# Deterministic conditionals

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In this section, we consider the instances of IF-THEN-ELSE and CASE that may not update primed variables. For the case, when the operators inside IF-THEN-ELSE or CASE can be used to do non-deterministic assignments, see Control Flow and Non-determinism.

Warning: Because frequent use of IF-THEN-ELSE is very common in most programming languages, TLA+ specification authors with programming experience often default to writing expressions such as IF A THEN B ELSE C. We encourage those authors to use this construct more sparingly. In our experience, the use of IF-THEN-ELSE is rarely required. Many things can be done with Boolean operators, which provide more structure in TLA+ code than in programming languages. We recommend using IF-THEN-ELSE to compute predicate-dependent values, not to structure code.

Warning 2: CASE is considered deterministic in this section, as it is defined with the CHOOSE operator in Specifying Systems, Section 16.1.4. For this reason, CASE should only be used when all of its guards are mutually exclusive. Given all the intricacies of CASE, we recommend using nested IF-THEN-ELSE instead.

## Deterministic IF-THEN-ELSE

Use it when choosing between two values, not to structure your code.

Notation: IF A THEN B ELSE C

LaTeX notation: the same

Arguments: a Boolean expression A and two expressions B and C

Apalache type: (Bool, a, a) => a. Note that a can be replaced with Bool. If a is Bool, and only in that case, the expression IF A THEN B ELSE C is equivalent to (A => B) /\ (~A => C).

Effect: IF A THEN B ELSE C evaluates to:

• The value of B, if A evaluates to TRUE.
• The value of C, if A evaluates to FALSE.

Determinism: This is a deterministic version. For the non-deterministic version, see Control Flow and Non-determinism.

Errors: If A evaluates to a non-Boolean value, the result is undefined. TLC raises an error during model checking. Apalache raises a type error when preprocessing. Additionally, if B and C may evaluate to values of different types, Apalache raises a type error.

Example in TLA+: Consider the following TLA+ expression:

IF x THEN 100 ELSE 0


As you most likely expected, this expression evaluates to 100, when x evaluates to TRUE; and it evaluates to 0, when x evaluates to FALSE.

Example in Python:

100 if x else 0


Note that we are using the expression syntax for if-else in python. This is because we write an expression, not a series of statements that assign values to variables!

## Deterministic CASE

Read the description and never use this operator

Notation:

CASE p_1 -> e_1
[] p_2 -> e_2
...
[] p_n -> e_n


LaTeX notation:

Arguments: Boolean expressions p_1, ..., p_n and expressions e_1, ..., e_n.

Apalache type: (Bool, a, Bool, a, ..., Bool, a) => a, for some type a. If a is Bool, then the case operator can be a part of a Boolean formula.

Effect: Given a state s, define the set I \subseteq 1..n as follows: The set I includes the index j \in 1..n if and only if p_j evaluates to TRUE in the state s. Then the above CASE expression evaluates to:

• the value of the expression e_i for some i \in I, if I is not empty; or
• an undefined value, if the set I is empty.

As you can see, when several predicates {p_i: i \in I} are evaluated to TRUE in the state s, then the result of CASE is equal to one of the elements in the set {e_i: i \in I}. Although the result should be stable, the exact implementation is unknown.

Whenever I is a singleton set, the result is easy to define: Just take the only element of I. Hence, when p_1, ..., p_n are mutually exclusive, the result is deterministic and implementation-agnostic.

Owing to the flexible semantics of simultaneously enabled predicates, TLC interprets the above CASE operator as a chain of IF-THEN-ELSE expressions:

  IF p_1 THEN e_1
ELSE IF p_2 THEN e_2
...
ELSE IF p_n THEN e_n
ELSE TLC!Assert(FALSE)


As TLC fixes the evaluation order, TLC may miss a bug in an arm that is never activated in this order!

Note that the last arm of the ITE-series ends with Assert(FALSE), as the result is undefined, when no predicate evaluates to TRUE. As the type of this expression cannot be precisely defined, Apalache does not support CASE expressions, but only supports CASE-OTHER expressions (see below), which it treats as a chain of IF-THEN-ELSE expressions.

Determinism. The result of CASE is deterministic, if there are no primes inside. For the non-deterministic version, see [Control Flow and Non-determinism]. When the predicates are mutually exclusive, the evaluation result is clearly specified. When the predicates are not mutually exclusive, the operator is still deterministic, but only one of the simultaneously enabled branches is evaluated. Which branch is evaluated depends on the CHOOSE operator, see [Logic].

Errors: If one of p_1, ..., p_n evaluates to a non-Boolean value, the result is undefined. TLC raises an error during model checking. Apalache raises a type error when preprocessing. Additionally, if e_1, ..., e_n may evaluate to values of different types, Apalache raises a type error.

Example in TLA+: The following expression classifies an integer variable n with one of the three strings: "negative", "zero", or "positive".

  CASE n < 0 -> "negative"
[] n = 0 -> "zero"
[] n > 0 -> "positive"


Importantly, the predicates n < 0, n = 0, and n > 0 are mutually exclusive.

The following expression contains non-exclusive predicates:

  CASE n % 2 = 0 -> "even"
[] (\A k \in 2..(1 + n \div 2): n % k /= 0) -> "prime"
[] n % 2 = 1 -> "odd"



Note that by looking at the specification, we cannot tell, whether this expression returns "odd" or "prime", when n = 17. We only know that the case expression should consistently return the same value, whenever it is evaluated with n = 17.

Example in Python: Consider our first example in TLA+. Similar to TLC, we give executable semantics for the fixed evaluation order of the predicates.

def case_example(n):
if n < 0:
return "negative"
elif n == 0:
return "zero"
elif n > 0:
return "positive"


## Deterministic CASE-OTHER

Better use IF-THEN-ELSE.

Notation:

CASE p_1 -> e_1
[] p_2 -> e_2
...
[] p_n -> e_n
[] OTHER -> e_0


LaTeX notation:

Arguments: Boolean expressions p_1, ..., p_n and expressions e_0, e_1, ..., e_n.

Apalache type: (Bool, a, Bool, a, ..., Bool, a, a) => a, for some type a. If a is Bool, then the case operator can be a part of a Boolean formula.

Effect: This operator is equivalent to the following version of CASE:

CASE p_1 -> e_1
[] p_2 -> e_2
...
[] p_n -> e_n
[] ~(p_1 \/ p_2 \/ ... \/ p_n) -> e_0


Both TLC and Apalache interpret this CASE operator as a chain of IF-THEN-ELSE expressions:

  IF p_1 THEN e_1
ELSE IF p_2 THEN e_2
...
ELSE IF p_n THEN e_n
ELSE e_0


All the idiosyncrasies of CASE apply to CASE-OTHER. Hence, we recommend using IF-THEN-ELSE instead of CASE-OTHER. Although IF-THEN-ELSE is a bit more verbose, its semantics are precisely defined.

Determinism. The result of CASE-OTHER is deterministic, if e_0, e_1, ..., e_n may not update primed variables. For the non-deterministic version, see [Control Flow and Non-determinism]. When the predicates are mutually exclusive, the semantics is clearly specified. When the predicates are not mutually exclusive, the operator is still deterministic, but only one of the simultaneously enabled branches is evaluated. The choice of the branch is implemented with the operator CHOOSE, see [Logic].

Errors: If one of p_1, ..., p_n evaluates to a non-Boolean value, the result is undefined. TLC raises an error during model checking. Apalache raises a type error when preprocessing. Additionally, if e_0, e_1, ..., e_n may evaluate to values of different types, Apalache raises a type error.