scholarly journals SOUR graphs for efficient completion

1998 ◽  
Vol Vol. 2 ◽  
Author(s):  
Christopher Lynch ◽  
Polina Strogova
Keyword(s):  
Data Structure ◽  
Term Structure ◽  
Standard Procedure ◽  
Critical Pair ◽  
Exponential Behavior ◽  
Rewriting Systems ◽  
Group Completion ◽  
International Audience ◽  
Local Graph ◽  
Rewrite Rules

International audience We introduce a data structure called \emphSOUR graphs and present an efficient Knuth-Bendix completion procedure based on it. \emphSOUR graphs allow for a maximal structure sharing of terms in rewriting systems. The term representation is a dag representation, except that edges are labelled with equational constraints and variable renamings. The rewrite rules correspond to rewrite edges, the unification problems to unification edges. The Critical Pair and Simplification inferences are recognized as patterns in the graph and are performed as local graph transformations. Our algorithm avoids duplicating term structure while performing inferences, which causes exponential behavior in the standard procedure. This approach gives a basis to design other completion algorithms, such as goal-oriented completion, concurrent completion and group completion procedures.

scholarly journals Groupoids and the algebra of rewriting in group presentations

2020 ◽  
Author(s):  
N. D. Gilbert ◽  
E. A. McDougall
Keyword(s):  
Reduced Form ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Group Presentations ◽  
Rewrite Rules ◽  
Fundamental Groupoid

Abstract Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form.

scholarly journals Analysis of some parameters for random nodes in priority trees

2008 ◽  
Vol Vol. 10 no. 2 (Analysis of Algorithms) ◽  
Author(s):  
Alois Panholzer
Keyword(s):  
Data Structure ◽  
Analysis Of Algorithms ◽  
Priority Queue ◽  
Limiting Distribution ◽  
Node Number ◽  
Random Node ◽  
International Audience ◽  
A Priority

Analysis of Algorithms International audience Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . . , n are equally likely to construct a priority tree of size n we study the following parameters in size-n trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.

scholarly journals Optimal Prefix and Suffix Queries on Texts

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Maxime Crochemore ◽  
Costas S. Iliopoulos ◽  
M. Sohel Rahman
Keyword(s):  
Data Structure ◽  
Suffix Tree ◽  
Query Time ◽  
Time And Space ◽  
Matching Problem ◽  
International Audience ◽  
Tree Data ◽  
Tree Data Structure ◽  
Restricted Pattern ◽  
Time And Space Complexity

International audience In this paper, we study a restricted version of the position restricted pattern matching problem introduced and studied by Mäkinen and Navarro [Position-Restricted Substring Searching, LATIN 2006]. In the problem handled in this paper, we are interested in those occurrences of the pattern that lies in a suffix or in a prefix of the given text. We achieve optimal query time for our problem against a data structure which is an extension of the classic suffix tree data structure. The time and space complexity of the data structure is dominated by that of the suffix tree. Notably, the (best) algorithm by Mäkinen and Navarro, if applied to our problem, gives sub-optimal query time and the corresponding data structure also requires more time and space.

scholarly journals BREP-INDEX: A MULTIDIMENSIONAL SPACE PARTITIONING TREE

1991 ◽  
Vol 01 (03) ◽  
pp. 243-261 ◽  
Author(s):  
GEORGE VANĚČEK
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Boundary Representation ◽  
Multidimensional Space ◽  
Classification Methods ◽  
Space Partitioning ◽  
Spatial Access ◽  
Finite Precision ◽  
Finite Precision Arithmetic ◽  
Rewrite Rules

In this paper we present the Brep-index, a multidimensional space partitioning data structure that provides quick spatial access to the vertices, edges and faces of a boundary representation (Brep), thus yielding a single unified representation for polyhedral solids. We give an algorithm for the construction of the Brep-index and prove its correctness. We show that its size is Ω(v+e+f), where v, e, and f are the number of vertices, edges, and faces of the Brep. The lower bound can be achieved for some Breps by compressing the structure using simple rewrite rules. We then demonstrate robust point and line/Brep classification methods given an implementation that uses finite-precision arithmetic.

scholarly journals Critical Pair Analysis in Nominal Rewriting

10.29007/7q54 ◽  
2018 ◽  
Author(s):  
Takaki Suzuki ◽  
Kentaro Kikuchi ◽  
Takahito Aoto ◽  
Yoshihito Toyama
Keyword(s):  
Order Term ◽  
Term Rewriting ◽  
Critical Pair ◽  
Binding Mechanism ◽  
First Order ◽  
Rewrite Rules ◽  
Critical Pair Analysis ◽  
Orthogonal Systems ◽  
Pair Analysis ◽  
Critical Pairs

Nominal rewriting (Fernández, Gabbay & Mackie, 2004;Fernández & Gabbay, 2007) is a framework that extendsfirst-order term rewriting by a binding mechanismbased on the nominal approach (Gabbay & Pitts, 2002;Pitts, 2003). In this paper, we investigate confluenceproperties of nominal rewriting, following the study oforthogonal systems in (Suzuki et al., 2015), but herewe treat systems in which overlaps of the rewrite rulesexist. First we present an example where choice ofbound variables (atoms) of rules affects joinability ofthe induced critical pairs. Then we give a detailedproof of the critical pair lemma, and illustrate someof its applications including confluence results fornon-terminating systems.

Promoting rewriting to a programming language: a compiler for non-deterministic rewrite programs in associative-commutative theories

2001 ◽  
Vol 11 (2) ◽  
pp. 207-251 ◽  
Author(s):  
HÉLÈNE KIRCHNER ◽  
PIERRE-ETIENNE MOREAU
Keyword(s):  
Data Structure ◽  
Evaluation Process ◽  
Restricted Class ◽  
Programming Environments ◽  
Matching Problems ◽  
First Order ◽  
Class A ◽  
Equational Theories ◽  
Rewrite Rules ◽  
Compilation Techniques

First-order languages based on rewrite rules share many features with functional languages, but one difference is that matching and rewriting can be made much more expressive and powerful by incorporating some built-in equational theories. To provide reasonable programming environments, compilation techniques for such languages based on rewriting have to be designed. This is the topic addressed in this paper. The proposed techniques are independent from the rewriting language, and may be useful to build a compiler for any system using rewriting modulo Associative and Commutative (AC) theories. An algorithm for many-to-one AC matching is presented, that works efficiently for a restricted class of patterns. Other patterns are transformed to fit into this class. A refined data structure, namely compact bipartite graph, allows encoding of all matching problems relative to a set of rewrite rules. A few optimisations concerning the construction of the substitution and of the reduced term are described. We also address the problem of non-determinism related to AC rewriting, and show how to handle it through the concept of strategies. We explain how an analysis of the determinism can be performed at compile time, and we illustrate the benefits of this analysis for the performance of the compiled evaluation process. Then we briefly introduce the ELAN system and its compiler, in order to give some experimental results and comparisons with other languages or rewrite engines.

scholarly journals A lightweight approach to datatype-generic rewriting

2010 ◽  
Vol 20 (3-4) ◽  
pp. 375-413 ◽  
Author(s):  
THOMAS VAN NOORT ◽  
ALEXEY RODRIGUEZ YAKUSHEV ◽  
STEFAN HOLDERMANS ◽  
JOHAN JEURING ◽  
BASTIAAN HEEREN ◽  
...  
Keyword(s):  
Term Rewriting ◽  
Term Rewriting Systems ◽  
Rewriting Systems ◽  
Fundamental Limitations ◽  
Rewrite Rules

AbstractTerm-rewriting systems can be expressed as generic programs parameterised over the shape of the terms being rewritten. Previous implementations of generic rewriting libraries require users to either adapt the datatypes that are used to describe these terms or to specify rewrite rules as functions. These are fundamental limitations: the former implies a lot of work for the user, while the latter makes it hard if not impossible to document, test, and analyze rewrite rules. In this article, we demonstrate how to overcome these limitations by making essential use of type-indexed datatypes. Our approach is lightweight in that it is entirely expressible in Haskell with GADTs and type families and can be readily packaged for use with contemporary Haskell distributions.

Cut rules and explicit substitutions

2001 ◽  
Vol 11 (1) ◽  
pp. 131-168 ◽  
Author(s):  
RENÉ VESTERGAARD ◽  
JOE WELLS
Keyword(s):  
Order Term ◽  
Term Rewriting ◽  
Cut Elimination ◽  
Cut Rule ◽  
Logical Deduction ◽  
Rewriting Systems ◽  
Explicit Substitution ◽  
Explicit Substitutions ◽  
Rewrite Rules ◽  
De Bruijn Indices

We introduce a method to associate calculi of proof terms and rewrite rules with cut elimination procedures for logical deduction systems (i.e., Gentzen-style sequent calculi) in the case of intuitionistic logic. We illustrate this method using two different versions of the cut rule for a variant of the intuitionistic fragment of Kleene's logical deduction system G3.Our systems are in fact calculi of explicit substitution, where the cut rule introduces an explicit substitution and the left-→ rule introduces a binding of the result of a function application. Cut propagation steps of cut elimination correspond to propagation of explicit substitutions, and propagation of weakening (to eliminate it) corresponds to propagation of index-updating operations. We prove various subject reduction, termination, and confluence properties for our calculi.Our calculi improve on some earlier calculi for logical deduction systems in a number of ways. By using de Bruijn indices, our calculi qualify as first-order term rewriting systems (TRS's), allowing us to use correctly certain results for TRS's about termination. Unlike in some other calculi, each of our calculi has only one cut rule and we do not need unusual features of sequents.We show that the substitution and index-updating mechanisms of our calculi work the same way as the substitution and index-updating mechanisms of Kamareddine and Ríos' λs and λt, two well-known systems of explicit substitution for the standard λ-calculus. By a change in the format of sequents, we obtain similar results for a known λ-calculus with variables and explicit substitutions, Rose's λbxgc.

scholarly journals Staircase diagrams and the enumeration of smooth Schubert varieties

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Edward Richmond ◽  
William Slofstra
Keyword(s):  
Data Structure ◽  
Finite Type ◽  
Schubert Varieties ◽  
International Audience ◽  
Combinatorial Data

International audience In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams.

Evolution of the microstructure of Cu(Ti)/SiO2 layers annealed in NH3

1995 ◽  
Vol 53 ◽  
pp. 452-453
Author(s):  
J. Liu ◽  
N. D. Theodore ◽  
D. Adams ◽  
S. Russell ◽  
T. L. Alford ◽  
...  
Keyword(s):  
Thermal Stability ◽  
Titanium Nitride ◽  
Large Scale ◽  
Standard Procedure ◽  
Alloy Film ◽  
Electron Beam Evaporation ◽  
Copper Metallization ◽  
Nominal Composition ◽  
Ti Alloy ◽  
Thermally Grown

Copper-based metallization has recently attracted extensive research because of its potential application in ultra-large-scale integration (ULSI) of semiconductor devices. The feasibility of copper metallization is, however, limited due to its thermal stability issues. In order to utilize copper in metallization systems diffusion barriers such as titanium nitride and other refractory materials, have been employed to enhance the thermal stability of copper. Titanium nitride layers can be formed by annealing Cu(Ti) alloy film evaporated on thermally grown SiO2 substrates in an ammonia ambient. We report here the microstructural evolution of Cu(Ti)/SiO2 layers during annealing in NH3 flowing ambient.The Cu(Ti) films used in this experiment were prepared by electron beam evaporation onto thermally grown SiO2 substrates. The nominal composition of the Cu(Ti) alloy was Cu73Ti27. Thermal treatments were conducted in NH3 flowing ambient for 30 minutes at temperatures ranging from 450°C to 650°C. Cross-section TEM specimens were prepared by the standard procedure.

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