Container Unification
Folkert de Vries, Sven-Bodo Scholz, Sjaak Smeters
september 2020
for uniqueness types
Uniqueness Types
Track in the
type
that a value has just one pointer to it
at runtime
System 1: Clean
System 2: UTS
- defined on the lambda calculus
- easier to add advanced type system features
Uniqueness
unique
exactly 1 reference
\texttt{MyType}^\bullet
non-unique
many references
\texttt{MyType}^\times
Identity
\texttt{identity} = \lambda x. x
Duplicate
\texttt{duplicate} = \lambda x. (x, x)
Uniqueness Relations
non-unique container
unique
element
UTS
idea: use logic expressions to encode relations
\bullet \text{: unique \& boolean True}
\times \text{: non-unique \& boolean False}
UTS
idea: use logic expressions to encode relations
\texttt{fst} :: (t^u, s^v)^{u \lor w} \to t^u
My Problem
\texttt{swap} :: (s^v, t^u)^{u \lor v \lor w} \to (t^u, s^v)^{w'}
\texttt{swap r = (snd r} \texttt{, fst r} \texttt{)}
\texttt{swap} :: (s^{(\lnot v \land u) \lor (\lnot v \land w) \lor (u1 \land u) \lor (u1 \land w)}, t^u)^{u \lor w} \to (t^u, s^{(\lnot v \land u) \lor (\lnot v \land w) \lor (u1 \land u) \lor (u1 \land w)})^{v1}
Boolean Unification
idea: when are two formulae equivalent?
finds the (most-general) unifier
u1 \mapsto (\neg u2 \land v1) \lor (\neg u2 \land v2) \lor (u1 \land v1) \lor (u1 \land v2) \\
u2 \mapsto (u2 \land v1) \lor (u2 \land v2)
u1 \lor u2 \doteq v1 \lor v2
Boolean Unification
claim: not a good fit in practice
😕
Toward Solutions
faster unification of uniqueness attributes
🚀
Disjunctions are for Containers
There is a hierarchy between variables
- \(u, v\) are lower bounds
- define \( w \) as an upper bound
- \(w\) dominates \(u\) and \(v\)
( t^u, s^v)^{u \lor v \lor w }
There is a hierarchy between variables
Disjunction & Unique
we don't know which variable will be \(\bullet\), therefore have to keep all options available in the unifier
u \lor v \lor w \doteq \bullet
w \geq u, v
\doteq \bullet
Disjunction & Unique
Disjunction & Disjunction
u_1 \geq u_2 \doteq v_1 \geq v_2
- unify dominators
- accumulate dominated
v_1 \geq v_2, u_2 \text{ where } [ u_1 \mapsto v_ 1 ]
Polymorphic records
\{ name :: String ~|~ a \} \doteq \{ age :: Int~|~ b \}
\{ age :: Int, name :: String ~|~ c \}
unifies to:
"at least field \(name\) and at least field \(age\)"
A Polymorphic Type System for Extensible Records and Variants (Gaster & Jones, 1996)
w \geq u, v \Rightarrow (w \geq \{ u, v | \alpha \})
Container Annotation
Record-Inspired Unification
(w_2, \{ u_1, \dots u_n, v_1, \dots v_m ~|~ \gamma \})
Container Unification
rules
Swap
with container unification
\texttt{swap} :: (s^v, t^u)^{(w \geq \{ u, v | \alpha \} )} \to (t^u, s^v)^{w'}
Conclusion
\texttt{swap} :: (s^v, t^u)^{(w \geq \{ u, v | \alpha \} )} \to (t^u, s^v)^{w'}
\texttt{swap} :: (s^{(\lnot v \land u) \lor (\lnot v \land w) \lor (u1 \land u) \lor (u1 \land w)}, t^u)^{u \lor w} \to (t^u, s^{(\lnot v \land u) \lor (\lnot v \land w) \lor (u1 \land u) \lor (u1 \land w)})^{v1}
- introduce hierarchy
- steal unification trick from records
IFL
By folkert de vries
