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37
The Convenience of Tilings
 In Complexity, Logic, and Recursion Theory
, 1997
"... Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling proble ..."
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Cited by 36 (0 self)
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Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling problems in subsequent reductions showing hardness of the combinatorial problems that really matter. We ilustrate our approach by means of three examples: a short reduction chain to the Knapsack problem followed by a Hilbert 10 reduction using similar ingredients. Finally we reprove the Deterministic Exponential Time lowerbound for satisfiablility in Propositional Dynamic Logic. The resulting reductions are relatively simple; they do however infringe on the principle of orthogonality of reductions since they abuse extra structure in the instances of the problems reduced from which results from the fact that these instances were generated by a master reduction previously. 1 Introduction This paper...
Incompleteness Theorems for Random Reals
, 1987
"... We obtain some dramatic results using statistical mechanicsthermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property tha ..."
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Cited by 32 (1 self)
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We obtain some dramatic results using statistical mechanicsthermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property that if one varies a parameter and asks whether the number of solutions is finite or infinite, the answer to this question is indistinguishable from the result of independent tosses of a fair coin. This yields a number of powerful Godel incompletenesstype results concerning the limitations of the axiomatic method, in which entropyinformation measures are used. c fl 1987 Academic Press, Inc. 2 G. J. Chaitin 1. Introduction It is now half a century since Turing published his remarkable paper On Computable Numbers, with an Application to the Entscheidungsproblem (Turing [15]). In that paper Turing constructs a universal Turing machine that can simulate any other Turing machine. He also use...
Hilbert’s tenth problem and Mazur’s conjecture for large subrings of Q
 J. Amer. Math. Soc
"... Abstract. We give the first examples of infinite sets of primes S such that Hilbert’s Tenth Problem over Z[S −1] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E ′ over Z[S −1] such that the topolog ..."
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Abstract. We give the first examples of infinite sets of primes S such that Hilbert’s Tenth Problem over Z[S −1] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E ′ over Z[S −1] such that the topological closure of E ′ (Z[S −1]) in E ′ (R) has infinitely many connected components. 1.
Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers
"... Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of . ..."
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Cited by 12 (2 self)
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Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of .
Asking Questions versus Verifiability
, 1992
"... this paper, # 0 , # 1 , # 2 , . . . denotes an acceptable programming system [17], also known as a Godel numbering of the partial recursive functions [15]. The function # e is said to be computed by the program e. ..."
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Cited by 10 (4 self)
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this paper, # 0 , # 1 , # 2 , . . . denotes an acceptable programming system [17], also known as a Godel numbering of the partial recursive functions [15]. The function # e is said to be computed by the program e.
HILBERT’S TENTH PROBLEM FOR FUNCTION FIELDS OF VARIETIES OVER NUMBER FIELDS AND pADIC
, 2006
"... Let k be a subfield of a padic field of odd residue characteristic, and let L be the function field of a variety of dimension n ≥ 1 over k. Then Hilbert’s Tenth Problem for L is undecidable. In particular, Hilbert’s Tenth Problem for function fields of varieties over number fields of dimension ≥ 1 ..."
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Cited by 7 (2 self)
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Let k be a subfield of a padic field of odd residue characteristic, and let L be the function field of a variety of dimension n ≥ 1 over k. Then Hilbert’s Tenth Problem for L is undecidable. In particular, Hilbert’s Tenth Problem for function fields of varieties over number fields of dimension ≥ 1 is undecidable.
CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSALEXISTENTIAL FORMULA
"... Abstract. We prove that Z in definable in Q by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Qmorphisms, whether there exists one that is surjective on rational points. We also giv ..."
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Cited by 5 (0 self)
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Abstract. We prove that Z in definable in Q by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Qmorphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers. 1. Introduction. 1.1. Background. D. Hilbert, in the 10th of his famous list of 23 problems, asked for an algorithm for deciding the solvability of any multivariable polynomial equation in integers. Thanks to the work of M. Davis, H. Putnam, J. Robinson [DPR61], and Y. Matijasevič [Mat70], we know that no such algorithm
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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Cited by 5 (0 self)
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
A Survey of Inductive Inference with an Emphasis on Queries
 Complexity, Logic, and Recursion Theory, number 187 in Lecture notes in Pure and Applied Mathematics Series
, 1997
"... this paper M 0 ; M 1 ; : : : is a standard list of all Turing machines, M ..."
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this paper M 0 ; M 1 ; : : : is a standard list of all Turing machines, M
DESCENT ON ELLIPTIC CURVES AND HILBERT’S TENTH PROBLEM
"... Abstract. Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert’s Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility seque ..."
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Abstract. Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert’s Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers. 1. Hilbert’s Tenth Problem In 1970, Matijasevič [10], building upon earlier work of Davis, Putnam and Robinson [5], resolved negatively Hilbert’s Tenth Problem for the ring Z, of rational integers. This means there is no general algorithm which will decide if a polynomial equation, in several variables, with integer coefficients has an integral solution. Equivalently, one says Hilbert’s Tenth Problem is undecidable for the integers. See [14, Chapter 1] for a full overview and background reading. The same problem, except now over the rational field Q, has not been resolved. In other words, it is not known if there is an algorithm which will decide if a polynomial equation with integer coefficients (or rational coefficients, it doesn’t matter) has a rational solution. Recently, Poonen [11] took a giant leap in this direction by proving the same negative result for some large subrings of Q. To make this precise, given a prime p of Z, let .p denote the usual padic absolute value. Let S denote a set of rational primes. Write ZS = Z[1/S] = {x ∈ Q: xp ≤ 1 for all p / ∈ S}, 1991 Mathematics Subject Classification. 11G05, 11U05. Key words and phrases. Elliptic curve, elliptic divisibility sequence, Hilbert’s