# log-e-sappho experimental type system implementation ## sappho grammar ``` s, t ::= s | t (disjunction, a.k.a. union) | s & t (conjunction, a.k.a. intersection) | s => t (implication) | s -> t (arrow, a.k.a. function) | forall x . t (forall, polymorphic) | box t (box, c.f. modular logic) | a [t..] (type operator application) | x (variable) | n (nominal) | true (true, a.k.a. top) | false (false, a.k.a. bottom) | { m : t } (member, a.k.a. atomic record) x ∈ var n ∈ nominal a ∈ opname ``` ## sappho subtyping rules ``` ---- [identity] s, γ ⊢ s, δ ---- [true-right] γ ⊢ true, δ ---- [false-left] false, γ ⊢ δ s, t, γ ⊢ δ ---- [conj-left] s & t, γ ⊢ δ γ ⊢ s, t, δ ---- [disj-right] γ ⊢ s | t, δ s, γ ⊢ δ t, γ ⊢ δ ---- [disj-left] s | t, γ ⊢ δ γ ⊢ s, δ γ ⊢ t, δ ---- [conj-right] γ ⊢ s & t, δ // XXX // remove s => t? always make progress s => t, s, t, γ ⊢ δ ---- [impl-left] s => t, s, γ ⊢ δ // discussion about implication below // check how "normal sequent calculus handles this" s, γ ⊢ t, δ ---- [impl-right] γ ⊢ s => t, δ // box works as a kind of "forall" for concrete types box t, t, γ ⊢ δ ---- [box-left] box t, γ ⊢ δ // (.)-- filters the contexts. γ-- ⊢ t, δ-- ---- [box-right] γ ⊢ box t, δ // using cyclic proof strategy, rule for dealing with aliases/type operators // becomes really simple. However, this requires well-formedness rules on types // to exclude typedefs like // type A = A // Or that our cycle detection excludes paths between "identical" states. γ ⊢ expand(a[t..]), δ ---- γ ⊢ a[t..], δ expand(a[t..]), γ ⊢ δ ---- a[t..], γ ⊢ δ // member types are filtered according to member name γ >> m ⊢ δ >> m ---- [member] γ ⊢ δ // foralls are unwrapped and variable substituted with fresh name // i.e. we treat the bound variable nominally n fresh γ [n] ⊢ δ [n] ---- [forall] γ ⊢ δ c, γ* ⊢ a_1, ..., a_n ∀ I ⊆ {1, ..., n}. c, γ* ⊢ { a_i | i ∈ I }, δ* OR { b_i | i ∈ ¬I }, γ* ⊢ d, δ* ---- [loads-of-fun] a_1 -> b_1, ..., a_n -> b_n, γ ⊢ c -> d, δ ``` ### context operators * box filtering ``` (t, γ)-- =def= t, γ-- if t == box s, for some s γ-- otherwise ``` * member unwrapping ``` (t, γ) >> m =def= s, (γ >> m) if t == { m : s }, for some s t, (γ >> m) if t == box s, for some s γ >> m otherwise ``` * forall-unwrapping and substitution ``` (t, γ) [n] =def= s [n/x], (γ [n]) if t == forall x . s, for some x, s t, (γ [n]) if t == box s, for some s γ [n] otherwise ``` ### Semantic function subtyping *Here we try to give an intuitive explainer of how to reason about function subtyping in the semantic subtyping domain.* Given function types ``` a_1 -> b_1, ..., a_n -> b_n ``` the conjunction (intersection) of `a_i -> b_i`, i.e. ``` intsec = (a_1 -> b_1) & ... & (a_n -> b_n) ``` represents the set of functions that combines the properties of all `a_i -> b_i` by themselves. To get an idea of what functions this describes, let's start with reasoning about values, that is, functions in `intsec`. First we note that any function in this set accepts any argument that are in either of the `a_i`s. That is, it can take any argument in the set represented by ``` a = a_1 | ... | a_n ``` Furthermore, a function `f` in `intsec` "promises" to map any value in `a_i` to a value in its corresponding `b_i`. Specifically, given `I ⊆ {1..n}`, a value `v` in the intersection ``` a_I_intsec = &[i ∈ I] a_i ``` is mapped to a value in ``` b_I_intsec = &[i ∈ I] b_i ``` that is, ``` v ∈ &[i ∈ I] a_i implies f(v) ∈ &[i ∈ I] b_i ``` *In the following we will "abuse notation" a bit. Try to look at it like talking to an old friend, and you just instinctively know exactly what they mean without explaining...* Now, given a type `c -> d`, how do we know if ``` intsec = &[i ∈ {1..n}] a_i -> b_i <: c -> d ``` ? Or equivalently, friend speaking, if ``` ∩[i ∈ {1..n}] a_i -> b_i ⊆ c -> d ``` ? Immediately, it should be obvious that ``` c ⊆ a (= ∪[i ∈ {1..}] a_i) ``` Furthermore, it is necessary that any function in `intsec` will have to map all values in `c` into `d`, specifically ``` f ∈ intsec, v ∈ c implies f(v) ∈ d ``` Note that this does not mean that any function in `a_i -> b_i` must map all values in `a_i` into `d`, since it's not necessary that `a_i \ c = ø`. Specifically, it is not necessary that `b_i ⊆ d`. However, taking `I ⊆ {1..n}`, assuming ``` a_I_intsec = &[i ∈ I] a_i ⊆ c ``` then, necessarily ``` b_I_intsec = &[i ∈ I] b_i ⊆ d ``` Conversely, if ``` b_I_intsect ⊆ d ``` then any function in `intsec` maps `a_I_intsec` into `d`. If `a_I_intsec` happens to intersect with `c`, it "covers" a part of `c`. I.e. the set `I` can be understood as promising to map a part of `c` into `d`. Defining this formally, we say that `I_cand ⊆ {1..n}` is a (d-)*candidate* if, ``` ∩[i ∈ I_cand] b_i ⊆ d ``` Now, letting `IC = {I_cand_1..I_cand_k}` be the set of candidates, it is fairly simple to see that ``` intsec <: c -> d ``` iff ``` c ⊆ ( ∪[I_cand ∈ IC] ∩[i ∈ I_cand] a_i ) ``` Using some pro gamer moves, we can see that this is equivalent to ``` c ⊆ ( ∩[I_cand ∈ IC] ∪[i ∉ I_cand] a_i ) ``` This gives us the final form for our subtyping rule ``` c, γ* ⊢ a_i | ... | a_n, δ* forall I ⊆ {1..n}. &[i ∈ I] b_i, γ* ⊢ d, δ* or c, γ* ⊢ |[i ∉ I] a_i, δ* ---- a_1 -> b_1, ..., a_n -> b_n, γ ⊢ c -> d, δ ``` ### Recursive subtyping Type aliases creates the possibility of recursive types, and thus we need to handle recursive subtyping Handling subtyping co-inductively would enable us to handle the relation ``` type A = { x : { y : A } } type B = { y : { x : B } } A <: { x : B } { x : B } <: A ``` We have made progress if we "switch context" If we make progress and end up at the same sequent, co-induction gives us the conclusion. Some actions/rules lead to equivalent contexts, i.e. does not switch context/make progress. ### Other True ~ { x : True } on the left False ~ { x : False } on the right or rather True ~ { x : True } in positive position False ~ { x : False } in negative position ### implication ``` γ ⊢ s => t, u => s, δ if s, then u => s holds if ¬s, then s => t holds ``` altho ``` γ ⊢ s => t, s, δ ``` Given the msph program ``` nominal n type Woops[e] = { member f : (e <: false => n) | (e <: false) } type Noooes = { member f : true } ``` we can prove that, for any type `t` ``` Noooes <: Woops[t] ``` The derivation looks like this: ``` ---- [identity] true, t <: false ⊢ n, (t <: false) ---- [impl-right] true ⊢ (t <: false => n), (t <: false) ---- [disj-right] true ⊢ (t <: false => n) | (t <: false) ---- [member] { f : true } ⊢ { f : (t <: false => n) | (t <: false) } ---- [expand-right] { f : true } ⊢ Woops[t] ---- [expand-left] Nooes ⊢ Woops[t] ---- [impl-right] ⊢ Nooes => Woops[t] ---- [box-right] ⊢ Nooes <: Woops[t] ``` This looks a bit dangerous at first. However, if we translate the implication to using disjunction and negation, we get ``` (t <: false => N) | (t <: false) == (¬(t <: false) | n) | (t <: false) ``` Considering the two cases ``` (1) t == false (2) t != false ``` we see that in (1) ``` (¬(t <: false) | n) | (t <: false) == (¬(true) | n) | (true) == true ``` and in (2) ``` (¬(t <: false) | n) | (t <: false) == (¬(false) | n) | false == (true | n) | false == true ``` ## What is true? Our "set interpretation" of types says `true` represents the full universe of "things". What are these things? For all intents and purposes, a programmer can look at them as the values of our language. What does this mean in practical terms? Well, it means something of type `true` is "a value". What is "a value" then? That's a good question. It can be anything really. An integer? Yep! A function? Yes, m'am! A solution to the halting problem? Well, yeah! But good luck constructing that. A... I think you get my point. A value can be anything from a simple integer to nuclear apocalypse. The thing is, from the point of view of our program, it means "we cannot really say anything here". But couldn't that be dangerous? Depends on what you mean by dangerous. I mean, to be able to give everything a type? Well, let me ask you this: do you like rat poison? What? To eat? Yeah! Nooo, that'd be dangerous! Yeah, exactly my point. What? You know what rat poison is obviously, despite it being dangerous? Well, yeah... Maybe you got it already, but my point is: rat poison is dangerous, but you still know about it, and its existance. So, something being dangerous does not mean you have this blank spot of knowledge that you just keep ignoring because of this perceived danger. ### interpretation of ⊢ what is the interpretation of γ ⊢ δ? given a concrete type k conjunction of γ hold for k then disjunction of δ hold for k