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Hilbert's Tenth Problem.
The American Mathematical Monthly
... Coordinate projections of semi-algebraic sets of lattice points are not in general again semialgebraic. Indeed, Matiyasevich's Theorem [16] (see Davis [10] and Matiyasevich [17]) states that all recursively enumerable sets of integer points are Diophantine sets. Matiyasevich proved that every Diophantine set has rank at most 9 (see [14], and most recently Sun [24]), that is, it is the projection of the integer points of some semi-algebraic set defined over Z with at most 9 additional variables. ...
... and (f 1 , f 2 ) is not one of the 13 pairs (12, 12), (14,13), (14,14), (15,15), (16,15), (17,16), (18,16), (18,18), (20,17), (21,19), (23,20), (24,20), (26,21). ...
... and (f 1 , f 2 ) is not one of the 13 pairs (12, 12), (14,13), (14,14), (15,15), (16,15), (17,16), (18,16), (18,18), (20,17), (21,19), (23,20), (24,20), (26,21). ...
Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the intersection of a semi-algebraic set with an integer lattice. Such "semi-algebraic sets of lattice points" have not received much attention, which is surprising in view of a close connection to Hilbert's Tenth problem, which deals with their projections. We develop proof techniques in order to show that, despite the observations above, some f-vector sets are NOT semi-algebraic sets of lattice points. This is then proved for the set of all pairs of 4-dimensional polytopes, the set of all f-vectors of simplicial d-polytopes for , and the set of all f-vectors of general d-polytopes for . For the f-vector set of all 4-polytopes this remains open.
... We then study the productiveness of the equivalence to the identically 0 function problem by investigating and extending the previous work in [9,10], as well as the undecidability result of Hilbert's tenth problem (HTP). To the best of our knowledge, HTP is the primary method used to show undecidable results for real fields (for example, see [11]). We show that HTP and the equivalence to the identically 0 function problem studied in [9,10] are not only undecidable but also productive; hence, they are not provable. ...
... We then briefly introduce HTP and review the standard proof of its undecidability as presented in [11,16]. The definitions and notations used here are mainly from the book [11]. ...
... We then briefly introduce HTP and review the standard proof of its undecidability as presented in [11,16]. The definitions and notations used here are mainly from the book [11]. A deeper analysis of this undecidability proof reveals that the set of false instances of HTP is productive. ...
This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep reexamination of Hilbert’s tenth problem and the equivalence to the identically 0 function problem, extending the undecidability results of these problems into the realm of productiveness. Additionally, we study the complexity of the equivalence to the identically 0 function problem over different domains. We then construct highly efficient many-one reductions to establish Rice-style theorems for the study of real functions. Specifically, we show that many predicates, including those related to continuity, differentiability, uniform continuity, right and left differentiability, semi-differentiability, and continuous differentiability, are as hard as the equivalence to the identically 0 function problem. Due to their high efficiency, these reductions preserve nearly any level of complexity, allowing us to address both complexity and productiveness results simultaneously. By demonstrating these results, which highlight a more nuanced and potentially more intriguing aspect of real function theory, we provide new insights into how various properties of real functions can be analyzed.
... and applying Kummer's theorem (cf. Matiyasevich [7,Appendix]), which asserts that HWpnq is equal to the dyadic valuation of`2 n n˘, he concluded that HWpnq has a representation as an arithmetic term (which is displayed in Appendix A). ...
... For every three integers q ą 1, r ě 0 and t ě 0, there are further useful arithmetic terms representing the so-called generalized geometric progression of the r-th kind (cf. Matiyasevich [7,Appendix]): ...
... as shown by Matiyasevich [7,Appendix]. ...
We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors function, Euler's totient function, the modular inverse, the integer part of the root, the integer part of the logarithm, the multiplicative order and the discrete logarithm. Although these are very complicated, they only involve elementary operations, and to our knowledge no other closed form of this kind is known for the aforementioned functions.
... Denote by Y ⊆ V a set of representatives of the cosets y + W ∈ V/W , and identify V/W with Y . Then By (2)(3)(4)(5)(6)(7)(8)(9)(10), it suffices to show that ...
... This is a set of inequalities of exactly the form (2-3), with V replaced by V V/W = (V/W, (V j /W j ), ([ϕ j ])). By Lemma 2.20, s ∈ P(V V/W ), and since dim(V/W) < dim(V), we conclude directly from the inductive hypothesis that (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) holds, concluding the proof of Lemma 2.21. ...
... Remark 4.7 Hilbert's Tenth Problem, proper, asks for an algorithm to determine solvability in integers of finite systems of equations over Z. From such an algorithm one could positively resolve Hilbert's Tenth Problem for Q. However, by the celebrated theorem of Matiyasevich-Davis-Putnam-Robinson [10], no such algorithm exists. The problem for the rationals remains open. ...
Holder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of L^p norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequalities for torsion-free discrete Abelian groups in three respects.The optimal constant in any such inequality is shown to equal 1 whenever it is finite.An algorithm that computes the admissible polyhedron of exponents is developed. It is shown that nonetheless, existence of an algorithm that computes the full list of inequalitiesin the Bennett-Carbery-Christ-Tao description of the admissible polyhedron for all data,is equivalent to an affirmative solution of Hilbert's Tenth Problem over the rationals.That problem remains open.
... The equation related to Fermat's Last Theorem, + = , has turned out to have no natural solutions for > 2 (Darmon et al., 1995;Wiles, 1995). In general, however, the problem of assessing whether a given Diophantine equation has a solution is in the class of undecidable problems (Matiyasevich, 1993). It has been proven that there cannot exist an algorithm for solving all Diophantine equations (Matiyasievich's theorem, (Matiyasevich, 1993)). ...
... In general, however, the problem of assessing whether a given Diophantine equation has a solution is in the class of undecidable problems (Matiyasevich, 1993). It has been proven that there cannot exist an algorithm for solving all Diophantine equations (Matiyasievich's theorem, (Matiyasevich, 1993)). ...
... An example of a function ( ) is shown in Fig. 4. Such a function grows exponentially in approximation (Matiyasevich, 1993). Numerical determination of the curve ( ) is subject to some inaccuracy. ...
The article presents the so-called coin bag problem, which is modeled by linear Diophantine equations. The problem in question involves assessing the contents of a set of coins based on its weight. Since this type of problem is undecidable, a special variant of the problem was proposed for which effective problem-solving algorithms can be developed. In this paper, an original heuristic is presented (an algorithm based on problem decomposition) which allows to solve the coin bag problem in fewer steps compared to a brute force algorithm. The proposed approach was verified in a series of computational experiments. Additionally, an authentication scheme making use of the approach was proposed as an example of potential practical use.
... For the proof and more information of this deep result see [6]. ...
... We will assume that the reader is familiar with the basics of language theory (see [1,7,8,11]) and recursively enumerable sets (see [6,9,10,12]). These references should be consulted for all unexplained notation and terminology. ...
... For the proof of Theorem 5 see the monograph [6]. ...
We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers established by Matiyasevich. These variants use linear equations with one unkwown instead of polynomial equations with several unknowns. As a corollary we get undecidability results for linear equations over morphism semigoups and over matrix semigroups.
... Mazzanti's approach, the hypercube method (described in § 2.2), makes clever usage of elementary arithmetic to count the number of solutions to Diophantine equations. Hilbert's 10th problem, which asked for a general algorithm that can determine if an arbitrary Diophantine equation has solutions in N, was shown to be unsolvable by Matiyasevich (for the details, see [21]). Thus, while the hypercube method offers a potential approach, there can be no general algorithm nor procedure for constructing arithmetic terms by its application. ...
... As described by Matiyasevich in the appendix of [21], G r (q, t) can be calculated via the formula ...
We present the first fixed-length elementary closed-form expressions for the prime-counting function, pi(n), and the n-th prime number, p(n). These expressions are represented as arithmetic terms, requiring only a fixed and finite number of elementary arithmetic operations from the set: addition, subtraction, multiplication, division with remainder, exponentiation. Mazzanti proved that every Kalmar function can be represented by arithmetic terms. We develop an arithmetic term representing the prime omega function, omega(n), which counts the number of distinct prime divisors of a positive integer n. From this term, we find immediately an arithmetic term for the prime-counting function, pi(n). We utilize these results, along with a new arithmetic term for binomial coefficients and new prime-related exponential Diophantine equations to construct an arithmetic term for the n-th prime number, p(n), thereby providing a constructive solution to a fundamental question in mathematics: Is there an order to the primes?
... Contrary to this, we propose that quantum computation may be able to compute the noncomputables, provided certain hamiltonian and its ground state can be physically constructed. We propose a quantum algorithm for the classically noncomputable Hilbert's tenth problem [6] which ultimately links to the halting problem for Turing machines in the computation of partial recursive functions. ...
... This decision problem for such polynomial equations, which are also known as Diophantine equations, has eventually been shown in 1970 by Matiyasevich to be undecidable [6,7] in the Turing sense. It is consequently noncomputable/undecidable in the most general sense if one accepts, as almost everyone does, the Church-Turing thesis of computability. ...
We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle--that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal--quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.
... Once the aperiodic and periodic denominator bounds d a , d p are calculated, and possibly an Ansatz d user for missing factors in the denominator has been set, one can search for the numerator contribution. In general, it has been shown in [166] based on [147] that this problem is unsolvable: given a homogeneous PLDE with polynomial coefficients, there does not exist an algorithm that can determine all polynomial solutions. Nevertheless, one can search for the desired polynomial solutions by taking as an Ansatz a general polynomial num(c i ) with undetermined coefficients c i where the polynomial degree is set sufficiently high. ...
... In general there is no algorithm by the Davis-Matiyasevich-Putnam-Robinson theorem[147] that can decide if there is an integer root (or even infinitely many integer roots). However, in our applications the polynomials are usually small, mostly even linear and thus such integers λ i can be determined.Content courtesy of Springer Nature, terms of use apply. ...
For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.
... The Riemann hypothesis states that the non-trivial zeros of ζ(s) = ∞ n=1 1 n s are all on the line Re (s) = 1/2. The Riemann hypothesis can be shown to be refutable by finite means, due to a result by Matiyasevich [11]. Therefore, the first bits of Ω could tell us if the hypothesis is true or not. ...
Algorithmic information theory roots the concept of information in computation rather than probability. These lecture notes were constructed in conjunction with the graduate course I taught at Universit\`a della Svizzera italiana in the spring of 2023. The course is intended for graduate students and researchers seeking a self-contained journey from the foundations of computability theory to prefix complexity and the information-theoretic limits of formal systems. My exposition ignores boundaries between computer science, mathematics, physics, and philosophy -- an approach I consider essential when explaining inherently multidisciplinary fields. Lecture recordings are available online. Among other topics, the notes cover bit strings, codes, Shannon information theory, computability theory, the universal Turing machine, the Halting Problem, Rice's Theorem, plain algorithmic complexity, the Invariance Theorem, incompressibility, Solomonoff's induction, self-delimiting Turing machines, prefix algorithmic complexity, the halting probability Omega, Chaitin's Incompleteness Theorem, The Coding Theorem, lower semi-computable semi-measures, and the chain rule for algorithmic complexity.
... • The Knapsack problem (hence Rational Subset Membership) is undecidable in large nilpotent groups, most notably H 3 (Z) k and N 2,k (the free 2-step nilpotent group of rank k) for k ≫ 1. [17,15,25] • Submonoid Membership is undecidable in H 3 (Z) k for k ≫ 1. [30] Both results rely on the negative solution to Hilbert's 10th problem: there exists no algorithm deciding whether a Diophantine equation (or system of equations) admits an integer solution [23]. On the positive side, the list of results is even shorter: ...
We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform 3-step nilpotent group has decidable Submonoid Membership.
Comment: v6. 25 pages, 5 figures. Published in the journal of Groups, Complexity, Cryptology
... In 1900 Hilbert asked the question, famously known as the "Hilbert's tenth problem", whether there exists an algorithm that can decide within finitely many steps if a given Diophantine equation has solutions in Z. In 1970, Matiyasevich [7] answered this negatively. Along a similar line, Mordell observed that the arithmetic behaviour of the points on a curve is quite closely related to the genus of it and conjectured that a curve over Q of genus at least 2 can have at most finitely many rational points. ...
We consider the parametric family of elliptic curves over of the form , where , and t are particular polynomial expressions in an integral variable m. In this paper, we investigate the torsion group , a lower bound for the Mordell-Weil rank and the 2-Selmer group under certain conditions on m. This extends the previous works done in this direction, which are mostly concerned with the Mordell-Weil ranks of various parametric families of elliptic curves.
... Nonlinear Integer Arithmetic (N IA) is the theory consisting of arbitrary Boolean combinations of Boolean variables and arithmetic atoms of the form of polynomial equalities and polynomial inequalities over integer variables. It is undecidable by Matiyasevich's theorem [38]. ...
The Model Constructing Satisfiability (MCSat) approach to the SMT problem extends the ideas of CDCL from the SAT level to the theory level. Like SAT, its search is driven by incrementally constructing a model by assigning concrete values to theory variables and performing theory-level reasoning to learn lemmas when conflicts arise. Therefore, the selection of values can significantly impact the search process and the solver's performance. In this work, we propose guiding the MCSat search by utilizing assignment values discovered through local search. First, we present a theory-agnostic framework to seamlessly integrate local search techniques within the MCSat framework. Then, we highlight how to use the framework to design a search procedure for (quantifier-free) Nonlinear Integer Arithmetic (NIA), utilizing accelerated hill-climbing and a new operation called feasible-sets jumping. We implement the proposed approach in the MCSat engine of the Yices2 solver, and empirically evaluate its performance over the N IA benchmarks of SMT-LIB.
... We know that exponential diophantine equations are undecidable over N, as proved by Davis, Putnam and Robinson in [7]. Also, we know that diophantine equations are undecidable over N as proved by Matiyasevich, see for example [8]. In order to prove (1), we may let the equation E = 0 vary over exponential diophantine equations or over diophantine equations. ...
In a previous paper of the author it was shown that the question whether systems of exponential diophantine equations are solvable in is undecidable. Now we show that the solvability of a conjunction of exponential diophantine equations in is equivalent to the solvability of just one such equation. It follows that the problem whether an exponential diophantine equation has solutions in is undecidable. We also show that two particular forms of exponential diophantine equations are undecidable.
... [Jon81,Sch82,Tun87] Given any f ∈ Z[x, y], we have that ∀x ∃y f (x, y) = 0 iff all three of the following conditions hold: The analogue of the JST Theorem over Z is essentially the same, save for the absence of condition (2), and the removal of the sign check in condition (1) [Tun87]. The study of the decidability of Diophantine prefixes dates back to [Mat73,MR74,Jon81], and [Mat93,Tun99,Roj99b,Roj00c] give some of the most recent results. Of course, as we have seen above, there is still much left to be done, and we hope that this paper sparks the interests of other researchers. ...
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set, (b) the height of the zero-dimensional part of an algebraic set over C, and (c) the number of connected components of a semi-algebraic set. We also present some results which significantly lower the complexity of deciding the emptiness of hypersurface intersections over C and Q, given the truth of the Generalized Riemann Hypothesis. Furthermore, we state some recent progress on the decidability of the prefixes \exists\forall\exists and \exists\exists\forall\exists, quantified over the positive integers. As an application, we conclude with a result connecting Hilbert's Tenth Problem in three variables and height bounds for integral points on algebraic curves. This paper is based on three lectures presented at the conference corresponding to this proceedings volume. The titles of the lectures were ``Some Speed-Ups in Computational Algebraic Geometry,'' ``Diophantine Problems Nearly in the Polynomial Hierarchy,'' and ``Curves, Surfaces, and the Frontier to Undecidability.''
... We will show that the extension of SL[replaceAll] with integer constraints entails undecidability, by a reduction from (a variant of) the Hilbert's 10th problem, which is well-known to be undecidable [Matiyasevich 1993]. For space reasons, all proofs appear in Appendix G. Intuitively, we want to find a solution to f (x 1 , · · · , x n ) = д(x 1 , · · · , x n ) in the natural numbers, where f and д are polynomials with positive coefficients. ...
Recently, it was shown that any theory of strings containing the string-replace function (even the most restricted version where pattern/replacement strings are both constant strings) becomes undecidable if we do not impose some kind of straight-line (aka acyclicity) restriction on the formulas. Despite this, the straight-line restriction is still practically sensible since this condition is typically met by string constraints that are generated by symbolic execution. In this paper, we provide the first systematic study of straight-line string constraints with the string-replace function and the regular constraints as the basic operations. We show that a large class of such constraints (i.e. when only a constant string or a regular expression is permitted in the pattern) is decidable. We note that the string-replace function, even under this restriction, is sufficiently powerful for expressing the concatenation operator and much more (e.g. extensions of regular expressions with string variables). This gives us the most expressive decidable logic containing concatenation, replace, and regular constraints under the same umbrella. Our decision procedure for the straight-line fragment follows an automata-theoretic approach, and is modular in the sense that the string-replace terms are removed one by one to generate more and more regular constraints, which can then be discharged by the state-of-the-art string constraint solvers. We also show that this fragment is, in a way, a maximal decidable subclass of the straight-line fragment with string-replace and regular constraints. To this end, we show undecidability results for the following two extensions: (1) variables are permitted in the pattern parameter of the replace function, (2) length constraints are permitted.
... Something is missing from the picture. Of course, the 'grand examples' are missing; for example, no important open problem except Hilbert's tenth problem, see [41], was proved to be unprovable. Other questions of interest include the source of incompleteness and how common the incompleteness phenomenon is. ...
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is invariant under the change of the G\"odel numbering. For this measure, the theorems of a finitely-specified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length n is provable in the theory tends to zero when n tends to infinity, while the probability that a sentence of length n is true is strictly positive.
... Regarding its solvability, there is no general method to determine by a finite number of operations whether the equation is solvable or not. [8] However these properties of Diophantine equation are also useful for encrypting message to keep it secret and have been applied to some public key cryptosystems. [9][10] For example, the cryptosystem proposed by Lin, et al. is based on that the Diophantine equation dealt with in the system is practically non-soluble. ...
One-way functions are widely used for encrypting the secret in public key cryptography, although they are regarded as plausibly one-way but have not been proven so. Here we discuss the public key cryptosystem based on the system of higher order Diophantine equations. In this system those Diophantine equations are used as public keys for sender and recipient, and sender can recover the secret from the Diophantine equation returned from recipient with a trapdoor. In general the system of Diophantine equations is hard to solve when it is positive-dimensional and it implies the Diophantine equations in this cryptosystem works as a possible one-way function. We also discuss some problems on implementation, which are caused from additional complexity necessary for constructing Diophantine equations in order to prevent from attacking by tamperers.
... Proof: By reduction from Hilbert's tenth problem, i.e., the problem of determining whether a polynomial with integer coefficients has solutions in the natural numbers. This was shown to be undecidable by Matiyasevich [Mat93]. ...
We investigate the decidability of model-checking logics of time, knowledge and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observed discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular, for a synchronous perfect recall, a generalization from temporal operators with finite reach to operators with infinite reach renders model checking undecidable. The case of the clock semantics is closely related to a monadic second order logic of time and probability that is known to be decidable, except on a set of measure zero. We show that two distinct extensions of this logic make model checking undecidable. One of these involves polynomial combinations of probability terms, the other involves monadic second order quantification into the scope of probability operators. These results explain some of the restrictions in previous work.
... Утверждения 6, 30 и 37 гласят, что у нас есть интерпретация ⟨N 0 , +, ×⟩ на YF * с определёнными операциями сложения и умножения над вершинами вида 1 n с помощью Π m формул. Неразрешимость положительной Σ 1 теории ⟨N 0 , +, ×⟩, доказанная Юрием Матиясевичем [3], влечёт за собой следующее: ...
For a poset we consider the first-order theory, that is defined by set P and relation . The problem of undecidability of combinatorial theories attracts significant attention. Recently A. Wires proved the undecidability of the elementary theory of Young lattice and also established the maximal definability property of this theory. The purpose of this article is to obtain the same results for another graded lattice, which has much in common with Young lattice: Young--Fibonacci lattice. As Wires does for Young lattice, for the proof of undecidability we define Arithmetic into this theory.
... In 1971 Matiyasevich (1993), using previous results by Davis, Putnam and ...
The study of undecidability in problems arising from physics has experienced a renewed interest, mainly in connection with quantum information problems. The goal of this review is to survey this recent development. After a historical introduction, we first explain the necessary results about undecidability in mathematics and computer science. Then we briefly review the first results about undecidability in physics which emerged mostly in the 80s and early 90s. Finally we focus on the most recent contributions, which we divide in two main categories: many body systems and quantum information problems.
... The microscopic derivation of the Yang-Mills mass gap from first principles has far-reaching phenomenological consequences across particle physics and cosmology [57][58][59][60][61][62][63]: -The mass gap controls the binding and spectra of hadrons, from light mesons and baryons to heavy quarkonia and exotic states. It sets the confinement scale and the transition between perturbative and nonperturbative dynamics. ...
I present a rigorous analytical solution to the Yang-Mills mass gapproblem on R4 by employing non-perturbative techniques, heat kernelmethods, and Ward identities. The existence of a mass gap ∆ > 0 isproved, and a mass gap formula ∆ = Ce−A/g2 relating ∆ to the gaugecoupling g is derived. The solution adheres to the criteria set by the ClayMathematics Institute and addresses the points raised by Michael Peskin.
... Solvability of Diophantine equations over the integers is undecidable; this is a fundamental theorem of Matijasevitch, who proved that computably enumerable sets are diophantine, which finally solved Hilbert's 10-th problem in the negative. See, e.g., the discussion in Manin (1977) and Matiyasevich (1993). ...
Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. Our dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion.
... In 1970, Matiyasevich, building on the work of Robinson [8] and Davis et al. [9], proved that all computable functions can be expressed as Diophantine equations [10]. Matiyasevich's results imply that there exists a Diophantine equation for calculating the n-th prime number [11]. However, no arithmetic term for the n-th prime is known [12]. ...
We conjecture new elementary formulas for computing the greatest common divisor (GCD) of two integers, alongside an elementary formula for extracting the prime factors of semiprimes. These formulas are of fixed-length and require only the basic arithmetic operations of: addition, subtraction, multiplication, division with remainder, and exponentiation. Our GCD formulas result from simplifying a formula of Mazzanti and are derived using Kronecker substitution techniques from our earlier research. By applying these GCD formulas together with our recent discovery of an arithmetic expression for , we are able to derive explicit elementary formulas for the prime factors of a semiprime .
... -la recherche d'une méthode de résolution des équations diophantiennes, dixième problème de la liste formulée au Congrès international des mathématiciens de 1900 (Hilbert, 1902) ; -le problème de la décision, formulé en 1928, qui consiste à établir une procédure permettant de déterminer si un énoncé donné est un théorème en logique du premier ordre (Hilbert & Ackermann, 1959). La résolution négative de ces deux problèmes, par Gödel (1931) pour le second et par Matiyasevich (1993) pour le premier, nécessite la formalisation de la notion de calcul. Plusieurs modèles de calcul sont proposés, notamment par Church (1941) avec le λ-calcul, Turing (1936) avec la machine qui porte son nom (évoquée plus précisément en section 3.A.), ou Herbrand (1930) avec les fonctions récursives qui sont au coeur des travaux de Gödel. ...
The question of the definition of what is an algorithm is recurrent. It is found in teaching, at different levels and particularly in secondary education because of the recent evolutions in high school, with immediate consequences in higher education. It is found in mediation, with the different meanings that the word "algorithm" is charged with in the media space. It is also found in research, with issues in different branches of computer science, from foundations in computability and complexity to applications in big data. Beyond the issue of definition, it is the raison d'{\^e}tre of the notion of algorithm that should be questioned: what do we want to do with it and what is at stake? It is by trying to specify this that we can identify didactic elements that are likely to help teach the algorithm, in interaction with mathematics or not, and to different audiences.
... CASU provides principled techniques for remapping computationally intractable arithmetic problems into more manageable sectors by exploiting physics insights into symmetry data and flows [49,50,51,52]. Arithmetic stratification into subvarieties cut out by arithmetic conditions isolating prime ideals, residue classes and divisor packet data, combined with dynamical evolution of D3-brane probes and isogeny-based spacetime surgery, opens up new pathways for tackling previously intractable problems like integer factorization. ...
The search for a unified theory reconciling all aspects of physical reality has been a driving force in theoretical physics and mathematics for over a century [72]. Despite remarkable progress, a complete unification of quantum mechanics and Einstein's general theory of relativity has remained elusive. The quest for unification has been a driving force in physics and mathematics for centuries, as chronicled in works like Weinberg's "Dreams of a Final Theory" [72]. CASU builds upon this long-standing tradition, while also drawing inspiration from the mysterious effectiveness of mathematics in describing physical reality, as famously explored by Wigner [73] and more recently by Tegmark [74].
... Diophantine equations have proved interesting in part due to the difficulty in predicting when integral solutions occur, and if so, how many exist. Matiyasevich proved that there is no general algorithm to predict when these solutions occur [13], though that does not mean we cannot work with specific cases. Andrew Wiles' proof of Fermat's Last Theorem is a famous example of this fact [14,15]. ...
Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. Such properties can be applied to a particular class of nonlinear Diophantine equations. Most importantly, we find that our restrictions enable an elementary proof of the unimodality of the nonzero terms of the generating function, which in general is quite hard. We also examine partitions with even numerators. We prove that there are 2ω(t) − 2 partitions of an integer t into fractions with the first x consecutive even integers for numerators and equal denominators of y, where 0 < y < x < t. We then use this to produce corollaries such as a series identity and an extension of the prime omega function to the complex plane.
... The expressiveness of existential formulas is even further from being understood. The undecidability proof in [HSZ17] reduces from solvability of Diophantine equations, i.e., polynomial equations over integers, which is a well-known undecidable problem [Mat93]. To this end, it is shown in [HSZ17] that the relations ADD = {(a m , a n , a m+n ) | m, n ∈ N} and MULT = {(a m , a n , a m·n ) | m, n ∈ N} are definable existentially using the subword ordering, if one has at least two letters. ...
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the (i.e., existential) fragment is undecidable, already for binary alphabets A. However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable. We show that if , then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments : If , then a relation is definable in if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the -fragments for of the pure logic, where the words of are not available as constants.
... for any integer value of n exceeding 2. Fermat posited the theorem in 1637, and its validation was accomplished by Andrew Wiles after a span of 358 years in 1995 ( [2], [3], [4]). This theorem has a long and interesting history. ...
... The problem of solving even a system of polynomial equations has been proven to be NP-hard [23]. Furthermore, it has also been proven [29] that no general algorithm exists for determining whether an integer solution exists for a polynomial equation with a finite number of unknowns and only integer coefficients. This is the famous 10th problem of Hilbert [15]. ...
Systems of nonlinear equations can be quite difficult to solve, even when the system is small. As the systems grow in size, the complexity can increase dramatically to find all solutions. This research discusses transforming the system into a global optimization problem and making use of a newly developed cloud-based optimization solver to efficiently find solutions. Examples on large systems are presented.
... As SMT solvers get stronger in quantified reasoning, it becomes more interesting to get a clear picture of decidability frontiers when arithmetic is used in a quantified SMT context. Some pure arithmetic theories are already undecidable, even in their quantifier-free fragment, e.g., Peano arithmetic [12], i.e., a first-order theory of the natural numbers with addition and multiplication. However, Presburger arithmetic, somehow the linear restriction of Peano arithmetic, is decidable even in the quantified case [10], but augmenting Presburger arithmetic with a single unary uninterpreted predicate already yields undecidability [7,11,19]. ...
First-order logic fragments mixing quantifiers, arithmetic, and uninterpreted predicates are often undecidable, as is, for instance, Presburger arithmetic extended with a single uninterpreted unary predicate. In the SMT world, difference logic is a quite popular fragment of linear arithmetic which is less expressive than Presburger arithmetic. Difference logic on integers with uninterpreted unary predicates is known to be decidable, even in the presence of quantifiers. We here show that (quantified) difference logic on real numbers with a single uninterpreted unary predicate is undecidable, quite surprisingly. Moreover, we prove that difference logic on integers, together with order on reals, combined with uninterpreted unary predicates, remains decidable.
... As SMT solvers get stronger in quantified reasoning, it becomes more interesting to get a clear picture of decidability frontiers when arithmetic is used in a quantified SMT context. Some pure arithmetic theories are already undecidable, even in their quantifier-free fragment, e.g., Peano arithmetic [11], i.e., a first-order theory of the natural numbers with addition and multiplication. However, Presburger arithmetic, somehow the linear restriction of Peano arithmetic, is decidable even in the quantified case [9], but augmenting Presburger arithmetic with a single unary uninterpreted predicate already yields undecidability [6,10,18]. ...
First-order logic fragments mixing quantifiers, arithmetic, and uninterpreted predicates are often undecidable, as is, for instance, Presburger arithmetic extended with a single uninterpreted unary predicate. In the SMT world, difference logic is a quite popular fragment of linear arithmetic which is less expressive than Presburger arithmetic. Difference logic on integers with uninterpreted unary predicates is known to be decidable, even in the presence of quantifiers. We here show that (quantified) difference logic on real numbers with a single uninterpreted unary predicate is undecidable, quite surprisingly. Moreover, we prove that difference logic on integers, together with order on reals, combined with uninterpreted unary predicates, remains decidable.
... rationals) thanks to the fundamental theory of algebra and Buchberger's algorithms for Gröbner basis computation [8,35]. Yet, when restricting the algorithmic study of solving polynomial equations over integers, the problem becomes undecidable [28]. ...
Non-linear polynomial systems over finite fields are used to model functional behavior of cryptosystems, with applications in system security, computer cryptography, and post-quantum cryptography. Solving polynomial systems is also one of the most difficult problems in mathematics. In this paper, we propose an automated reasoning procedure for deciding the satisfiability of a system of non-linear equations over finite fields. We introduce zero decomposition techniques to prove that polynomial constraints over finite fields yield finite basis explanation functions. We use these explanation functions in model constructing satisfiability solving, allowing us to equip a CDCL-style search procedure with tailored theory reasoning in SMT solving over finite fields. We implemented our approach and provide a novel and effective reasoning prototype for non-linear arithmetic over finite fields.
... The papers [11,8] saw in Davis' approach a strategy to affirmatively answer the question: "Is there a finitefold (or better a singlefold) polynomial Diophantine definition of b n = c ?". Recently, the interest in that finite-foldness issue was aroused again in [13,10], and [14] reiterated the relevance of Davis' approach for solving it. ...
By following the same construction pattern which Martin Davis proposed in a 1968 paper of his, we have obtained six quaternary quartic Diophantine equations that candidate as `rule-them-all' equations: proving that one of them has only a finite number of integer solutions would suffice to ensure that each recursively enumerable set admits a finite-fold polynomial Diophantine representation.
... Brown [57], who hypothesised that it would converge to the value of a zero-sum game. Julia Robinson [58] (yes, the Julia Robinson from Hilbert's tenth problem [59] ...
Reinforcement learning is an area of Machine Learning. The three primary types of machine learning are supervised learning, unsupervised learning, and reinforcement learning (RL). Pre-training a model on a labeled dataset is known as supervised learning. The model is trained on unlabeled data in unsupervised learning, on the other hand. Instead of being driven by labels, RL is motivated by assessing feedback. By interacting with the environment and choosing the best course of action in each circumstance in order to maximize the reward, the agent learns the best way to solve sequential decision-making issues. The RL agent chooses how to carry out tasks on its own. Furthermore, since there are no training data, the agent learns by gaining experience. In order to make subsequent judgments, RL aids agents in efficiently interacting with their surroundings. In this essay, state-of-the-art RL is thoroughly reviewed in the literature. Applications for reinforcement learning (RL) may be found in a wide range of industries, including smart grids, robots, computer vision, healthcare, gaming, transportation, finance, and engineering.
... The expressiveness of existential formulas is even further from being understood. The undecidability proof in [17] reduces from solvability of Diophantine equations, i.e., polynomial equations over integers, which is a well-known undecidable problem [29]. To this end, it is shown in [17] that the relations ADD = {(a m , a n , a m+n ) | m, n ∈ N} and MULT = {(a m , a n , a m·n ) | m, n ∈ N} are definable existentially using the subword ordering, if one has at least two letters. ...
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the (i.e., existential) fragment is undecidable, already for binary alphabets A. However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable. We show that if , then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments : If , then a relation is definable in if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the -fragments for of the pure logic, where the words of are not available as constants.
We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
Subsumption resolution is an expensive but highly effective simplifying inference for first-order saturation theorem provers. We present a new SAT-based reasoning technique for subsumption resolution, without requiring radical changes to the underlying saturation algorithm. We implemented our work in the theorem prover Vampire , and show that it is noticeably faster than the state of the art.
We show that SCL(FOL) can simulate the derivation of non-redundant clauses by superposition for first-order logic without equality. Superposition-based reasoning is performed with respect to a fixed reduction ordering. The completeness proof of superposition relies on the grounding of the clause set. It builds a ground partial model according to the fixed ordering, where minimal false ground instances of clauses then trigger non-redundant superposition inferences. We define a respective strategy for the SCL calculus such that clauses learned by SCL and superposition inferences coincide. From this perspective the SCL calculus can be viewed as a generalization of the superposition calculus.
We present an Isabelle/HOL formalization of Simple Clause Learning for first-order logic without equality: SCL(FOL). The main results are formal proofs of soundness, non-redundancy of learned clauses, termination, and refutational completeness. Compared to the unformalized version, the formalized calculus is simpler and more general, some results such as non-redundancy are stronger and some results such as non-subsumption are new. We found one bug in a previously published version of the SCL Backtrack rule. Compared to related formalizations, we introduce a new technique for showing termination based on non-redundant clause learning.
This paper describes the formal verification of NP-hardness reduction functions of two key problems relevant in algebraic lattice theory: the closest vector problem and the shortest vector problem, both in the infinity norm. The formalization uncovered a number of problems with the existing proofs in the literature. The paper describes how these problems were corrected in the formalization. The work was carried out in the proof assistant Isabelle.KeywordsverificationNP-hardnesslattice problemsinteger programming
On the undergraduate level, this book is an introduction to the Theory of Computation. A public purpose of terminology must protect a combination of different classifications. Natural languages, programming languages, mathematical languages, etc. The notation of a natural language is like English, Hindi, etc. Informal language can be defined as a system suitable for expressing specific ideas, facts, or concepts, including symbols and rules to manipulate these. The language we consider for our discussion is an abstraction of natural languages. The theory of Computation is based on Logic, Algorithms, and Theorem-proving. A machine is used to perform the user's essential tasks (requirements). So, it is equally important to know the computing models that compare the study. The model tells us how the machine will work together to do Computation. The theory of Computation abstractly provides computational models.
This thesis contains three parts:
1) l find all integer solutions of A Pillai type problem.
2) l solved Mersenne version of Brocard-Ramanujan equation.
3) l introduces some new Fermat type equation
Expressions of type and , where are natural and are prime numbers, are studied.
In this paper, we consider the relationship between the theory of hypercomputing and some problems of modern physics, in particular, the theory of black holes.
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