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Journal articles on the topic 'Classes of recursive functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Wainer, Stanley S. "Accessible Recursive Functions." Bulletin of Symbolic Logic 5, no. 3 (September 1999): 367–88. http://dx.doi.org/10.2307/421185.

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AbstractThe class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from “within”. On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been “coded” at previous levels. The question is: how can a recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The “accessible” recursive functions thus generated turn out to be those provably recursive in ( –CA)0.

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2

Stephan, Frank, and Thomas Zeugmann. "Learning classes of approximations to non-recursive functions." Theoretical Computer Science 288, no. 2 (October 2002): 309–41. http://dx.doi.org/10.1016/s0304-3975(01)00405-4.

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3

Mazzanti, Stefano. "Plain Bases for Classes of Primitive Recursive Functions." MLQ 48, no. 1 (January 2002): 93–104. http://dx.doi.org/10.1002/1521-3870(200201)48:1<93::aid-malq93>3.0.co;2-8.

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4

SEMIGRODSKIKH, A. P. "On Closed Classes of Primitive Recursive Functions, II." Multiple-Valued Logic 8, no. 2 (January 1, 2002): 183–91. http://dx.doi.org/10.1080/10236620215292.

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5

Kummer, Martin. "A learning-theoretic characterization of classes of recursive functions." Information Processing Letters 54, no. 4 (May 1995): 205–11. http://dx.doi.org/10.1016/0020-0190(95)00036-c.

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6

Volkov, S. A. "Generating some classes of recursive functions by superpositions of simple arithmetic functions." Doklady Mathematics 76, no. 1 (August 2007): 566–67. http://dx.doi.org/10.1134/s1064562407040217.

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7

AUSIELLO, G., and M. PROTASI. "LIMITING POLYNOMIAL APPROXIMATION OF COMPLEXITY CLASSES." International Journal of Foundations of Computer Science 01, no. 02 (June 1990): 111–22. http://dx.doi.org/10.1142/s0129054190000096.

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The concept of limiting approximation, formerly introduced by Gold for recursive functions, is applied to the polynomial level of complexity in order to determine meaningful characterizations of classes of functions and sets which are not (or which are not known to be) polynomially computable. In particular, characterizations of NP, PSPACE and other classes of elementary functions are provided in terms of limiting polynomial approximation. In addition trade-offs between the space required by the approximating functions and the rate of convergence of the approximation are shown.

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8

Zhukov, Vladimir V., and Sergey A. Lozhkin. "Asymptotically best method for synthesis of Boolean recursive circuits." Discrete Mathematics and Applications 30, no. 2 (April 28, 2020): 137–46. http://dx.doi.org/10.1515/dma-2020-0013.

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AbstractModels of multi-output and scalar recursive Boolean circuits of bounded depth in an arbitrary basis are considered. Methods for lower and upper estimates for the Shannon function for the complexity of circuits of these classes are provided. Based on these methods, an asymptotic formula for the Shannon function is put forward. Moreover, in the above classes of recursive circuits, upper estimates for the complexity of implementation of some functions and systems of functions used in applications are obtained.

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9

Calude, Cristian, and Gabriel Istrate. "Determining and stationary sets for some classes of partial recursive functions." Theoretical Computer Science 82, no. 1 (May 1991): 151–55. http://dx.doi.org/10.1016/0304-3975(91)90178-5.

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10

Rout, Ranjeet Kumar, Pabitra Pal Choudhury, and Sudhakar Sahoo. "Classification of Boolean Functions Where Affine Functions Are Uniformly Distributed." Journal of Discrete Mathematics 2013 (October 31, 2013): 1–12. http://dx.doi.org/10.1155/2013/270424.

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The present paper on classification of -variable Boolean functions highlights the process of classification in a coherent way such that each class contains a single affine Boolean function. Two unique and different methods have been devised for this classification. The first one is a recursive procedure that uses the Cartesian product of sets starting from the set of one variable Boolean functions. In the second method, the classification is done by changing some predefined bit positions with respect to the affine function belonging to that class. The bit positions which are changing also provide us information concerning the size and symmetry properties of the classes/subclasses in such a way that the members of classes/subclasses satisfy certain similar properties.

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11

Provotar, O. I., and O. O. Provotar. "Elements of concrete algorithmics: computability and solvability." PROBLEMS IN PROGRAMMING, no. 2-3 (September 2020): 198–207. http://dx.doi.org/10.15407/pp2020.02-03.198.

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An approach to proving the fundamental results of the theory of recursive functions using specific algorithms is consider. For this, the basic constructions of the algorithm are describing exactly and Church's thesis for more narrow classes of algorithmically computational functions is specified (concretized). Using this approach, the belonging of functions to classes of algorithmically computable is argued by the construction of the corresponding algorithms.

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12

JONES, NEIL D. "The expressive power of higher-order types or, life without CONS." Journal of Functional Programming 11, no. 1 (January 2001): 55–94. http://dx.doi.org/10.1017/s0956796800003889.

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Compare first-order functional programs with higher-order programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In particular, higher-order values may be first-order simulated by use of the list constructor ‘cons’ to build function closures. This paper uses complexity theory to prove some expressivity results about small programming languages that are less than Turing complete. Complexity classes of decision problems are used to characterize the expressive power of functional programming language features. An example: second-order programs are more powerful than first-order, since a function f of type [Bool]-〉Bool is computable by a cons-free first-order functional program if and only if f is in PTIME, whereas f is computable by a cons-free second-order program if and only if f is in EXPTIME. Exact characterizations are given for those problems of type [Bool]-〉Bool solvable by programs with several combinations of operations on data: presence or absence of constructors; the order of data values: 0, 1, or higher; and program control structures: general recursion, tail recursion, primitive recursion.

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13

Brus, Adam, Jiří Hrivnák, and Lenka Motlochová. "Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions." Entropy 20, no. 12 (December 6, 2018): 938. http://dx.doi.org/10.3390/e20120938.

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Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented.

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14

Ignjatović, Aleksandar. "Delineating classes of computational complexity via second order theories with weak set existence principles. I." Journal of Symbolic Logic 60, no. 1 (March 1995): 103–21. http://dx.doi.org/10.2307/2275511.

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AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

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15

Vollmer, Heribert, and Klaus W. Wagner. "Recursion theoretic characterizations of complexity classes of counting functions." Theoretical Computer Science 163, no. 1-2 (August 1996): 245–58. http://dx.doi.org/10.1016/0304-3975(95)00237-5.

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16

Burr, Wolfgang. "Fragments of Heyting arithmetic." Journal of Symbolic Logic 65, no. 3 (September 2000): 1223–40. http://dx.doi.org/10.2307/2586698.

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AbstractWe define classes Φn of formulae of first-order arithmetic with the following properties:(i) Every φ ϵ Φn is classically equivalent to a Πn-formula (n ≠ 1, Φ1 := Σ1).(ii) (iii) IΠn and iΦn (i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φn both under existential and universal quantification (we call these classes Θn) the corresponding theories iΘn still prove the same Π2-formulae. In a second part we consider iΔ0 plus collection-principles. We show that both the provably recursive functions and the provably total functions of are polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence .

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17

Qu, Zhihua, and Darren M. Dawson. "Lyapunov Direct Design of Robust Control for Electrical-Mechanical Systems Composed of Cascaded Nonlinear Uncertain Subsystems." Journal of Dynamic Systems, Measurement, and Control 117, no. 1 (March 1, 1995): 54–62. http://dx.doi.org/10.1115/1.2798523.

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Robust control design of nonlinear uncertain systems is investigated. A system under consideration consists of finite nonlinear systems which are cascaded and have significant uncertainties. Such a system arises naturally from many real physical systems, especially mechanical systems. An important feature of these systems is that they do not satisfy the assumption of the standard matching conditions required by most existing robust control results. General classes of cascaded uncertain systems are identified for which robust controllers are obtained explicitly in terms of the bounding functions of the uncertainties. The resulting robust controllers guarantee stability of global uniform ultimate boundedness or global exponential convergence to zero. The controls are designed by a two-step systematic design procedure. First, design fictitious robust controllers for input of individual subsystem as if every subsystem had an independent control. Then, a recursive mapping is proposed which maps the individual fictitious controls recursively into the only control of the overall system.

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18

Cenzer, D., V. W. Marek, and J. B. Remmel. "On the complexity of index sets for finite predicate logic programs which allow function symbols." Journal of Logic and Computation 30, no. 1 (January 2020): 107–56. http://dx.doi.org/10.1093/logcom/exaa005.

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Abstract We study the recognition problem in the metaprogramming of finite normal predicate logic programs. That is, let $\mathcal{L}$ be a computable first-order predicate language with infinitely many constant symbols and infinitely many $n$-ary predicate symbols and $n$-ary functions symbols for all $n \geq 1$. Then we can effectively list all the finite normal predicate logic programs $Q_0,Q_1,\ldots $ over $\mathcal{L}$. Given some property $\mathcal{P}$ of finite normal predicate logic programs over $\mathcal{L}$, we define the index set $I_{\mathcal{P}}$ to be the set of indices $e$ such that $Q_e$ has property $\mathcal{P}$. We classify the complexity of the index set $I_{\mathcal{P}}$ within the arithmetic hierarchy for various natural properties of finite predicate logic programs. For example, we determine the complexity of the index sets relative to all finite predicate logic programs and relative to certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having no recursive stable models, (iii) having at least one stable model, (iv) having at least one recursive stable model, (v) having exactly $c$ stable models for any given positive integer $c$, (vi) having exactly $c$ recursive stable models for any given positive integer $c$, (vii) having only finitely many stable models, (viii) having only finitely many recursive stable models, (ix) having infinitely many stable models and (x) having infinitely many recursive stable models.

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19

VĪKSNA, JURIS. "INDUCTIVE INFERENCE OF LIMITING PROGRAMS WITH BOUNDED NUMBER OF MIND CHANGES." International Journal of Foundations of Computer Science 07, no. 03 (September 1996): 187–208. http://dx.doi.org/10.1142/s0129054196000154.

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We consider inductive inference of total recursive functions in the case, when produced hypotheses are allowed some finite number of times to change “their mind” about each value of identifiable function. Such type of identification, which we call inductive inference of limiting programs with bounded number of mind changes, by its power lies somewhere between the traditional criteria of inductive inference and recently introduced inference of limiting programs. We consider such model of inductive inference for EX and BC types of identification, and we study • tradeoffs between the number of allowed mind changes and the number of anomalies, and • relations between classes of functions identifiable with different probabilities. For the case of probabilistic identification we establish probabilistic hierarchies which are quite unusual for EX and BC types of inference.

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20

Grädel, Erich, and Yuri Gurevich. "Tailoring recursion for complexity." Journal of Symbolic Logic 60, no. 3 (September 1995): 952–69. http://dx.doi.org/10.2307/2275767.

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AbstractWe design functional algebras that characterize various complexity classes of global functions. For this purpose, classical schemata from recursion theory are tailored for capturing complexity. In particular we present a functional analog of first-order logic and describe algebras of the functions computable in nondeterministic logarithmic space, deterministic and nondeterministic polynomial time, and for the functions computable by AC1 -circuits.

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21

BURGIN, MARK. "DECIDABILITY AND UNIVERSALITY IN THE AXIOMATIC THEORY OF COMPUTABILITY AND ALGORITHMS." International Journal of Foundations of Computer Science 23, no. 07 (November 2012): 1465–80. http://dx.doi.org/10.1142/s012905411240059x.

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In this paper, we study to what extent decidability is connected to universality. A natural context for such a study is provided by the axiomatic theory of computability, automata and algorithms. In Section 2, we introduce necessary concepts, constructions, axioms, postulates, conditions and problems. Section 3 contains the main results of the paper. In particular, it is demonstrated (Theorems 1 and 2) that undecidability of algorithmic problems does not depend on existence of universal algorithms and may be caused by weaker conditions. At the same time, results of Theorems 2 and 3 demonstrate that decidability is incompatible with universality. It is also proved (Theorem 5) that sufficiently big classes of total algorithms/automata, such as the class of all primitive recursive functions, cannot have universal algorithms/automata.

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22

CAPORASO, SALVATORE, EMANUELE COVINO, and GIOVANNI PANI. "A predicative approach to the classification problem." Journal of Functional Programming 11, no. 1 (January 2001): 95–116. http://dx.doi.org/10.1017/s0956796800003853.

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We harmonize many time-complexity classes DTIMEF(f(n)) (f(n) [ges ] n) with the PR functions (at and above the elementary level) in a transfinite hierarchy of classes of functions [Tscr ]α. Class [Tscr ]α is obtained by means of unlimited operators, namely: a variant Π of the predicative or safe recursion scheme, introduced by Leivant, and by Bellantoni and Cook, if α is a successor; and constructive diagonalization if α is a limit. Substitution (SBST) is discarded because the time complexity classes are not closed under this scheme. [Tscr ]α is a structure for the PR functions finer than [Escr ]α, to the point that we have [Tscr ]ε0 = [Escr ]3 (elementary functions). Although no explicit use is made of hierarchy functions, it is proved that f(n) ∈ [Tscr ]α implies f(n) [les ] nGα(n), where Gα belongs to the slow growing hierarchy (of functions) studied, in particular, by Girard and Wainer.

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23

Yuan, Fang, Gan Liu, Xiwen Yang, Shunfang Wang, and Xueren Wang. "Prediction of oxidoreductase subfamily classes based on RFE-SND-CC-PSSM and machine learning methods." Journal of Bioinformatics and Computational Biology 17, no. 04 (August 2019): 1950029. http://dx.doi.org/10.1142/s021972001950029x.

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Oxidoreductase is an enzyme that widely exists in organisms. It plays an important role in cellular energy metabolism and biotransformation processes. Oxidoreductases have many subclasses with different functions, creating an important classification task in bioinformatics. In this paper, a dataset of 2640 oxidoreductase sequences was used to perform an analysis and comparison. The idea of dipeptides was introduced to process the Position Specific Score Matrix (PSSM), since each dipeptide consists of two amino acids and each column of PSSM corresponds to the information of one amino acid. Two kinds of dipeptide scores were proposed, the Standardization Normal Distribution PSSM (SND-PSSM) and the Correlation Coefficient PSSM (CC-PSSM). Recursive Feature Elimination (RFE) is used to extract features from the SND-PSSM and CC-PSSM, and the two sets of extracted features are combined to form a new feature matrix, the RFE-SND-CC-PSSM. The results show that, with the proposed method and a kernel-based nonlinear SVM classifier, the accuracy can reach 95.56% by the Jackknife test. Our method greatly improves the accuracy of oxidoreductase subclass prediction. Using this method to predict the categories of the 6 major types of enzymes effectively improves its prediction accuracy to 94.54%, indicating that this method has general applicability to other protein problems. The results show that our method is effective and universally applicable, and might be complementary to the existing methods.

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Milic, Ljiljana, and Tapio Saramaki. "Power-complementary IIR filter pairs with an adjustable crossover frequency." Facta universitatis - series: Electronics and Energetics 16, no. 3 (2003): 295–304. http://dx.doi.org/10.2298/fuee0303295m.

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This paper introduces two classes of power-complementary recursive low-pass/high-pass filter pairs with an adjustable crossover frequency in such a way that the stopband attenuation of both filters remains the same. For each class, the filter pair is constructed using two all-pass sub-filters as building blocks. Based on the properties elliptic minimal Q-factors transfer functions, simple expressions are derived for evaluating the coefficients in all-pass sections in order to achieve the desired crossover frequency. The design procedures are developed for synthesizing power-complementary filter pairs implemented as a parallel connection of two all-pass sub-filters and for the tapped cascaded interconnections of two identical all-pass sub-filters. The direct parallel connection has both the power-complementary and all-pass complementary property. The second class of filters constructed using several identical copies of the two all-pass filters possesses the power-complementary property.

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25

Hofmann, Martin. "An Application of Category-Theoretic Semantics to the Characterisation of Complexity Classes Using Higher-Order Function Algebras." Bulletin of Symbolic Logic 3, no. 4 (December 1997): 469–86. http://dx.doi.org/10.2307/421100.

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AbstractWe use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PVω defines exactly the PTIME-iunctions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types.By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with -induction over PVω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion.

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Liu, Kefeng, Motohico Mulase, and Hao Xu. "Recursions and asymptotics of intersection numbers." International Journal of Mathematics 27, no. 09 (August 2016): 1650072. http://dx.doi.org/10.1142/s0129167x16500725.

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We establish the asymptotic expansion of certain integrals of [Formula: see text] classes on moduli spaces of curves [Formula: see text], when either the [Formula: see text] or [Formula: see text] goes to infinity. Our main tools are cut-join type recursion formulae from the Witten–Kontsevich theorem, as well as asymptotics of solutions to the first Painlevé equation. We also raise a conjecture on large genus asymptotics for [Formula: see text]-point functions of [Formula: see text] classes and partially verify the positivity of coefficients in generalized Mirzakhani’s formula of higher Weil–Petersson volumes.

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27

Mercer, P., S. Sheldon, T. Kotsos, and W. O’Neill. "Detection of Juvenile Sleep Deprivation by Stochastic Optimization of Pupillographic Records." Methods of Information in Medicine 42, no. 03 (2003): 282–86. http://dx.doi.org/10.1055/s-0038-1634362.

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Summary Objective: To address the challenging problem of measuring juvenile sleep deprivation, we test the hypothesis that a pupillographic method found successful for adult narcoleptics might also discriminate between sleep deprived juveniles acting as their own controls. Methods: A linear, nonstationary model relating pupillary diameter and a random photic stimulus are estimated by recursive regressions from pupillographic records of 8 juveniles of median age 7 years acting as their own rested controls. The estimated pupillary impulse response noise functions are stochastically optimized using the Kullback divergence measure to maximally separate the sleep deprived records from the control records. Results: Both the average and covariance statistics of the estimated pupillary noise functions exhibit statistically significant differences between sleep deprived and rested subjects. The main result is that sleep deprivation decreases pupillary noise variance; a finding consistent with a previous study of adult narcoleptics. Further, it was found that virtually the same stochastic parameters were optimal for the juvenile sleep deprived data and for the previous adult narcoleptic study. Conclusions: Although our results are preliminary, the consistent reduction of pupillary noise appears to justify a comprehensive clinical trial across a broad range of age classes. In addition, the finding that the same parameters stochastically optimze both juvenile and adult recordings suggests the procedure holds promise as a clinical test which could produce sleep deprivation measures simultaneous with data collection.

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Raghunath, Bindiya Lakshmi, Claudio Mulatti, Michelle Jin-Yee Neoh, Marc H. Bornstein, and Gianluca Esposito. "The Associations between Imageability of Positive and Negative Valence Words and Fear Reactivity." Psychiatry International 2, no. 1 (February 9, 2021): 32–47. http://dx.doi.org/10.3390/psychiatryint2010003.

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This study investigated the associations of imageability with fear reactivity. Imageability ratings of four word classes: positive and negative (i) emotional and (ii) propriosensitive, neutral and negative (iii) theoretical and (iv) neutral concrete filler, and fear reactivity scores—degree of fearfulness towards different situations (Total Fear (TF) score) and total number of extreme fears and phobias (Extreme Fear (EF) score), were obtained from 171 participants. Correlations between imageability, TF and EF scores were tested to analyze how word categories and their valence were associated with fear reactivity. Imageability ratings were submitted to recursive partitioning. Participants with high TF and EF scores had higher imageability for negative emotional and negative theoretical words. The correlations between imageability of negative emotional words and negative theoretical words for EF score were significant. Males showed stronger correlations for imageability of negative emotional words for EF and TF scores. High imageability for positive emotional words was associated with lower fear reactivity in females. These findings were discussed with regard to negative attentional bias theory of anxiety, influence on emotional systems, and gender-specific coping styles. This study provides insight into cognitive functions involved in mental imagery, semantic competence for mental imagery in relation to fear reactivity, and a potential psycholinguistic instrument assessing fear reactivity.

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Choudhury, Gagan L., Kin K. Leung, and Ward Whitt. "An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models." Advances in Applied Probability 27, no. 04 (December 1995): 1104–43. http://dx.doi.org/10.1017/s0001867800047819.

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In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.

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Choudhury, Gagan L., Kin K. Leung, and Ward Whitt. "An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models." Advances in Applied Probability 27, no. 4 (December 1995): 1104–43. http://dx.doi.org/10.2307/1427936.

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In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.

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31

Jain, Sanjay, and Arun Sharma. "The structure of intrinsic complexity of learning." Journal of Symbolic Logic 62, no. 4 (December 1997): 1187–201. http://dx.doi.org/10.2307/2275636.

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AbstractLimiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of “intrinsic” complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification.The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages.Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open.

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Van Ness, Brian G., Christine Ramos, Vipin Kumar, Michael Steinbach, Brian GM Durie, John Crowley, Bart Barlogie, et al. "Analytical Approaches for the BOAC SNP Panel Association with Progression Free Survival in Myeloma." Blood 112, no. 11 (November 16, 2008): 2715. http://dx.doi.org/10.1182/blood.v112.11.2715.2715.

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Abstract The Bank On A Cure (BOAC) has established DNA banks from multiple cooperative and institutional clinical trials, and platforms for examining the association of genetic variations (SNPs) with disease risk and outcomes in myeloma. We have previously described the development and content of a novel custom SNP panel that contains 3,404 SNPs in 983 genes, representing cellular functions and pathways that may influence disease response, toxicities, complications, and survival. Although survival certainly varies according to tumor heterogeneity (ie. chromosomal abnormalities, gene expression variations) germline variations that influence the microenvironment, drug distribution, drug transport and metabolism, may also have an association with event free survival outcomes. To explore SNP associations with progression free survival (PFS) we compared the BOAC SNP profiles of short term PFS (less than 1 year, n=70) versus long term PFS (greater than 3 years, n=73) in two phase III clinical trials (ECOG E9487 and SWOG S9321). A variety of analytical approaches was undertaken including univariate rank ordering, recursive partitioning, and support vector machine learning tools (SVM). Each of these approaches has advantages and limitations in dealing with type I false positive errors as well as sensitivity and specificity. We included subset validation approaches and randomization of classes to address how robust and predictive different approaches were. From our analysis we conclude germline genomic variations do have an impact on progression free survival, with a subset of SNPs from the panel reaching 76% predictive association and hazard ratios of PFS of 9.6 (CI 4.5, 20.5), p<0.001, using SVM analysis. Based on univariate approaches, we find the most significant variations associated with PFS differences were genes that could be functionally categorized as pharmacologic. The presentation will focus on the analytical approaches, and refinements necessary to assure predictive value compared to random associations. Notwithstanding the clear importance of tumor cell variations in genetic deregulation, we conclude that various functions within the bone marrow and drug response likely interplay as a complex influence on disease progression, response, and survival. This suggests combining gene expression profiles of the tumors with germline SNP profiles may provide more accurate prognosis. These combined analytical approaches are currently being developed with BOAC data bases, and examples will be discussed.

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Жихарев and L. Zhikharev. "Generalization to Three-Dimensional Space Fractals of Pythagoras and Koch. Part I." Geometry & Graphics 3, no. 3 (November 30, 2015): 24–37. http://dx.doi.org/10.12737/14417.

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Fractals are geometric objects, each part of which is similar to the whole object, so that if we take a part and increase its size to the size of the whole object, it would be impossible to notice a difference. In other words, fractals are sets having scale invariance. In mathematics, they are associated primarily with non-differentiable functions. The concept of "fractal" (from the Latin "Fractus" meaning «broken») had been introduced by Benoit Mandelbrot (1924–2010), French and American mathematician, physicist, and economist. Mandelbrot had found that seemingly arbitrary fluctuations in price of goods have a certain tendency to change: it turned out that daily fluctuations are symmetrical with long-term price fluctuations. In fact, Benoit Mandelbrot applied his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number of fractal curves and flat objects have been created; and methods for their application have been developed. From geometrical point of view, the most interesting fractals are "Koch snowflake" and "Pythagoras Tree". Two classes of analogues of the volumetric fractals were created with modern three-dimensional modeling program: "Fractals of growth” – like Pythagoras Tree, “Fractals of separation” – like Koch snowflake; the primary classification was developed, their properties were studied. Empiric data was processed with basic arithmetic calculations as well as with computer software. Among other things, for fractals of separation the task was to create an object with an infinite surface area, which in the future might acquire great importance for the development of the chemical and other industries.

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Olas, Andrzej. "Recursive Lyapunov Functions." Journal of Dynamic Systems, Measurement, and Control 111, no. 4 (December 1, 1989): 641–45. http://dx.doi.org/10.1115/1.3153107.

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The paper presents the concept of recursive Lyapunov function. The concept is applied to investigation of asymptotic stability problem of autonomous systems. The sequence of functions {Uα(i)} and corresponding performance measures λ(i) are introduced. It is proven that λ(i+1) ≤ λ(i) and in most cases the inequality is a strong one. This fact leads to a concept of a recursive Lyapunov function. For the very important applications case of exponential stability the procedure is effective under very weak conditions imposed on the function V = U(0). The procedure may be particularly applicable for the systems dependent on parameters, when the Lyapunov function determined from one set of parameters may be employed at the first step of the procedure.

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Chesebro, Eric, Cory Emlen, Kenton Ke, Denise LaFontaine, Kelly McKinnie, and Catherine Rigby. "Farey recursive functions." Involve, a Journal of Mathematics 14, no. 3 (July 17, 2021): 439–61. http://dx.doi.org/10.2140/involve.2021.14.439.

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FRADET, PASCAL, and JULIEN MALLET. "Compilation of a specialized functional language for massively parallel computers." Journal of Functional Programming 10, no. 6 (November 2000): 561–605. http://dx.doi.org/10.1017/s0956796800003816.

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We propose a parallel specialized language that ensures portable and cost-predictable implementations on parallel computers. The language is basically a first-order, recursion-less, strict functional language equipped with a collection of higher-order functions or skeletons. These skeletons apply on (nested) vectors and can be grouped into four classes: computation, reorganization, communication and mask skeletons. The compilation process is described as a series of transformations and analyses leading to SPMD-like functional programs which can be directly translated into real parallel code. The language restrictions enforce a programming discipline whose benefit is to allow a static, symbolic and accurate cost analysis. The parallel cost takes into account both load balancing and communications, and can be statically evaluated even when the actual size of vectors or the number of processors are unknown. It is used to automatically select the best data distribution among a set of standard distributions. Interestingly, this work can be seen as a cross-fertilization between techniques developed within the FORTRAN parallelization, skeleton and functional programming communities.

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DONG, YunMei. "Multi-type recursive functions." SCIENTIA SINICA Mathematica 45, no. 9 (September 1, 2015): 1549–62. http://dx.doi.org/10.1360/n012015-00053.

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Kneuss, Etienne, Ivan Kuraj, Viktor Kuncak, and Philippe Suter. "Synthesis modulo recursive functions." ACM SIGPLAN Notices 48, no. 10 (November 12, 2013): 407–26. http://dx.doi.org/10.1145/2544173.2509555.

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BECKMANN, ARNOLD, SAMUEL R. BUSS, and SY-DAVID FRIEDMAN. "SAFE RECURSIVE SET FUNCTIONS." Journal of Symbolic Logic 80, no. 3 (July 22, 2015): 730–62. http://dx.doi.org/10.1017/jsl.2015.26.

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AbstractWe introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

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Ho, Chun-Kuen. "Beyond Recursive Real Functions." Information and Computation 124, no. 2 (February 1996): 113–26. http://dx.doi.org/10.1006/inco.1996.0009.

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Severin, Daniel E. "Unary primitive recursive functions." Journal of Symbolic Logic 73, no. 4 (December 2008): 1122–38. http://dx.doi.org/10.2178/jsl/1230396909.

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AbstractIn this article, we study some new characterizations of primitive recursive functions based on restricted forms of primitive recursion, improving the pioneering work of R. M. Robinson and M. D. Gladstone. We reduce certain recursion schemes (mixed/pure iteration without parameters) and we characterize one-argument primitive recursive functions as the closure under substitution and iteration of certain optimal sets.

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Beckmann, Arnold, Sam Buss, Sy-David Friedman, Moritz Müller, and Neil Thapen. "Cobham recursive set functions." Annals of Pure and Applied Logic 167, no. 3 (March 2016): 335–69. http://dx.doi.org/10.1016/j.apal.2015.12.005.

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Jockusch, C. G., A. Lewis, and J. B. Remmel. "Π01-classes and Rado's selection principle." Journal of Symbolic Logic 56, no. 2 (June 1991): 684–93. http://dx.doi.org/10.2307/2274710.

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There are several areas in recursive algebra and combinatorics in which bounded or recursively bounded -classes have arisen. For our purposes we may define a -class to be a set Path(T) of all infinite paths through a recursive tree T. Here a recursive tree T is just a recursive subset of ω<ω, the set of all finite sequences of the natural numbers ω = {0,1,2,…}, which is closed under initial segments. If the tree T is finitely branching, then we say the -class Path(T) is bounded. If T is highly recursive, i.e., if there exists a partial recursive function f: T→ω such that for each node ηЄ T, f(η) equals the number of immediate successors of η, then we say that the -class Path(T) is recursively bounded (r.b.). For example, Manaster and Rosenstein in [6] studied the effective version of the marriage problem and showed that the set of proper marriages for a recursive society S was always a bounded -class and the set of proper marriages for a highly recursive society was always an r.b. -class. Indeed, Manaster and Rosenstein showed that, in the case of the symmetric marriage problem, any r.b. -class could be represented as the set of symmetric marriages of a highly recursive society S in the sense that given any r.b. Π1-class C there is a society Sc such that there is a natural, effective, degree-preserving 1:1 correspondence between the elements of C and the symmetric marriages of Sc. Jockusch conjectured that the set of marriages of a recursive society can represent any bounded -class and the set of marriages of a highly recursive society can represent any r.b. -class. These conjectures remain open. However, Metakides and Nerode [7] showed that any r.b. -class could be represented by the set of total orderings of a recursive real field and vice versa that the set of total orderings of a recursive real field is always an r.b. -class.

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Borie, Richard B., R. Gary Parker, and Craig A. Tovey. "Deterministic Dcomposition of Recursive Graph Classes." SIAM Journal on Discrete Mathematics 4, no. 4 (November 1991): 481–501. http://dx.doi.org/10.1137/0404043.

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Manolopoulos, Yannis. "On the number of recursive calls of recursive functions." ACM SIGCSE Bulletin 37, no. 2 (June 2005): 61–64. http://dx.doi.org/10.1145/1083431.1083465.

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FU, CHENG, and BOGDAN J. FALKOWSKI. "TERNARY RECURSIVE FAST TRANSFORMS." Journal of Circuits, Systems and Computers 16, no. 02 (April 2007): 155–68. http://dx.doi.org/10.1142/s0218126607003459.

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This paper introduces two new classes of recursive fast transforms over GF (3). They are based on recursive equations using Kronecker products that allows to obtain simple corresponding fast transforms and regular butterfly diagrams. The computational costs to calculate both classes of new transforms and the experimental results comparing introduced transforms with ternary Reed–Muller transform are also presented.

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Boza Cordero, Juan. "Recursive formulas for partition functions." Revista de Matemática: Teoría y Aplicaciones 4, no. 1 (March 29, 2012): 21. http://dx.doi.org/10.15517/rmta.v4i1.137.

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Krauss, Alexander. "Recursive Definitions of Monadic Functions." Electronic Proceedings in Theoretical Computer Science 43 (December 21, 2010): 1–13. http://dx.doi.org/10.4204/eptcs.43.1.

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Sabadini, Nicoletta, Sebastiano Vigna, and Robert F. C. Walters. "A note on recursive functions." Mathematical Structures in Computer Science 6, no. 2 (April 1996): 127–39. http://dx.doi.org/10.1017/s0960129500000918.

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In this paper, we propose a new and elegant definition of the class of recursive functions, which is analogous to Kleene's definition but differs in the primitives taken, thus demonstrating the computational power of the concurrent programming language introduced in Walters (1991), Walters (1992) and Khalil and Walters (1993).The definition can be immediately rephrased for any distributive graph in a countably extensive category with products, thus allowing a wide, natural generalization of computable functions.

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Solov'ev, V. D. "Program algebras of recursive functions." Russian Mathematical Surveys 51, no. 4 (August 31, 1996): 705–22. http://dx.doi.org/10.1070/rm1996v051n04abeh002966.

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