In the Map-Reduce programming model for data parallel computation in a cloud environment, the reducer phase is responsible for computing an output key-value pair, given a sequence of input values associated with a particular key. Because of non-determinism in transmitting key-value pairs over the network, the inputs may not arrive at a reducer in a fixed order. This gives rise to the reducer co...

then we term the elements x and y n-commutative. It is not difficult to verify that »-commutativity and (1—»)-commutativity are equivalent properties of the elements x and y, that ( — ̂ -commutativity implies ordinary commutativity, and that commuting elements are »-commutative. From any concept and property involving the fact that certain elements [or functions of elements] commute, one may der...

How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural an...

Double semigroups have two associative operations ◦, • related by the interchange relation: (a • b) ◦ (c • d) ≡ (a ◦ c) • (b ◦ d). Kock [13] (2007) discovered a commutativity property in degree 16 for double semigroups: associativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with ...

We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories de ne functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as con uence and su cient completeness are modular for so-called fair extensions. This enables us to incrementally check these p...

This paper introduces an analysis technique, commutativity analysis, for automatically parallelizing computations that manipulate dynamic, pointer-based data structures. Commutativity analysis views computations as composed of operations on objects. It then analyzes the program to discover when operations commute, i.e. leave the objects in the same state regardless of the order in which they ex...