The paper investigates whether it is possible to learn every enumerable classes of recursive functions from "typical" examples. "Typical" means, there is a computable family of finite sets, such th...
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ISBN:
(纸本)3540667482
The paper investigates whether it is possible to learn every enumerable classes of recursive functions from "typical" examples. "Typical" means, there is a computable family of finite sets, such that for each function in the class there is one set of examples that can be used in any suitable hypothesis space for this class of functions. As it will turn out, there are enumerable classes of recursive functions that are not learnable from "typical" examples. The learnable classes are characterized. The results are proved within an abstract model of learning from examples, introduced by Freivalds, Kinber and Wiehagen. Finally, the results are interpreted and possible connections of this theoretical work to the situation in real life classrooms are pointed out.
Many useful calculation rules, such as fusion and tupling, rely on well-structured functions, especially in terms of inputs and outputs. For instance, fusion requires that well-produced outputs should be connected to ...
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ISBN:
(纸本)3540356312
Many useful calculation rules, such as fusion and tupling, rely on well-structured functions, especially in terms of inputs and outputs. For instance, fusion requires that well-produced outputs should be connected to well-consumed inputs, so that unnecessary intermediate data structures can be eliminated. These calculation rules generally fail to work unless functions are well-structured. In this paper, we propose a new calculation rule called 10 swapping. 10 swapping exchanges call-time computations (occurring in the arguments) and return-time computations (occurring in the results) of a function, while guaranteeing that the original and resulting function compute the same value. 10 swapping enables us to rearrange inputs and outputs so that the existing calculation rules can be applied. We present new systematic derivations of efficient programs for detecting palindromes, and a method of higher-order removal that can be applied to defunctionalize function arguments, as two concrete applications.
Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted Vir...
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ISBN:
(纸本)9783319284750;9783319284743
Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted Virus Machines are unbounded Virus Machines, in the sense that no restriction on the number of hosts, the number of instructions and the number of viruses contained in any host along any computation is placed on them. The computational completeness of these machines has been obtained by simulating register machines. In this paper, Virus Machines as function computing devices are considered. Then, the universality of non-restricted virus machines is proved by showing that they can compute all partial recursive functions.
The article discusses the algebraic structure of primitive recursive (PR) functions. It cites Julia Robinson's works on the characterizations of the class PR wherein it can be obtained from two suitable functions ...
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The article discusses the algebraic structure of primitive recursive (PR) functions. It cites Julia Robinson's works on the characterizations of the class PR wherein it can be obtained from two suitable functions using two operators. It details the endomorphisms of the PR structure and presents a corollary to prove the importance of the theorem that there are only to endomorphisms of PR.
Path coverage is of critical importance in software testing and verification, and further, path explosion is a well-known challenge for automatic software analysis techniques like symbolic execution [7]. Asymptotic Pa...
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ISBN:
(纸本)9798350322637
Path coverage is of critical importance in software testing and verification, and further, path explosion is a well-known challenge for automatic software analysis techniques like symbolic execution [7]. Asymptotic Path Complexity (APC), a code complexity metric developed in my research lab, formalizes the quantitative measurement of path explosion.
Let gE(m, n)=o mean that n is the Gödel-number of the shortest derivation from E of an equation of the form φ{symbol}(m)=k. Hao Wang suggests that the condition for general recursiveness ∀m∃n(gE(m, n)=o) can be ...
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The present paper surveys some results from the inductive inference of recursive functions, which are related to the characterization of inferrible function classes in terms of complexity theory, and in terms of recur...
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ISBN:
(数字)9783030514662
ISBN:
(纸本)9783030514662;9783030514655
The present paper surveys some results from the inductive inference of recursive functions, which are related to the characterization of inferrible function classes in terms of complexity theory, and in terms of recursive numberings. Some new results and open problems are also included.
Recently, functions over the reals that extend elementarily computable functions over the integers have been proved to correspond to the smallest class of real functions containing some basic functions and closed by c...
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ISBN:
(纸本)3540252614
Recently, functions over the reals that extend elementarily computable functions over the integers have been proved to correspond to the smallest class of real functions containing some basic functions and closed by composition and linear integration. We extend this result to all computable functions: functions over the reals that extend total recursive functions over the integers are proved to correspond to the smallest class of real functions containing some basic functions and closed by composition, linear integration and a very natural unique minimization schema.
In this paper, we report some empirical studies of students evaluating recursive functions defined according to the rules of the functional programming language Miranda, and describe the misconceptions and processing ...
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In this paper, we report some empirical studies of students evaluating recursive functions defined according to the rules of the functional programming language Miranda, and describe the misconceptions and processing strategies observed. We then discuss the implications of these observations as regards teaching content.
Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. We could obt...
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ISBN:
(纸本)9783540710653
Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. We could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second evaluates them, and the last computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor. Every partial recursive function is associated with an inductive domain predicate;evaluation of the function requires a proof that the input values satisfy the predicate. We give a constructive justification for the new construct by interpreting it into the base type theory. This shows that the extended theory is consistent and constructive.
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