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Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view. The technique ofvectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases. Furthermore, a rank-based monomialrepresentation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorizedcomparisons. Extensive benchmark tests with the Computer Algebra System SINGULAR are used to evaluate these concepts.
Groups can be studied using methods from different fields such as combinatorial group theory or string rewriting. Recently techniques from Gröbner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to special polynomials called binomials. In the same mood, the aim of this paper is to discuss the relation between Nielsen reduced sets of generators and the Todd-Coxeter coset enumeration procedure on the one side and the Gröbner basis theory for free group rings on the other. While it is well-known that there is a strong relationship between Buchberger's algorithm and the Knuth-Bendix completion procedure, and there are interpretations of the Todd-Coxeter coset enumeration procedure using the Knuth-Bendix procedure for special cases, our aim is to show how a verbatim interpretation of the Todd-Coxeter procedure can be obtained by linking recent Gröbner techniques like prefix Gröbner bases and the FGLM algorithm as a tool to study the duality of ideals. As a side product our procedure computes Nielsen reduced generating sets for subgroups in finitely generated free groups.
This paper describes a tableau-based higher-order theorem prover HOT and an application to natural language semantics. In this application, HOT is used to prove equivalences using world knowledge during higher-order unification (HOU). This extended form of HOU is used to compute the licensing conditions for corrections.
Simultaneous quantifier elimination in sequent calculus is an improvement over the well-known skolemization. It allows a lazy handling of instantiations as well as of the order of certain reductions. We prove the soundness of a sequent calculus which incorporates a rule for simultaneous quantifier elimination. The proof is performed by semantical arguments and provides some insights into the dependencies between various formulas in a sequent.
Rewriting techniques have been applied successfully to various areas of symbolic computation. Here we consider the notion of prefix-rewriting and give a survey on its applications to the subgroup problem in combinatorial group theory. We will see that for certain classes of finitely presented groups finitely generated subgroups can be described through convergent prefix-rewriting systems, which can be obtained from a presentation of the group considered and a set of generators for the subgroup through a specialized Knuth-Bendix style completion procedure. In many instances a finite presentation for the subgroup considered can be constructed from such a convergent prefix-rewriting system, thus solving the subgroup presentation problem. Finally we will see that the classical procedures for computing Nielsen reduced sets of generators for a finitely generated subgroup of a free group and the Todd-Coxeter coset enumeration can be interpreted as particular instances of prefix-completion. Further, both procedures are closely related to the computation of prefix Gr"obner bases for right ideals in free group rings.
Todd and Coxeter's method for enumerating cosets of finitely generated subgroups in finitely presented groups (abbreviated by Tc here) is one famous method from combinatorial group theory for studying the subgroup problem. Since prefix string rewriting is also an appropriate method to study this problem, prefix string rewriting methods have been compared to Tc. We recall and compare two of them briefly, one by Kuhn and Madlener [4] and one by Sims [15]. A new approach using prefix string rewriting in free groups is derived from the algebraic method presented by Reinert, Mora and Madlener in [14] which directly emulates Tc. It is extended to free monoids and an algebraic characterization for the "cosets" enumerated in this setting is provided.
We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).
On a family F of probability measures on a measure space we consider the Hellinger and Kullback-Leibler distances. We show that under suitable regulari ty conditions Jeffreys' prior is proportional to the k-dimensional Hausdorff measure w.r.t. Hellinger dis tance respectively to the k2 -dimensional Hausdorff measure w.r.t. Kullback-Leibler distance. The proof i s based on an area-formula for the Hausdorff measure w.r.t. to generalized distances.
The paper studies differential and related properties of functions of a real variable with values in the space of signed measures. In particular the connections between different definitions of differentiability are described corresponding to different topologies on the measures. Some conditions are given for the equivalence of the measures in the range of such a function. These conditions are in terms of socalled logarithmic derivatives and yield a generalization of the Cameron-Martin-Maruyama-Girsanov formula. Questions of this kind appear both in the theory of differentiable measures on infinite-dimensional spaces and in the theory of statistical experiments.