pinecone/core-2020-05-10-deduplication huggingface dataset:
Cut-elimination and Redundancy-elimination by Resolution

AbstractA new cut-elimination method for Gentzen’s LK is defined. First cut-elimination is generalized to the problem of redundancy-elimination. Then the elimination of redundancy in LK-proofs is performed by a resolution method in the following way. A set of clauses C is assigned to an LK-proof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LK-proof ψ′ with atomic cuts;ψ′ can be constructed from the resolution proof and ψ by a projection method. In the final step the atomic cuts are eliminated and a cut-free proof is obtained. The complexity of the method is analyzed and it is shown that a non-elementary speed-up over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudo-cuts) can be transformed into formal ones by the method of redundancy-elimination; moreover, the method can even be used to transform incorrect proofs into correct ones
Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models

AbstractA new method for building infinite models for first-order formulae is presented. The method combines enumeration techniques with existing deductive (in a broad sense) ones. Its soundness and completeness w.r.t. the class of models that can be represented by equational constraints are proven. This shows that the use of enumeration techniques strictly increases the power of existing methods for building Herbrand models that are not complete in this sense. Some strategies are proposed to reduce the search space. We give examples and show how to use this approach for building interactively a model of a formula introduced by Goldfarb in his proof of the undecidability of the Gödel class with identity. This formula is satisfiable but has no finite model
Speeding up Algorithms on Atomic Representations of Herbrand Models via New Redundancy Criteria

AbstractThe importance of models within automated deduction is generally acknowledged both in constructing countermodels (rather than just giving the answer “NO", if a given formula is found to be not a theorem) and in speeding up the deduction process itself (e.g. by semantic resolution refinement). However, so far little attention has been paid to the efficiency of algorithms to actually work with models. There are two fundamental decision problems as far as models are concerned, namely: the equivalence of two models and the truth evaluation of an arbitrary clause within a given model. This paper focuses on the efficiency of algorithms for these problems in the case of Herbrand models given through atomic representations. Both problems have been shown to be coNP-hard by Gottlob and Pichler (1999), so there is a certain limit to the efficiency that we can possibly expect. Nevertheless, what we can do is find out the real “source" of complexity and make use of this theoretical result for devising an algorithm which, in general, has a considerably smaller upper bound on the complexity than previously known algorithms, e.g. the partial saturation method in Fermüller and Leitsch (1996)and the transformation into equational problems in Caferra and Zabel (1991). The main results of this paper are algorithms for these two decision problems, where the complexity depends non-polynomially on the number of atoms (rather than on the total size) of the input model equivalence problem or clause evaluation problem, respectively. Hence, in contrast to the above-mentioned algorithms, the complexity of the expressions involved (e.g. the arity of the predicate symbols and, in particular, the term depth of the arguments) only has polynomial influence on the overall complexity of the algorithms
Goals and Benchmarks for Automated Map Reasoning

AbstractTarski–Givant’s map calculus is briefly reviewed, and a plan of research is outlined aimed at investigating applications of this ground equational formalism in the theorem-proving field. The main goal is to create synergy between first-order predicate calculus and the map calculus. Techniques for translating isolated sentences, as well as entire theories, from first-order logic into map calculus are designed, or in some cases simply brought nearer through the exercise of specifying properties of a few familiar structures (natural numbers, nested lists, finite sets, lattices). It is also highlighted to what extent a state-of-the-art theorem-prover for first-order logic, namely Otter, can be exploited not only to emulate, but also to reason about, map calculus. Issues regarding “safe" forms of map reasoning are singled out, in sight of possible generalizations to the database area
Controlled Use of Clausal Lemmas in Connection Tableau Calculi

AbstractProof procedures based on model elimination or the connection tableau calculus have become more and more successful. But these procedures still suffer from long proof lengths as well as from a rather high degree of redundant search effort in comparison with resolution-style search procedures. In order to overcome these problems we investigate the use of clausal lemmas. We develop a method to augment a given set of clauses by a lemma set in a preprocessing phase and discuss the ability of this technique to reduce proof lengths and depths and to provide an appropriate reordering of the search space. We deal with the basic connection tableau calculus as well as with several calculus refinements and extensions. In order to control the use of lemmas we develop techniques for selecting relevant lemmas based on partial evaluation techniques. Experiments with the prover Setheo performed in several domains demonstrate the high potential of our approach
Connection Calculus Theorem Proving with Multiple Built-in Theories

AbstractGradually more applications of automated reasoning are discovered. This development has the consequence that deduction systems need to be increasingly flexible. They should exhibit a behavior appropriate to a given problem. One way to achieve this behavior is the integration of different systems or calculi. This leads to the so-called hybrid reasoning (Stickel, 1985; Frisch, 1991; Baumgartner, 1992; Petermann, 1993a) which describes the integration of a general purpose foreground reasoner with one specialized theory reasoner. The aim of this paper is to go a step further, i.e. to treat the theory reasoner as a hybrid system itself. The framework proposed below is suitable for building multiple theories into theorem provers. Those theories can be given syntactically but also semantically. Here, semantical reasoning is understood as reasoning, or rather computing, under a theory given by a class of models, whereas syntactical reasoning means reasoning under a theory given by first-order axioms. The presented approach is a generalization of previous attempts of combining syntactical reasoning under the empty theory with semantical reasoning (Bürckert, 1994; Baumgartner and Stolzenburg, 1995), of combining different theories given syntactically (Petermann, 1997) or just theory (or hybrid) reasoning. The paper formulates sufficient criteria for the construction of complete calculi which enable reasoning under hybrid theories combined from sub-theories given semantically and those given syntactically and briefly reports experimental work
Generic Polynomials with Few Parameters

AbstractWe call a polynomial g(t1, . . . , tm,X ) over a field K generic for a group G if it has Galois group G as a polynomial inX , and if every Galois field extension N/L withK⊆L and Gal(N/L) ≤G arises as the splitting field of a suitable specializationg (λ1, . . . , λm, X) withλi∈L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU2(q2) andGO3 (q)
Computing Global Extension Modules for Coherent Sheaves on a Projective
  Scheme

Let X be a projective scheme; let M and N be two coherent O_X-modules. Given
an integer m, we present an algorithm for computing the global extension module
Ext^m(X;M,N). In particular, this allows one to compute the sheaf cohomology
H^m(X,N) and to construct the sheaf corresponding to an element of the module
Ext^1(X;M,N). This algorithm can be implemented using only the computation of
Grobner bases ans syzygies, and it has been implemented in the computer algebra
system Macaulay2.Comment: 18 pages. amsLaTeX. no figures. Also available at
  http://math.berkeley.edu/~ggsmit
Weyl Closure of a Linear Differential Operator

AbstractWe study the Weyl closure Cl(L) =K(x)〈∂〉L∩D for an operator L of the first Weyl algebra D=K〈x, ∂〉. We give an algorithm to compute Cl(L) and we describe its initial ideal under the order filtration. Our main application is an algorithm for constructing a Jordan–Hölder series for a holonomic D -module and a formula for its length. Using the closure, we also reproduce a result ofStrömbeck (1978), who described the initial ideals of left ideals of D under the order filtration, and a result ofCannings and Holland (1994), who described the isomorphism classes of right ideals of D
Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials

AbstractWe define a special type of reduction in a free left module over a ring of difference–differential operators and use the idea of the Gröbner basis method to develop a technique that allows us to determine the Hilbert function of a finitely generated difference–differential module equipped with the natural double filtration. The results obtained are applied to the study of difference–differential field extensions and systems of difference–differential equations. We prove a theorem on difference–differential dimension polynomial that generalizes both the classical Kolchin’s theorem on dimension polynomial of a differential field extension and the corresponding author’s result for difference fields. We also determine invariants of a difference–differential dimension polynomial and consider a method of computation of the dimension polynomial associated with a system of linear difference–differential equations
A Fast Algorithm for Gröbner Basis Conversion and its Applications

AbstractThe Gröbner walk method converts a Gröbner basis by partitioning the computation of the basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of equations. The method works with ideals of zero-dimension as well as positive dimension. Typically, the target point of the walking path lies on the intersection of very many cones, which ends up with initial forms of a considerable number of terms. Therefore, it is crucial to the performance of the conversion to change the target point since we have to compute a Gröbner basis with respect to the elimination term order of such large initial forms. In contrast to heuristic methods found in the literature, in this paper the author presents a deterministic method to vary the target point in order to ensure the generality of the position, i.e. we always have just a few terms in the initial forms. It turns out that this theoretical result brings a dramatic speed-up in practice. We have implemented the Gröbner walk method together with the deterministic method for varying the target point in the kernel of Mathematica. Our experiments show the superlative performance of our improved Gröbner walk method in comparison with other known methods. Our best performance is 3 × 104times faster than the direct computation of the reduced Gröbner basis with respect to pure lexicographic term order (using the Buchberger algorithm and the sugar cube strategy). We also discuss the complexity of the conversion algorithm and prove a degree bound for polynomials in the target Gröbner basis. In the second part of the paper, we present some applications of the conversion method for implicitization and geometric reasoning. We compare the efficiency of the improved Gröbner walk method with other methods for elimination such as multivariate resultant methods
Gröbner Bases Applied to Finitely Generated Field Extensions

AbstractUsing a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (x)/k(g), resp. how to determine the (separable) degree if k(x)/k(g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(x)/k(g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the field-ideal correspondence can also be used to determine the elements of the Galois group
Embeddings ofPSL(2, 41) andPSL(2, 49) inE8(C)

AbstractWe classify embeddings of the finite simple groups PSL(2, 41) andPSL (2, 49) into algebraic groups of type E8in characteristic not dividing the orders of these respective groups. Lie theory and finite group theory point to families of elements in a group of typeE8 over a finite field which might satisfy a presentation for our finite simple group. Techniques from computational representation theory identify all suitable systems of generators. We describe a number of methods of computation in Chevalley groups of type E8
On the Deterministic Complexity of Factoring Polynomials

AbstractThe paper focuses on the deterministic complexity of factoring polynomials over finite fields assuming the extended Riemann hypothesis (ERH). By the works of Berlekamp (1967, 1970) and Zassenbaus (1969), the general problem reduces deterministically in polynomial time to finding a proper factor of any squarefree and completely splitting polynomial over a prime field Fp. Algorithms are designed to split such polynomials. It is proved that a proper factor of a polynomial can be found deterministically in polynomial time, under ERH, if its roots do not satisfy some stringent condition, called super square balanced. It is conjectured that super square balanced polynomials do not exist
Some Applications of Magma in Designs and Codes: Oval Designs, Hermitian Unitals and Generalized Reed–Muller Codes

AbstractWe describe three applications of Magma to problems in the area of designs and the associated codes: • Steiner systems, Hadamard designs and symmetric designs arising from an oval in an even-order plane, leading in the classical case to bent functions and difference-set designs; • the Hermitian unital as a 2-(q3+ 1, q+ 1, 1) design, and the code overFp where p divides q+ 1; • a basis of minimum-weight vectors for the code over Fpof the design of points and hyperplanes of the affine geometry AGd(Fp), where p is a prime
Irregular Primes and Cyclotomic Invariants to 12 Million

AbstractComputations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes Shokrollahi (1996). The latter idea reduces the problem to that of finding zeros of a polynomial overFpof degree < (p− 1)/2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives anO (p log2p loglog p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schönhage–Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for whichp− 1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer–Vandiver conjecture and for computing the cyclotomic invariants of a prime
An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals, and Applications to Commutative Semigroups

AbstractIt is known that the reduced Gröbner basis of general polynomial ideals can be computed in exponential space. The algorithm, obtained by Kühnle and Mayr, is, however, based on rather complex parallel computations, and, above that, makes extensive use of the parallel computation thesis. In this paper, we exhibit an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals which can be implemented without any complex parallel computations. This result is then applied to derive space optimal decision procedures for the finite enumeration and subword problems for commutative semigroups
Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal

AbstractWe present an algorithm that computes an unmixed-dimensional decomposition of a finitely generated perfect differential ideal I. Each Iiin the decompositionI=I1∩⋯ ∩Ikis given by its characteristic set. This decomposition is a generalization of the differential case of Kalkbrener’s decomposition. We use a different approach. The basic operation in our algorithm is the computation of the inverse of an algebraic polynomial with respect to a finite set of algebraic polynomials. No factorization is needed. Some of the main problems in polynomial ideal theory can be solved by means of this decomposition: we show how the radical membership can be decided, a characteristic set of a prime differential ideal can be selected, and the differential dimension with a parametric set of a differential ideal can be read. The algorithm has been implemented in the computer algebra system MAPLE and has been tested successfully on many examples
On Solutions of Linear Ordinary Difference Equations in their Coefficient Field

AbstractWe extend the notion of monomial extensions of differential fields, i.e. simple transcendental extensions in which the polynomials are closed under differentiation, to difference fields. The structure of such extensions provides an algebraic framework for solving generalized linear difference equations with coefficients in such fields. We then describe algorithms for finding the denominator of any solution of those equations in an important subclass of monomial extensions that includes transcendental indefinite sums and products. This reduces the general problem of finding the solutions of such equations in their coefficient fields to bounding their degrees. In the base case, this yields in particular a new algorithm for computing the rational solutions of q -difference equations with polynomial coefficients
The Projective Noether Maple Package: Computing the Dimension of a Projective Variety

AbstractRecent theoretical advances in elimination theory use straight-line programs as a data structure to represent multivariate polynomials. We present here the Projective Noether Package which is a Maple implementation of one of these new algorithms, yielding as a byproduct a computation of the dimension of a projective variety. Comparative results on benchmarks for time and space of several families of multivariate polynomial equation systems are given and we point out both weaknesses and advantages of different approaches
Challenges of Symbolic Computation: My Favorite Open Problems

AbstractThe success of the symbolic mathematical computation discipline is striking. The theoretical advances have been continuous and significant: Gröbner bases, the Risch integration algorithm, integer lattice basis reduction, hypergeometric summation algorithms, etc. From the beginning in the early 1960s, it has been the tradition of our discipline to create software that makes our ideas readily available to scientists, engineers, and educators: SAC-1, Reduce, Macsyma, etc. The commercial viability of our system products is proven by Maple and Mathematica. Today’s user communities of symbolic computation systems are diverse: educators, engineers, stock market analysts, etc. The mathematics and computer science in the design and implementation of our algorithms are sophisticated. The research challenges in symbolic computation at the close of the twentieth century are formidable. I state my favorite eight open problems in symbolic computation. They range from problems in symbolic/numeric computing, symbolic algorithm synthesis, to system component construction. I have worked on seven of my problems and borrowed one from George Collins. I present background to each of my problems and a clear-cut test that evaluates whether a proposed attack has solved one of my problems. An additional ninth open problem by Rob Corless and David Jeffrey on complex function semantics is given in an appendix
Finding a Basis of a Linear System with Pairwise Distinct Discrete Valuations on an Algebraic Curve

AbstractUnder the assumption that we have defining equations of an affine algebraic curve in special position with respect to a rational place Q, we propose an algorithm computing a basis of L(D) of a divisor D from an ideal basis of the ideal L(D+∞Q) of the affine coordinate ring L(∞Q) of the given algebraic curve, where L(D+∞Q) :=∪i=1∞L(D+iQ). Elements in the basis produced by our algorithm have pairwise distinct discrete valuations at Q, which is convenient in the construction of algebraic geometry codes. Our method is applicable to a curve embedded in an affine space of arbitrary dimension, and involves only the Gaussian elimination and the division of polynomials by the Gröbner basis for the ideal defining the curve
Boundary of Hurwitz Spaces and Explicit Patching

AbstractWe describe a general method for the study and computation of Hurwitz spaces of curves of any genus. It is based on a careful combinatorial study of the associated Jacobian. The key tool is an adapted cell decomposition of the cohomology of a graph (used here for the intersection graphs of special curves). We illustrate this method in the context of modular curves to produce modular units. We also give a detailed simple example and show how the algebraic difficulty of Hurwitz spaces computation can be reduced to its minimum
An Algorithm of Katz and its Application to the Inverse Galois Problem

AbstractIn this paper we present a new and elementary approach for proving the main results of Katz (1996) using the Jordan–Pochhammer matrices of Takano and Bannai (1976) and Haraoka (1994). We find an explicit version of the middle convolution of Katz (1996) that connects certain tuples of matrices in linear groups. From this, Katz’ existence algorithm for rigid tuples in linear groups can easily be deduced. It can further be shown that the convolution operation on tuples commutes with the braid group action. This yields a new approach in inverse Galois theory for realizing subgroups of linear groups regularly as Galois groups over Q. This approach is then applied to realize numerous series of classical groups regularly as Galois groups over Q. In the Appendix we treat an additive version of the convolution