Structured Data

Structures

Sequences cover tuples and arrays, now we need something to represent string hashmaps. In TLA+, this is the struct:

struct == [a |-> 1, b |-> {}]

You index structs the same way you do sequences so struct["a"] = 1. You can also write it in a more “object” like with a dot instead: struct.a = 1.

Troubleshooting

If you get

Encountered “|->” at line X, column Y and token “key”

It’s because you wrote ["key" |-> val] instead of [key |-> val].

Unsurprisingly, structs are mainly used to represent organized collections of data. For example, a BankTransaction might consist of an account, an amount, and whether it was a deposit or withdrawal.

Just like sequences, sets, and primitive values, we want some way to generate sets of structs, so we can stick them in our type invariants. Here’s how:

\* Accounts is a set of model values
BankTransactionType == [acct: Accounts, amnt: 1..10, type: {"deposit", "withdraw"}]

This is the set of all structures where s.acct \in Accounts, s.amnt \in 1..10, etc.

Tip

With complex types especially, it’s good idea to regularly estimate how many elements are in the set. Here, for in a model with three accounts, BankTransactionType will have \(3 \cdot 10 \cdot 2 = 60\) elements.

Troubleshooting

If you get

Attempted to compute the number of elements in the string “val”

at model-checking time, it’s because you wrote [key: "val"] instead of [key: {"val"}].

Getting a Struct’s Keys

If I wanted to find all of the values of a sequence, I could get it like this:

RangeSeq(seq) == {seq[i]: i \in 1..Len(seq)}

How can I get all the values of a structure? Len isn’t defined for structures. Instead, we have the special keyword DOMAIN. DOMAIN struct is the set of all keys of a structure.

RangeStruct(struct) == {struct[key]: key \in DOMAIN struct}

Now for the fun bit. What happens if we pass a sequence into RangeStruct?

>>> RangeStruct(<<"a", "b", "a">>)

{"a", "b"}

It looks like DOMAIN seq == 1..Len(seq)! In fact, it’s actually the other way around: Len is defined in terms of DOMAIN!

Here’s the punchline: both sequences and structures are just syntactic sugar. TLA+ has only two “real” collections: sets and functions. Sequences and structures are both particular classes of functions, the ones that we as programmers are most familiar with. It’s time to finally introduce the true data type.

Functions

First of all, throw away the programming definition of “function”. The closest thing TLA+ has to a programmatic function are operators. A function follows the mathematical definition, a mapping of values in one set to another set.

F == [x \in S |-> expr]

The set we’re mapping from, S, is the domain of the function, and can be retreived by writing DOMAIN F. That’s why we could also use DOMAIN with sequences and structures:

  1. A sequence is just a function where the domain is 1..n.

  2. A struct is just a function where the domain is a set of strings.

But functions are a more general than that, and can map any set of values. For example, we can have pairs of numbers in the domain of the function.

Prod ==
  LET S == 1..10 IN
  [p \in S \X S |-> p[1] * p[2]]

\* Prod[<<3, 5>>] = 15

Tip

You can also write that as Prod == [x \in S, y \in S |-> x * y], or G == [x, y \in S |-> x * y]. You can also call the function with Prod[3, 5] and leave out the angle brackets.

(Internally, TLA+ will represent it as a tuple, so DOMAIN F = S \X T.)

I like using functions to show me the results of an expression for various inputs. For what values of P and Q is P => Q true?

TruthTable == [p, q \in BOOLEAN |-> p => q]

If you run this in scratch, you’ll get the results, though they’ll be in an unusual format:

>>> TruthTable

( <<FALSE, FALSE>> :> TRUE @@
<<FALSE, TRUE>> :> TRUE @@
<<TRUE, FALSE>> :> FALSE @@
<<TRUE, TRUE>> :> TRUE )

This is in “expanded form”: x :> y is the single-valued function mapping x to y (so [s \in {x} |-> y]), and @@ merges two functions. If the two functions share a key, then @@ keeps the value on the left.

Note

@@ and :> are only available in your spec if you extend TLC.

With this, we can see how sequences and structures are just functions:

>>> 1 :> "a" @@ 2 :> "b"
<<"a", "b">>

>>> "a" :> 1 @@ "b" :> 2
[a |-> 1, b |-> 2]

Example: Zip

Python has a function called zip. It takes two iterables and returns a single sequence, where the elements are pairs of elements from the two inputs. If one is larger than the other, it only does up to the length of the shorter.

>>> list(zip([1, 2], ["a", "b", "c"]))
[(1, 'a'), (2, 'b')]

Normally programming languages implement zip with iteration or recursion. We don’t need that here because we can “see” the entire sequence at once.

Zip1(seq1, seq2) ==
  LET Min(a, b) == IF a < b THEN a ELSE b
      N == Min(Len(seq1), Len(seq2))
  IN
    [i \in 1..N |-> <<seq1[i], seq2[i]>>]

Another way we could write this would be to notice that the intersection of 1..a and 1..b is 1..Min(a,b). So we can simplify Zip to:

Zip2(seq1, seq2) ==
  LET N == (DOMAIN seq1) \intersect (DOMAIN seq2)
  IN
    [i \in N |-> <<seq1[i], seq2[i]>>]

We can check that these are equivalent by writing a quantifier check:

LET
  S == 1..4
  Input == (S \X S \X S) \union (S \X S)
IN
  \A s1, s2 \in Input:
    Zip1(s1, s2) = Zip2(s1, s2)

Using Functions

Why functions over operators? We rarely use functions for computations— operators are far superior for that. Functions are important as values. We can assign them to variables and manipulate them like any other value.

In a spec I once wrote, I had to assign tasks to CPUs. Some tasks needed to be assigned to many CPUs, but each CPU should only have one task. In that spec, the best solution was to store assignments as functions, where each task mapped to a set of CPUs.

variables
  assignments = [t \in Tasks |-> {}]

Then I could write assignment[t] := assignment[t] \union {cpu} to assign cpu to task t. For my invariant, I said no two tasks shared a CPU assignment.

OnlyOneTaskPerCpu ==
  \A t1, t2 \in Tasks, c \in CPU:
    /\ (t1 # t2)
    /\ c \in assignments[t1]
    => c \notin assignments[t2]

We could also write this invariant by noticing that “tasks don’t share cpus” is the same as saying “assignment sets are disjoint”:

OnlyOneTaskPerCpu ==
  \A t1, t2 \in Tasks:
    (t1 # t2)
    => assignments[t1] \intersect assignments[t2] = {}

Function sets

You know the drill: new class of value, new need for a way to generate sets of that value. We need to add function values to our type invariants, too!

The syntax for function sets is [S -> T] and is “every function where the domain is S and all of the values are in T.”1 In the prior task example, assignments was always a function in the function set [Tasks -> SUBSET CPUs].

Tip

A function set of form [A -> B] will have \(\#B^{\#A}\) elements in it. If there were two tasks and three CPUs, that would be :math`(2^3)^2 = 64` possible functions.

A good way to remember this: [1..n -> BOOLEAN] is the set of all binary strings of length n, and we know there are \(2^n\) such strings.

I can also use set maps and filters here. Let’s say a task can only be assigned to at most two CPUs. If I wanted to, I could fold that into the type invariant, using a function set:

TypeInvariant ==
  \* ...
  /\ assignments \in
    LET LeqTwoCPUs == {set \in SUBSET CPUs: Cardinality(set) <= 2}
    IN [Tasks -> LeqTwoCPUs]

Though in this case I’d prefer to keep the type invariant simple and write a second invariant with the additional restriction:

TypeInvariant ==
  /\ assignments \in [Tasks -> SUBSET CPUs]

AnotherInvariant ==
  \A t \in Tasks: Cardinality(assignments[t]) <= 2

Some more examples of function sets:

  1. We have a set of servers, which can have one of three states. Then status \in [Server -> {"online", "booting", "offline"}].

  2. We represent a directed graph as a function on pairs of points, which is true iff there’s an edge between the two points. Then graph \in [Node \X Node -> BOOLEAN].

  3. If we define the previous set as the operator GraphType, we could get the set of all undirected graphs with {g \in GraphType: \A n1, n2 \in Node: g[n1,n2] = g[n2,n1]}.

  4. If we have a set of users and resources, the set of all possible allocations could be [Resource -> User]. If some resources could be unallocated, it would instead be [Resource -> User \union {NULL}] (where NULL is a model value).

Troubleshooting

If you get

Encountered “|->” in line X, column Y

In a function set, then you probably wrote [S |-> T] instead of [S -> T]. Similarly, if you get

Encountered “->” in line X, column Y

In a function, then you probably wrote [x \in S -> T] instead of [x \in S |-> T]. Don’t worry, everybody gets the two mixed up at some point.

Example: Sorting

Let’s put function sets to good use. If we want to test if a sequence is sorted in ascending order, we can write it like this:

IsSorted(seq) ==
  \A i, j \in 1..Len(seq):
    i < j => seq[i] <= seq[j]

Now what about an operator that sorts as a sequence? Specifically, one such that IsSorted(SortSeq(seq)) is always true. That’s easy:

Sort(seq) ==
  <<>>

We had to tweak the definition a bit and make sure that the output sequence has all the same elements, too.

Now you might remember from our discussion of CHOOSE that instead of manually constructing the sequence with the desired properties, it’s easier to instead take a set of sequences and pluck out the one that has the properties we want.

We know we can get the set of elements in a sequence this way:

Range(f) == {f[x] : x \in DOMAIN f}

Then [DOMAIN seq -> Range(seq)] is the set of all sequences which have the same elements as seq. Our operator will then look something like this:

Sort(seq) ==
  CHOOSE sorted \in [DOMAIN seq -> Range(seq)]:
    /\ \* sorted has the same number of each element as seq
    /\ IsSorted(sorted)

To figure out if two sequences have the same number of each element, let’s define a CountMatching(f, val) operator that tells us the number of inputs matching val. To get the size of a set, we need Cardinality from the FiniteSets module.

CountMatching(f, val) ==
  Cardinality({key \in DOMAIN f: f[key] = val})

Then we just need to test to check this over every element in the sequence:

Sort(seq) ==
  CHOOSE sorted \in [DOMAIN seq -> Range(seq)]:
    /\ \A i \in DOMAIN seq:
      CountMatching(seq, seq[i]) = CountMatching(sorted, seq[i])
    /\ IsSorted(sorted)

Let’s try this on some input:

>>> Sort(<<8, 2, 7, 4, 3, 1, 3>>)
<<1, 2, 3, 3, 4, 7, 8>>

Perfect!

The Duplicate Checker Again

Last time, I promise.

Our last version of the duplicate checker was this:

Note

S <- 1..10 for all these examples.

---- MODULE duplicates ----
EXTENDS Integers, Sequences, TLC, FiniteSets
CONSTANT S
ASSUME Cardinality(S) >= 4

(*--algorithm dup
  variable seq \in S \X S \X S \X S;
  index = 1;
  seen = {};
  is_unique = TRUE;

define
  TypeInvariant ==
    /\ is_unique \in BOOLEAN
    /\ seen \subseteq S
    /\ index \in 1..Len(seq)+1
    
  IsUnique(s) == 
    \A i, j \in 1..Len(s): 
      i # j => seq[i] # seq[j] 

  IsCorrect == pc = "Done" => is_unique = IsUnique(seq)
end define; 

begin
  Iterate:
    while index <= Len(seq) do
      if seq[index] \notin seen then
        seen := seen \union {seq[index]};
      else
        is_unique := FALSE;
      end if;
      index := index + 1;
    end while;
end algorithm; *)
====

Currently we can control the value of S per model, it would be good if we could control the length of seq too. Then we can test both 2-element sequences and 20-element sequences. But currently the length is hardcoded by the number of \X cross-products we used.

We can simplify this with function sets. S \X S \X S is going to be a set of 3-tuples. We now know that a 3-tuple is a function with domain 1..3. Then [1..3 -> S] = S \X S \X S: the set of all 3-tuples where each element of each tuple is a value in S.

From this, extending this to five-element sequences is trivial:

 ASSUME Cardinality(S) >= 4
 
 (*--algorithm dup
-  variable seq \in S \X S \X S \X S;
+variable
+  seq \in [1..5 -> S];
   index = 1;
   seen = {};
   is_unique = TRUE;
800000 states / 700000 distinct spec

Notice now that, while S \X S \X S has a hardcoded length, [1..3 -> S] is based on a value — the size of the domain set. This means we can pull it into a constant!

 ---- MODULE duplicates ----
 EXTENDS Integers, Sequences, TLC, FiniteSets
-CONSTANT S
+CONSTANT S, Size
 ASSUME Cardinality(S) >= 4
+ASSUME Size > 0
 
 (*--algorithm dup
 variable
-  seq \in [1..5 -> S];
+  seq \in [1..Size -> S];
   index = 1;
   seen = {};
   is_unique = TRUE;

Tip

State sweeping is when we use an initial starting state variable to control the parameters for other variables. For example, we could have one variable determine the length of an input sequence, or the maximum size of a bounded buffer.

 
 (*--algorithm dup
 variable
-  seq \in [1..Size -> S];
+  n \in 1..Size;
+  seq \in [1..n -> S];
   index = 1;
   seen = {};
   is_unique = TRUE;
876540 states / 765430 distinct spec

Now, instead of checking all length 5 sequences, we’re checking all length 5 or smaller sequences! This is a useful specifying trick known as state sweeping.

Strictly speaking, sweeping isn’t necessary: we can, with sufficient cleverness, construct a complex operator that does the same thing. Sweeping, however, is often much easier than doing that, and frees up your brainpower for the actual process of specification.

Summary

  • Functions map a set of values to another set of values. They are written [x \in set |-> Expr(x)] and called with f[value].

    • Functions can also be written [x, y \in Set1, z \in Set2 |-> P(x, y, z) and called with f[a, b, c] (or f[<<a, b, c>>]).

  • The domain of a function, the set we’re mapping from, is DOMAIN f.

    • a :> b is the function [x \in {a} |-> b].

    • f @@ g merges f and g, preferring keys in f.

  • Sequences are just a special kind of function, where the domain is 1..n.

  • Structures are another special kind of function, written [key1 |-> val1, key2 |-> val2]. They are called with struct["key1"] (or struct.key1).

  • Functions and structures both have special set syntax. For structures, it is [key1: set1]. For functions, it’s [A -> B].

1

T is sometimes referred to as the “codomain”.