2,725 research outputs found
Expectations for Associative-Commutative Unification Speedups in a Multicomputer Environment
An essential element of automated deduction systems is unification algorithms which identify general substitutions and when applied to two expressions, make them identical. However, functions which are associative and commutative, such as the usual addition and multiplication functions, often arise in term rewriting systems, program verification, the theory of abstract data types and logic programming. The introduction to the associative and commutative equality axioms together with standard unification brings with it problems of termination and unreasonably large search spaces. One way around these problems is to remove the troublesome axioms from the system and to employ a unification algorithm which unifies modulo the axioms of associativity and commutativity. Unlike standard unification, the associative-commutative (AC) unification of two expressions can lead to the formation of many most general unifiers. A report is presented on a hybrid AC unification algorithm which has been implemented to run in parallel on an Intel iPSC/
Associative-commutative unification
Résumé disponble dans les fichiers attaché
Higher-Order Equational Pattern Anti-Unification
We consider anti-unification for simply typed lambda terms in associative, commutative, and associative-commutative theories and develop a sound and complete algorithm which takes two lambda terms and computes their generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of generalizations contains finitely many elements. We define the notion of optimal solution and investigate special fragments of the problem for which the optimal solution can be computed in linear or polynomial time
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
Non-Associative Geometry and the Spectral Action Principle
Chamseddine and Connes have argued that the action for Einstein gravity,
coupled to the SU(3)\times SU(2)\times U(1) standard model of particle physics,
may be elegantly recast as the "spectral action" on a certain "non-commutative
geometry." In this paper, we show how this formalism may be extended to
"non-associative geometries," and explain the motivations for doing so. As a
guiding illustration, we present the simplest non-associative geometry (based
on the octonions) and evaluate its spectral action: it describes Einstein
gravity coupled to a G_2 gauge theory, with 8 Dirac fermions (which transform
as a singlet and a septuplet under G_2). This is just the simplest example: in
a forthcoming paper we show how to construct more realistic models that include
Higgs fields, spontaneous symmetry breaking and fermion masses.Comment: 24 pages, no figures, matches JHEP versio
COMMENTS ABOUT HIGGS FIELDS, NONCOMMUTATIVE GEOMETRY AND THE STANDARD MODEL
We make a short review of the formalism that describes Higgs and Yang Mills
fields as two particular cases of an appropriate generalization of the notion
of connection. We also comment about the several variants of this formalism,
their interest, the relations with noncommutative geometry, the existence (or
lack of existence) of phenomenological predictions, the relation with Lie
super-algebras etc.Comment: pp 20, LaTeX file, no figures, also available via anonymous ftp at
ftp://cpt.univ-mrs.fr/ or via gopher gopher://cpt.univ-mrs.fr
Nominal C-Unification
Nominal unification is an extension of first-order unification that takes
into account the \alpha-equivalence relation generated by binding operators,
following the nominal approach. We propose a sound and complete procedure for
nominal unification with commutative operators, or nominal C-unification for
short, which has been formalised in Coq. The procedure transforms nominal
C-unification problems into simpler (finite families) of fixpoint problems,
whose solutions can be generated by algebraic techniques on combinatorics of
permutations.Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
The Role of Term Symmetry in E-Unification and E-Completion
A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix [070] or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied.
The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings.
The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply plugging in a unification module for the new theory
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