# Top-level operator definitions

[Back to user operators]

## Quick example

Here is a quick example of a top-level user operator (which has to be defined in a module) and of its application:

----------------------- MODULE QuickTopOperator -------------------------------
...
Abs(i) == IF i >= 0 THEN i ELSE -i
...
B(k) == Abs(k)
===============================================================================


As you most probably guessed, the operator Abs expects one argument i. Given an integer j, then the result of computing Abs(j) is the absolute value of j. The same applies, when j is a natural number or a real number.

## Syntax of operator definitions

In general, operators of n arguments are defined as follows:

\* an operator without arguments (nullary)
Opa0 == body_0

\* an operator of one argument (unary)
Opa1(param1) == body_1

\* an operator of two arguments (binary)
Opa2(param1, param2) == body_2
...


In this form, the operator arguments are not allowed to be operators. If you want to receive an operator as an argument, see the syntax of Higher-order operators.

Here are concrete examples of operator definitions:

----------------------------- MODULE FandC ------------------------------------
EXTENDS Integers
...

ABSOLUTE_ZERO_IN_CELCIUS ==
-273

Fahrenheit2Celcius(t) ==
(t - 32) * 10 / 18

Max(s, t) ==
IF s >= t THEN s ELSE t
...
===============================================================================


What is their arity (number of arguments)?

If you are used to imperative languages such as Python or Java, then you are probably surprised that operator definitions do not have any return statement. The reason for that is simple: TLA+ is not executed on any hardware. To understand how operators are evaluated, see the semantics below.

## Syntax of operator applications

Having defined an operator, you can apply it inside another operator as follows (in a module):

----------------------------- MODULE FandC ------------------------------------
EXTENDS Integers
VARIABLE fahrenheit, celcius
\* skipping the definitions of
\* ABSOLUTE_ZERO_IN_CELCIUS, Fahrenheit2Celcius, and Max
...

UpdateCelcius(t) ==
celcius' = Max(ABSOLUTE_ZERO_IN_CELCIUS, Fahrenheit2Celcius(t))

Next ==
/\ fahrenheit' \in -1000..1000
/\ UpdateCelcius(fahrenheit')
...
===============================================================================


In the above example, you see examples of four operator applications:

1. The nullary operator ABSOLUTE_ZERO_IN_CELCIUS is applied without any arguments, just by its name. Note how a nullary operator does not require parentheses (). Yet another quirk of TLA+.

2. The one-argument operator Fahrenheit2Celcius is applied to t, which is a parameter of the operator UpdateCelcius.

3. The two-argument operator Max is applied to ABSOLUTE_ZERO_IN_CELCIUS and Fahrenheit2Celcius(t).

4. The one-argument operator UpdateCelcius is applied to fahrenheit', which is the value of state variable fahrenheit in the next state of the state machine. TLA+ has no problem applying the operator to fahrenheit' or to fahrenheit.

Technically, there are more than four operator applications in our example. However, all other operators are the standard operators. We do not focus on them here.

Note on the operator order. As you can see, we are applying operators after they have been defined in a module. This is a general rule in TLA+: A name can be only referred to, if it has been defined in the code before. TLA+ is not the first language to impose that rule. For instance, Pascal had it too.

Note on shadowing. TLA+ does not allow you to use the same name as an operator parameter, if it has been defined in the context of the operator definition. For instance, the following is not allowed:

-------------------------- MODULE NoShadowing ---------------------------------
VARIABLE x

\* the following operator definition produces a semantic error:
\* the parameter x is shadowing the state variable x
IsZero(x) == x = 0
===============================================================================


There are a few tricky cases, where shadowing can actually happen, e.g., see the operator dir in SlidingPuzzles. However, we recommend to keep things simple and avoid shadowing at all.

## Semantics of operator application

Precise treatment of operator application is given on page 320 of Specifying Systems. In a nutshell, operator application in TLA+ is a Call by macro expansion, though it is a bit smarter: It does not blindly mix names from the operator's body and its application context. For example, the following semantics by substitution is implemented in the Apalache model checker.

Here we give a simple explanation for non-recursive operators. Consider the definition of an n-ary operator A and its application in the definition of another operator B:

A(p_1, ..., p_n) == body_of_A
...
B(p_1, ..., p_k) ==
...
A(e_1, ..., e_n)
...


The following three steps allow us to replace application of the operator A in B:

1. Change the names in the definition of A in such a way such they do not clash with the names in B (as well as with other names that may be used in B). This is the well-known technique of Alpha conversion in programming languages. This may also require renaming of the parameters p_1, ..., p_n. Let the result of alpha conversion be the following operator:
uniq_A(uniq_p_1, ..., uniq_p_n) == body_of_uniq_A

1. Substitute the expression A(e_1, ..., e_n) in the definition of B with body_of_uniq_A.

2. Substitute the names uniq_p_1, ..., uniq_p_n with the expressions e_1, ..., e_n, respectively.

The above transformation is usually called Beta reduction.

Example. Let's go back to the module FandC, which we considered above. By applying the substitution approach several times, we transform Next in several steps as follows:

First, by substituting the body of UpdateCelsius:

  Next ==
/\ fahrenheit' \in -1000..1000
/\ celcius' = Max(ABSOLUTE_ZERO_IN_CELCIUS, Fahrenheit2Celcius(fahrenheit'))


Second, by substituting the body of Max:

  Next ==
/\ fahrenheit' \in -1000..1000
/\ celcius' =
IF ABSOLUTE_ZERO_IN_CELCIUS >= Fahrenheit2Celcius(fahrenheit')
THEN ABSOLUTE_ZERO_IN_CELCIUS
ELSE Fahrenheit2Celcius(fahrenheit')


Third, by substituting the body of Fahrenheit2Celcius (twice):

  Next ==
/\ fahrenheit' \in -1000..1000
/\ celcius' =
IF ABSOLUTE_ZERO_IN_CELCIUS >= (fahrenheit' - 32) * 10 / 18
THEN ABSOLUTE_ZERO_IN_CELCIUS
ELSE (fahrenheit' - 32) * 10 / 18


You could notice that we applied beta reduction syntactically from top to bottom, like peeling an onion. We could do it in another direction: First starting with the application of Fahrenheit2Celcius. This actually does not matter, as long as our goal is to produce a TLA+ expression that is free of user-defined operators. For instance, Apalache applies Alpha conversion and Beta reduction to remove user-defined operator and then translates the TLA+ expression to SMT.