scholarly journals Rewriting with generalized nominal unification

2020 ◽  
Vol 30 (6) ◽  
pp. 710-735
Author(s):  
Yunus Kutz ◽  
Manfred Schmidt-Schauß
Keyword(s):  
Rewrite Rules ◽  
Rewrite System ◽  
Critical Pairs

AbstractWe consider matching, rewriting, critical pairs and the Knuth–Bendix confluence test on rewrite rules in a nominal setting extended by atom-variables. We utilize atom-variables instead of atoms to formulate and rewrite rules on constrained expressions, which is an improvement of expressiveness over previous approaches. Nominal unification and nominal matching are correspondingly extended. Rewriting is performed using nominal matching, and computing critical pairs is done using nominal unification. We determine the complexity of several problems in a quantified freshness logic. In particular we show that nominal matching is $$\prod _2^p$$ -complete. We prove that the adapted Knuth–Bendix confluence test is applicable to a nominal rewrite system with atom-variables, and thus that there is a decidable test whether confluence of the ground instance of the abstract rewrite system holds. We apply the nominal Knuth–Bendix confluence criterion to the theory of monads and compute a convergent nominal rewrite system modulo alpha-equivalence.

scholarly journals OORS: An object-oriented rewrite system

2007 ◽  
Vol 4 (2) ◽  
pp. 2-26
Author(s):  
Gernot Gebhard ◽  
Philipp Lucas
Keyword(s):  
Code Generation ◽  
Graphics Processing Units ◽  
Object Oriented ◽  
Graphics Hardware ◽  
Code Optimization ◽  
Target Architecture ◽  
Rewrite Rules ◽  
Graphics Processing ◽  
Traditional Approaches ◽  
Rewrite System

Retargeting a compiler?s back end to a new architecture is a time-consuming process. This becomes an evident problem in the area of programmable graphics hardware (graphics processing units, GPUs) or embedded processors, where architectural changes are faster than elsewhere. We propose the object-oriented rewrite system OORS to overcome this problem. Using the OORS language, a compiler developer can express the code generation and optimization phase in terms of cost-annotated rewrite rules supporting complex non-linearmatching and replacing patterns. Retargetability is achieved by organizing rules into profiles, one for each supported target architecture. Featuring a rule and profile inheritance mechanism, OORS makes the reuse of existing specifications possible. This is an improvement regarding traditional approaches. Altogether OORS increases the maintainability of the compiler?s back end and thus both decreases the complexity and reduces the effort of the retargeting process. To show the potential of this approach, we have implemented a code generation and a code optimization pattern matcher supporting different target architectures using the OORS language and introduced them in a compiler of a programming language for CPUs and GPUs.

Modularity of strong normalization in the algebraic-λ-cube

1997 ◽  
Vol 7 (6) ◽  
pp. 613-660 ◽  
Author(s):  
FRANCO BARBANERA ◽  
MARIBEL FERNÁNDEZ ◽  
HERMAN GEUVERS
Keyword(s):  
Higher Order ◽  
Strong Normalization ◽  
First Order ◽  
Order Algebraic ◽  
General Schema ◽  
Rewrite Rules ◽  
Modular Property ◽  
Critical Pairs

In this paper we present the algebraic-λ-cube, an extension of Barendregt's λ-cube with first- and higher-order algebraic rewriting. We show that strong normalization is a modular property of all the systems in the algebraic-λ-cube, provided that the first-order rewrite rules are non-duplicating and the higher-order rules satisfy the general schema of Jouannaud and Okada. We also prove that local confluence is a modular property of all the systems in the algebraic-λ-cube, provided that the higher-order rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.

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.

Improved separation of C12-C22 fatty acid phenacyl esters by reversed phase column liquid chromatography

1986 ◽  
Vol 51 (12) ◽  
pp. 2722-2726 ◽  
Author(s):  
Tomáš Haniš ◽  
Miroslav Smrž ◽  
Pavel Klír ◽  
Karel Macek ◽  
Zdeněk Deyl
Keyword(s):  
Volume Concentration ◽  
Gradient Elution ◽  
Reversed Phase ◽  
Isocratic Elution ◽  
Water Gradient ◽  
Phenacyl Esters ◽  
Acetonitrile Concentration ◽  
Phase Column ◽  
Reversed Phase Column ◽  
Critical Pairs

Phenacyl esters of C12-C22 fatty acids were separated on Separon SGX C18 column, using a gradient elution with methanol-acetonitrile-water. The proposed gradient showed better resolution of the critical pairs C18:3-C14:0, C16:1-C20:4, and C16:0-C18:1 than the gradient elution with methanol-water or acetonitrile-water, or than the isocratic elution with methanol-acetonitrile-water. The optimum volume concentration (83%) of the sum of both methanol and acetonitrile was maintained constant for 35 min; in this period the acetonitrile concentration decreased linearly from the initial 42-60% to 0% while the methanol concentration increased from the initial 41-23% to 83% at the same rate. After 35 min the elution was completed with a methanol-water gradient. The whole analysis can be performed within 63 min at a flow rate 1 ml/min.

scholarly journals The Size of the Giant Joint Component in a Binomial Random Double Graph

10.37236/8846 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Mark Jerrum ◽  
Tamás Makai
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Random Graphs ◽  
Spanning Tree ◽  
Branching Process ◽  
Graph Model ◽  
First Order ◽  
Blue Edge ◽  
Critical Pairs

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices.  A joint component is a maximal set of vertices that supports both a red and a blue spanning tree.  We show that there are critical pairs of red and blue edge densities at which a giant joint component appears.  In contrast to the standard binomial graph model, the phase transition is first order:  the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point.  We connect this phenomenon to the properties of a certain bicoloured branching process. 

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