Martin Davis. Hilbert's Tenth Problem is Unsolvable. American Mathematical Monthly, 80(3):233-269, March 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....4) finds a pair ( 3 ; 4 ) such that p = 4 . Finally, square root of 1 modulo p can be found efficiently. ut Exponential Relation is in PD. Matiyasevich was the first to describe an explicit representing polynomial for the exponential relation. Alternative polynomial were later found in [Dav73,JSWW76], but none of these polynomials is really practical for our purposes. As an example, a polynomial proposed in [MR75] was used in [AM76] to show that exponential relation is in D, but nevertheless one cannot call the results really practical due to the relatively long witnesses. Next, we construct ....

Martin Davis. Hilbert's Tenth Problem is Unsolvable. American Mathematical Monthly, 80(3):233--269, March 1973.

....results on the decidability of unification problems indicate the difficulty of this problem. The undecidability results in this area have been based on the undecidability of Hilbert s 10 th problem, solving Diophantine equa tions over the integers, which was shown undecidable by Maiyasevi [Davis 73] JAmborg 85] and [Szabo 78] investigate combinations of theories with noradisjoint function symbols by studying the lattice of sub theories that are consistent with Peano arithmetic. Szabo 78] shows the undecidability of the associative theory with two sided distributivity, AD, while [Arnborg ....

M. Davis, "Hilbert's Tenth Problem is Unsolvable," American Mathematical Monthly 80(3):233-269, 1973.

....polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z This question, otherwise known as Hilbert s 10th problem, has been answered negatively in the work of M. Davis, H. Putnam, J. Robinson and Yu. Matijasevich. See [3] and [4] Since the time when this result was obtained, similar questions have been raised for other fields and rings. Arguably the two most interesting and di#cult problems in the area are the questions of Diophantine decidability of Q and the rings of algebraic integers of arbitrary number ....

Martin Davis. Hilbert's tenth problem is unsolvable. American Mathematical Monthly, 80:233--269, 1973.

....polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z This question, otherwise known as Hilbert s 10th problem, has been answered negatively in the work of M. Davis, H. Putnam, J. Robinson and Yu. Matijasevich. See [2] and [3] Since the time when this result was obtained, similar questions have been raised for other fields and rings. In other words, let R be a recursive ring. Then, given an arbitrary polynomial equation in several variables over R, is there a uniform algorithm to determine whether such an ....

Martin Davis. Hilbert's tenth problem is unsolvable. American Mathematical Monthly, 80:233--269, 1973.

....polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z This question, otherwise known as Hilbert s 10th problem, has been answered negatively in the work of M. Davis, H. Putnam, J. Robinson and Yu. Matijasevich. See [2] and [3] Since the time when this result was obtained, similar questions have been raised for other fields and rings. Arguably the two most interesting and di#cult problems in the area are the questions of Diophantine decidability of Q and the rings of algebraic integers of arbitrary number ....

M. Davis, Hilbert's Tenth Problem Is Unsolvable, American Mathematical Monthly, 80(1973), p. 233-269.

....polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z This question, otherwise known as Hilbert s 10th problem, has been answered negatively in the work of M. Davis, H. Putnam, J. Robinson and Yu. Matijasevich. See [6] and [7] Since the time when this result was obtained, similar questions have been raised for other fields and rings. Arguably the two most interesting and di#cult problems in the area are the questions of Diophantine decidability of Q and the rings of algebraic integers of arbitrary number ....

Martin Davis. Hilbert's tenth problem is unsolvable. American Mathematical Monthly, 80:233--269, 1973.

....2 = 1; 2) and obtain representations similar to that in [9] For example, see [3, 8, 14] This allows us to prove in a simple way that exponentiation is Diophantine, i.e. the set of triples f ha; b; ci : a = b c g is Diophantine. The reader may find a detailed exposition of this technique in [3, 4, 8, 12, 13]. Up to now, all known ways of proving the fact that exponentiation is Diophantine are based on Diophantine representations of second order recurrences. But this is insufficient for some applications. Thus, we have need for a larger collection of sequences of exponential growth that allow for ....

M. Davis, "Hilbert's tenth problem is unsolvable," Am. Math. Monthly, 80, No. 3, 233--269 (1973).

.... (ABS X) Y) MINUS (PLUS (ABS X) ABS Y) ZTIMES X Y) IF (AND (NUMBERP X) NUMBERP Y) TIMES X Y) IF (AND (NUMBERP X) MINUSP Y) MINUS (TIMES X (ABS Y) IF (AND (MINUSP X) NUMBERP Y) MINUS (TIMES (ABS X) Y) 49 It was proved by Matiyacevic, a Russian mathematician [13]. SOME THEORETICAL ASPECTS OF THE EGS EXAMPLE GENERATION 105 (TIMES (ABS X) ABS Y) ZEXP X Y) IF (ZEROP Y) 1 (ZTIMES X (ZEXP X (SUB1 Y) The functions ZPLUS, ZTIMES, and ZEXP actually extend PLUS, TIMES, and EXP over the domain of integers instead of the domain of natural numbers. ....

Davis, M. "Hilbert's Tenth Problem is Unsolvable". American Mathematical Monthly 80 (1973), 233-269.

....admissible concrete domains (in the sense of [3] that satisfy the preconditions of Theorem 1. Examples are the (nonnegative) integers, reals, and rationals together with the predicates P=1;P = P 6= P 6= 1 . Sketch of the proof of Theorem 1: The proof is by reduction of Hilbert s 10th problem [5] to satisfiability of concepts, i.e. for given polynomials P;Q 2 IN#x1; x m #, we construct an ALC#D agg # concept CP;Q that is satisfiable iff the polynomial equation P #x1; x m#=Q #x 1 ; x m# (1) has a solution in IN m . Undecidability of subsumption follows because C is ....

M. Davis, `Hilbert's tenth problem is unsolvable', American Mathematical Monthly, 80, 233--269, (1973).

....a polynomial in several 6 variables, with integer coe#cients, and we wish to decide whether it has an integer solution. Building on previous work by Davis and Robinson, this problem was proved undecidable in [Matiyasevich, 1970] An elementary exposition of Matiyasevich s Theorem can be found in [Davis, 1973]. A third example, which is remarkable in its simplicity, is Post s correspondence problem. We are given finitely many pairs of words 2 (x 1 , y 1 ) x n , y n ) and we wish to decide whether there exists some n # 1 and a finite sequence (i 1 , i m ) of integers in the range ....

Davis, M. (1973). Hilbert's tenth problem is unsolvable, American Mathematical Monthly, 80, 233-269.

.... Let Theta( x) and Phi( x) be two polynomials with natural number coefficients over n variables (represented by x) The equation Theta( x) Phi( x) is referred to as a Diophantine equation and the problem of determining whether there exists a solution in the natural numbers is undecidable [6]. The next lemma states that Diophantine equations 5 The form of relative information capacity we use in this work was first studied in [8] and [10] Information capacity has also been applied to a number of translation and integration problems [1, 12, 15, 16, and others] Information Systems, ....

M. Davis. Hilbert's Tenth Problem is Unsolvable. American Mathematical Monthly, 8(3):233--269, Mar. 1973.

....mention without proof an example that shows particularly clearly the relationship between the number of bits of axioms that are assumed and what can be deduced. This example is based on the work of M. Davis, Ju. V. Matisjasevic, H. Putnam, and J. Robinson that settled Hilbert s tenth problem (cf. [30]) There is a polynomial P in k 2 variables with integer coefficients that has the following property. Consider the infinite string whose ith bit is 1 or 0 depending on whether or not the set S i = fn 2 Nj9x 1 ; x k 2 N P (i; n; x 1 ; x k ) 0g is infinite. Here N denotes the ....

Davis, M. Hilbert's tenth problem is unsolvable. Amer. Math. Mon. 80 (1973), 233--269.

....concrete domains (in the sense of [3] that satisfy the preconditions of Theorem 1. Examples are the (nonnegative) integers, reals, and rationals together with the predicates P=1 ; P= P 6= P 6= 1 . Sketch of the proof of Theorem 1: The proof is by reduction of Hilbert s 10th problem [5] to satisfiability of concepts, i.e. for given polynomials P; Q 2 IN[x1 ; xm ] we construct an ALC(D agg ) concept CP;Q that is satisfiable iff the polynomial equation P (x1 ; xm) Q(x1 ; xm ) 1) has a solution in IN m . Undecidability of subsumption follows ....

M. Davis. Hilbert's tenth problem is unsolvable. American Mathematical Monthly, 80:233--269, 1973.

.... ] and in the quantifier free linear case one even has incremental methods for deciding satisfiability [ Jaakola,1990; Jaffar et al. 1992 ] The concrete domain Z is not admissible since Hilbert s Tenth Problem one of the most prominent undecidable problems [ Matijacevic,1970; Davis,1973 ] is a special case of its satisfiability problem. In [ Baader and Hanschke,1991 ] it is shown how the tableau based reasoning algorithm for ALC can be extended to ALC(D) provided that D is admissible. Theorem 3.4 If D is an admissible concrete domain, then the subsumption problem is ....

M. Davis. Hilbert's tenth problem is unsolvable. Am. Math. Monthly, 80:239--269, 1973.

.... in the equalities and inequalities have to be linear) there exist more efficient methods (see e.g. Weispfenning, 1988; Loos and Weispfenning, 1990 ] The concrete domain Z is not admissible since Hilbert s Tenth Problem one of the most prominent undecidable problems [ Matijacevic, 1970; Davis, 1973 ] is a special case of its satisfiability problem. Sometimes the adequate modeling of a problem domain could be facilitated if reference to more than one concrete domain would be possible in a terminology. Therefore, we show how two disjoint admissible concrete domains D 1 and D 2 can be ....

M. Davis. Hilbert's tenth problem is unsolvable. Am. Math. Monthly, 80:239--269, 1973.

....among D. Hilbert s famous 23 problems it was not until 1970 when Y. Matijasevich could prove that it is unsolvable or undecidable [Matijasevich, 1970] There is no recursive computable predicate for diophantine equations which holds if a solution in ZZ (or IIN) exists and fails otherwise [Davis, 1973, Theorem 7.4] 3.2 THE NETWORK ARCHITECTURE The construction of a neural network N for each diophantine D is now straight forward (see Fig.1) It is just a three step construction. First, each variable x i of D is represented in N by a small sub network. The structure of these modules is quite ....

Martin Davis. Hilbert's tenth problem is unsolvable. Amer. Math. Monthly, 80:233--269, March 1973.

....order logic t 1 and t 2 , a unifier is a substitution such that t 1 =t 2 . Unification is the process of finding the most general unifier (mgu) The earliest reference for unification of general terms goes back to 1920 when E. Post mentioned in his diary and notes, partially published in [14], a hint of the concept of a unification algorithm that computes a most general representation as opposite to all possible instantiations. The first published unification algorithm was given in J. Herbrand s thesis in 1930 [23] In his algorithm, Herbrand described three properties related to the ....

M. Davis. Hilbert's tenth problem is unsolvable. American Mathematical Monthly, 80, 1973.

....that determining whether the reachable markings for two arbitrary Petri nets are identical, called the Petri net equality problem, is undecidable. This is proved using a reduction of Hilbert s Tenth Problem to the Petri net equality problem; Hilbert s Tenth Problem has been shown to be unsolvable [Dav73] Grabowski goes on to prove that the problem becomes decidable if the Petri nets in question have no unbounded places; we note that safe Petri nets have no unbounded places. Mayr and Meyer prove in [MM81] that, if the reachability sets of two Petri nets are finite (which is true of safe Petri ....

Martin Davis. Hilbert's tenth problem is unsolvable. The American Mathematical Monthly, 80(3):233-269, March 1973.

.... two polynomials with natural number coefficients over n variables ranging over the natural numbers (represented by x) The equation Theta( x) Phi( x) is referred to as a Diophantine equation and the problem of determining whether there exists a solution in the natural numbers is undecidable [Dav73]. The next lemma states that Diophantine equations without constant terms may be encoded by a SIG schema. Specifically, given a Diophantine equation Theta( x) Phi( x) we construct a schema S such that every valid instance of S corresponds to a solution to the Diophantine equation and vice ....

M. Davis. Hilbert's Tenth Problem is Unsolvable. American Mathematical Monthly, 8(3):233--269, March 1973.

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Martin Davis. Hilbert's Tenth Problem is Unsolvable. American Mathematical Monthly, 80(3):233-269, March 1973.

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Davis, M.: Hilbert's tenth problem is unsolvable. Amer. Math. Monthly 80 (1973), 233-269.

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Davis, M. Hilbert's tenth problem is unsolvable. Amer. Math. Mon. 80 (1973), 233--269.

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M. Davis, "Hilbert's tenth problem is unsolvable," American Mathematical Monthly, vol. 80, pp. 233-269, 1973.

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M. Davis. Hilbert's tenth problem is unsolvable. American Mathematical Monthly, 80:233-- 269, 1973.

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Davis, M., Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80 (1973) 233--269.

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