Quantitative program reasoning with graded modal types

In programming, data is often considered to be infinitely copiable, arbitrarily discardable, and universally unconstrained. However this view is naive: some data encapsulates resources that are subject to protocols (e.g., file and device handles, channels); some data should not be arbitrarily copied or communicated (e.g., private data). Linear types provide a partial remedy by delineating data in two camps: "resources" to be used but never copied or discarded, and unconstrained values. However, this binary distinction is too coarse-grained. Instead, we propose the general notion of graded modal types, which in combination with linear and indexed types, provides an expressive type theory for enforcing fine-grained resource-like properties of data. We present a type system drawing together these aspects (linear, graded, and indexed) embodied in a fully-fledged functional language implementation, called Granule. We detail the type system, including its metatheoretic properties, and explore examples in the concrete language. This work advances the wider goal of expanding the reach of type systems to capture and verify a broader set of program properties