# ADR-014: Precise type inference for records and variants

authorslast revised
Igor Konnov2021-12-12

## 1. Summary

Virtually every user of Snowcat has faced the issue of record type checking being imprecise. Some people call it "unsound", though soundness depends on the type system. This is due to our decision to support the variant pattern that can be found in untyped TLA+ specifications. In this ADR, we are proposing a plan of action for introducing precise type inference for records and variants (discriminated unions) in the type checker. This would deliver the most asked feature. On the downside, we would have to:

1. Increase the complexity of the type checker.

2. Slightly update the rewriting rules in the model checker.

3. Require the users to modify their specs to use the variant operators.

As much as possible, we have tried to make the type annotations non-intrusive and compatible with TLC. After precisely specifying the requirements for the variant type, we have found that it would be impossible to do sound type checking without introducing additional operators.

We believe that the benefits outweigh the downsides in the long run. Moreover, it will improve user experience, as this is the most requested feature.

## 2. Context

As discussed in ADR002, the type checker is not checking the record types precisely. Consider the following operator:

Foo ==
LET S == {[ type |-> "A", a |-> 1], [ type |-> "B", b |-> 2 ]} IN
\E m \in S:
m.a > m.b


The type checker assigns the type Set([type: Str, a: Int, b: Int]) to S. As a result, one can write the expression m.a > m.b, which does not make a lot of sense. This may lead to unexpected results in a large specification. In the above example, the model checker would just produce some values for m.a or m.b, which will probably result in a spurious counterexample.

Further, multiple related issues and potential solutions were underlined in #401 and #789.

There are two main patterns of record use in TLA+:

1. Plain records. A record with a fixed number of fields is passed around.

2. Variants. Records of various shapes are collected in a single set and passed around. The precise record shape is controlled with a special field (discriminator), which is usually called type in TLA+ specs.

### 2.1. Untyped plain records

When it comes to records, it is clear that users expect the type checker to complain about missing record fields. Indeed, it is very easy to introduce a spurious record field by mistyping the field name. It happened to all of us.

Interestingly, plain records are used less often than variants. Perhaps, if records are required, the specification is quite complex already, so it would also need variants.

Occurences in tlaplus-examples:

LamportMutex is an interesting borderline case, in which the spec uses the variant pattern, but it could be typed with a single record type. Here are the interesting pieces in this spec:

ReqMessage(c) == [type |-> "req", clock |-> c]
AckMessage == [type |-> "ack", clock |-> 0]
RelMessage == [type |-> "rel", clock |-> 0]

Message == {AckMessage, RelMessage} \union {ReqMessage(c) : c \in Clock}

...
[r \in Proc |-> IF s=r THEN network[s][r] ELSE Append(network[s][r], m)]
...
Request(p) ==
...
/\ network' = [network EXCEPT ![p] = Broadcast(p, ReqMessage(clock[p]))]
...


Although, every message record is accompanied with the field type, all records have the same shape, namely, they have two fields: A string field type and an integer field clock.

### 2.2. Untyped variants

Most of the benchmarks stem from the Paxos specification. They all follow the same pattern. Messages are represented with records of various shapes. Every record carries the field type that characterizes the record shape. For instance, here is how records are used in Paxos:

Message ==      [type : {"1a"}, bal : Ballot]
\cup [type : {"1b"}, acc : Acceptor, bal : Ballot,
mbal : Ballot \cup {-1}, mval : Value \cup {None}]
\cup [type : {"2a"}, bal : Ballot, val : Value]
\cup [type : {"2b"}, acc : Acceptor, bal : Ballot, val : Value]
...
Send(m) == msgs' = msgs \cup {m}
...
Phase1b(a) == /\ \E m \in msgs :
/\ m.type = "1a"
/\ m.bal > maxBal[a]
...
/\ Send([type |-> "1b", acc |-> a, bal |-> m.bal,
mbal |-> maxVBal[a], mval |-> maxVal[a]])
...


Occurences in tlaplus-examples:

More advanced code patterns over variants can be found in Raft:

Receive(m) ==
LET i == m.mdest
j == m.msource
IN \* Any RPC with a newer term causes the recipient to advance
\* its term first. Responses with stale terms are ignored.
\/ UpdateTerm(i, j, m)
\/ /\ m.mtype = RequestVoteRequest
/\ HandleRequestVoteRequest(i, j, m)
\/ /\ m.mtype = RequestVoteResponse
/\ \/ DropStaleResponse(i, j, m)
\/ HandleRequestVoteResponse(i, j, m)
\/ /\ m.mtype = AppendEntriesRequest
/\ HandleAppendEntriesRequest(i, j, m)
\/ /\ m.mtype = AppendEntriesResponse
/\ \/ DropStaleResponse(i, j, m)
\/ HandleAppendEntriesResponse(i, j, m)
...
Next ==
...
\/ \E m \in DOMAIN messages : Receive(m)
\/ \E m \in DOMAIN messages : DuplicateMessage(m)
\/ \E m \in DOMAIN messages : DropMessage(m)
...


### 3. Other issues

As can be seen from the Paxos example, we should take care of sets of records that are used as sets of variants:

Message ==      [type : {"1a"}, bal : Ballot]
\cup [type : {"1b"}, acc : Acceptor, bal : Ballot,
mbal : Ballot \cup {-1}, mval : Value \cup {None}]
\cup [type : {"2a"}, bal : Ballot, val : Value]
\cup [type : {"2b"}, acc : Acceptor, bal : Ballot, val : Value]


## 4. Options

There are several solutions to the issue of precise record typing.

### 4.1. Support only plain records

In this case, we would only allow to mix records that have exactly the same shape. As a result, when we use variants, we would have to add extra fields to all of them. This solution is not very different from the current implementation of record type checking, though it would allow us to quickly detect spelling errors.

Blocker. This does not look like a real solution, as it would immediately render the existing examples invalid. Moreover, they would be no obvious way to repair these examples.

### 4.2. Support plain records and variants, but no row typing

In this scenario, the type checker would issue an error, if a record expression accesses a field that is outside of its type:

FieldAccess ==
LET m == [ a |-> 2, b |-> "B" ] IN
/\ m.a > 1        \* type OK
/\ m.b = "B"      \* type OK
/\ m.c = { 1, 2 } \* should flag a type error


Moreover, the following example would require a type annotation:

RowAccess(m) ==
m.a > 0         \* should flag a type error


In the above example, the type checker would not be able to infer the type of m and would require an explicit type annotation:

\* @type: [ a: Int, b: Str ];
RowAccessAnnotated(m) ==
m.a > 0         \* should not flag a type error


Blocker. It is not clear to me, what type we would assign to the record access operator. Although the type of m in FieldAccess is obvious to a human reader, as it simply requires us to do the top-to-the-bottom type propagation, Snowcat constructs a set of equality constraints that are solved by unification. This approach requires that we capture the record access operator . as an equality over type variables and types.

### 4.3. Support plain records and variants, including row typing

This solution is inspired by the approach outlined by Leijen05, but is much more limited. The following features discussed in Leijen05 are neither needed needed nor supported:

• extension for records
• extension for variants
• record restriction
• scoped labels

Our use is limited to a few special cases that give support for subtyping and incremental inference of anonymous record types, in which we cannot know the full set of fields up front.

## 5. Solution

In the following, we present row types as type terms. We discuss the user-facing syntax of the type system later in the text.

### 5.1. Plain records

By using Row types, we should be able to infer a polymorphic record type for m in the unannotated RowAccess operator:

Rec(RowCons("a", Int, z))


In this example, RowCons("a", Int, z) indicates a row indicating that the type of the record enclosing it has the field a of type Int. On top of that, this row extends a parametric type z, which either contains a non-empty sequence of rows, or is an empty sequence, that is, RowNil. Importantly, RowCons is wrapped with the term Rec, so no additional fields can be added to the type.

The example FieldAccess contains a record constructor [ a |-> 2, b |-> "B" ]. We can write a general type inference rule for it:

e_1: t_1, ..., e_n: t_n
-------------------------------- [rec]
[ f_1 |-> e_1, ..., f_n |-> e_n]:
Rec(
RowCons(f_1, t_1,
RowCons(f_2, t_2,
...
RowCons(f_n, t_n, RowNil)
...)
)
)


In FieldAccess, we use row types to construct a series of type equations (over free type variables k, ..., q):

// from LET m == [ a |-> 2, b |-> "B" ] IN
m_type = Rec("a", Int, RowCons("b", Str, RowNil))
// from m.a > 1
m_type = Rec(RowCons("a", k, l))
k = Int
// from m.b = "B"
m_type = Rec(RowCons("b", m, n))
m = Str
// from m.c = { 1, 2 }
m_type = Rec(RowCons("c", p, q))
p = Set(Int)


To solve the above equations, one has to apply unification rules. Precise unification rules for rows are given in the paper by Leijen05. Importantly, their unification rules allow RowCons(f1, t1, RowCons(f2, t2, r3)) to be unified with RowCons(f2, t2, RowCons(f1, t1, r3)). Hence, fields may bubble up to the head, and it should be possible to isolate a single field and assign the rest to a type variable. By partially solving the above equation, we would arrive at contradiction. This supports our intuition that the operator FieldAccess is ill-typed.

Hence, we formulate the type inference rule for record access in our type system as follows:

r: Rec(RowCons("f", t_1, t_2))
-------------------------------- [rec_acc]
r.f: t_1


Note that the above rule can be rewritten into a series of equalities over types variables and type terms, which is how this would be implemented in the type checker.

If we only had to deal with records, that would be a complete solution. Unfortunately, variants introduce additional complexity.

### 5.2. Variants

Example 5.2.1. Now we have to figure out how to deal with TLA+ expressions like:

  { [ tag |-> "1a", bal |-> 3 ], [ tag |-> "2a", bal |-> 4, val |-> 0 ] }


Obviously, we cannot fit both of the records into a single plain record type, provided that we want to precisely track the fields that are present in a record. So the type checker should report a type error, if we only implement type inference for the case explained in Section 5.1. To support this important pattern, we introduce variants. They are similar to unions in TypeScript. In contrast to TypeScript, we fix one field to designate the record type in a variant. Also, we are using the word "variant", to avoid any confusion with the TLA+ operators UNION and \union.

We reserve the field name tag for the record discriminator.

Variant constructor. We introduce a special TLA+ operator Variant that extracts the tag from a record and wraps the record into a variant. We need this operator to distinguish between plain records and records that belong to a variant. The operator Variant is defined in TLA+ as follows:

  Variant(r) ==
\* fallback untyped implementation
r


This operator does not change its argument, but it provides the type checker with a hint that the record should be treated as a member of a variant.

Consider the record constructor of n+1 fields, one of them being the field "tag":

  [ tag |-> "<TAG>", f_1 |-> e_1, ..., f_n |-> e_n ]


Here, "<TAG>" stands for a string literal such as "1a" or "2a". The general rule for Variant([ tag |-> "<TAG>", f_1 |-> e_1, ..., f_n |-> e_n ]) looks as follows:

e_1: t_1, ..., e_n: t_n
z is a fresh type variable
------------------------------------------------------------ [variant]
Variant([ tag |-> "<TAG>", f_1 |-> e_1, ..., f_n |-> e_n ]):
Variant(
RowCons("<TAG>",
Rec(
RowCons("tag", Str,
RowCons(f_1, t_1,
...
RowNil)...)
),
z
)
)


According to the rule [variant], the operator Variant wraps a record constructor that contains a string literal for the field tag. The variant contains the record that was passed in the constructor, whereas the other alternatives of the variant are captured with a fresh type variable z, which must be a row.

Importantly, we use rows at two levels:

1. To construct a single record, whose shape is defined precisely.

2. To construct a variant, whose only record is known at the time, while the rest is captured with the type variable z.

Going back to Example 5.2.1, the set constructor would produce a set of equalities:

  a = Set(b)
a = Set(d)
b = Variant(RowCons("1a",
Rec(RowCons("tag", Str, RowCons("bal", Int, RowNil))),
c))
d = Variant(RowCons("2a",
Rec(RowCons("tag", Str,
RowCons("bal", Int,
RowCons("val", Int, RowNil)))),
e))


By solving these equalities with unification, we will arrive at the following variant:

  Set(Variant(RowCons(
"1a",
Rec(RowCons("tag", Str, RowCons("bal", Int, RowNil))),
RowCons(
"2a",
Rec(RowCons("tag", Str,
RowCons("bal", Int,
RowCons("val", Int, RowNil)))),
t
))))


Note that we still do not know the precise shape of the variant, as it closes with the type variable t. This is actually what we expect, as the set may be combined with records of other shapes. Normally, the final shape of a variant propagates via state variables of the TLA+ specification.

Type annotations for variants. Snowcat requires that all state variables are annotated. What shall we write for variants? We introduce the common type notation for variants that separates records with a pipe, that is, |. For instance, consider the following variable declaration in a TLA+ specification:

VARIABLES
\* @type: Set([ tag: "1a", bal: Int ] | [ tag: "2a", bal: Int, val: Int ]);
msgs


Note that even though the syntax of individual elements of a variant is very similar to that of a record, there is small difference: The tag field is not declared as a string type, but carries the values of the tag itself.

As variants can grow large very quick, it is more convenient to introduce them via a type alias. For instance:

VARIABLES
(*
@typeAlias: MESSAGE =
[ tag: "1a", bal: Int ]
| [ tag: "2a", bal: Int, val: Int ]);
@type: Set(MESSAGE);
*)
msgs


Filter a set of variants. As we have seen, the following pattern is quite common in TLA+ specifications, e.g., it is met in Paxos:

  \E m \in msgs:
/\ m.type = "1a"
...
LET Q1b == { m \in msgs : m.type = "1b" /\ ... }


We introduce the operator FilterByTag that is a type-safe version of this pattern:

FilterByTag(Set, tag) == { e \in Set: e.tag = tag }


We introduce a special type inference rule for FilterByTag:

set: Set(Variant(RowCons("<TAG>", r, z)))
------------------------------------------- [variant_filter]
FilterByTag(set, "<TAG>"): Set(r)


Importantly, FilterByTag returns a set of records that carry the tag <TAG>, so we can access record fields of every individual record in the set.

Match by tag. In rare cases, we do not have a set to filter. For instance, we could have a sequence of log messages:

VARIABLE
\* @type: Seq([ tag: "EventA", val: Int ] | [ tag: "EventB", src: Str ]);
log


In this case, we would not be able to easily use FilterByTag. Of course, we could wrap a variant into a singleton set and then apply FilterByTag to it and CHOOSE on top of it. However, this looks clunky and does not guarantee type safety. A much simpler solution is to introduce another special operator:

MatchTag(variant, tag, ThenOper(_), ElseOper(_)) ==
\* fallback untyped implementation
IF variant.tag = tag
THEN ThenOper(variant)
ELSE ElseOper(variant)


The idea of MatchTag is that it passes the extracted record to ThenOper, when its tag value matches tag; otherwise, it passes the reduced variant to ElseOper. This is precisely captured by the inference rule:

variant: Variant(RowCons("<TAG>", r, t))
ThenOper: r => z
ElseOper: Variant(t) => z
------------------------------------------------ [variant_match]
MatchTag(variant, "<TAG>", ThenOper, ElseOper):
(Variant(RowCons("<TAG>", r, t)),
Str,
r => z,
Variant(t) => z
) => z


The above rule looks menacing. Here is an example of matching a record in the above example with the variable log:

IsDefined(eventAOrB) ==
LET ElseB(onlyB) ==
MatchTag(onlyB, "EventB", LAMBDA b: b.src /= "", LAMBDA elseValue: FALSE)
IN
MatchTag(event, "EventA", LAMBDA a: a.val /= -1, ElseB)


The operator ElseB looks redundant, as we know that onlyB is a singleton variant. To this end, we introduce the operator MatchOnly:

MatchOnly(variant, ThenOper(_)) ==
\* fallback untyped implementation
ThenOper(variant)


For completeness, we give the inference rule for MatchOnly:

variant: Variant(RowCons("<TAG>", r, RowNil))
ThenOper: r => z
------------------------------------------------ [variant_match_only]
MatchOnly(variant, ThenOper):
(Variant(RowCons("<TAG>", r, RowNil)),
r => z
) => z


It looks like the solution with MatchTag and MatchOnly introduce a lot of boilerplate. However, this is probably the best solution that we can have, unless we can extend the grammar of TLA+.

### 5.3. Changes in the model checker

Having precise types for variants, we have two options:

1. Keep the current encoding, that is, a variant is encoded as a super-record that contains all possible fields of the member records. The type checker will guarantee that we do not access the fields of the super-record that are not present in the actual record type.

2. Implement the suggestion by Shon Feder.

Although Option 2 looks nicer, we prefer keeping Option 1. The reason is that the current implementation introduces the minimal number of constraints by mashing all possible fields into a super-record. The alternative solution (option 2) would introduce additional constraints, when the spec requires us to extract an element from a set.

Recall the example with a set of messages:

VARIABLES
(*
@typeAlias: MESSAGE =
[ tag: "1a", bal: Int ]
| [ tag: "2a", bal: Int, val: Int ]);
@type: Set(MESSAGE);
*)
msgs


Consider an existential quantifier over the variable msgs:

Next ==
\E m \in msgs:
P


In the current encoding, m is a super-record that contains three fields: tag, bal, and val, even if some of these fields are not required by the actual type of m. In the alternative encoding, m is a tuple tup, which equals to one of the following tuples, depending on the value of the field m.tag:

tup = IF m.tag = "1a"
THEN << { [ tag |-> "1a", bal |-> b ] }, {} >>
ELSE << {}, [ tag |-> "2a", bal |-> b, val |-> v ] }, {} >>


Since it is impossible to statically compute the actual type of m, the rewriting rules would have to replicate the structure of both elements of the tuple tup. This would lead to a blow-up in the number of constraints.

Given the decisions in Section 5.3, we additionally require that all records in a variant type have compatible field types. In more detail, if a variant type contains two record types [ tag: "A", value: a, ... ] and [ tag: "B", value: b, ...], then the types a and b must be unifiable. In practice, this often implies that a and b are simply the same type.
We introduce a new module that is called Variants.tla. It contains the operators Variant, FilterByTag, MatchTag, and MatchOnly. This module will be distributed with Apalache. As is custom in the TLA+ community, the users should be also able to copy Variants.tla next to their specification.