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In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner bases theory. Our algorithm also requires computing computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated K-algebras.
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a smooth number.
It is known that multiplication of linear differential operators over ground fields of characteristic zero can be reduced to a constant number of matrix products. We give a new algorithm by evaluation and interpolation which is faster than the previously-known one by a constant factor, and prove that in characteristic zero, multiplication of differential operators and of matrices are computationally equivalent problems. In positive characteristic, we show that differential operators can be multiplied in nearly optimal time. Theoretical results are validated by intensive experiments.
In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit. In the linear case, we apply basic "symbolic" methods from linear algebra to the system of differential equations which is used to model the analog circuit. For nonlinear circuits and their corresponding nonlinear differential equations, we show how to employ computer algebra tools implemented in Maple, which are based on differential algebra.
This paper describes a series of model-checking experiments to verify optimistic replication algorithms based on Operational Transformation (OT) approach used for supporting collaborative edition. We formally define, using tool UPPAAL, the behavior and the main consistency requirement (i.e. convergence property) of the collaborative editing systems, as well as the abstract behavior of the environment where these systems are supposed to operate. Due to data replication and the unpredictable nature of user interactions, such systems have infinitely many states. So, we show how to exploit some features of the UPPAAL specification language to attenuate the severe state explosion problem. Two models are proposed. The first one, called concrete model, is very close to the system implementation but runs up against a severe explosion of states. The second model, called symbolic model, aims to overcome the limitation of the concrete model by delaying the effective selection and execution of editing operations until the construction of symbolic execution traces of all sites is completed. Experimental results have shown that the symbolic model allows a significant gain in both space and time. Using the symbolic model, we have been able to show that if the number of sites exceeds 2 then the convergence property is not satisfied for all OT algorithms considered here. A counterexample is provided for every algorithm.
We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this approach with explicit examples. In particular, we describe a short program to determine whether a given linear exterior differential system is involutive.
In this paper, we informally introduce dynamic mind-maps that represent a new approach on the basis of a dynamic construction of connectionist structures during the processing of a data stream. This allows the representation and processing of recursively defined structures and avoids the problem of a more traditional, fixed-size architecture with the processing of input structures of unknown size. For a data stream analysis with association discovery, the incremental analysis of data leads to results on demand. Here, we describe a framework that uses symbolic cells to calculate associations based on transactional data streams as it exists in e.g. bibliographic databases. We follow a natural paradigm of applying simple operations on cells yielding on a mind-map structure that adapts over time.
In a previous work, the authors proposed a Grammatical Evolution algorithm to automatically generate Lindenmayer Systems which represent fractal curves with a pre-determined fractal dimension. This paper gives strong statistical evidence that the probability distributions of the execution time of that algorithm exhibits a heavy tail with an hyperbolic probability decay for long executions, which explains the erratic performance of different executions of the algorithm. Three different restart strategies have been incorporated in the algorithm to mitigate the problems associated to heavy tail distributions: the first assumes full knowledge of the execution time probability distribution, the second and third assume no knowledge. These strategies exploit the fact that the probability of finding a solution in short executions is non-negligible and yield a severe reduction, both in the expected execution time (up to one order of magnitude) and in its variance, which is reduced from an infinite to a finite value.
This paper shows how to use Computational Algebra techniques, namely the decomposition of rational functions in one variable, to explore a certain set of modular functions, called replicable functions, that arise in Monstrous Moonshine. In particular, we have computed all the rational relations with coefficients in Z between pairs of replicable functions. ----- En este articulo mostramos como usar tecnicas de Algebra Computacional, concretamente la descomposcion de funciones racionales univariadas, para estudiar un cierto conjunto de funciones modulares, llamadas funciones replicables, que aparecen en Monstrous Moonshine. En concreto, hemos calculado todas las relaciones racionales con coeficientes en Z entre pares de funciones replicables.
Let G=Aut_K (K(x)) be the Galois group of the transcendental degree one pure field extension K(x)/K. In this paper we describe polynomial time algorithms for computing the field Fix(H) fixed by a subgroup H < G and for computing the fixing group G_f of a rational function f in K(x).
In this paper we present an algorithm to compute all unirational fields of transcendence degree one containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate rational function f of the form f=g(h), where g is a univariate rational function and h a multivariate one.
It is proposed the algorithm that find a basis of the ideal and a basis of the space of all root functionals by using the extension operation for bounded root functionals, when the number of polynomials is equal to the number of variables, if it is known that the ideal of polynomials is 0-dimensional. The asyptotic complexity of this algorithm is d^{O(n)} operations, where n is the number of polynomials and the number of variables, d is the maximal degree of polynomials. The extension operation has connection with the multivariate Bezoutian construction.
Requirements about the quality of clinical guidelines can be represented by schemata borrowed from the theory of abductive diagnosis, using temporal logic to model the time-oriented aspects expressed in a guideline. Previously, we have shown that these requirements can be verified using interactive theorem proving techniques. In this paper, we investigate how this approach can be mapped to the facilities of a resolution-based theorem prover, Otter, and a complementary program that searches for finite models of first-order statements, Mace. It is shown that the reasoning required for checking the quality of a guideline can be mapped to such fully automated theorem-proving facilities. The medical quality of an actual guideline concerning diabetes mellitus 2 is investigated in this way.
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial polynomials. Finally, we discuss usage of recursive subresultants in approximate algebraic computation, which motivates the present work.
In this paper, we develop a novel approach to model the Internet round trip time using a recently proposed symbolic type neural network model called symbolic function network. The developed predictor is shown to have good generalization performance and simple representation compared to the multilayer perceptron based predictors.
The paper is organized as a self-contained literate Prolog program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The complete Prolog code is available at http://logic.csci.unt.edu/tarau/research/2008/pHFS.zip . First, ranking and unranking functions for some "mathematically elegant" data types in the universe of Hereditarily Finite Sets with Urelements are provided, resulting in arithmetic encodings for powersets, hypergraphs, ordinals and choice functions. After implementing a digraph representation of Hereditarily Finite Sets we define {\em decoration functions} that can recover well-founded sets from encodings of their associated acyclic digraphs. We conclude with an encoding of arbitrary digraphs and discuss a concept of duality induced by the set membership relation. In the process, we uncover the surprising possibility of internally sharing isomorphic objects, independently of their language level types and meanings.
A "pairing function" J associates a unique natural number z to any two natural numbers x,y such that for two "unpairing functions" K and L, the equalities K(J(x,y))=x, L(J(x,y))=y and J(K(z),L(z))=z hold. Using pairing functions on natural number representations of truth tables, we derive an encoding for Binary Decision Diagrams with the unique property that its boolean evaluation faithfully mimics its structural conversion to a a natural number through recursive application of a matching pairing function. We then use this result to derive {\em ranking} and {\em unranking} functions for BDDs and reduced BDDs. The paper is organized as a self-contained literate Prolog program, available at http://logic.csci.unt.edu/tarau/research/2008/pBDD.zip Keywords: logic programming and computational mathematics, pairing/unpairing functions, encodings of boolean functions, binary decision diagrams, natural number representations of truth tables
Using specializations of unfold and fold on a generic tree data type we derive unranking and ranking functions providing natural number encodings for various Hereditarily Finite datatypes. In this context, we interpret unranking operations as instances of a generic anamorphism and ranking operations as instances of the corresponding catamorphism. Starting with Ackerman's Encoding from Hereditarily Finite Sets to Natural Numbers we define pairings and tuple encodings that provide building blocks for a theory of Hereditarily Finite Functions. The more difficult problem of ranking and unranking Hereditarily Finite Permutations is then tackled using Lehmer codes and factoradics. The self-contained source code of the paper, as generated from a literate Haskell program, is available at \url{http://logic.csci.unt.edu/tarau/research/2008/fFUN.zip}. Keywords: ranking/unranking, pairing/tupling functions, Ackermann encoding, hereditarily finite sets, hereditarily finite functions, permutations and factoradics, computational mathematics, Haskell data representations