Feature Specification: Upper and Lower Bounds for Extreme Types

Owner: eernst@

Status: Background material, normative language now in dartLangSpec.tex.

Version: 0.2 (2018-05-22)

This document is a Dart 2 feature specification which specifies how to compute the standard upper bound and standard lower bound (SUB and SLB) of a pair of types when at least one of the operands is an extreme type: Object, dynamic, void, or bottom.


In order to motivate the rules for upper and lower bounds of a pair of types, we will focus on concrete examples that embody upper bounds very directly, namely conditional expressions, and extend that to lower bounds using functions.

For example, can we use the result from the evaluation of b ? print('Hello!') : 42? In Dart 2, an expression with static type void is considered to have a value which is not valid, and it is a compile-time error to use it in most situations. When it is one of two possible branches in a conditional expression (here: print('Hello!')) we would expect to consider the whole value to be not valid, because otherwise we could receive the value of “the void branch” and thus inadvertently use a value which is not valid. Based on this kind of reasoning we have chosen to give expressions like b ? print('Hello!') : 42 the type void.

Similarly, can we do (b ? 42 : f()).isEven if f has return type dynamic? In this case the result from evaluating the conditional expression (?:) may have any type whatsoever, so isEven cannot be assumed to exist in the interface of that object. On the other hand, isEven could be safely invoked on the result from the branch 42, and the other branch would admit arbitrary member access operations (such as f().isEven), even though there is no static evidence for the existence of any particular members (except for a few members which are declared in Object and hence inherited or overridden for every object). So you could say “it is OK for both branches, so it must be OK for the expression as a whole.”

Then how about (b ? 42 : f()).fooBar() where f is again assumed to have the return type dynamic? In this situation we would accept f().fooBar() because dynamic receivers admit all member accesses, but 42.fooBar() would be rejected. Hence, we might say that “it is not OK for both branches, hence it is not OK for the conditional expression”.

We could give b ? 42 : f() the type int, which would allow us to accept (b ? 42 : f()).isEven and reject (b ? 42 : f()).fooBar(). This would effectively mean that dynamic would be considered to be a subtype of all other types during computations of upper and lower bounds.

However, we consider that to be such a serious anomaly relative to the rest of the Dart 2 type system that we have not taken that approach.

Instead, we have chosen to accept that a dynamic branch in a conditional expression will make the whole conditional expression dynamic.

A set of situations with the opposite polarity arise when we consider types in a contravariant position, e.g., b ? (T1 t1) => e1 : (T2 t2) => e2, where we need to consider various combinations of types as the values of T1 and T2 in order to compute the type of the whole expression.

Ignoring “voidness” and “dynamicness” for a moment and focusing on the pure subtyping relationships, we apply the standard upper bound function to the types of the two branches in the former situation (like b ? e1 : e2), and for the latter situation (where an intervening function literal reverses the polarity, that is, the types in question occur in contravariant locations) we use the standard lower bound function.

These bound functions take exactly two arguments, so we may also call them ‘operators’ and the arguments ‘operands’. We call these functions ‘standard’ rather than ‘least’ and ‘greatest’ because the Dart 2 type language cannot express a true least upper bound and greatest lower bound of all pairs of types, but it is still useful to choose an approximation thereof in many cases. We abbreviate the function names to SUB and SLB.

As long as we are concerned with non-extreme types (everything except the top and bottom types), these bound functions deliver an approximation of the least upper bound and the greatest lower bound of its operands. For instance, SUB(int, num) is num, and SLB(int, num) is int, so we get the type Object for b ? new Object() : 42, and the type num Function(int) for b ? (num n) => 41 : (int o) => 4.1.

This specification is concerned with combining the treatment of the pure subtyping related properties and the other properties like “dynamicness” and “voidness”. We achieve that by means of a specification of the values of SUB and SLB when at least one of their operands is an extreme type.


The grammar is unaffected by this feature.

Static Analysis

An extreme type is one of the types Object, dynamic, void, and bottom.

Consider a pair of types such that at least one of them is an extreme type. The value of the functions SUB and SLB on that pair of types is then determined by the following rules:

SUB(T, T) == T, for all T.
SUB(S, T) == SUB(T, S), for all S, T.
SUB(void, T) == void, for all T.
SUB(dynamic, T) == dynamic, for all T != void.
SUB(Object, T) == Object, for all T != void, dynamic.
SUB(bottom, T) == T, for all T.

SLB(T, T) == T, for all T.
SLB(S, T) == SLB(T, S), for all S, T.
SLB(void, T) == T, for all T.
SLB(dynamic, T) == T, for all T != void.
SLB(Object, T) == T, for all T != void, dynamic.
SLB(bottom, T) == bottom, for all T.

Note that this is the same outcome as we would have had if Object were a proper subtype of dynamic and dynamic were a proper subtype of void. Hence, an easy way to recall these rules would be to think Object < dynamic < void. Here, < is a “micro subtype” relationship which is able to distinguish between the top types, as opposed to the subtype relationship <: which considers the top types to be the same type. For any relationship involving a non-top type, < is the same thing as <:.


We considered a different set of rules as well:

SUB(T, T) == T, for all T.
SUB(S, T) == SUB(T, S), for all S, T.
SUB(void, T) == void, for all T.
SUB(dynamic, T) == Object, for all T != void, dynamic.
SUB(Object, T) == Object, for all T != void.
SUB(bottom, T) == T, for all T != dynamic.

SLB(T, T) == T, for all T.
SLB(S, T) == SLB(T, S), for all S, T.
SLB(void, dynamic) == Object.
SLB(void, T) == T, for all T != dynamic.
SLB(dynamic, T) == T, for all T != void.
SLB(Object, T) == T, for all T != void, dynamic.
SLB(bottom, T) == bottom, for all T.

This set of rules cannot be reduced to any “micro subtype” relationship where we simply make a choice of how to order the top types and then get all other results as a consequence of that choice. These alternative rules are more strict on the propagation of “dynamicness” in a way which may be helpful for developers. Here is how it works:

The ‘Motivation’ section mentioned a number of pragmatic reasons why it may be meaningful to let SUB(dynamic, int) be some other type than dynamic.

If we take the stance that the relaxed type checks on member accesses that we apply to dynamic receivers are error-prone and hence shouldn't propagate very far implicitly, we may chose to eliminate the special treatment of dynamic, unless that type is present in all branches.

This means that we would make the invocation (b ? 42 : f()).isEven a compile-time error: We could say that “it is not a dynamic invocation because some branches do not deliver a dynamic receiver, and there is no static guarantee that the isEven method exists, so the expression is a compile-time error”. This is achieved by means of one of the rules shown above: SUB(dynamic, T) == Object, for all T != void, dynamic.

In order to show that the alternative set of rules has an internal structure (as opposed to being a random mixture of decisions), we can describe them in the following manner. First we translate all types into a tuple-representation:

void      (1, 1, 0)
dynamic   (1, 0, 1)
Object    (1, 0, 0)
T         (T, 0, 0), for all non-extreme types T
bottom    (0, 0, 0)

In this tuple, the first component is the core type (where “voidness” and “dynamicness” have been erased), where 0 is bottom, 1 is top (that is, the types Object, dynamic, and void), and every other (non-extreme) type is itself, e.g., int is int.

The second component is the “voidness”: 1 means that the type is a void type (there is only void), and 0 means non-void.

The third component is the “dynamicness”: 1 means that the type is dynamic (there is only dynamic, at least for now), and 0 means non-dynamic.

This means that the tuple contains one “core type” and two “bits” (boolean components). With that, we can compute the functions using simple operations:

SUB((t1, v1, d1), (t2, v2, d2)) = (lub(t1,t2), v1 || v2, d1 && d2)
SLB((t1, v1, d1), (t2, v2, d2)) = (glb(t1,t2), v1 && v2, d1 && d2)

We may also specify the same thing more concisely in a curried form:

SUB = (lub, lub, glb)
SLB = (glb, glb, glb)

where lub and glb are specialized for the domain of types and booleans, respectively, but are basically “least upper bound” and “greatest lower bound”: For types we rely on an underlying notion of upper and lower bounds for all non-extreme types, and for booleans it is simply the indicated operators above (|| and &&, respectively).

It may seem tempting, for symmetry, to change the definitions such that we get SLB = (glb, glb, lub), but this would introduce types of the form (T, 0, 1), that is, types like dynamic(int) and dynamic(int Function(String)) that we have considered but not yet decided to introduce. Given that the only effect this change would have is to change some types in a contravariant location from Object to dynamic, and given that this is generally not detectable for clients (e.g., we don‘t care about, and actually can’t even detect, the difference between calling a function of type int Function(Object) and a function of type int Function(dynamic)), so this choice is not likely to matter much. Also the fact that the use of min yields fewer occurrences of the type dynamic seems to be consistent with the nature of Dart 2 typing.

Note that the operations are reflexive and symmetric by construction, and they are also likely to be associative, because both lub and glb are associative for booleans, and for types we will need to consider the actual underlying mechanism, but it ought to be associative if at all possible:

lub(lub(a,b),c) = lub(a,lub(b,c))
glb(glb(a,b),c) = glb(a,glb(b,c))


  • May 22nd 2018, version 0.2: Adjusted to use Object < dynamic < void as a “subtyping micro-structure” (which produces a simpler set of rules) and mention the rules from version 0.1 merely as a possible alternative ruleset.

  • May 1st 2018, version 0.1: Initial version of this feature specification created, based on discussions in SDK issue 28513.