1 research outputs found
The Semantics of Rank Polymorphism
Iverson's APL and its descendants (such as J, K and FISh) are examples of the
family of "rank-polymorphic" programming languages. The principal control
mechanism of such languages is the general lifting of functions that operate on
arrays of rank (or dimension) to operate on arrays of any higher rank . We present a core, functional language, Remora, that captures this
mechanism, and develop both a formal, dynamic semantics for the language, and
an accompanying static, rank-polymorphic type system for the language.
Critically, the static semantics captures the shape-based lifting mechanism of
the language. We establish the usual progress and preservation properties for
the type system, showing that it is sound, which means that "array shape"
errors cannot occur at run time in a well-typed program. Our type system uses
dependent types, including an existential type abstraction which permits
programs to operate on arrays whose shape or rank is computed dynamically;
however, it is restricted enough to permit static type checking.
The rank-polymorphic computational paradigm is unusual in that the types of
arguments affect the dynamic execution of the program -- they are what drive
the rank-polymorphic distribution of a function across arrays of higher rank.
To highlight this property, we additionally present a dynamic semantics for a
partially erased variant of the fully-typed language and show that a
computation performed with a fully-typed term stays in lock step with the
computation performed with its partially erased term. The residual types thus
precisely characterise the type information that is needed by the dynamic
semantics, a property useful for the (eventual) construction of efficient
compilers for rank-polymorphic languages