## User-defined Conversions ## C's implicit "usual arithmetic conversions" define a structure among the built-in types consisting of _unsafe_ narrowing conversions and a hierarchy of _safe_ widening conversions. There is also a set of _explicit_ conversions that are only allowed through a cast expression. Based on Glen's notes on conversions [1], I propose that safe and unsafe conversions be expressed as constructor variants, though I make explicit (cast) conversions a constructor variant as well rather than a dedicated operator. Throughout this article, I will use the following operator names for constructors and conversion functions from `From` to `To`: void ?{} ( To*, To ); // copy constructor void ?{} ( To*, From ); // explicit constructor void ?{explicit} ( To*, From ); // explicit cast conversion void ?{safe} ( To*, From ); // implicit safe conversion void ?{unsafe} ( To*, From ); // implicit unsafe conversion [1] http://plg.uwaterloo.ca/~cforall/Conversions/index.html Glen's design made no distinction between constructors and unsafe implicit conversions; this is elegant, but interacts poorly with tuples. Essentially, without making this distinction, a constructor like the following would add an interpretation of any two `int`s as a `Coord`, needlessly multiplying the space of possible interpretations of all functions: void ?{}( Coord *this, int x, int y ); That said, it would certainly be possible to make a multiple-argument implicit conversion, as below, though the argument above suggests this option should be used infrequently: void ?{unsafe}( Coord *this, int x, int y ); An alternate possibility would be to only count two-arg constructors `void ?{} ( To*, From )` as unsafe conversions; under this semantics, safe and explicit conversions should also have a compiler-enforced restriction to ensure that they are two-arg functions (this restriction may be valuable regardless). Regardless of syntax, there should be a type assertion that expresses `From` is convertable to `To`. If user-defined conversions are not added to the language, `void ?{} ( To*, From )` may be a suitable representation, relying on conversions on the argument types to account for transitivity. On the other hand, `To*` should perhaps match its target type exactly, so another assertion syntax specific to conversions may be required, e.g. `From -> To`. ### Constructor Idiom ### Basing our notion of conversions off otherwise normal Cforall functions means that we can use the full range of Cforall features for conversions, including polymorphism. Glen [1] defines a _constructor idiom_ that can be used to create chains of safe conversions without duplicating code; given a type `Safe` which members of another type `From` can be directly converted to, the constructor idiom allows us to write a conversion for any type `To` which `Safe` converts to: forall(otype To | { void ?{safe}( To*, Safe ) }) void ?{safe}( To *this, From that ) { Safe tmp = /* some expression involving that */; *this = tmp; // uses assertion parameter } This idiom can also be used with only minor variations for a parallel set of unsafe conversions. What selective non-use of the constructor idiom gives us is the ability to define a conversion that may only be the *last* conversion in a chain of such. Constructing a conversion graph able to unambiguously represent the full hierarchy of implicit conversions in C is provably impossible using only single-step conversions with no additional information (see Appendix A), but this mechanism is sufficiently powerful (see [1], though the design there has some minor bugs; the general idea is to use the constructor idiom to define two chains of conversions, one among the signed integral types, another among the unsigned, and to use monomorphic conversions to allow conversions between signed and unsigned integer types). ### Appendix A: Partial and Total Orders ### The `<=` relation on integers is a commonly known _total order_, and intuitions based on how it works generally apply well to other total orders. Formally, a total order is some binary relation `<=` over a set `S` such that for any two members `a` and `b` of `S`, `a <= b` or `b <= a` (if both, `a` and `b` must be equal, the _antisymmetry_ property); total orders also have a _transitivity_ property, that if `a <= b` and `b <= c`, then `a <= c`. If `a` and `b` are distinct elements and `a <= b`, we may write `a < b`. A _partial order_ is a generalization of this concept where the `<=` relation is not required to be defined over all pairs of elements in `S` (though there is a _reflexivity_ requirement that for all `a` in `S`, `a <= a`); in other words, it is possible for two elements `a` and `b` of `S` to be _incomparable_, unable to be ordered with respect to one another (any `a` and `b` for which either `a <= b` or `b <= a` are called _comparable_). Antisymmetry and transitivity are also required for a partial order, so all total orders are also partial orders by definition. One fairly natural partial order is the "subset of" relation over sets from the same universe; `{ }` is a subset of both `{ 1 }` and `{ 2 }`, which are both subsets of `{ 1, 2 }`, but neither `{ 1 }` nor `{ 2 }` is a subset of the other - they are incomparable under this relation. We can compose two (or more) partial orders to produce a new partial order on tuples drawn from both (or all the) sets. For example, given `a` and `c` from set `S` and `b` and `d` from set `R`, where both `S` and `R` both have partial orders defined on them, we can define a ordering relation between `(a, b)` and `(c, d)`. One common order is the _lexicographical order_, where `(a, b) <= (c, d)` iff `a < c` or both `a = c` and `b <= d`; this can be thought of as ordering by the first set and "breaking ties" by the second set. Another common order is the _product order_, which can be roughly thought of as "all the components are ordered the same way"; formally `(a, b) <= (c, d)` iff `a <= c` and `b <= d`. One difference between the lexicographical order and the product order is that in the lexicographical order if both `a` and `c` and `b` and `d` are comparable then `(a, b)` and `(c, d)` will be comparable, while in the product order you can have `a <= c` and `d <= b` (both comparable) which will make `(a, b)` and `(c, d)` incomparable. The product order, on the other hand, has the benefit of not prioritizing one order over the other. Any partial order has a natural representation as a directed acyclic graph (DAG). Each element `a` of the set becomes a node of the DAG, with an arc pointing to its _covering_ elements, any element `b` such that `a < b` but where there is no `c` such that `a < c` and `c < b`. Intuitively, the covering elements are the "next ones larger", where you can't fit another element between the two. Under this construction, `a < b` is equivalent to "there is a path from `a` to `b` in the DAG", and the lack of cycles in the directed graph is ensured by the antisymmetry property of the partial order. Partial orders can be generalized to _preorders_ by removing the antisymmetry property. In a preorder the relation is generally called `<~`, and it is possible for two distict elements `a` and `b` to have `a <~ b` and `b <~ a` - in this case we write `a ~ b`; `a <~ b` and not `a ~ b` is written `a < b`. Preorders may also be represented as directed graphs, but in this case the graph may contain cycles. ### Appendix B: Building a Conversion Graph from Un-annotated Single Steps ### The short answer is that it's impossible. The longer answer is that it has to do with what's essentially a diamond inheritance problem. In C, `int` converts to `unsigned int` and also `long` "safely"; both convert to `unsigned long` safely, and it's possible to chain the conversions to convert `int` to `unsigned long`. There are two constraints here; one is that the `int` to `unsigned long` conversion needs to cost more than the other two (because the types aren't as "close" in a very intuitive fashion), and the other is that the system needs a way to choose which path to take to get to the destination type. Now, a fairly natural solution for this would be to just say "C knows how to convert from `int` to `unsigned long`, so we just put in a direct conversion and make the compiler smart enough to figure out the costs" - this is the approach taken by the existing compipler, but given that in a user-defined conversion proposal the users can build an arbitrary graph of conversions, this case still needs to be handled. We can define a preorder over the types by saying that `a <~ b` if there exists a chain of conversions from `a` to `b` (see Appendix A for description of preorders and related constructs). This preorder corresponds roughly to a more usual type-theoretic concept of subtyping ("if I can convert `a` to `b`, `a` is a more specific type than `b`"); however, since this graph is arbitrary, it may contain cycles, so if there is also a path to convert `b` to `a` they are in some sense equivalently specific. Now, to compare the cost of two conversion chains `(s, x1, x2, ... xn)` and `(s, y1, y2, ... ym)`, we have both the length of the chains (`n` versus `m`) and this conversion preorder over the destination types `xn` and `ym`. We could define a preorder by taking chain length and breaking ties by the conversion preorder, but this would lead to unexpected behaviour when closing diamonds with an arm length of longer than 1. Consider a set of types `A`, `B1`, `B2`, `C` with the arcs `A->B1`, `B1->B2`, `B2->C`, and `A->C`. If we are comparing conversions from `A` to both `B2` and `C`, we expect the conversion to `B2` to be chosen because it's the more specific type under the conversion preorder, but since its chain length is longer than the conversion to `C`, it loses and `C` is chosen. However, taking the conversion preorder and breaking ties or ambiguities by chain length also doesn't work, because of cases like the following example where the transitivity property is broken and we can't find a global maximum: `X->Y1->Y2`, `X->Z1->Z2->Z3->W`, `X->W` In this set of arcs, if we're comparing conversions from `X` to each of `Y2`, `Z3` and `W`, converting to `Y2` is cheaper than converting to `Z3`, because there are no conversions between `Y2` and `Z3`, and `Y2` has the shorter chain length. Also, comparing conversions from `X` to `Z3` and to `W`, we find that the conversion to `Z3` is cheaper, because `Z3 < W` by the conversion preorder, and so is considered to be the nearer type. By transitivity, then, the conversion from `X` to `Y2` should be cheaper than the conversion from `X` to `W`, but in this case the `X` and `W` are incomparable by the conversion preorder, so the tie is broken by the shorter path from `X` to `W` in favour of `W`, contradicting the transitivity property for this proposed order. Without transitivity, we would need to compare all pairs of conversions, which would be expensive, and possibly not yield a minimal-cost conversion even if all pairs were comparable. In short, this ordering is infeasible, and by extension I believe any ordering composed solely of single-step conversions between types with no further user-supplied information will be insufficiently powerful to express the built-in conversions between C's types.