Nigel Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Finally, if we take as model the IC, then the computable functions are given by corollary 13. Moreover, by adding more power to the model, we can generate a full hierarchy having as bottom the class of computable functions. For example, in the classical case, we can consider the jump operation [5]. This is done by considering Turing machines with oracles. It is well known that the Halting Problem is not computable by a Turing machine and, hence, if we use an oracle that solves the problem, we will get more power. This defines the class next to the computable functions. By taking a similar ....

N. J. Cutland. Computability: An introduction to Recursive Function Theory, Cambridge University Press, 1980.

....functions and, therefore, can be characterized in terms of standard computational complexity. Key words: Continuous time computation, di#erential equations, recursion theory, computational complexity. 1 Introduction Recursive function theory provides the standard notion of computable function [Cut80,Odi89]. Moreover, many time and space complexity classes have recursive characterizations [Clo99] As far as we know, Moore [Moo96] was the first to extend recursion theory to real valued functions. We will explore this and show that all main concepts in recursion theory like basic functions, operators, ....

N. J. Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

....an initial starting point is essential. A stream with continuous time is often called a signal. There are well established notions of computability on lt and and hence on our chosen models of discrete and continuous time. Classical models of com putability on the natural numbers are well known [9, 15, 40]. Computability models on have been developed since the 1950 s [7, 17, 29, 37] Our data set A is in general simply a set or an algebra. In order to discuss computability of streams and stream transformers we need to have notions of computability on our data algebras as well. For discrete ....

N.J. CUTLAND, Computability: An Introduction to Recursive Function Theory, Cambridge University Press, 1980.

....a function from the set of statements to the set of values, such that we can manipulate values as though, they were statements. Such a function # : Stm V al is known as a Godel numbering, after Kurt Godel, who was the first to use such a coding in [3] Often this function is a bijection (e.g. in [2]) but for our purpose it suffices to have a coding which is one to one. 2.3.1 The Godel Numbering of Statements. Due to the way the value set was defined in 2.1.1, the encoding of statements is very straightforward. As usual we have to treat the expressions first: CHAPTER 2. THE WHILE ....

....demand, that the codings are one to one. 9 Because of this, the one to one theorem will play an important role in dealing with the self halting set in 5.3. 1 Chapter 3 Defining Computability The definition of computability given in this chapter will be a seemingly restricted version of those in [2] and [10] The restriction will be that I will demand that a statement, which computes a function, should be tidy. That is, it should always terminate in a state, where only the variable 0 holds a value different from the standard value (the notion of a tidy statement is taken from [10] This ....

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Nigel J. Cutland. Computability: an introduction to recursive functions theory. Cambrigde University Press, 1980.

....For instance, f(x,y) x 1 is a PRF and is described by the expression C(U(1,2) S) The function f(x,y) x y is also a PRF described by the expression R(U(1,1) C(U(3,3) S) In fact, it can be shown that every Turing computable function is a PRF. More details on PRF theory can be found in [1] or [2]. 3. Coding PRF into ARNNs Finding a systematic way of generating ARNNs from given descriptions of PRF s greatly simplifies the task of producing neural nets to perform certain specific tasks. Furthermore, it also gives a proof that neural nets can effectively compute all Turing computable ....

Cutland, N., Computability -- An introduction to recursive function theory, Cambridge University Press, 1980.

....is devoted to showing that control state unreachability is undecidability is undecidable for data independent programs with at least two resetable arrays. 4 can not be extended to more than one. Using only two arrays we are able to simulate an arbitrary universal register machine (URM) see [2], for example) We are also able to show the impossibility of solving decision problems in a wider range of situations for arrays of type X ## X rather than X ## Y . The rest of the paper is devoted to showing how, within the limits implied by the negative result for two infinite types, we are ....

N. Cutland, Computability: An Introduction to Recursive Function Theory, Cambridge University Press, 1980.

....the primitive recursive functions introduced by Kalmar [Kal43] which is closed under the operations of forming bounded sums and products. This class contains virtually any function that can be computed in a practical sense, as well as the important number theoretic and metamathematical functions [Cut80,Ros84]. Thus we seem to have found a natural analog description of the elementary functions. To generalize this further, we recall that Grzegorczyk [Grz53] proposed a hierarchy of computable functions that strati es the set of primitive recursive functions. The elementary functions are simply the third ....

....composition, for each m the m times iterated exponential exp [m] x) is in E , where exp [m 1] x) 2 exp [m] x) and exp [0] x) x. In fact, no elementary function can grow faster than exp [m] for some m, and many of our results will depend on the following bound on their growth [Cut80]: Proposition 1. If f(x) 2 E, there is a number m such that, for all x, f(x) exp [m] kxk) where kxk = max i x i . The elementary functions are exactly the functions computable in elementary time [Cut80] The class E is therefore very large, and many would argue that it contains all ....

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N. J. Cutland. Computability: an introduction to recursive function theory. Cambridge University Press, 1980.

....a comparative study of different theorem proving approaches and a brief discussion regarding theorem proving in HOL follows the description of the mechanisation. 1 Introduction The theory of computation is a field which has been widely explored in mathematical and computer science literature [4, 12, 13] and several approaches to a standard model of computation have been attempted. However, each exposition of the theory centres on the basic notion of a computable function, and as such, one of the main objectives of a mechanisation of computability in a theorem prover is the formal definition of ....

....study, a brief discussion regarding theorem proving in HOL is given after the description of the implementation. This particular mechanisation of the theory is based on the URM model [11] of computation and much of the implementation is based on the definitions and results described in [4]. The next section gives a brief discussion on URM computability, however it is strongly suggested that the interested reader consults the literature ( 4, 12, 13] The rest of the paper illustrates the actual mechanisation and includes the specification of the notion of a computable function in ....

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N.J. Cutland. Computability: An introduction to recursive function theory. Cambridge University Press, 1980.

....an unlucky case is at most ffi . Thus, the learned computable hypothesis probably approximates the Halting Problem. 1. 4 RELATED WORK The literature in both computability and learnability spans decades (Blum and Blum 1975; Gold 1967; Herken 1988; Turing 1936) The reader is referred to references (Cutland 1980; Hopcroft and Ullman 1979; Lewis and Papadimitriou 1981; Minsky 1967) for theory of computation and to references (Angluin and Smith 1983; Kearns 1990; Valiant 1984; Pitt 1990) for formal machine learning, among others. The formal approach has been widely explored by the machine learning ....

Cutland, N. (1980). Computability: An Introduction to Recursive Function Theory, Cambridge University Press, Cambridge, UK.

....offers an extensive case study for the analysis of the two approaches of mechanical verification. The implementation illustrated in this report is based on a model of computation similar to the definition of partial recursive functions found in the literature on computation (see for instance [3, 10, 12]. The next section introduces the definition of partial recursive functions and section 3 gives a brief overview of the Coq theorem prover. A model of computation based on partial recursive functions and its formalisation in Coq is then given in section 4. In section 5, the key notion of a ....

....a proof that partial recursive functions are URM computable. This mechanisation is illustrated separately in [13] The results of the comparative study will be published in [14] 2 Partial Recursive Functions The set of partial recursive functions is defined in the literature (see for instance [3, 10]) as the smallest set of n ary partial functions on natural numbers which contains the three basic types of functions: ffl the zero functions: 8n; x 0 ; xn Gamma1 :Zn (x 0 ; xn Gamma1 ) 0, ffl the successor function: 8x 0 :S(x 0 ) x 0 1, ffl and the projection functions: ....

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N.J. Cutland. Computability: An introduction to recursive function theory. Cambridge University Press, 1980.

.... 1987) we use here the classical real analysis as the framework to state properties of computable real numbers y , as in Rice fundamental paper (Rice, 1954) In this section we will use the Godel Kleene representation of recursive functions as recursive functions (Weihrauch, 1987; Cutland, 1980; Hermes, 1965; Hennie, 1977; Cori and Lascar, 1993) 2.1. Definitions Let us note f Q N a recursive bijection from Q to N. We define first the notion of recursive Cauchy sequence for rational numbers and for intervals with rational bounds. Definition 2.1. Recursive Cauchy sequence) 1. A ....

Cutland, N. (1980). Computability: an introduction to recursive function theory. Cambridge University Press.

....example, Feferman 1992 [x6.6] wrote, the mathematical theory of computability in the guise of recursion theory. For several decades there has been a gradual tendency to replace the name recursion theory by computability theory especially in titles or subtitles of books, for example, Cutland 1980 or the subtitle of Soare 1987, A Study of Computable Functions and Computably Generated Sets. The purpose of this paper is to examine the meaning, origin, and history of the concepts recursive and computable with an eye toward re examining how we use them in practice. The ultimate aim is to ....

....298 ROBERT I. SOARE and Turing s Thesis one breaks the demonstration into smaller and smaller pieces until it becomes evident. 10 3.5. Register machines. Closely related to Turing machines is the formalism proposed much later of register machines by Shepherdson and Sturgis 1963. See also Cutland 1980, or Shoenfield 1991. These have the advantage of more closely resembling modern digital computers which manipulate data and instructions stored in various registers rather than having to go back and forth through the data on a single tape. In the version of Cutland [1980, p. 9] the register ....

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Nigel Cutland, Computability: An introduction to recursive function theory, Cambridge University Press, Cambridge, England.

....can (and should) be re examined under the light of the proposed computability requirements on theories. We assume that the reader knows about institutions and their applications. We also assume some basic knowledge of computability theory (recursive functions with oracles: see for instance [3]) and category theory (up to adjunctions: see for instance [2] The paper is organized as follows. In section 2 we introduce the (plain) theory spaces and their relationship to (plain) consequence systems. In section 3 we bring in the compactness requirement, leading to algebraic theory spaces ....

N. Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980. 13

....(f(x 1 , x n ,y) 0) for instance f(x,y) x 1 is a PRF and is given by the expression C(U(1,2) S) The function f(x,y) x y is also a PRF given by the expression R(U(1,1) C(U(3,3) S) In fact, it is shown that every Turing computable function is a PRF. More details on PRF theory can be found in [Cutland 80] and [Boolos 80] Least y such that f(x 1 , x n ,y) 0 and zy, f(x 1 , x n ,z) ##there is no such y. 3. Coding PRF into ARNNs Finding a systematic way of generating ARNN s from given descriptions of PRF s greatly simplifies the task of producing neural nets to perform certain specific ....

Cutland Nigel, Computability -- An introduction to recursive function theory, Cambridge University Press.

....an unlucky case is at most ffi . Thus, the learned computable hypothesis probably approximates the Halting Problem. 1. 4 RELATED WORK The literature in both computability and learnability spans decades (Blum and Blum 1975; Gold 1967; Herken 1988; Turing 1936) The reader is referred to references (Cutland 1980; Hopcroft and Ullman 1979; Lewis and Papadimitriou 1981; Minsky 1967) for theory of computation and to references (Angluin and Smith 1983; Kearns 1990; Valiant 1984; Pitt 1990) for formal machine learning, among others. The formal approach has been widely explored by the machine learning ....

Cutland, N. (1980). Computability: An Introduction to Recursive Function Theory, Cambridge University Press, Cambridge, UK.

....errors are usually undecidable, and this makes the type approach relevant. For the polyadic # calculus [8] this is also the case. Herein we show that the notion of communication errors in the polyadic # calculus is undecidable. The proof follows a general pattern of undecidability results [4]: we reduce the problem of deciding whether a lambda term has a normal form [5] to the problem of deciding whether a process is an error. More precisely, we define a computable function ## from # terms into # terms, and show that the decidability of #M##Err implies the decidability of M# , ....

Nigel Cutland. Computability: An introduction to recursive function theory. Cambridge University Press, 1980.

....exponential exp [m] x) is in E , where exp [0] x) x and exp [m 1] x) 2 exp [m] x) In fact, these are the fastest growing functions in E , in the sense that no elementary function can grow faster than exp [m] for some xed m. The following bound will be useful to us below [Cut80]: Proposition 1 If f 2 E, there is a number m such that, for all x, f(x) exp [m] kxk) where kxk = max i x i . The elementary functions also correspond to a natural time complexity class: Proposition 2 The elementary functions are exactly the functions computable by a Turing machine in ....

....the connectives of propositional calculus, functions for coding and decoding sequences of natural numbers such as the prime numbers and factorizations, and most of the useful number theoretic and metamathematical functions. It is also closed under limited recursion and bounded minimization [Cut80,Ros84]. However, E does not contain all recursive functions, or even all primitive recursive ones. For instance, Proposition 1 shows that it does not contain the iterated exponential exp [m] x) where the number of iterations m is a variable, since any function in E has an upper bound where m is xed. ....

[Article contains additional citation context not shown here]

N. J. Cutland. Computability: an introduction to recursive function theory. Cambridge University Press, 1980.

....exponential exp [m] x) is in E , where exp [0] x) x and exp [m 1] x) 2 exp [m] x) In fact, these are the fastest growing functions in E , in the sense that no elementary function can grow faster than exp [m] for some fixed m. The following bound will be useful to us below [Cut80]: Proposition 1 If f # E, there is a number m such that, for all x, f(x) # exp [m] #x#) where #x# = max i x i . The elementary functions also correspond to a natural time complexity class: Proposition 2 The elementary functions are exactly the functions computable by a Turing machine ....

....the connectives of propositional calculus, functions for coding and decoding sequences of natural numbers such as the prime numbers and factorizations, and most of the useful number theoretic and metamathematical functions. It is also closed under limited recursion and bounded minimization [Cut80,Ros84]. However, E does not contain all recursive functions, or even all primitive recursive ones. For instance, Proposition 1 shows that it does not contain the iterated exponential exp [m] x) where the number of iterations m is a variable, since any function in E has an upper bound where m is ....

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N. J. Cutland. Computability: an introduction to recursive function theory. Cambridge University Press, 1980.

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Nigel Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

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N. J. Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

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Nigel Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

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Cutland, N., J.: Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

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N. J. Cutland, Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

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[Cutland, 1980 ] Nigel Cutland, Computability: An introduction to recursive function theory, Cambridge Univ. Press, Cambridge, Engl., 1980.

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N.J. Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.

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