27,924 research outputs found
Termination orderings for associative-commutative rewriting systems
In this paper we describe a new class of orderings—associative path orderings—for proving termination of associative-commutative term rewriting systems .These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to E - congruence classes, where E is an equational theory consisting of associativity and commutativity axioms. Associative path orderings are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition,the associative path condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings,boolean algebra,etc.) and, in addition, point out ways to handle situations where the associative path condition is too restrictive
Ordering constraints on trees
We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature
On an easy transition from operator dynamics to generating functionals by Clifford algebras
Clifford geometric algebras of multivectors are treated in detail. These
algebras are build over a graded space and exhibit a grading or multivector
structure. The careful study of the endomorphisms of this space makes it clear,
that opposite Clifford algebras have to be used also. Based on this
mathematics, we give a fully Clifford algebraic account on generating
functionals, which is thereby geometric. The field operators are shown to be
Clifford and opposite Clifford maps. This picture relying on geometry does not
need positivity in principle. Furthermore, we propose a transition from
operator dynamics to corresponding generating functionals, which is based on
the algebraic techniques. As a calculational benefit, this transition is
considerable short compared to standard ones. The transition is not injective
(unique) and depends additionally on the choice of an ordering. We obtain a
direct and constructive connection between orderings and the explicit form of
the functional Hamiltonian. These orderings depend on the propagator of the
theory and thus on the ground state. This is invisible in path integral
formulations. The method is demonstrated within two examples, a non-linear
spinor field theory and spinor QED. Antisymmetrized and normal-ordered
functional equations are derived in both cases.Comment: 23p., 76kB, plain LaTeX, [email protected]
The first-order theory of lexicographic path orderings is undecidable
We show, under some assumption on the signature, that the *This formula not viewable on a Text-Browser* fragment of the theory of any lexicographic path ordering is undecidable. This applies to partial and to total precedences. Our result implies in particular that the simplification rule of ordered completion is undecidable
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
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