closed or neither) of the formwhere each Qi (if there be any) is either the universal or the existential quantifier and each αi, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form.

scholarly journals DOMINIONS AND PRIMITIVE POSITIVE FUNCTIONS

2018 ◽  
Vol 83 (1) ◽  
pp. 40-54 ◽  
Author(s):  
MIGUEL CAMPERCHOLI
Keyword(s):  
Partial Function ◽  
Finite Algebras ◽  
Positive Formula ◽  
Primitive Positive Formula ◽  
Near Unanimity Term ◽  
Finite Set ◽  
Image Position ◽  
Positive Functions ◽  
Near Unanimity

AbstractLetA≤Bbe structures, and${\cal K}$a class of structures. An elementb∈BisdominatedbyArelative to${\cal K}$if for all${\bf{C}} \in {\cal K}$and all homomorphismsg,g':B → Csuch thatgandg'agree onA, we havegb=g'b. Our main theorem states that if${\cal K}$is closed under ultraproducts, thenAdominatesbrelative to${\cal K}$if and only if there is a partial functionFdefinable by a primitive positive formula in${\cal K}$such thatFB(a1,…,an) =bfor somea1,…,an∈A. Applying this result we show that a quasivariety of algebras${\cal Q}$with ann-ary near-unanimity term has surjective epimorphisms if and only if$\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$has surjective epimorphisms. It follows that if${\cal F}$is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by${\cal F}$has surjective epimorphisms.

The equation A(t, u(t))′ + B(t, u(t)) = 0

1976 ◽  
Vol 79 (3) ◽  
pp. 545-561 ◽  
Author(s):  
Bruce Calvert
Keyword(s):  
Differential Operator ◽  
Banach Spaces ◽  
Lower Order ◽  
Kdv Equation ◽  
Initial Value ◽  
Order Elliptic Operator ◽  
The Kdv Equation ◽  
Unbounded Subset ◽  
Image Position ◽  
Bbm Equation

The initial value problem for the equationhas been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equationwhere At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .

On the reduction of the decision problem. Third paper. Pepis prefix, a single binary predicate

1950 ◽  
Vol 15 (3) ◽  
pp. 161-173 ◽  
Author(s):  
László Kalmár ◽  
János Surányi
Keyword(s):  
Decision Problem ◽  
Predicate Calculus ◽  
Existential Quantifier ◽  
First Order ◽  
Binary Predicate ◽  
Image Position ◽  
Analogous Theorem ◽  
Order Predicate Calculus ◽  
Predicate Variable

It has been proved by Pepis that any formula of the first-order predicate calculus is equivalent (in respect of being satisfiable) to another with a prefix of the formcontaining a single existential quantifier. In this paper, we shall improve this theorem in the like manner as the Ackermann and the Gödel reduction theorems have been improved in the preceding papers of the same main title. More explicitly, we shall prove theTheorem 1. To any given first-order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.An analogous theorem, but producing a prefix of the formhas been proved in the meantime by Surányi; some modifications in the proof, suggested by Kalmár, led to the above form.

Selection in abstract recursion theory

1976 ◽  
Vol 41 (1) ◽  
pp. 153-158 ◽  
Author(s):  
L. A. Harrington ◽  
D. B. Macqueen
Keyword(s):  
General Result ◽  
Recursion Theory ◽  
Normal Type ◽  
Arbitrary Integer ◽  
Selection Theorem ◽  
Arbitrary Type ◽  
Direct Generalization ◽  
Existential Quantification

In [1] Gandy established the following selection theorem for recursion in type-2 objects.Theorem. Let F be a normal type-2 object. Then it is possible to select (uniformly and effectively in F) an integer from each nonempty set of integers semirecursive in F.Notice that this really asserts that the predicates semirecursive in F are closed under existential quantification over type-0. Moschovakis [6] has essentially proven this theorem for F of arbitrary type.In [2] Grilliot stated a powerful generalization of Gandy's result, namely:Grilliot's Selection Theorem. Let F be a normal type-(n + 2) object (n an arbitrary integer). Then it is possible to select (uniformly and effectively in F) a nonempty recursive in F subset of each nonempty semirecursive in F set oftype-(n − 1) objects.Notice again that this actually says that predicates semirecursive in F are closed under quantification over type-(n − 1) objects.Despite the similarity of these two results, Gandy and Grilliot proposed rather different methods of proof. Furthermore, the proof that Grilliot presented in [2] contains an error which cannot easily be corrected. (We will comment on the nature of this error at the end of §1.) Fortunately, however, Grilliot's theorem is valid. We will present a proof of Grilliot's selection theorem which is a direct generalization of the proof of Gandy's theorem given in [6]. In fact, we will prove a general result (the theorem stated in §2) which subsumes both Gandy's and Grilliot's results.

Solution to a Problem of Spector

1971 ◽  
Vol 23 (2) ◽  
pp. 247-256 ◽  
Author(s):  
A. H. Lachlan
Keyword(s):  
Point Of View ◽  
Partial Function ◽  
Natural Numbers ◽  
Image Position

In [6, p. 586] Spector asked whether given a number e there exists a unary partial function from the natural numbers into {0, 1} with coinfinite domain such that for any function ƒ into {0, 1} extending it is the case thatWe answer this question affirmatively in Corollary 1 below and show that can be made partial recursive (p.r.) with recursive domain. The reader who is familiar with Spector's paper [6] will find the new trick that is required in the first paragraph of the proof of Lemma 2 below.From one point of view, this is a theorem about trees which branch twice at every node. We shall formulate a generalization which applies to trees which branch n times at every node.

Kleene's Amazing Second Recursion Theorem

2010 ◽  
Vol 16 (2) ◽  
pp. 189-239 ◽  
Author(s):  
Yiannis N. Moschovakis
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Short Note ◽  
Partial Function ◽  
Modern Notation ◽  
Recursion Theorem ◽  
Image Position

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong (uniform) form, it reads as follows:Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with.(a) Every n-ary recursive partial function with values in V is for some e.(b) For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let .We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler formin this case (and the proof sets ž = S(e, e)).

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

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