The Decision Problem for Exponential Diophantine Equations
About: This article is published in Journal of Symbolic Logic.The article was published on 1970-03-01. It has received 247 citations till now. The article focuses on the topics: Diophantine equation & Diophantine set.
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31 Jul 2013TL;DR: The Lambda Calculus has been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs.
Abstract: This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.
927Â citations
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TL;DR: It is proved that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.
Abstract: We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant of a 4-tensor is NP-, #P-, and VNP-hard. We shall argue that our results provide another view of the boundary separating the computational tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones.
649Â citations
Cites background or methods from "The Decision Problem for Exponentia..."
...To give the reader a sense of the generality of nonlinear equation solving, we .rst explain, in very simpli.ed form, the connection of quadratic systems to the Halting Problem established in the seminal works [Davis et al. 1961; Matijasevi. c 1970]....
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...he reader a sense of the generality of nonlinear equation solving, we rst explain, in very simplied form, the connection of quadratic systems to the Halting Problem established in the seminal works [Davis et al. 1961; Matijasevic 1970]. Collectively, these papers resolve (in the negative) Hilbert’s 10th Problem [Hilbert 1902]: Is there a nite procedure to decide the solvability of general polynomial equations ov...
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TL;DR: In this article, Hilbert's Tenth Problem is shown to be unsolvable, and the paper concludes that Hilbert's problem cannot be solved by any number of solutions, e.g.,
Abstract: (1973). Hilbert's Tenth Problem is Unsolvable. The American Mathematical Monthly: Vol. 80, No. 3, pp. 233-269.
402Â citations
TL;DR: In this paper, the authors give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. They also give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logariths can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied.
Abstract: This paper deals with the problem of telling whether a given elementary function, in the sense of analysis, has an elementary indefinite integral. In ?1 of this work, we give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. By using functions of a complex, rather than a real variable, we can limit ourselves to exponentiation, taking logs, and algebraic operations in defining the elementary functions, since sin, tan- 1, etc., can be expressed in terms of these three. Following Ostrowski [9], we use the concept of a differential field. We strengthen the classical Liouville theorem and derive a number of consequences. ?2 uses the terminology of mathematical logic to discuss formulations of the problem of integration in finite terms. ?3 (the major part of this paper) uses the previously developed theory to give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logarithms can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied. The man who established integration in finite terms as a mathematical discipline was Joseph Liouville (1809-1882), whose work on this subject appeared in the years 1833-1841. The Russian mathematician D. D. Mordoukhay-Boltovskoy (1876-1952) wrote much on this and related matters. The present writer received his introduction to this subject through the book [10] by the American J. F. Ritt
321Â citations
TL;DR: Let E be a set of expressions representing real, single valued, partially defined functions of one real variable and E * be the set of functions represented by expressions in E .
Abstract: Let E be a set of expressions representing real, single valued, partially defined functions of one real variable. E* will be the set of functions represented by expressions in E.If A is an expression in E, A(x) is the function denoted by A.It is assumed that E* contains the identity function and the rational numbers as constant functions and that E* is closed under addition, subtraction, multiplication and composition.
265Â citations