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 usual termination. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into terminating part and 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.
The decreasing diagram technique @cite_0 is a powerful technique to obtain many confluence criteria. In @cite_20 , the technique is extended to obtain a criterion for CRM.
@cite_0 @cite_20
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The decreasing diagrams technique is a complete method to reduce confluence of a rewrite relation to local confluence. Whereas previous presentations have focussed on the proofthe technique is correct, here we focus on applicability. We present a simple but powerful generalisation of the technique, requiring peaks to be closed only by conversions instead of valleys, which is demonstrated to further ease applicability. The persistence diagram of a filtered simplicial complex is usually computed by reducing the boundary matrix of the complex. We introduce a simple optimization technique: by processing the simplices of the complex in decreasing dimension, we can “kill” columns (i.e., set them to zero) without reducing them. This technique completely avoids reduction on roughly half of the columns. We demonstrate that this idea significantly improves the running time of the reduction algorithm in practice. We also give an output-sensitive complexity analysis for the new algorithm which yields to sub-cubic asymptotic bounds under certain assumptions. We study communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties, such as: 1) fairness; 2) rate of convergence; and 3) throughput, can be characterized by certain nonnegative matrices. We demonstrate that these results can be used to develop tools for analyzing the behavior of AIMD communication networks. The accuracy of the models is demonstrated by several NS studies. This papers analyzes a class of nonlinear additive-increase multiplicative-decrease (AIMD) protocols that are widely deployed in communication networks. It is demonstrated that the use of these protocols guarantees that the system has a unique stable outcome to which it converges geometrically under all starting points. The development is based on a contraction argument and the derivation of explicit bounds on the contraction coefficient of corresponding operators in terms of the network parameters. In particular, bounds on the corresponding rate of convergence are obtained, improving upon known bounds for standard (linear) AIMD networks.