New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized

The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation fails to identify two `obviously' equal terms. If only the machine could decide a richer theory! We propose a way to decide theories which supplement evaluation with `$\nu$-rules', rearranging the neutral parts of normal forms, and report a successful initial experiment. We study a simple -calculus with primitive fold, map and append operations on lists and develop in Agda a sound and complete decision procedure for an equational theory enriched with monoid, functor and fusion laws

Multifocal: a strategic bidirectional transformation language for XML schemas

Lenses are one of the most popular approaches to define bidirectional transformations between data models. However, writing a lens transformation typically implies describing the concrete steps that convert values in a source schema to values in a target schema. In contrast, many XML-based languages allow writing structure-shy programs that manipulate only specific parts of XML documents without having to specify the behavior for the remaining structure. In this paper, we propose a structure-shy bidirectional two-level transformation language for XML Schemas, that describes generic type-level transformations over schema representations coupled with value-level bidirectional lenses for document migration. When applying these two-level programs to particular schemas, we employ an existing algebraic rewrite system to optimize the automatically-generated lens transformations, and compile them into Haskell bidirectional executables. We discuss particular examples involving the generic evolution of recursive XML Schemas, and compare their performance gains over non-optimized definitions.Fundação para a Ciência e a Tecnologi

Algebraic Property Graphs

In this paper, we use algebraic data types to define a formal basis for the property graph data models supported by popular open source and commercial graph databases. Developed as a kind of inter-lingua for enterprise data integration, algebraic property graphs encode the binary edges and key-value pairs typical of property graphs, and also provide a well-defined notion of schema and support straightforward mappings to and from non-graph datasets, including relational, streaming, and microservice data commonly encountered in enterprise environments. We propose algebraic property graphs as a simple but mathematically rigorous bridge between graph and non-graph data models, broadening the scope of graph computing by removing obstacles to the construction of virtual graphs

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.Comment: 28 page