A jump operator for subrecursion theories

1999 ◽  
Vol 64 (2) ◽  
pp. 460-468
Author(s):  
A.J. Heaton
Keyword(s):  
Elementary Function ◽  
Recursion Theory ◽  
Jump Operator ◽  
Fast Growing ◽  
Recursive Functions ◽  
Ordinal Notations ◽  
Independence Results ◽  
Image Position ◽  
Hyperarithmetic Hierarchy

In classical recursion theory, the jump operator is an important concept fundamental in the study of degrees. In particular, it provides a way of defining the hyperarithmetic hierarchy by iterating over Kleene's O. In subrecursion theories, hierarchies (variants of the fast growing hierarchy) are also important in underlying the central concepts, e.g. in classifying provably recursive functions and associated independence results for given theories (see, e.g. [BW87], [HW96], [R84] and [Z77]). A major difference with the hyperarithmetic hierarchy is in the way each level of a subrecursive hierarchy is crucially dependent upon the system of ordinal notations used (see [F62]). This has been perhaps the major stumbling block in finding a classification of all general recursive functions using such hierarchies.Here we present a natural subrecursive analogue of the jump operator and prove that the hierarchy based on the ”subrecursive jump” is elementarily equivalent to the fast growing hierarchy.The paper is organised as follows. First the preliminary definitions are given together with a statement of the main theorem and a brief outline of its proof. The proof of the theorem is then given, with the more technical parts separated out as facts which are proven afterwards.We let {e}g(x) denote computation of the e-th partial recursion with oracle g, on input x. Furthermore [e] denotes the e-th elementary recursive function, defined so thatSimilarly, for a given oracle g the e-th relativized elementary function is denoted by [e]g.

On a question of G. E. Sacks

1970 ◽  
Vol 35 (1) ◽  
pp. 46-50
Author(s):  
Kempachiro Ohashi
Keyword(s):  
Recursion Theory ◽  
Finite System ◽  
Fundamental Question ◽  
System Of Equations ◽  
Jump Operator ◽  
Partial Functions ◽  
Image Position

In [1], Sacks points out that there is one fundamental question: which true statements of ordinary recursion theory remain true when appropriately extended to metarecursion theory?A particular interest is taken in the question [1]:Q6. How does one define the jump operator for metarecursion theory? (A satisfactory definition should have the property that if A is metarecursive in B, then the jump of A is metarecursive in the jump of B.)In [2], Kreisel and Sacks give some definitions of predicates and functions analogous to those of Kleene as follows:The T-predicate of [2] is analogous to that of Kleene [3]. Its definition iswhere e is the Gödel number of a finite system of equations E and t(e, s) is a special metarecursive function which indexes “deductions” from E.U(e, s) is a metarecursive function such that if t(e, s) = 〈e, M, N, x, y〉, then U(e,s) = y.Two partial functions {e}s and {e} areThen e is intrinsically consistent if for all x, s1 and s2, if t(e, S1) = 〈e, M1N1, x, y1〉, t(e, s2) = 〈e, M2, N2, x, y2〉 and (M1 ∪ M2) ∩ (N1 ∪ N2) = ∅, then y1 = y2.

How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: a case study

1998 ◽  
Vol 63 (4) ◽  
pp. 1348-1370 ◽  
Author(s):  
Andreas Weiermann
Keyword(s):  
Recursive Function ◽  
Term Rewriting ◽  
Careful Analysis ◽  
Recursion Theory ◽  
Cut Elimination ◽  
Complexity Bounds ◽  
Recursive Functions ◽  
Collapsing Function ◽  
Primitive Recursive

AbstractInspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε0-recursive function []0: T → ω so that a reduces to b implies [a]0 > [b]0. The construction of [ ]0 is based on a careful analysis of the Howard-Schütte treatment of Gödel's T and utilizes the collapsing function ψ: ε0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [ ]0 is also crucially based on ideas developed in the 1995 paper “A proof of strongly uniform termination for Gödel's T by the method of local predicativity” by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper.Indeed, for given n let Tn be the subsystem of T in which the recursors have type level less than or equal to n + 2. (By definition, case distinction functionals for every type are also contained in Tn.) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the Tn-derivation lengths in terms of ω+2-descent recursive functions. The derivation lengths of type one functionals from Tn (hence also their computational complexities) are classified optimally in terms of <ωn+2 -descent recursive functions.In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T0 is primitive recursive, thus any type one functional a in T0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T1 in terms of multiple recursion.As proof-theoretic corollaries we reobtain the classification of the IΣn+1-provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO(ε0) ⊢ Π20 − Refl(PA) and PRA + PRWO(ωn+2) ⊢ Π20 − Refl(IΣn+1), hence PRA + PRWO(ε0) ⊢ Con(PA) and PRA + PRWO(ωn+2) ⊢ Con(IΣn+1).For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity.

Classifying Predicates and Languages

1997 ◽  
Vol 08 (01) ◽  
pp. 15-41 ◽  
Author(s):  
Carl H. Smith ◽  
Rolf Wiehagen ◽  
Thomas Zeugmann
Keyword(s):  
Learning Theory ◽  
Classification Problems ◽  
Attention In Learning ◽  
Recursive Functions ◽  
Standard Classification ◽  
Resource Bound ◽  
Multi Classification

The present paper studies a particular collection of classification problems, i.e., the classification of recursive predicates and languages, for arriving at a deeper understanding of what classification really is. In particular, the classification of predicates and languages is compared with the classification of arbitrary recursive functions and with their learnability. The investigation undertaken is refined by introducing classification within a resource bound resulting in a new hierarchy. Furthermore, a formalization of multi-classification is presented and completely characterized in terms of standard classification. Additionally, consistent classification is introduced and compared with both resource bounded classification and standard classification. Finally, the classification of families of languages that have attracted attention in learning theory is studied, too.

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &lt; γ &lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

Classification of singular solutions of porous media equations with absorption

2005 ◽  
Vol 135 (3) ◽  
pp. 563-584 ◽  
Author(s):  
Xinfu Chen ◽  
Yuanwei Qi ◽  
Mingxin Wang
Keyword(s):  
Porous Media ◽  
Singular Solution ◽  
Singular Solutions ◽  
Constant Independent ◽  
Porous Media Equation ◽  
Porous Media Equations ◽  
Self Similar ◽  
Image Position ◽  
Very Singular Solution

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in Rn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

scholarly journals The algebraic structure of relativistic wave equations

1978 ◽  
Vol 20 (4) ◽  
pp. 446-486 ◽  
Author(s):  
A. Cant ◽  
C. A. Hurst
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Mass Spectra ◽  
Wave Equations ◽  
Relativistic Wave Equations ◽  
Relativistic Wave ◽  
Partial Classification ◽  
The Family ◽  
Image Position

The algebraic structure of relativistic wave equations of the formis considered. This leads to the problem of finding all Lie algebrasLwhich contain the Lorentz Lie algebraso(3, 1) and also contain a “four-vector” αμa such anLgives rise to a family of wave equations. The simplest possibility is the Bhabha equations whereL≅so(5). Some authors have claimed that this is theonlyone, but it is shown that there are many other possibilities still in accord with physical requirements. Known facts about representations, along with Dynkin's theory of the embeddings of Lie algebras, are used to obtain a partial classification of wave equations. The discrete transformationsC, P, Tare also discussed, along with reality properties. Finally, a simple example of a family of wave equations based onL=sp(12) is considered in detail. Theso(3, 1) content and mass spectra are given for the low order members of the family, and the problem of causality is briefly discussed.

Recursive functions in basic logic

1956 ◽  
Vol 21 (4) ◽  
pp. 337-346 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Basic Logic ◽  
Recursive Functions ◽  
Numerical Equality ◽  
Partial Recursive Functions ◽  
Image Position ◽  
Primitive Recursive ◽  
Double Arrow

1.1 The system K* of basic logic, as presented in a previous paper, will be shown to be formalizable in an alternative way according to which the rule [E],is replaced by the rule [F],1.2. General recursive functions will be shown to be definable in K* in a way that retains functional notation, so that the equation,will be formalized in K* by the formula,where ‘f’, ‘p1’, … ‘pn’ respectively denote φ, x1, …, xn, and where ‘≐’ plays the role of numerical equality. Partial recursive functions may be handled in a similar way. The rule [E] is not required for dealing with primitive recursive functions by this method.1.3. An operator ‘G’ will be defined such that ‘[Ga ≐ p]’ is a theorem of K* if and only if ‘p’ denotes the Gödel number of ‘a’.1.4. In reformulating K* we assume ‘o0’, ‘o1,’ ‘o2’, …, have been so chosen that we can determine effectively whether or not a given U-expression is the mth member of the above series. The revised rules for K* are then as follows. (Double-arrow forms of these rules are derivable, except in the case of rule [V].)

AN ANALYTICAL METHOD FOR PARALLELIZATION OF RECURSIVE FUNCTIONS

2000 ◽  
Vol 10 (04) ◽  
pp. 359-370 ◽  
Author(s):  
JOONSEON AHN ◽  
TAISOOK HAN
Keyword(s):  
Analytical Method ◽  
Parallel Programs ◽  
Data Parallelism ◽  
Complex Data ◽  
Recursive Functions ◽  
Parallel Skeletons ◽  
Data Flows ◽  
Recursive Data Structures ◽  
Recursive Data

Programming with parallel skeletons is an attractive framework because it encourages programmers to develop efficient and portable parallel programs. However, extracting parallelism from sequential specifications and constructing efficient parallel programs using the skeletons are still difficult tasks. In this paper, we propose an analytical approach to transforming recursive functions on general recursive data structures into compositions of parallel skeletons. Using static slicing, we have defined a classification of subexpressions based on their data-parallelism. Then, skeleton-based parallel programs are generated from the classification. To extend the scope of parallelization, we have adopted more general parallel skeletons which do not require the associativity of argument functions. In this way, our analytical method can parallelize recursive functions with complex data flows.

A Hausdorff measure classification of polar sets for the heat equation

1985 ◽  
Vol 97 (2) ◽  
pp. 325-344 ◽  
Author(s):  
S. J. Taylor ◽  
N. A. Watson
Keyword(s):  
Heat Equation ◽  
Hausdorff Measure ◽  
Image Position

Our main purpose is to give criteria for determining which subsets of Rn+1 are polar relative to the heat equation

The Perfect Tense Ending K(-) in the Spoken Arabic of Ta'izz

1974 ◽  
Vol 37 (2) ◽  
pp. 439-442
Author(s):  
Theodore Prochazka
Keyword(s):  
Western Slope ◽  
General Classification ◽  
Arabic Dialects ◽  
Image Position ◽  
Spoken Arabic

The occurrence of -k- in the conjugation of the verb in the perfect, instead of the expected -t- which marks the 2s., 2 pl., and 1s. in other Arabic dialects, was noted and discussed in two studies by Ettore Rossi. His aim was to offer a general classification of the Yemeni dialects, and he treated the occurrence of the personal suffixes in -k- in terms of their geographical extent. He found these suffixes in the area of the western slope of the Yemen plateau, and the parts stretching southward to Aden. As an illustration he gave for the dialect of Raymah:This type of suffixation, as Rossi noted, falls into line with that of Ethiopic, and the Modern South Arabian languages, such as Mahri and Socotri.

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