# Booleans

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You find these operators in every programming language and every textbook on logic. These operators form propositional logic.

## Constants

TLA+ contains three special constants: TRUE, FALSE, and BOOLEAN. The constant BOOLEAN is defined as the set {FALSE, TRUE}.

In Apalache, TRUE, FALSE, and BOOLEAN have the types Bool, Bool, and Set(Bool), respectively.

A note for set-theory purists: In theory, TRUE and FALSE are also sets, but in practice they are treated as indivisible values. For instance, Apalache and TLC will report an error, if you try to treat FALSE and TRUE as sets.

## Operators

Warning: Below, we discuss Boolean operators in terms of the way they are usually defined in programming languages. However, it is important to understand that the disjunction operator F \/ G induces a nondeterministic effect when F or G contain the prime operator ('), or when they are used inside the initialization predicate Init. We discuss this effect Control Flow and Non-determinism.

### And (conjunction)

Notation: F /\ G or F \land G

LaTeX notation: Arguments: Two or more arbitrary expressions.

Apalache type: (Bool, Bool) => Bool

Effect: The binary case F /\ G evaluates to:

• TRUE, if both F and G evaluate to TRUE.

• FALSE, if F evaluates to FALSE, or F evaluates to TRUE and G evaluates to FALSE.

The general case F_1 /\ ... /\ F_n can be understood by evaluating the expression F_1 /\ (F_2 /\ ... /\ (F_{n-1} /\ F_n)...).

Determinism: Deterministic, if the arguments are deterministic. Otherwise, the possible effects of non-determinism of each argument are combined. See Control Flow and Non-determinism.

Errors: In pure TLA+, the result is undefined if either conjunct evaluates to a non-Boolean value (the evaluation is lazy). In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.

Example in TLA+:

TRUE  /\ TRUE    \* TRUE
FALSE /\ TRUE    \* FALSE
TRUE  /\ FALSE   \* FALSE
FALSE /\ FALSE   \* FALSE
FALSE /\ 1       \* FALSE in TLC, type error in Apalache
1 /\ FALSE   \* error in TLC, type error in Apalache


Example in Python:

>>> True  and True
True
>>> False and True
False
>>> True  and False
False
>>> False and False
False
>>> False and 1 # 1 is cast to True
False
>>> 1 and False # 1 is cast to True
False



Special syntax form: To minimize the number of parentheses, conjunction can be written in the indented form:

  /\ F_1
/\ G_1
...
/\ G_k
/\ F_2
...
/\ F_n


Similar to scopes in Python, the TLA+ parser groups the expressions according to the number of spaces in front of /\. The formula in the above example is equivalent to:

  F_1 /\ (G_1 /\ ... /\ G_k) /\ F_2 /\ ... /\ F_n


### Or (disjunction)

Notation: F \/ G or F \lor G

LaTeX notation: Arguments: Two or more Boolean expressions.

Apalache type: (Bool, Bool) => Bool

Effect:

The binary case F \/ G evaluates to:

• FALSE, if both F and G evaluate to FALSE.

• TRUE, if F evaluates to TRUE, or F evaluates to FALSE and G evaluates to TRUE.

The general case F_1 \/ ... \/ F_n can be understood by evaluating the expression F_1 \/ (F_2 \/ ... \/ (F_{n-1} \/ F_n)...).

Determinism: deterministic, if the arguments may not update primed variables. If the arguments may update primed variables, disjunctions may result in non-determinism, see Control Flow and Non-determinism.

Errors: In pure TLA+, the result is undefined, if a non-Boolean argument is involved in the evaluation (the evaluation is lazy). In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.

Example in TLA+:

TRUE  \/ TRUE    \* TRUE
FALSE \/ TRUE    \* TRUE
TRUE  \/ FALSE   \* TRUE
FALSE \/ FALSE   \* FALSE
TRUE  \/ 1       \* TRUE in TLC, type error in Apalache
1 \/ TRUE    \* error in TLC, type error in Apalache


Example in Python:

>>> True  or True
True
>>> False or True
True
>>> True  or False
True
>>> False or False
False



Special syntax form: To minimize the number of parentheses, disjunction can be written in the indented form:

  \/ F_1
\/ G_1
...
\/ G_k
\/ F_2
...
\/ F_n


Similar to scopes in Python, the TLA+ parser groups the expressions according to the number of spaces in front of \/. The formula in the above example is equivalent to:

  F_1 \/ (G_1 \/ ... \/ G_k) \/ F_2 \/ ... \/ F_n


The indented form allows you to combine conjunctions and disjunctions:

  \/ /\ F
/\ G
\/ \/ H
\/ J


The above formula is equivalent to:

  (F /\ G) \/ (H \/ J)


### Negation

Notation: ~F or \neg F or \lnot F

LaTeX notation: Arguments: One argument that should evaluate to a Boolean value.

Apalache type: Bool => Bool

Effect:

The value of ~F is computed as follows:

• if F is evaluated to FALSE, then ~F is evaluated to TRUE,
• if F is evaluated to TRUE, then ~F is evaluated to FALSE.

Determinism: Deterministic.

Errors: In pure TLA+, the result is undefined, if the argument evaluates to a non-Boolean value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.

Example in TLA+:

~TRUE    \* FALSE
~FALSE   \* TRUE
~(1)     \* error in TLC, type error in Apalache


Example in Python:

>>> not True
False
>>> not False
True



### Implication

Notation: F => G

LaTeX notation: Arguments: Two arguments. Although they can be arbitrary expressions, the result is only defined when both arguments are evaluated to Boolean values.

Apalache type: (Bool, Bool) => Bool. Note that the => operator at the type level expresses the relation of inputs types to output types for operators, and as opposed to the => expressing the implication relation at the value level.

Effect: F => G evaluates to:

• TRUE, if F evaluates to FALSE, or F evaluates to TRUE and G evaluates to TRUE.

• FALSE, if F evaluates to TRUE and G evaluates to FALSE.

Determinism: Deterministic.

Errors: In pure TLA+, the result is undefined, if one of the arguments evaluates to a non-Boolean value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.

Example in TLA+:

FALSE => TRUE    \* TRUE
TRUE  => TRUE    \* TRUE
FALSE => FALSE   \* TRUE
TRUE  => FALSE   \* FALSE
FALSE => 1       \* TRUE in TLC, type error in Apalache
TRUE  => 1       \* runtime error in TLC, type error in Apalache
1     => TRUE    \* runtime error in TLC, type error in Apalache


Example in Python:

Recall that A => B is equivalent to ~A \/ B.

>>> (not False) or True
True
>>> (not True)  or True
True
>>> (not False) or False
True
>>> (not True)  or False
False



### Equivalence

Notation: F <=> G or F \equiv G

LaTeX notation: or Arguments: Two arguments. Although they can be arbitrary expressions, the result is only defined when both arguments are evaluated to Boolean values.

Apalache type: (Bool, Bool) => Bool

Effect: F <=> G evaluates to:

• TRUE, if both F and G evaluate to TRUE, or both F and G evaluate to FALSE.

• FALSE, if one of the arguments evaluates to TRUE, while the other argument evaluates to FALSE.

How is F <=> G different from F = G? Actually, F <=> G is equality that is defined only for Boolean values. In other words, if F and G are evaluated to Boolean values, then F <=> G and F = G are evaluated to the same Boolean value. We prefer F <=> G to F = G, as F <=> G clearly indicates the intended types of F and G and thus makes the logical structure more obvious.

Determinism: Deterministic.

Errors: In pure TLA+, the result is undefined, if one of the arguments evaluates to a non-Boolean value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.

Example in TLA+:

FALSE <=> TRUE   \* FALSE
TRUE  <=> TRUE   \* TRUE
FALSE <=> FALSE  \* TRUE
TRUE  <=> FALSE  \* TRUE
FALSE <=> 1      \* runtime error in TLC, type error in Apalache
1     <=> TRUE   \* runtime error in TLC, type error in Apalache


Example in Python:

Assuming that both expressions are Boolean, F <=> G is equivalent to F = G.

>>> False == True
False
>>> True  == True
True
>>> False == False
True
>>> True  == False
False