In their classical papers, Lerman, Lachlan, and Lebeuf developed the embedding method, which provides constructions of initial segments of Turing degrees isomorphic to various partially ordered structures. We analyze ...
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In their classical papers, Lerman, Lachlan, and Lebeuf developed the embedding method, which provides constructions of initial segments of Turing degrees isomorphic to various partially ordered structures. We analyze this method and prove that there is a nonzero degree below each decreasing chain of degrees uniform in 0'. This imposes restrictions on the application of the embedding method.
Ackermann functions are examples of nonprimitive recursive function which grow so fast with respect to their arguments. In this paper, we present Ackermann functions in detail and introduce its influence on theoretica...
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Ackermann functions are examples of nonprimitive recursive function which grow so fast with respect to their arguments. In this paper, we present Ackermann functions in detail and introduce its influence on theoretical computer science.
LFC is a functional language based on recursive functions defined in context-free languages. In this paper, a new pattern matching algorithm for LFC is presented, which can represent a sequence of patterns as an integ...
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LFC is a functional language based on recursive functions defined in context-free languages. In this paper, a new pattern matching algorithm for LFC is presented, which can represent a sequence of patterns as an integer by an encoding method. It is a rather simple method and produces efficient case-expressions for pattern matching definitions of LFC. The algorithm can also be used for other functional languages, but for nested patterns it may become complicated and further studies are needed.
We introduce the basic theory of quantization of radiation field in quantum physics and explain how it relates to the theory of recursive functions in computer science. We outline the basic differences between quantum...
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We introduce the basic theory of quantization of radiation field in quantum physics and explain how it relates to the theory of recursive functions in computer science. We outline the basic differences between quantum mechanics (QM) and quantum field theory (QFT) and explain why QFT is better suited for a computational paradigm-based on algorithmic requirement, countably infinite degrees of freedom and the creation of macroscopic output objects. The quanta of the radiation field correspond to the non-negative integers and the harmonic oscillator spectra correspond to the recursive computation with the creation and annihilation operators, respectively, playing the same role as the successor and predecessor in computability theory. Accordingly, this approach relates the classical computational model and the quantum physical model more directly than the Turing machine approach used earlier. Also, the application of Lambda calculus formalism and the associated denotational semantics (that is widely used in the classical computational paradigm involving recursive functions) for applications to computational paradigm based on quantum field theory is described. Finally, we explain where QFT and conventional paradigm depart from each other, and examine the concept of fixed points, phase transitions, programmability, emergent computation and related open problems.
A function Definition Language (FDL) is presented. Though de-signed for describing specifications, FDL is also a general-purpose functional pro-gramming language. It uses context-free language as data type, supports p...
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A function Definition Language (FDL) is presented. Though de-signed for describing specifications, FDL is also a general-purpose functional pro-gramming language. It uses context-free language as data type, supports patternmatching definition of functions, offers several function definition forms, and is exe-cutable. It is shown that FDL has strong expressiveness, is easy to use and describesalgorithms concisely and naturally. An interpreter of FDL is introduced. Experi-ments and discussion are included.
This paper challenges two orthodox theses: (a) that computational processes must be algorithmic;and (b) that all computed functions must be Turing-computable. Section 2 advances the claim that the works in computabili...
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This paper challenges two orthodox theses: (a) that computational processes must be algorithmic;and (b) that all computed functions must be Turing-computable. Section 2 advances the claim that the works in computability theory, including Turing's analysis of the effective computable functions, do not substantiate the two theses. It is then shown (Section 3) that we can describe a system that computes a number-theoretic function which is not Turing-computable. The argument against the first thesis proceeds in two stages. It is first shown (Section 4) that whether a process is algorithmic depends on the way we describe the process. It is then argued (Section 5) that systems compute even if their processes are not described as algorithmic. The paper concludes with a suggestion for a semantic approach to computation.
An efficient method for calculation of collocation constants in the orthogonal collocation method was discussed. To make it Possible to calculate the collocation constants of arbitrary order using the same subroutine,...
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An efficient method for calculation of collocation constants in the orthogonal collocation method was discussed. To make it Possible to calculate the collocation constants of arbitrary order using the same subroutine, an algorithm utilizing their recursive properties was proposed. The first and second-order collocation constants calculated based on the algorithm were confirmed to have the same high accuracies as those by the method previously reported. From the result of investigation on accuracies of the approximate values to high-order derivatives of a trigonometric function, the proposed algorithm was shown to also be useful in the calculation of higher-order collocation constants. The execution time required to calculate a set of collocation constants was almost the same as that of the previous method.
We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and"(general) recursiv...
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We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and"(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas.(1) After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section 7 about preserving the intensional differences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be called "computable" rather than "recursive;" we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions;the name of the subject should be "Computability Theory" or simply Computability rather than "recursive function Theory."
In this note we consider register-machines with symbol manipulation capabilities. They can form words over a given alphabet in their registers by appending symbols to the strings already stored. These machines are sim...
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In this note we consider register-machines with symbol manipulation capabilities. They can form words over a given alphabet in their registers by appending symbols to the strings already stored. These machines are similar to Post's normal systems and the related machine-models discussed in the literature. But unlike the latter devices they are deterministic and are not allowed to read symbols from the front of the registers. Instead they can compare registers and erase them. At first glance it is surprising that in general these devices are as powerful as the seemingly stronger models. Here we investigate the borderline of universality for these machines.
An on-line vertex coloring algorithm receives the vertices of a graph in some externally determined order, and, whenever a new vertex is presented, the algorithm also learns to which of the previously presented vertic...
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An on-line vertex coloring algorithm receives the vertices of a graph in some externally determined order, and, whenever a new vertex is presented, the algorithm also learns to which of the previously presented vertices the new vertex is adjacent. As each vertex is received, the algorithm must make an irrevocable choice of a color to assign the new vertex, and it makes this choice without knowledge of future vertices. A class of graphs Gamma is said to be on-line x-bounded if there exists an on-line algorithm A and a function f such that A uses at most f(w(G)) colors to properly color any graph G in Gamma. If H is a graph, let Forb(H) denote the class of graphs that do not induce H. The goal of this paper is to establish that Forb(T) is on-line X-bounded for every radius-2 tree T. As a corollary, the authors answer a question of Schmerl's;the authors show that every recursive cocomparability graph can be recursively colored with a number of colors that depends only on its clique number.
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