pinecone/core-2020-05-10-deduplication huggingface dataset:
Design and Implementation of MP, a Protocol for Efficient Exchange of Mathematical Expressions

AbstractThe Multi Project is an ongoing research effort at Kent State University aimed at providing an environment for distributed scientific computing. An integral part of this environment is the Multi Protocol (MP) which is designed to support efficient communication of mathematical data between scientifically-oriented software tools. MP exchanges data in the form of linearized annotated syntax trees. Syntax trees provide a simple, flexible and tool-independent way to represent and exchange data, and annotations provide a powerful and generic expressive facility for transmitting additional information. At a level above the data exchange protocol, dictionaries provide definitions for operators and constants, providing shared semantics across heterogeneous packages. A clear distinction between MP-defined and user-defined entities is enforced. Binary encodings are used for efficiency. Commonly used values and blocks of homogeneous data are further optimized. The protocol is independent of the underlying communication paradigm and can support parallel computation, distributed problem-solving environments, and the coupling of tools for specific applications
The Use of Proof Planning for Co-operative Theorem Proving

AbstractWe describebarnacle: a co-operative interface to theclaminductive theorem proving system. For the foreseeable future, there will be theorems which cannot be proved completely automatically, so the ability to allow human intervention is desirable; for this intervention to be productive the problem of orienting the user in the proof attempt must be overcome. There are many semi-automatic theorem provers: we call our style of theorem provingco-operative, in that the skills of both human and automaton are used each to their best advantage, and used together may find a proof where other methods fail. The co-operative nature of thebarnacleinterface is made possible by the proof planning technique underpinningclam. Our claim is that proof planning makes new kinds of user interaction possible.Proof planning is a technique for guiding the search for a proof in automatic theorem proving. Common patterns of reasoning in proofs are identified and represented computationally as proof plans, which can then be used to guide the search for proofs of new conjectures. We have harnessed the explanatory power of proof planning to enable the user to understand where the automatic prover got to and why it is stuck. A user can analyse the failed proof in terms ofclam's specification language, and hence override the prover to force or prevent the application of a tactic, or discover a proof patch. This patch might be to apply further rules or tactics to bridge the gap between the effects of previous tactics and the preconditions needed by a currently inapplicable tactic
Interactive Theorem Proving: An Empirical Study of User Activity

AbstractIn this paper the interaction between users and the interactive theorem prover HOL is investigated from a human–computer interaction perspective. First, we outline three possible views of interaction, and give a brief survey of some current interfaces and how they may be described in terms of these views. Second, we describe and present the results of an empirical study of intermediate and expert HOL users. The results are analysed for evidence in support of the proposed view of proof activity in HOL. We believe that this approach provides a principled basis for the assessment and design of interfaces to theorem provers
Decomposing Polynomial Systems into Simple Systems

AbstractA simple system is a pair of multivariate polynomial sets (one set for equations and the other for inequations) ordered in triangular form, in which every polynomial is squarefree and has non-vanishing leading coefficient with respect to its leading variable. This paper presents a method that decomposes any pair of polynomial sets into finitely many simple systems with an associated zero decomposition. The method employs top-down elimination with splitting and the formation of subresultant regular subchains as basic operation
Gröbner Bases and Normal Forms in a Subring of the Power Series Ring on Countably Many Variables

AbstractIfKis a field, let the ringR′consist of finite sums of homogeneous elements inR=K[[x1,x2,x3,…]]. Then,R′contains M, the free semi-group on the countable set of variables {x1,x2,x3,…}. In this paper, we generalize the notion of admissible order from finitely generated sub-monoids of M to M itself; assume that > is such an admissible order on M. We show that we can define leading power products, with respect to >, of elements inR′, and thus the initial ideal gr(I) of an arbitrary idealI⊂R′. IfIis what we call a locally finitely generated ideal, then we show that gr(I) is also locally finitely generated; this implies thatIhas a finite truncated Gröbner basis up to any total degree. We give an example of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic initial order >lexon M
Symbolic Asymptotics: Functions of Two Variables, Implicit Functions

AbstractA number of recent papers have been concerned with algorithms to decide the limiting behaviour of functions of a single variable. Here we make a corresponding study of a class of functions of two variables, namely the exp-log functions. As in the one-variable case, we need to make certain assumptions regarding the handling of constants.Two of the main tools in the one-variable case are Hardy fields and nested forms. Here, we show how to compute some asymptotic estimates for two-variable exp-log functions. This method is then used to give an algorithm for computing the nested forms of real implicit functions
Generation and Verification of Algorithms for Symbolic-Numeric Processing

AbstractSome large scale physical computations require algorithms performing symbolic computations with a particular class of algebraic formulas in a numerical code. Developing and implementing such algorithms in a numerical programming language is a tedious and error prone task. The algorithms can be developed in a computer algebra system and their correctness can be checked by comparison with build-in facilities of the system so that the system is used as an advanced debugging tool. After that a numerical code for the algorithms is automatically generated from the same source code. The proposed methodology is explained in detail on a simple example. Real applications to calculation of matrix elements of Coulomb interaction and two-centre exchange integrals needed in atomic collision codes, are described. The method makes the developing and debugging of such algorithms easier and faster
Completion of Rewrite Systems with Membership Constraints Part I: Deduction Rules

AbstractWe consider a constrained equational logic where the constraints are membership conditions t∈swheresis interpreted as a regular tree language. Our logic includes a fragment of second-order equational logic (without projections) where second-order variables range over regular sets of contexts. The problem with constrained equational logics is the failure of the critical pair lemma. This is the reason why we propose new deduction rules for which the critical pair lemma is restored. Computing critical pairs requires, however, solving some constraints in a second-order logic with membership constraints. In a second paper we give a terminating set of transformation rules for these formulas, which decides the existence of a solution, thus showing a new term scheme unification algorithm.Since an order-sorted signature is nothing but a bottom–up tree automaton, order-sorted equational logic falls into the scope of our study; our results show how to perform order-sorted completion without regularity and without sort-decreasingness. It also shows how to perform unification in the order-sorted case, with some higher-order variables (without any regularity assumption)
Dynamically Typed Computations for Order-sorted Equational Presentations

AbstractEquational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with few restrictions on the allowed presentations. We adopt the G-algebra framework, where equational, membership and existence formulas can be expressed, and this provides a complete deduction calculus which incorporates the interaction between all these formulas.To practically deal with this calculus, we introduce an operational semantics for G-algebra using rewrite systems over so-called decorated terms, that has assertions concerning the sort membership of any subterm in its head node. Decorated rewrite rules perform equational replacement, decoration rewrite rules enrich the decorations and record sort information. Therefore we use the semantic sort principle, i.e. equal terms belong to equal sorts, rather than the syntactic sort principle that does not use the equational part of a presentation.In order to have a complete and decidable unification on decorated terms, we restrict to sort-inheritance theories. Then a completion procedure on decorated terms is designed to compute all interactions between equational and membership formulas. When the completion terminates, the resulting set of rewrite rules provides a way to decide equational theorems of the form (t=t′) and typing theorems of the form (t:A).The sort inheritance property is undecidable in general but we propose a test to check it for a given presentation. The test provides information on how to extend the presentation in a model conservative way, in order to obtain sort inheritance
A Semantic Approach to Order-sorted Rewriting

AbstractOrder-sorted rewriting builds a nice framework to handle partially defined functions and subtypes. To be able to prove a critical-pair lemma and Birkhoff's completeness theorem, order-sorted rewriting was restricted to sort decreasing term rewriting systems. However, natural examples show that this approach is too restrictive.To solve this problem, we generalize well-sorted terms to semantically well-sorted terms and well-sorted substitutions to a restricted form of semantically well-sorted substitutions. Semantically well-sorted terms with respect to a set of equationsEare terms that denote well-defined elements in every algebra satisfyingE.We prove a critical-pair lemma and Birkhoff's completeness theorem for so-called range-unique signatures and arbitrary order-sorted rewriting systems. A transformation is given which allows us to obtain an equivalent range-unique signature from each non-range-unique one. We also show decidability and undecidability results
The Search for the Maximum of a Polynomial

AbstractFor a real polynomialf(X) ofKvariables the problem of finding maxX∈RKf(X) is investigated by reducing it to that of searching for the real roots of the univariate polynomial F(z):=∏j(z−f(Λj)), where the product is extended over all the critical points Λjoff(X). Employment of the Hermite method of separation of real solutions of an algebraic equation system permits one to construct along withF(z) its Sturm series, and to restore the coordinates of the corresponding critical point. The problem of finding the maxfin the set defined by the real polynomial inequalityG(X)≥0 is also discussed
Systems of Algebraic Equations with Bad Reduction

AbstractThis article proposes a new method for finding certain solutions of systems of algebraic equations, a method successfully applied to huge systems where general methods usually fail. We usep-adic approximation, and combine a trivial but effective method for solving systems modulopthat takes advantage of a smallpwith a new lifting scheme able to cope with bad reduction, this being a frequent phenomenon with small primes. The method works particularly good for solutions of small (or at least bounded) algebraic degree. Such solutions are interesting in Constructive Galois Theory where they yield polynomials with a prescribed Galois Group
The Theory of Involutive Divisions and an Application to Hilbert Function Computations

AbstractGeneralising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more flexible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov.Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Gröbner basis and that it is straightforward to compute the (affine) Hilbert function of an idealIfrom an arbitrary involutive basis of alI
A Deterministic Lazy Narrowing Calculus

AbstractIn this paper we study the non-determinism between the inference rules of the lazy narrowing calculusLNC(Middeldorpet al., 1996,Theoret. Comput. Sci.,167, 95–130. We show that all non-determinism can be removed without losing the important completeness property by restricting the underlying term rewriting systems to left-linear confluent constructor systems and interpreting equality as strict equality. For the subclass of orthogonal constructor systems the resulting narrowing calculus is shown to have the nice property that solutions computed by different derivations starting from the same goal are incomparable
Advances on the Simplification of Sine–Cosine Equations

AbstractIn this paper we contribute several results to the approach initiated by Hommel and Kovács (well documented with applications in a recent book by Kovács (1993)) on the symbolic simplification of sine–cosine polynomials that arise, for instance, as determining equations for joint values in robotics inverse kinematic problems. We present, taking into consideration for the first time sine–cosine polyomials, fast algorithms for the functional decomposition and factorization problems, reducing the solving of suchs–cequations to a sequence of lower degree ones. Moreover, we show that triangularization of a given sine–cosine equation provides a conceptual understanding of the conditions that yield extraneous roots in the half-angle tangent substitution (and therefore that imply a reduction of the degree in the determining equation of a givens–csystem)
Modular Arithmetic for Linear Algebra Computations in the Real Field

AbstractThe aim of this work is to decrease the bit precision required in computations without affecting the precision of the answer, whether this is computed exactly or within some tolerance. By precision we understand the number of bits in the binary representation of the values involved in the computation, hence a smaller precision requirement leads to a smaller complexity. We achieve this by combining the customary numerical techniques of rounding the least significant bits with the algebraic technique of reduction modulo an integer, which we extend to the reduction modulo a positive number. In particular, we show that, if the sum of several numbers has small magnitude, relative to the magnitude of the summands, then the precision used in the computation of this sum can be decreased without affecting the precision of the answer. Furthermore, if the magnitude of the inner product of two vectors is small and if one of them is filled with “short” binary numbers, then again we may decrease the precision of the computation. The method is applied to the iterative improvement algorithm for a linear system of equations whose coefficients are represented by “short” binary numbers, as well as to the solution of PDEs by means of multigrid methods. Some results of numerical experiments are presented to demonstrate the power of the method
An Algorithm for Solving the Factorization Problem in Permutation Groups

AbstractThe factorization problem in permutation groups is to represent an elementgof some permutation groupGas a word over a given setSof generators ofG. For practical purposes, the word should be as short as possible, but must not be minimal. Like many other problems in computational group theory, the problem can be solved from a strong generating set (SGS) and a base ofG. Different algorithms to compute an SGS and a base have been published. The classical algorithm is the Schreier–Sims method. However, for factorization an SGS is needed that has all its elements represented as words overS. The existing methods are not suitable, because they lead to an exponential growth of word lengths. This article presents a simple algorithm to solve the factorization problem. It is based on computing an SGS with elements represented by relatively short words over the generators
Negation Elimination in Empty or Permutative Theories

AbstractAn equational formula is a first-order formula over an alphabet F of function symbols and the equality predicate. Such formulae are interpreted in the algebraT(F) of ground terms. A unification problem is an equational formula which does not contain negation (in particular no disequation). We give a terminating set of transformation rules such that a formula is semantically equivalent to a unification problem iff its irreducible form is a unification problem. More precisely, our set of transformation rules computes a finite complete set of most general unifiers for an equational formula each time such a finite set exists, thus extending the results of Lassez and Marriott on explicit representation of terms defined by counter examples. We extend the above results also to equational formulae interpreted inT(F)/=E, the quotient of the free algebra by a congruence generated by a setEof shallow permutative equations (commutative axioms are a particular case)
Sparsification of Rectangular Matrices

AbstractConsider the problem of sparsifying a rectangular matrix with more columns than rows. This means forming linear combinations of the rows, while preserving the rank, such that the result contains as many zero entries as possible. A combinatorial search method is presented which sparsifies a matrix with exponentially many arithmetic operations in the worst case. Moreover, a method is presented which substantially reduces the combinatorial search space if the matrix gives rise to a non-trivial block structure
A Parallel Implementation of Buchberger's Algorithm overZpforp≤ 31991

AbstractGröbner bases of ideals of polynomials are known to have many applications. They have been applied to problems in commutative algebra, statistics, graph theory, robotics and differential equations. Their use as a research tool, however, is limited by their computational complexity. These two facts have inspired numerous attempts to parallelize Buchberger's algorithm to compute them.In this paper, we describe a parallel implementation developed on the Cray T3D using the extensions to C provided by ac. The program is based on the publicly available package Macaulay which computes Gröbner bases of homogeneous ideals overZpfor primesp≤ 31991. The efficiency is nearly 100% on up to 16 processors for moderately sized problems. Above 16 processors, the efficiency drops
An Algebraic Computational Approach to the Identifiability of Fourier Models

AbstractComputer algebra and in particular Gröbner bases are powerful tools in experimental design (Pistone and Wynn, 1996,Biometrika83, 653–666). This paper applies this algebraic methodology to the identifiability of Fourier models. The choice of the class of trigonometric models forces one to deal with complex entities and algebraic irrational numbers. By means of standard techniques we have implemented a version of the Buchberger algorithm that computes Gröbner bases over the complex rational numbers and other simple algebraic extensions of the rational numbers. Some examples are fully carried out
Algorithms for Trigonometric Curves (Simplification, Implicitization, Parameterization)

AbstractA trigonometric curve is a real plane curve where each coordinate is given parametrically by a truncated Fourier series. The trigonometric curves frequently arise in various areas of mathematics, physics, and engineering. Some trigonometric curves can also be represented implicitly by bivariate polynomial equations. In this paper, we give algorithms for (a) simplifying a given parametric representation, (b) computing an implicit representation from a given parametric representation, and (c) computing a parametric representation from a given implicit representation
Computing Representations of a Lie Group via the Universal Enveloping Algebra

AbstractWe present an algorithm for the computation of representations of a Lie algebra acting on its universal enveloping algebra. This is a new algorithm which permits the effective computation of these representations and of the matrix elements of the corresponding Lie group. The approach is based on a mathematical formulation originated by the authors. An interesting feature is the efficient computation of the adjoint representation of the corresponding Lie group. The methods are implemented using a symbolic computation program such as MAPLE
Finding Possible Permutation Characters

AbstractWe describe three different methods to compute all those characters of a finite group that have certain properties of transitive permutation characters. First, a combinatorial approach can be used to enumerate vectors of multiplicities. Secondly, these characters can be found as certain integral solutions of a system of inequalities. Thirdly, they are calculated via Gaussian elimination. The methods are used to determine these characters for some finite groups and runtimes are listed. In the final section, a permutation character of the Lyons group is constructed
Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems

AbstractAn algorithm is described which uses a finite number of differentiations and linear operations to determine the Cartan structure of a transitive Lie pseudogroup from its infinitesimal defining equations. In addition, an algorithm is presented for determining from the infinitesimal defining system whether a Lie pseudogroup has essential invariants. If such invariants exist, the pseudogroup is intransitive. These methods make feasible the calculation of the Cartan structure of infinite Lie pseudogroups of symmetries of differential equations. The structure of the symmetry pseudogroup of the KP equation is presented
The Schur–Cohn Algorithm Revisited

AbstractThe aim of the Schur–Cohn algorithm is to compute the number of roots of a complex polynomial in the open unit disk, each root counted with its multiplicity. Unfortunately, in its original form, it does not work with all polynomials. Using technics similar to those of the sub-resultants, we construct a new sequence of polynomials, the Schur–Cohn sub-transforms. For this, we propose an algorithm in only O(d2) arithmetical operations (dbeing the degree of the polynomial studied), which is well adapted to computer algebra and supports specialization. We then show how to use bezoutians and hermitian forms to compute the number of roots in the unit disk with the help of the Schur–Cohn sub-transforms we have built
Strategies for Computing Minimal Free Resolutions

AbstractIn the present paper we study algorithms based on the theory of Gröbner bases for computing free resolutions of modules over polynomial rings. We propose a technique which consists in the application of special selection strategies to the Schreyer algorithm. The resulting algorithm is efficient and, in the graded case, allows a straightforward minimalization algorithm. These techniques generalize to factor rings, skew commutative rings, and some non-commutative rings. Finally, the proposed approach is compared with other algorithms by means of an implementation developed in the new system Macaulay2