Register machine proof of the theorem on exponential diophantine representation of enumerable sets

1984 ◽  
Vol 49 (3) ◽  
pp. 818-829 ◽  
Author(s):  
J. P. Jones ◽  
Y. V. Matijasevič
Keyword(s):  
Natural Number ◽  
Polynomial Time ◽  
Simple Proof ◽  
Diophantine Equations ◽  
Polynomial Equation ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Exponential Relation

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

Universal diophantine equation

1982 ◽  
Vol 47 (3) ◽  
pp. 549-571 ◽  
Author(s):  
James P. Jones
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
Positive Integers ◽  
Martin Davis ◽  
Exponential Relation

In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the formHere P is a polynomial with integer coefficients and the variables range over positive integers.In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved [11] that the exponential relation y = 2x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the formFrom this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets.Now it is well known that the recursively enumerable sets W1, W2, W3, … can be enumerated in such a way that the binary relation x ∈ Wv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,

Some strongly undecidable natural arithmetical problems, with an application to intuitionistic theories

2003 ◽  
Vol 68 (1) ◽  
pp. 262-266
Author(s):  
Panu Raatikainen
Keyword(s):  
Number Theory ◽  
Essential Role ◽  
Diophantine Equations ◽  
Formal Learning ◽  
Trial And Error ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Formal Learning Theory ◽  
Martin Davis ◽  
Ordinary Number

Although Church and Turing presented their path-breaking undecidability results immediately after their explication of effective decidability in 1936, it has been generally felt that these results do not have any direct bearing on ordinary mathematics but only contribute to logic, metamathematics and the theory of computability. Therefore it was such a celebrated achievement when Yuri Matiyasevich in 1970 demonstrated that the problem of the solvability of Diophantine equations is undecidable. His work was building essentially on the earlier work by Julia Robinson, Martin Davis and Hilary Putnam (1961), who had showed that the problem of solvability of exponential Diophantine equations is undecidable. One should note, however, that although it was only Matiyasevich's result which finally solved Hilbert's tenth problem, already the earlier result had provided a perfectly natural problem of ordinary number theory which is undecidable.Nevertheless, both the set of Diophantine equations with solutions and the set of exponential Diophantine equations with solutions are still semi-decidable, that is, recursively enumerable (i.e., Σ10); if an equation in fact has a solution, this can be eventually verified. More generally, they are — as are their complements, the sets of equations with no solutions, which are Π10, — also Trial and Error decidable (Putnam [1965]), or decidable in the limit (Shoenfield [1959]), for every Δ20 set is (and conversely). This last-mentioned natural “liberalized” notion of decidability has begun more recently to play an essential role e.g., in so-called Formal Learning Theory (see e.g., Osherson, Stob, and Weinstein [1986], Kelly [1996]).

On the Sieve of Eratosthenes

1987 ◽  
Vol 39 (5) ◽  
pp. 1107-1122 ◽  
Author(s):  
M. Ram Murty ◽  
S. Saradha
Keyword(s):  
Natural Number ◽  
Simple Proof ◽  
Classical Theorem ◽  
Natural Numbers ◽  
Normal Order ◽  
Prime Factors ◽  
Image Position

Let v(n) denote the number of distinct prime factors of a natural number n. A classical theorem of Hardy and Ramanujan states that the normal order of v(n) is log log n. That is, given any , the number of natural numbers not exceeding x which fail to satisfy the inequality1is o(x) as x → ∞. A very simple proof of this was subsequently given by Turán. He showed that2

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

scholarly journals RAMSEY’S THEOREM FOR PAIRS ANDKCOLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

2017 ◽  
Vol 82 (2) ◽  
pp. 737-753
Author(s):  
STEFANO BERARDI ◽  
SILVIA STEILA
Keyword(s):  
Natural Number ◽  
Ramsey's Theorem ◽  
Recursively Enumerable ◽  
Ramsey’S Theorem ◽  
Heyting Arithmetic ◽  
Classical Principle ◽  
Image Position ◽  
Intuitionistic Arithmetic

AbstractThe purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and some branch with infinitely many children of indexi.

A theorem on shortening the length of proof in formal systems of arithmetic

1975 ◽  
Vol 40 (3) ◽  
pp. 398-400 ◽  
Author(s):  
Robert A. di Paola
Keyword(s):  
Ab Initio ◽  
Natural Number ◽  
A Priori ◽  
Relative Length ◽  
Constant Term ◽  
Recursive Functions ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Image Position ◽  
Standard Enumeration

This note is concerned with an aspect of the length of proof of formulas in recursively enumerable theories T adequate for recursive arithmetic. In particular, we consider the relative length of proof of formulas in the theories T and T(S), where F represents an r.e. set A in T and T(S) is the theory obtained from T by adjunction, as a new axiom, of a sentence S undecidable in T.Throughout the sequel T is a consistent, r.e. theory with standard formalization [7] in which all recursive functions of one variable are definable, and in which there is a binary formula x ≤ satisfying the well-known conditions [7]:Here is the constant term corresponding to the natural number n. Wn is the nth r.e. set in a standard enumeration of the r.e. sets. Also, we assume an a priori Gödel numbering of our formalism satisfying the usual conditions, so that all formulas are numbers ab initio.In the more common applications of the theorem below, if F is a k-ary formula of T, is a natural number that measures in some way the length of the shortest proof of in T.

scholarly journals Two exponential Diophantine equations

1997 ◽  
Vol 39 (2) ◽  
pp. 231-232 ◽  
Author(s):  
Erik Dofs
Keyword(s):  
Prime Number ◽  
Diophantine Equations ◽  
Rational Integer ◽  
Open Problems ◽  
Advanced Method ◽  
Exponential Diophantine Equations ◽  
Analytical Results ◽  
Image Position

In [3], two open problems were whether either of the diophantine equationswith n ∈ Z and f a prime number, is solvable if ω > 3 and 3 √ ω, but in this paper we allow f to be any (rational) integer and also 3 | ω. Equations of this form and more general ones can effectively be solved [5] with an advanced method based on analytical results, but the search limits are usually of enormous size. Here both equations (1) are norm equations in K (√–3): N(a + bp) = fw with p = (√–1 + –3)/2 which makes it possible to treat them arithmetically.

Residual Gas Reactions in the Electron Microscope: I. Qualitative Observations on the Water Gas Reaction

1968 ◽  
Vol 26 ◽  
pp. 292-293
Author(s):  
Richard E. Hartman ◽  
Roberta S. Hartman ◽  
Peter L. Ramos
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Gas Phase ◽  
Absolute Temperature ◽  
Residual Gas ◽  
Gas Reactions ◽  
Image Position ◽  
The Right ◽  
Water Gas ◽  
Thinning Rate

The action of water and the electron beam on organic specimens in the electron microscope results in the removal of oxidizable material (primarily hydrogen and carbon) by reactions similar to the water gas reaction .which has the form:The energy required to force the reaction to the right is supplied by the interaction of the electron beam with the specimen.The mass of water striking the specimen is given by:where u = gH2O/cm2 sec, PH2O = partial pressure of water in Torr, & T = absolute temperature of the gas phase. If it is assumed that mass is removed from the specimen by a reaction approximated by (1) and that the specimen is uniformly thinned by the reaction, then the thinning rate in A/ min iswhere x = thickness of the specimen in A, t = time in minutes, & E = efficiency (the fraction of the water striking the specimen which reacts with it).

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

scholarly journals FRAGMENTS OF APPROXIMATE COUNTING

2014 ◽  
Vol 79 (2) ◽  
pp. 496-525 ◽  
Author(s):  
SAMUEL R. BUSS ◽  
LESZEK ALEKSANDER KOŁODZIEJCZYK ◽  
NEIL THAPEN
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Image Position

AbstractWe study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

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