Alan M Turing. On computable numbers, with an application to the entscheidungs problem. Proc. Lond. Math. Soc., 43(2), 1936.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we use the constructor term t as a synonym for its unique value eval (t) 5 Related Work and Results Gurevich s initial program for ASMs is pure mathematical: a mathematically defined dynamic system, which allows one to model arbitrary algorithms. His thesis [13] is that unlike Turing machines [28] his machines would allow to model algorithm without encoding data structures and splitting execution steps. He observed that every conceivable data structure can be modeled as a Tarski structure, and every possible state change of the algorithm can be modeled by a set of explicit, pointwise ....

A. M. Turing. On computable numbers, with an application to the Entscheidungs problem. Proc. London Math. Soc., 2(42):230--265, 1936. (Corrections on volume 2(43):544--546).

....be speculated. Suffice it to say that the prospects are good. Modeling For arbitrary programs with potentially unbounded capacity to store and retrieve information, no algorithmic techniques can exist for mechanically proving all properties of interest. In this form, the problem is undecidable [T36]. If we can put a finite bound on the possible memory use of a program, we obtain a system with a finite number of possible states (i.e. configurations of memory) that can in theory be enumerated. We can conceive of constructing the execution graph of such a program, to capture the ....

A.M. Turing, On computable numbers, with an application to the Entscheidungs problem. Proc. London Mathematical Soc., Ser. 2-42, pp. 230-265 (see p. 247), 1936.

....code for a new telephone switch. 1. Introduction The construction of a reliable logic verification system for software systems has long been an elusive goal. It is well known that in general even simple properties, such as program termination or system deadlock, are formally undecidable [23]. Efforts that target the development of mechanical verification systems therefore usually concentrate on carefully defined subsets of programs. Such subsets can, for instance, be obtained by imposing syntactic restrictions on general programming languages. These restrictions, however, are usually ....

A.M. Turing, On computable numbers, with an application to the Entscheidungs problem. Proc. London Mathematical Soc., Ser. 2-42, pp. 230-265 (see p. 247), 1936.

....Introduction The present work brings together two ideas, namely type two computability, and p adic fields. We start with a brief review of the notion a computable number and of a p adic field as a background. The basic idea for computable real numbers is contained in Turing s fundamental paper [20], where he introduced the notion of a computable (or recursive) real number, and initiated the study of the relevant concepts and notions. He described the recursive real numbers as a subset of the real numbers obtained by imposing restrictions on the definition of a real number. A real number can ....

....of recursive analysis, i.e. a real or p adic number is viewed as infinite object, and is understood as the limit of finite objects, namely rational numbers. This approach for the case of real numbers has been extensively developed by various authors. A partial list of related work follows: [1, 2, 7, 8, 6, 10, 13, 15, 16, 17, 18, 14, 20]. The main issue in the present work is to study the analogous problems in the case of p adic numbers and verify which ones carry over and which ones fail and the reason for the failures. The present approach to the nature of a real or a p adic number should be contrasted with the ....

A. M. Turing. On computable numbers, with an application to the entscheidungs problem. Proc. London Math. Society, 42:230--265, 1937. 22

....findings. 2. Mechanized model extraction It is known that it is not possible to devise an algorithm that could prove arbitrary properties of arbitrary C or C programs. It is not even possible to mechanically prove a single specific, property such as program termination for arbitrary programs [T36][S65] So if we want to be able to render proofs, we have no choice but to restrict ourselves to a smaller class of programs. An example of such a class is the set of all finite state programs: programs that on any given input can generate only a finite number of possible program states (i.e. ....

Turing, A.M., On computable numbers, with an application to the Entscheidungs problem. Proc. London Mathematical Soc., Ser. 2-42, pp. 230-265 (see p. 247), 1936.

....is the limit of the computation. 2. Digit Notations for Reals. 2 Lazy functional languages are not the only possible approach to computability on infinite streams. Several authors used the digit notation for reals and a different approach to the computability, as for example Turing machine in [15], 17] 18] or approximations spaces in [14] The standard digit notation is not suitable for the lazy computation described above. The usual solution to this problem consists in adding new digits to the notation, often negative digits. In this work we present an alternative solution. It consists ....

A.M. Turing. On computable numbers, with an application to the entscheidungs problem. In Proc. London Math. Soc. 42, pages 230--265, 1937.

....uniquely identifies the particular root of the polynomial. Note that, given such data, an arbitrarily good rational approximation can be easily computed, say, by Newton s iteration. This is the approach taken by Lov asz in [11] and it is consistent with Turing s notion of a computable real number [15]. In fact, in terms of computational complexity, the fact that quintic equations may not have radical expressions for their roots is largely irrelevant; it simply rules out one mode of expression. Of course, as we will see later, the complexity of the Galois group itself will enter the picture. ....

A. M. Turing, On computable real numbers, with an application to the Entscheidung problems, Proc. London Math. Soc., 42, 230--265.

.... t 4 elements of the alphabet V (including to the left and to the right) 8 A simplified universal Turing machine with three tapes From the similarity theorem is possible to see that if we translate a UTM with the characteristics we have defined (for instance the one defined by Turing in [8]) we obtain a universal deterministic extended mH system. However we have obtained a translation from a UTM with three tapes (described in [5] demonstrating in this way the extreme versatility of the splicing operation. Now a short summary of the UTM we have translated is given and some parts ....

A. M. Turing, On computable numbers, with an application to the Entscheidungs problem, Proc. London. Math. Soc., Series 2, Vol. 24, 230--265, 1936 14

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Alan M Turing. On computable numbers, with an application to the entscheidungs problem. Proc. Lond. Math. Soc., 43(2), 1936.

No context found.

Alan M Turing. On computable numbers, with an application to the entscheidungs problem. Proc. Lond. Math. Soc., 43(2), 1936.

No context found.

A. M. Turing. On Computable Numbers, with an Application to the Entscheidungs problem. Proceedings of the London Mathematical Society, Series 2(42):230-265, 1936-1937.

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A. Turing. On computable numbers, with an application to the Entscheidungs problem. In Proc. Lond. Math. Soc., volume 2, pages 230--365, 1936.

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A. M. Turing, On computable numbers, with an application to the entscheidungs problem, Proc. London Math. Soc. (2) 442 (1936), 230265.

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