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21st Asia-Pacific Software Engineering Conference, APSEC 2014

作者： Chen, Yu-Fang Wang, Bow-Yaw Yang, Kai-Chun Institute of Information Science Academia Sinica Taiwan

ISBN: (纸本)9781479974252

We describe a learning-based approach for verifying recursive functions. The Boolean formula learning algorithm CDNF is used to automatically infer function summaries for recursive functions. In contrast to traditional iterative fixpoint computation-based approaches, ours can quickly guess summaries and verify purported summaries. When purported summaries are incorrect, the learning algorithm refines them by posing queries. We solve examples that are unattainable by a mature model checker for recursive programs. © 2014 IEEE.

关键词： recursive functions

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Optimising recursive functions yielding multiple results in tuples in a lazy functional language 11th

Optimising recursive functions yielding multiple results in ...

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11th International Workshop on Implementation of Functional Languages, IFL 1999

作者： van Groningen, John H. G. University of Nijmegen Department of Computer Science Toernooiveld 1 Nijmegen6525 ED Netherlands

ISBN: (纸本)3540678646

We discuss a new optimisation for recursive functions yielding multiple results in tuples for lazy functional languages, like Clean and Haskell. This optimisation improves the execution time of such functions and also reduces the amount of memory allocated in the heap by these functions, which reduces garbage collection costs. In some cases execution time is improved by more than a factor of two and allocation costs by a factor of four. Furthermore, the space leak that is caused by selector nodes is removed. This is achieved by reusing nodes allocated in the previous iteration of the recursion to create the nodes for the next iteration, by updating these nodes. Only the parts of the nodes that have changed are updated. Because of these updates, the code that is used to select an element of a tuple is not executed anymore for many selections, because the selector node was overwritten with a new selector node or the result before it is evaluated. © Springer-Verlag Berlin Heidelberg 2000.

关键词： recursive functions

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Linear recursive functions

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作者： Alves, Sandra Fernández, Maribel Florido, Mário Mackie, Ian University of Porto Department of Computer Science LIACC R. do Campo Alegre 823 4150-180 Porto Portugal King's College London Department of Computer Science Strand London WC2R 2LS United Kingdom LIX CNRS UMR 7161 École Polytechnique 91128 Palaiseau Cedex France

ISBN: (纸本)9783540731467

With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turing-complete computation model. However, erasing and copying don't need to be explicitly included in Turing-complete computation models: in this paper we show that the class of partial recursive functions that are syntactically linear (that is, partial recursive functions where no argument is erased or copied) is Turing-complete. © Springer-Verlag Berlin Heidelberg 2007.

关键词： recursive functions

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A Note on Conjectures of Calude About the Topological Size of Sets of Partial recursive functions

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Mathematical Logic Quarterly 1985年第14-18期31卷 279-280页

作者： Schäfer, Gisela Technische Hochschule Aachen Aachen D‐5100 Büchel 29–31 Germany

The article discusses the conjectures of Calude on the topological size of the set of partial recursive functions. It says that the set of partial recursive functions is nowhere dense under f, g and is considered meagre of first Baire category if there exists a sequence and two r.e. sets. It mentions that the set of total recursive functions is a residual in the topological space of the set of partial recursive functions.

关键词： TOPOLOGY COMPUTABLE functions recursive functions SET theory BAIRE spaces

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Approximate Degree Composition for recursive functions 27

Approximate Degree Composition for Recursive Functions

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27th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2024 and the 28th International Conference on Randomization and Computation, RANDOM 2024

作者： Chakraborty, Sourav Kayal, Chandrima Mittal, Rajat Paraashar, Manaswi Saurabh, Nitin Indian Statistical Institute Kolkata India Indian Institute of Technology Kanpur India University of Copenhagen Denmark Indian Institute of Technology Hyderabad India

ISBN: (纸本)9783959773485

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e. functions obtained by composing a base function with itself a number of times. Let hd denote the standard d-fold composition of the base function h. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: The outer function f : {0, 1}n → {0, 1} is a recursive function of the form hd, with h being any base function and d = Ω(log log n). The inner function is a recursive function of the form hd, with h being any constant arity base function (other than AND and OR) and d = Ω(log log n), where n is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be efficiently eliminated if the inner or outer function is a recursive function. © Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, and Nitin Saurabh.

关键词： recursive functions

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Defining recursive functions in HOL

Defining recursive functions in HOL

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1991 International Tutorial and Workshop on the HOL Theorem Proving System and Its Applications

作者： Ploegaerts, W. Claesen, L. De Man, H. IMEC Vzw Kapeldreef 75 LeuvenB-3001 Belgium Belgian National Fund for Scientific Research Belgium K.U.Leuven Belgium

ISBN: (纸本)0818624604

The limited support for defining recursive functions and generating induction schemes is a major shortcoming of the HOL system. In this paper, a theory of primitive recursive functions and a tool that minimizes the proof effort required for the definition of a wide class of primitive recursive functions are presented. Due to its flexibility and degree of automation, this work offers a useful extension of the HOL system. © 1991 Institute of Electrical and Electronics Engineers Inc. All rights reserved.

关键词： recursive functions

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The foundations of intuitionistic mathematics : especially in relation to recursive functions /

The foundations of intuitionistic mathematics : especially i...

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1965年

作者： Kleene Stephen Cole

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Symbolic test case generation for primitive recursive functions

Symbolic test case generation for primitive recursive functi...

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4th International Workshop on Formal Approaches to Software Testing, FATES 2004

作者： Brucker, Achim D. Wolff, Burkhart Information Security ETH Zürich ETH Zentrum CH-8092 Zürich Switzerland

We present a method for the automatic generation of test cases for HOL formulae containing primitive recursive predicates. These test cases can be used for the animation of specifications as well as for black-box testing of external programs. Our method is two-staged: first, the original formula is partitioned into test cases by transformation into a Horn-clause normal form (HCNF). Second, the test cases are analyzed for instances with constant terms satisfying the premises of the clauses. Particular emphasis is put on the control of test hypotheses and test hierarchies to avoid intractability. We applied our method to several examples, including AVL-trees and the red-black tree implementation in the standard library from SML/NJ. © Springer-Verlag Berlin Heidelberg 2005.

关键词： recursive functions

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Type-2 computability and Moore's recursive functions

Type-2 computability and Moore's recursive functions

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作者： Kawamura, Akitoshi Department of Computer Science Grad. Sch. Info. Sci. and Technol. The University of Tokyo Japan

Regarding effectivity of functions on the reals, there have been several proposed models of analog, continuous-time computation, as opposed to the digital, discrete nature of the type-2 computability. We study one of them, Moore's real (primitive) recursive functions, whose definition mimics the classical characterization of recursive functions on N by the closure properties. We show that the class of type-2 computable real functions falls between Moore's classes of primitive recursive and recursive functions. © 2005 Elsevier B.V.

关键词： recursive functions

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Enumerable classes of total recursive functions: Complexity of inductive inference 4th

Enumerable classes of total recursive functions: Complexity ...

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4th International Workshop on Analogical and Inductive Inference, AII 1994 and 5th International Workshop on Algorithmic Learning Theory, ALT 1994

作者： Ambainis, Andris Smotrovs, Juris Institute of Mathematics and Computer Science University of Latvia Raina bulv. 29 Riga Latvia

ISBN: (纸本)9783540585206

This paper includes some results on complexity of inductive inference for enumerable classes of total recursive functions, where enumeration is considered in more general meaning than usual recursive enumeration. The complexity is measured as the worst-case mindchange (error) number for the first n functions of the given class. Three generalizations are considered. First: the numbering is computed in limit (with a fixed number of mind- changes). Then the complexity can be arbitrary fast growing recursive function. Second: a fixed number of functions are given by the enumbering function wrongly. In this case only universal strategies have large complexity function. Third: every function given by the enumbering function can differ in a fixed number of points from the corresponding genuine function of the class. Two cases are considered: functions given by the enumbering function can be only partially defined or they must be total. In the first case there are unidentifiable classes. In the second case there are logarithmic algorithms for prediction and EX-identifying and linear algorithms for identifying of τ-indices. © 1994, Springer Verlag. All Rights Reserved.

关键词： recursive functions

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