Maude: specification and programming in rewriting logic

Maude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude's language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude

A Symbolic Transformation Language and its Application to a Multiscale Method

The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple package for rule-based programming, rewriting strategies and their combination with standard Maple code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.Comment: 36 page

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

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more citations, to appear in Mathematische Zeitschrif