# NBEP 6: Typing Recursion

Author:

  Siu Kwan Lam

Date:

  Sept 2016

Status:

  Draft

## Introduction

This document proposes an enhancement to the type inference algorithm to support recursion without explicitly annotating the function signature. As a result, the proposal enables numba to type-infer both self-recursive and mutual-recursive functions under some limitations. In practice, these limitations can be easily overcome by specifying a compilation order.

## The Current State

Recursion support in numba is currently limited to self-recursion with explicit type annotation for the function. This limitation comes from the inability to determine the return type of a recursive call. This is because the callee is either the current function (for self-recursion) or a parent function (mutual-recursion) and its type inference process has been suspended while waiting for the function-type of its callee. This results in the formation of a cyclic dependency. For example, given a function `foo()` that calls `bar()`, which in turns call `foo()`:

    def foo(x):
        if x > 0:
            return bar(x)
        else:
            return 1

    def bar(x):
        return foo(x - 1)

The type inference process of `foo()` depends on that of `bar()`, which depends on `foo()`. Therefore `foo()` depends on itself and the type inference algorithm cannot terminate.

## The Solution

The proposed solution has two components:

1. The introduction of a compile-time callstack that tracks the compiling functions.

2. The allowance of a partial type inference on functions by leveraging the return type on non-recursive control-flow paths.

The compile-time callstack stores typing information of the functions being compiled. Like an ordinary callstack, it pushes a new record every time a function is “called”. Since this occurs at compile-time, a “call” triggers a compilation of the callee.

To detect recursion, the compile-time callstack is searched bottom-up (stack grows downward) for a record that matches the callee. As the record contains a reference to the type inference state, the type inference process can be resumed to determine the return type.

Recall that the type inference process cannot be resumed normally because of the cyclic dependency of the return type. In practice, we can assume that a useful program must have a terminating condition, a path that does not recurse. So, the type inference process can make an initial guess for the return-type at the recursive call by using the return-type determined by the non-recursive paths. This allows type information to propagate on the recursive paths to generate the final return type, which is used to refine the type information by the subsequent iteration in the type inference process.

The following figure illustrates the compile-time callstack when the compiler reaches the recursive call to `foo()` from `bar()`:

At this time, the type inference process of `foo()` is suspended and that of `bar()` is active. The compiler can see that the callee is already compiling by searching the callstack. Knowing that it is a recursive call, the compiler can resume the type-inference on `foo()` by ignoring the paths that contain recursive calls. This means only the `else` branch is considered and we can easily tell that `foo()` returns an `int` in this case. The compiler will then set the initial return type of `foo()` and `bar()` to `int`. The subsequent type propagation can use this information to complete the type inference of both functions, unifying the return-type of all returning paths.

## Limitations

For the proposed type inference algorithm to terminate, it assumes that at least one of the control path leads to a return-statement without undertaking a recursive call. Should this not be the case, the algorithm will raise an exception indicating a potential runaway recursion.

For example:

    @jit
    def first(x):
        # The recursing call must have a path that is non-recursing.
        if x > 0:
            return second(x)
        else:
            return 1

    @jit
    def second(x):
        return third(x)

    @jit
    def third(x):
        return first(x - 1)

The `first()` function must be the compiled first for the type inference algorithm to complete successfully. Compiling any other function first will lead to a failure in type inference. The type inference algorithm will treat it as a runaway recursion due to the lack of a non-recursive exit in the recursive callee.

For example, compiling `second()` first will move the recursive call to `first()`. When the compiler tries to resume the type inference process of `second()`, it will fail to find a non-recursive path.

This is a small limitation and can be overcome easily by code restructuring or precompiling in a specific order.