On size vs. efficiency for programs admitting speed-ups

1971 ◽  
Vol 36 (1) ◽  
pp. 21-27 ◽  
Author(s):  
John Helm ◽  
Paul Young
Keyword(s):  
Computational Complexity ◽  
Recursive Function ◽  
Recursion Theory ◽  
Typical Application ◽  
Recursion Theorem ◽  
Efficient Program ◽  
Do So

Since the publication in 1967 of the two papers [1] and [2] by Manuel Blum, the techniques and results of “pure” recursion theory, particularly the recursion theorem and priority methods, have come to play an increasingly important role in studies of computational complexity. This paper gives a typical application of the recursion theorem with a fairly intricate diagonalization to answer a question raised by Blum in [3]. Roughly, we prove the existence of functions which have the property that if we are given any program for computing the function and want to pass to a program which computes the function much more efficiently, then we can only do so at the expense of obtaining a much larger program: the function which describes the necessary increase in the size of the more efficient program must grow more rapidly than any recursive function.

J. C. Shepherdson. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 285–308. - J. C. Shepherdson. Computational complexity of real functions. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 309–315. - A. J. Kfoury. The pebble game and logics of programs. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 317–329. - R. Statman. Equality between functionals revisited. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 331–338. - Robert E. Byerly. Mathematical aspects of recursive function theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 339–352.

1990 ◽  
Vol 55 (2) ◽  
pp. 876-878
Author(s):  
J. V. Tucker
Keyword(s):  
New York ◽  
Computational Complexity ◽  
Recursive Function ◽  
Function Theory ◽  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Real Functions ◽  
Pebble Game ◽  
Logics Of Programs

On generalized computational complexity

1977 ◽  
Vol 42 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Barry E. Jacobs
Keyword(s):  
Computational Complexity ◽  
Set Theory ◽  
Initial Segment ◽  
Ordinal Number ◽  
Recursion Theory ◽  
Universal Function ◽  
Closure Properties ◽  
Numbering Scheme ◽  
Ordinal Numbers ◽  
Recursion Theorem

If one regards an ordinal number as a generalization of a counting number, then it is natural to begin thinking in terms of computations on sets of ordinal numbers. This is precisely what Takeuti [22] had in mind when he initiated the study of recursive functions on ordinals. Kreisel and Sacks [9] too developed an ordinal recursion theory, called metarecursion theory, which specialized to the initial segment of the ordinals bounded by(the first nonconstructive ordinal).The notion of admissibility was introduced by Kripke [11] and Platek [14] and served to generalize metarecursion theory. Kripke called ordinal α admissible if it satisfied certain closure properties of infinitary computations. It was shown that admissibility could be equivalently formulated in terms of the replacement schema of ZF set theory and that α =is an admissible ordinal. The study of a recursion theory on an initial segment of the ordinals bounded by some arbitrary admissible α became known as α-recursion theory.Kripke [10] employed a Gödel numbering scheme to perform an arithmetiza-tion of α -recursion theory and created an analogue to Kleene'sT-predicate (cf. [8]) of ordinary recursion theory (o.r.t.). TheT-predicate then served as the basis for showing that analogues of the major results of unrelativized o.r.t. held in α-recursion theory; namely, the α-Enumeration Theorem,T Theorem, α-Recursion Theorem, and α-Universal Function Theorem.

scholarly journals MRILDU: An Improvement to ILUT Based on Incomplete LDU Factorization and Dropping in Multiple Rows

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jian-Ping Wu ◽  
Huai-Fa Ma
Keyword(s):  
Computational Complexity ◽  
Linear Systems ◽  
Numerical Weather Prediction ◽  
Sparse Matrix ◽  
Weather Prediction ◽  
Concrete Sample ◽  
Storage Requirement ◽  
Sparse Linear Systems ◽  
Numerical Weather ◽  
Do So

We provide an improvement MRILDU to ILUT for general sparse linear systems in the paper. The improvement is based on the consideration that relatively large elements should be kept down as much as possible. To do so, two schemes are used. Firstly, incomplete LDU factorization is used instead of incomplete LU. Besides, multiple rows are computed at a time, and then dropping is applied to these rows to extract the relatively large elements in magnitude. Incomplete LDU is not only fairer when there are large differences between the elements of factorsLandU, but also more natural for the latter dropping in multiple rows. And the dropping in multiple rows is more profitable, for there may be large differences between elements in different rows in each factor. The provided MRILDU is comparable to ILUT in storage requirement and computational complexity. And the experiments for spare linear systems from UF Sparse Matrix Collection, inertial constrained fusion simulation, numerical weather prediction, and concrete sample simulation show that it is more effective than ILUT in most cases and is not as sensitive as ILUT to the parameterp, the maximum number of nonzeros allowed in each row of a factor.

Turing reducibility and Turing degrees

2018 ◽  
Author(s):  
Harold Hodes
Keyword(s):  
Computational Complexity ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Natural Numbers ◽  
Mathematical Objects ◽  
Classical Setting ◽  
Turing Reducibility

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.

Minimal α-recursion theoretic degrees

1973 ◽  
Vol 38 (1) ◽  
pp. 18-28 ◽  
Author(s):  
John M. MacIntyre
Keyword(s):  
Recursive Function ◽  
Regular Cardinal ◽  
Recursion Theory ◽  
Minimal Degree ◽  
The Other ◽  
Ordinal Numbers ◽  
Definition Of

This paper investigates the problem of extending the recursion theoretic construction of a minimal degree to the Kripke [2]-Platek [5] recursion theory on the ordinals less than an admissible ordinal α, a theory derived from the Takeuti [11] notion of a recursive function on the ordinal numbers. As noted in Sacks [7] when one generalizes the recursion theoretic definition of relative recursiveness to α-recursion theory for α > ω the two usual definitions give rise to two different notions of reducibility. We will show that whenever α is either a countable admissible or a regular cardinal of the constructible universe there is a subset of α whose degree is minimal for both notions of reducibility. The result is an excellent example of a theorem of ordinary recursion theory obtainable via two different constructions, one of which generalizes, the other of which does not. The construction which cannot be lifted to α-recursion theory is that of Spector [10]. We sketch the reasons for this in §3.

scholarly journals Alternating Turing machines and the analytical hierarchy

10.29007/t77g ◽  
2018 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Recursion Theory ◽  
Turing Machines ◽  
Arithmetical Hierarchy ◽  
Definition Of ◽  
Alternating Turing Machines ◽  
Insight Into ◽  
Computably Enumerable

We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy..The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

Community Interests in International Adjudication

2018 ◽  
Author(s):  
Eyal Benvenisti
Keyword(s):  
Crucial Role ◽  
Recursive Function ◽  
Dispute Settlement ◽  
Ad Hoc ◽  
Focal Points ◽  
International Courts ◽  
Legal Obligations ◽  
International Adjudication ◽  
Do So

The chapter examines the extent to which international courts and tribunals can take community interests into consideration and develop community obligations. It explores the significance of this distinction between the ad hoc dispute-settlement tribunals and standing courts with jurisdiction to adjudicate multiple cases, and argues that the recursive function transforms international courts into global lawmakers that weave together a system of norms with secondary rules of recognition. International tribunals serve a crucial role of coordinating the behavior of state and nonstate actors by creating focal points that define the parties’ legal obligations and stabilize expectations. Moreover, the chapter argues that because of this function international courts are uniquely situated to take community interests into account, and they often, if not always, do so. This implies that if properly insulated from pressures and prejudices, international adjudicators are institutionally inclined to promote community obligations.

Recursion-theoretic hierarchies

2018 ◽  
Author(s):  
Harold Hodes
Keyword(s):  
Computational Complexity ◽  
Model Theory ◽  
Recursion Theory ◽  
Bottom Up ◽  
Basic Class

In mathematics, a hierarchy is a ‘bottom up’ system classifying entities of some particular sort, a system defined inductively, starting with a ‘basic’ class of such entities, with further (‘higher’) classes of such entities defined in terms of previously defined (‘lower’) classes. Such a classification reflects complexity in some respect, one entity being less complex than another if it appears ‘earlier’ (‘lower’) then that other. Many of the hierarchies studied by logicians construe complexity as complexity of definition, placing such hierarchies within the purview of model theory; but even such notions of complexity are closely tied to species of computational complexity, placing them also in the purview of recursion theory.

scholarly journals Fixed points and self-reference

1984 ◽  
Vol 7 (2) ◽  
pp. 283-289 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Recursion Theory ◽  
Diagonal Argument ◽  
Recursion Theorem ◽  
Self Reference

It is shown how Gödel's famous diagonal argument and a generalization of the recursion theorem are derivable from a common construation. The abstract fixed point theorem of this article is independent of both metamathematics and recursion theory and is perfectly comprehensible to the non-specialist.

Type two partial degrees

1978 ◽  
Vol 43 (4) ◽  
pp. 623-629
Author(s):  
Ko-Wei Lih
Keyword(s):  
Recursive Function ◽  
Natural Extension ◽  
Degree Theory ◽  
Recursion Theory ◽  
Natural Numbers ◽  
Partial Functions ◽  
Reduction Procedure ◽  
Turing Reducibility

Roughly speaking partial degrees are equivalence classes of partial objects under a certain notion of relative recursiveness. To make this notion precise we have to state explicitly (1) what these partial objects are; (2) how to define a suitable reduction procedure. For example, when the type of these objects is restricted to one, we may include all possible partial functions from natural numbers to natural numbers as basic objects and the reduction procedure could be enumeration, weak Turing, or Turing reducibility as expounded in Sasso [4]. As we climb up the ladder of types, we see that the usual definitions of relative recursiveness, equivalent in the context of type-1 total objects and functions, may be extended to partial objects and functions in quite different ways. First such generalization was initiated by Kleene [2]. He considers partial functions with total objects as arguments. However his theory suffers the lack of transitivity, i.e. we may not obtain a recursive function when we substitute a recursive function into a recursive function. Although Kleene's theory provides a nice background for the study of total higher type objects, it would be unsatisfactory when partial higher type objects are being investigated. In this paper we choose the hierarchy of hereditarily consistent objects over ω as our universe of discourse so that Sasso's objects are exactly those at the type-1 level. Following Kleene's fashion we define relative recursiveness via schemes and indices. Yet in our theory, substitution will preserve recursiveness, which makes a degree theory of partial higher type objects possible. The final result will be a natural extension of Sasso's Turing reducibility. Due to the abstract nature of these objects we do not know much about their behaviour except at the very low types. Here we pay our attention mainly to type-2 objects. In §2 we formulate basic notions and give an outline of our recursion theory of partial higher type objects. In §3 we introduce the definitions of singular degrees and ω-consistent degrees which are two important classes of type-2 objects that we are most interested in.

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