Version control systems typically rely on a patch language , heuristic patch synthesis algorithms like diff , and three-way merge algorithms . Standard patch languages and merge algorithms often fail to identify conflicts correctly when there are multiple edits to one line of code or code is relocated. This paper introduces Grove, a collaborative structure editor calculus that eliminates patch synthesis and three-way merge algorithms entirely. Instead, patches are derived directly from the log of the developer’s edit actions and all edits commute, i.e. the repository state forms a commutative replicated data type (CmRDT). To handle conflicts that can arise due to code relocation, the core datatype in Grove is a labeled directed multi-graph with uniquely identified vertices and edges. All edits amount to edge insertion and deletion, with deletion being permanent. To support tree-based editing, we define a decomposition from graphs into groves , which are a set of syntax trees with conflicts–including local, relocation, and unicyclic relocation conflicts–represented explicitly using holes and references between trees. Finally, we define a type error localization system for groves that enjoys a totality property, i.e. all editor states in Grove are statically meaningful, so developers can use standard editor services while working to resolve these explicitly represented conflicts. The static semantics is defined as a bidirectional marking system in line with recent work, with gradual typing employed to handle situations where errors and conflicts prevent type determination. We then layer on a unification-based type inference system to opportunistically fill type holes and fail gracefully when no solution exists. We mechanize the metatheory of Grove using the Agda theorem prover. We implement these ideas as the Grove Workbench , which generates the necessary data structures and algorithms in OCaml given a syntax tree specification.

In higher order settings, control-flow analysis aims to model the propagation of both data and control by finitely approximating program behaviors across all possible executions. The polyvariance of an analysis describes the number of distinct abstract representations, or variants, for each syntactic entity (e.g., functions, variables, or intermediate expressions). Monovariance, one of the most basic forms of polyvariance, maintains only a single abstract representation for each variable or expression. Other polyvariant strategies allow a greater number of distinct abstractions and increase analysis complexity with the aim of increasing analysis precision. For example, k -call sensitivity distinguishes flows by the most recent k call sites, k -object sensitivity by a history of allocation points, and argument sensitivity by a tuple of dynamic argument types. From this perspective, even a concrete operational semantics may be thought of as an unboundedly polyvariant analysis. In this paper, we develop a unified methodology that fully captures this design space. It is easily tunable and guarantees soundness regardless of how tuned. We accomplish this by extending the method of abstracting abstract machines, a systematic approach to abstract interpretation of operational abstract-machine semantics. Our approach permits arbitrary instrumentation of the underlying analysis and arbitrary tuning of an abstract-allocation function. We show that the design space of abstract allocators both unifies and generalizes existing notions of polyvariance. Simple changes to the behavior of this function recapitulate classic styles of analysis and yield novel combinations and variants.