The Role of Commutativity in Constraint Propagation Algorithms

Constraint propagation algorithms form an important part of most of the constraint programming systems. We provide here a simple, yet very general framework that allows us to explain several constraint propagation algorithms in a systematic way. In this framework we proceed in two steps. First, we introduce a generic iteration algorithm on partial orderings and prove its correctness in an abstract setting. Then we instantiate this algorithm with specific partial orderings and functions to obtain specific constraint propagation algorithms. In particular, using the notions commutativity and semi-commutativity, we show that the {\tt AC-3}, {\tt PC-2}, {\tt DAC} and {\tt DPC} algorithms for achieving (directional) arc consistency and (directional) path consistency are instances of a single generic algorithm. The work reported here extends and simplifies that of Apt \citeyear{Apt99b}.Comment: 35 pages. To appear in ACM TOPLA

Monads, partial evaluations, and rewriting

Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of "evaluating an expression partially": for example, "2+3" can be obtained as a partial evaluation of "2+2+1". This construction can be given for any monad, and it is linked to the famous bar construction, of which it gives an operational interpretation: the bar construction induces a simplicial set, and its 1-cells are partial evaluations. We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms. In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.Comment: Originally written for the ACT Adjoint School 2019. To appear in Proceedings of MFPS 202

Multi-Centered First Order Formalism

We propose a first order formalism for multi-centered black holes with flat tree-dimensional base-space, within the stu model of N=2, D=4 ungauged Maxwell-Einstein supergravity. This provides a unified description of first order flows of this universal sector of all models with a symmetric scalar manifold which can be obtained by dimensional reduction from five dimensions. We develop a D=3 Cartesian formalism which suitably extends the definition of central and matter charges, as well as of black hole effective potential and first order "fake" superpotential, in order to deal with not necessarily axisimmetric solutions, and thus with multi-centered and/or (under-)rotating extremal black holes. We derive general first order flow equations for composite non-BPS and almost BPS classes, and we analyze some of their solutions, retrieving various single-centered (static or under-rotating) and multi-centered known systems. As in the t^3 model, the almost BPS class turns out to split into two general branches, and the well known almost BPS system is shown to be a particular solution of the second branch.Comment: 1+22 pages; v2 : some typos fixes and Refs. adde

Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards efficiency is to evaluate operators in a matrix-free way with sum-factorization kernels. The method allows for general curved geometries and variable coefficients. Temporal discretization is carried out by low-storage explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For ADER, we propose a flexible basis change approach that combines cheap face integrals with cell evaluation using collocated nodes and quadrature points. Additionally, a degree reduction for the optimized cell evaluation is presented to decrease the computational cost when evaluating higher order spatial derivatives as required in ADER time stepping. We analyze and compare the performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with the proposed optimizations. ADER involves fewer operations and additionally reaches higher throughput by higher arithmetic intensities and hence decreases the required computational time significantly. Comparison of Runge-Kutta and ADER at their respective CFL stability limit renders ADER especially beneficial for higher orders when the Butcher barrier implies an overproportional amount of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown to take a substantial amount of the computational time due to their memory intensity