5,131 research outputs found
A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems
We give a method to prove confluence of term rewriting systems that contain
non-terminating rewrite rules such as commutativity and associativity. Usually,
confluence of term rewriting systems containing such rules is proved by
treating them as equational term rewriting systems and considering E-critical
pairs and/or termination modulo E. In contrast, our method is based solely on
usual critical pairs and it also (partially) works even if the system is not
terminating modulo E. We first present confluence criteria for term rewriting
systems whose rewrite rules can be partitioned into a terminating part and a
possibly non-terminating part. We then give a reduction-preserving completion
procedure so that the applicability of the criteria is enhanced. In contrast to
the well-known Knuth-Bendix completion procedure which preserves the
equivalence relation of the system, our completion procedure preserves the
reduction relation of the system, by which confluence of the original system is
inferred from that of the completed system
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule
A driving question in (quantum) cohomology of flag varieties is to find
non-recursive, positive combinatorial formulas for expressing the product of
two classes in a particularly nice basis, called the Schubert basis. Bertram,
Ciocan-Fontanine and Fulton provided a way to compute quantum products of
Schubert classes in the Grassmannian of k-planes in complex n-space by doing
classical multiplication and then applying a combinatorial rim hook rule which
yields the quantum parameter. In this paper, we provide a generalization of
this rim hook rule to the setting in which there is also an action of the
complex torus. Combining this result with Knutson and Tao's puzzle rule then
gives an effective algorithm for computing all equivariant quantum
Littlewood-Richardson coefficients. Interestingly, this rule requires a
specialization of torus weights modulo n, suggesting a direct connection to the
Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in
Algebraic Combinatoric
Tactics for Reasoning modulo AC in Coq
We present a set of tools for rewriting modulo associativity and
commutativity (AC) in Coq, solving a long-standing practical problem. We use
two building blocks: first, an extensible reflexive decision procedure for
equality modulo AC; second, an OCaml plug-in for pattern matching modulo AC. We
handle associative only operations, neutral elements, uninterpreted function
symbols, and user-defined equivalence relations. By relying on type-classes for
the reification phase, we can infer these properties automatically, so that
end-users do not need to specify which operation is A or AC, or which constant
is a neutral element.Comment: 16
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
On Link Homology Theories from Extended Cobordisms
This paper is devoted to the study of algebraic structures leading to link
homology theories. The originally used structures of Frobenius algebra and/or
TQFT are modified in two directions. First, we refine 2-dimensional cobordisms
by taking into account their embedding into the three space. Secondly, we
extend the underlying cobordism category to a 2-category, where the usual
relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is
called an extended quantum field theory (EQFT). We show that the Khovanov
homology, the nested Khovanov homology, extracted by Stroppel and Webster from
Seidel-Smith construction, and the odd Khovanov homology fit into this setting.
Moreover, we prove that any EQFT based on a Z_2-extension of the embedded
cobordism category which coincides with Khovanov after reducing the
coefficients modulo 2, gives rise to a link invariant homology theory
isomorphic to those of Khovanov.Comment: Lots of figure
On Isomorphism of "Functional" Intersection and Union Types
Type isomorphism is useful for retrieving library components, since a
function in a library can have a type different from, but isomorphic to, the
one expected by the user. Moreover type isomorphism gives for free the coercion
required to include the function in the user program with the right type. The
present paper faces the problem of type isomorphism in a system with
intersection and union types. In the presence of intersection and union,
isomorphism is not a congruence and cannot be characterised in an equational
way. A characterisation can still be given, quite complicated by the
interference between functional and non functional types. This drawback is
faced in the paper by interpreting each atomic type as the set of functions
mapping any argument into the interpretation of the type itself. This choice
has been suggested by the initial projection of Scott's inverse limit
lambda-model. The main result of this paper is a condition assuring type
isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
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