Type-checking Linear Dependent Types

Linear indexed type systems have been used to ensure safety properties of programs with respect to dierent kinds of resources; examples include usage analysis [10], implicit complexity [3], and more. Linear indexed types use a type-level index language to describe resources and linear types to reason about the program's resource usage in a compositional way. A limitation of current analysis techniques for such systems is that resource usage is inferred independently of the control ow of a programe.g., the typing rule for branching usually approximates resources by taking the maximal usage of one of the branches. To make this analysis more precise, some authors have proposed extending adding dependent types, considering both resource usage and the size information of a program's input. This signicantly enriches the resulting analysis by allowing resource usage to depend on runtime information. Linear dependent type systems have been used in several domains, such as implicit complexity [1] and others. Of course, there is a price to be paid for the increase in expressiveness: type checking and type inference inevitably become more complex. In linear indexed type systems, these tasks are often done in two stages: a standard Hindley-Milner-like pass, followed by a constraint-solving procedure. In some cases, the generated constraints can be solved automatically with custom algorithms [6] or o-the-shelf SMT solvers [4]. However, the constraints are specic to the index language, and richer index languages often lead to more complex constraints. In this work we consider the type-checking problem for a particular system with linear dependent types, DFuzz. DFuzz was born out of Fuzz [9], a language where types are used to reason about sensitivity of programs, which measures the distance between outputs on nearby inputs. Fuzz uses real numbers as indices for the linear types, which provide an upper bound on the sensitivity of the program. As shown by [4], type-checking Fuzz programs can be done eciently by using an SMT solver to discharge the numeric proof obligations arising from the type system. The same approach works for type inference, which infers the minimal sensitivity of a function. DFuzz [5] was introduced to overcome a fundamental limitation of Fuzz: sensitivity information cannot depend on runtime information, such as the size of a data structure. This is done by enriching Fuzz with a limited form of dependent types, whose index language combines information about the size of data structures and the sensitivity of functions. …