Logic
In this section, you find the operators that – together with Sets – form the foundation of TLA+. It is a bit strange that we call this section "Logic", as the whole language of TLA+ is a logic. However, the operators of this section are often seen in firstorder logic, as opposed to propositional logic (see Booleans).
Note that the special form \E y \in S: x' = y
is often used to express
nondeterminism in TLA+. See Control Flow and Nondeterminism. In this
section, we only consider the deterministic use of the existential quantifier.
Bounded universal quantifier
Notation: \A x \in S: P
LaTeX notation:
Arguments: At least three arguments: a variable name, a set, and an expression. As usual in TLA+, if the second argument is not a set, the result is undefined. You can also use multiple variables and tuples, see Advanced syntax.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
\A x \in S: P
has the typeBool
.
Effect: This operator evaluates to a Boolean value. We explain semantics only for a single variable:
\A x \in S: P
evaluates toTRUE
, if for every elemente
ofS
, the expressionP
evaluates toTRUE
against the binding[x > e]
. Conversely,
\A x \in S: P
evaluates toFALSE
, if there exists an elemente
ofS
that makes the expressionP
evaluate toFALSE
against the binding[x > e]
.
Importantly, when S = {}
, the expression \A x \in S: P
evaluates to
TRUE
, independently of what is written in P
. Likewise, when {x \in S: P} = {}
, the expression \A x \in S: P
evaluates to TRUE
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible with the type
of x
that is expected in the predicate P
.
Advanced syntax: Instead of a single variable x
, you can use the tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write \A <<x, y>> \in S: P
. In this case, for every
element e
of S
, the variable x
is bound to e[1]
and y
is bound to
e[2]
. The predicate P
is evaluated against this binding.
Moreover, instead of introducing one variable, one can quantify over several
sets. For instance, you can write: \A x \in S, y \in T: P
. This form is
simply syntax sugar for the form with nested quantifiers: \A x \in S: \A y \in T: P
. Similarly, \A x, y \in S: P
is syntax sugar for
\A x \in S: \A y \in S: P
.
Example in TLA+:
\A x \in {1, 2, 3, 4}:
x > 0
\* TRUE
\A x \in {1, 2, 3, 4}:
x > 2
\* FALSE
\* check the section on tuples to understand the following syntax
\A <<x, y>> \in { 1, 2 } \X { 3, 4 }:
x < y
\* TRUE
Example in Python: Python conveniently offers us a concise syntax:
>>> S = {1, 2, 3, 4}
>>> all(x > 0 for x in S)
True
>>> all(x > 2 for x in S)
False
>>> T2 = {(x, y) for x in [1, 2] for y in [3, 4]}
>>> all(x < y for (x, y) in T2)
True
Bounded existential quantifier
Notation: \E x \in S: P
LaTeX notation:
Arguments: At least three arguments: a variable name, a set, and an expression. As usual in TLA+, if the second argument is not a set, the result is undefined.You can also use multiple variables and tuples, see Advanced syntax.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
\E x \in S: P
has the typeBool
.
Effect: This operator evaluates to a Boolean value. We explain semantics only for a single variable:
\E x \in S: P
evaluates toTRUE
, if there is an elemente
ofS
that makes the expressionP
evaluate toTRUE
against the binding[x > e]
. Conversely,
\E x \in S: P
evaluates toFALSE
, if for all elementse
ofS
, the expressionP
evaluate toFALSE
against the binding[x > e]
.
Importantly, when S = {}
, the expression \E x \ in S: P
evaluates to
FALSE
, independently of what is written in P
. Likewise, when {x \in S: P} = {}
, the expression \E x \ in S: P
evaluates to FALSE
.
As you probably have noticed, \E x \in S: P
is equivalent to ~(\A x \in S: ~P)
, and \A x \in S: P
is equivalent to ~(\E x \in S: ~P)
. This is called
duality in logic. But take care! If \E x \in S: P
may act as a
nondeterministic assignment, duality does not work anymore! See Control
Flow and Nondeterminism.
Determinism: Deterministic when P
contains no action operators (including
the prime operator '
). For the nondeterministic case, see Control Flow and
Nondeterminism.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible in the context of P
when an element of S
is bound to x
.
Advanced syntax: Instead of a single variable x
, you can use the tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write \E <<x, y>> \in S: P
. In this case, for every
element e
of S
, the variable x
is bound to e[1]
and y
is bound to
e[2]
. The predicate P
is evaluated against this binding.
Moreover, instead of introducing one variable, one can quantify over several
sets. For instance, you can write: \E x \in S, y \in T: P
. This form is
simply syntax sugar for the form with nested quantifiers: \E x \in S: \E y \in T: P
. Similarly, \E x, y \in S: P
is syntax sugar for \E x \in S: \E y \in S: P
.
Example in TLA+:
\E x \in {1, 2, 3, 4}:
x > 0
\* TRUE
\E x \in {1, 2, 3, 4}:
x > 2
\* TRUE
\* check the section on tuples to understand the following syntax
\E <<x, y>> \in { 1, 2 } \X { 3, 4 }:
x < y
\* TRUE
Example in Python: Python conveniently offers us a concise syntax:
>>> S = {1, 2, 3, 4}
>>> any(x > 0 for x in S)
True
>>> any(x > 2 for x in S)
True
>>> T2 = {(x, y) for x in [1, 2] for y in [3, 4]}
>>> any(x < y for (x, y) in T2)
True
Equality
A foundational operator in TLA+
Notation: e_1 = e_2
LaTeX notation:
Arguments: Two arguments.
Apalache type: (a, a) => Bool
, for some type a
.
Effect: This operator evaluates to a Boolean value. It tests the values
of e_1
and e_2
for structural equality. The exact effect depends on the
values of e_1
and e_2
. Let e_1
and e_2
evaluate to the values
v_1
and v_2
. Then e_1 = e_2
evaluates to:

If
v_1
andv_2
are Booleans, thene_1 = e_2
evaluates tov_1 <=> v_2
. 
If
v_1
andv_2
are integers, thene_1 = e_2
evaluates toTRUE
if and only ifv_1
andv_2
are exactly the same integers. 
If
v_1
andv_2
are strings, thene_1 = e_2
evaluates toTRUE
if and only ifv_1
andv_2
are exactly the same strings. 
If
v_1
andv_2
are sets, thene_1 = e_2
evaluates toTRUE
if and only if the following expression evaluates toTRUE
:v_1 \subseteq v_2 /\ v_2 \subseteq v_1

If
v_1
andv_2
are functions, tuples, records, or sequences, thene_1 = e_2
evaluates toTRUE
if and only if the following expression evaluates toTRUE
:DOMAIN v_1 = DOMAIN v_2 /\ \A x \in DOMAIN v_1: v_1[x] = v_2[x]

In other cases,
e_1 = e_2
evaluates toFALSE
if the values have comparable types. 
TLC and Apalache report an error, if the values have incomparable types.
Determinism: Deterministic, unless e_1
has the form x'
, which can be
interpreted as an assignment to the variable x'
. For the nondeterministic
case, see Control Flow and Nondeterminism.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if e_1
and e_2
evaluate to incomparable values.
Apalache produces a static type error, if the types of e_1
and e_2
do not
match.
Example in TLA+:
FALSE = FALSE \* TRUE
FALSE = TRUE \* FALSE
10 = 20 \* FALSE
15 = 15 \* TRUE
"Hello" = "world" \* FALSE
"Hello" = "hello" \* FALSE
"Bob" = "Bob" \* TRUE
{ 1, 2 } = { 2, 3} \* FALSE
{ 1, 2 } = { 2, 1} \* TRUE
{ 1 } \ { 1 } = { "a" } \ { "a" } \* TRUE in pure TLA+ and TLC,
\* type error in Apalache
{ { 1, 2 } } = { { 1, 2, 2, 2 } } \* TRUE
<<1, "a">> = <<1, "a">> \* TRUE
<<1, "a">> = <<1, "b">> \* FALSE
<<1, FALSE>> = <<2>> \* FALSE in pure TLA+ and TLC,
\* type error in Apalache
<<1, 2>> = <<1, 2, 3>> \* FALSE in pure TLA+ and TLC,
\* FALSE in Apalache, when both values
\* are treated as sequences
[ a > 1, b > 3 ] = [ a > 1, b > 3 ] \* TRUE
[ a > 1, b > 3 ] = [ a > 1 ] \* FALSE
[ x \in 2..2 > x + x ] = [ x \in {2} > 2 * x ] \* TRUE
[ x \in 2..3 > x + x ] = [ x \in {2, 3} > 2 * x ] \* TRUE
Example in Python: The standard data structures also implement
structural equality in Python, though we have to be careful to
use ==
instead of =
:
>>> False == False
True
>>> False == True
False
>>> 10 == 20
False
>>> 15 == 15
True
>>> "Hello" == "world"
False
>>> "Hello" == "hello"
False
>>> "Bob" == "Bob"
True
>>> { 1, 2 } == { 2, 3 }
False
>>> { 1, 2 } == { 2, 1 }
True
>>> { 1 }  { 1 } == { "a" }  { "a" }
True
>>> { frozenset({ 1, 2 }) } == { frozenset({ 1, 2, 2, 2 }) }
True
>>> (1, "a") == (1, "a")
True
>>> (1, "a") == (1, "b")
False
>>> (1, False) == (2, )
False
>>> (1, 2) == (1, 2, 3)
False
>>> { "a": 1, "b": 3 } == { "a": 1, "b": 3 }
True
>>> { "a": 1, "b": 3 } == { "a": 1 }
False
>>> { x: (x + x) for x in { 2 } } == { x: (x * x) for x in { 2 } }
True
>>> { x: (x + x) for x in { 2, 3 } } == { x: 2 * x for x in { 2, 3 } }
True
Inequality
Notation: e_1 /= e_2
or e_1 # e_2
LaTeX notation:
Arguments: Two arguments.
Apalache type: (a, a) => Bool
, for some type a
.
Effect: This operator is syntax sugar for ~(e_1 = e_2)
. Full stop.
Bounded Choice
This operator causes a lot of confusion. Read carefully!
Notation: CHOOSE x \in S: P
LaTeX notation:
Arguments: Three arguments: a variable name, a set, and an expression.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
CHOOSE x \in S: P
has the typea
.
Effect: This operator implements a blackbox algorithm that somehow picks
one element from the set {x \in S: P}
. Is it an algorithm? Yes! CHOOSE x \in S: P
is deterministic. When you give it two equal sets and two equivalent
predicates, CHOOSE
produces the same value. Formally, the only known property
of CHOOSE
is as follows (which is slightly more general than what we wrote
above):
{x \in S: P} = {y \in T: Q} =>
(CHOOSE x \in S: P) = (CHOOSE y \in T: Q)
Importantly, when {x \in S: P} = {}
, the expression CHOOSE x \ in S: P
evaluates to an undefined value.
How does CHOOSE
actually work? TLA+ does not fix an algorithm for CHOOSE
by
design. Maybe it returns the first element of the set? Sets are not ordered, so
there is no first element.
Why should you use CHOOSE
? Actually, you should not. Unless you have no other
choice 🎀
There are two common use cases, where the use of CHOOSE
is well justified:

Use case 1: Retrieving the only element of a singleton set. If you know that
Cardinality({x \in S: P}) = 1
, thenCHOOSE x \in S: P
returns the only element of{x \in S: P}
. No magic. For instance, see: Min and Max in FiniteSetsExt. 
Use case 2: Enumerating set elements in a fixed but unknown order. For instance, see: ReduceSet in FiniteSetsExt.
In other cases, we believe that CHOOSE
is bound to do Program synthesis.
So TLC does some form of synthesis by brute force when it has to evaluate
CHOOSE
.
Determinism: Deterministic. Very much deterministic. Don't try to model
nondeterminism with CHOOSE
. For nondeterminism, see:
Control Flow and Nondeterminism.
Apalache picks a set element that satisfies the predicate P
, but it does not
guarantee the repeatability property of CHOOSE
. It does not guarantee
nondeterminism either. Interestingly, this behavior does not really make a
difference for the use cases 1 and 2. If you believe that this causes a problem
in your specification, open an issue...
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible with the type of x
as expected in P
.
Example in TLA+:
CHOOSE x \in 1..3: x >= 3
\* 3
CHOOSE x \in 1..3:
\A y \in 1..3: y >= x
\* 1, the minimum
CHOOSE f \in [ 1..10 > BOOLEAN ]:
\E x, y \in DOMAIN f:
f[x] /\ ~f[y]
\* some Boolean function from 1..10 that covers FALSE and TRUE
Example in Python: Python does not have anything similar to CHOOSE
.
The closest possible solution is to sort the filtered set by the string values
and pick the first one (or the last one). So we have introduced a particular
way of implementing CHOOSE, see choose.py:
# A fixed implementation of CHOOSE x \in S: TRUE
# that sorts the set by the string representation and picks the head
def choose(s):
lst = sorted([(str(e), e) for e in s], key=(lambda pair: pair[0]))
(_, e) = lst[0]
return e
if __name__ == "__main__":
s = { 1, 2, 3}
print("CHOOSE {} = {}".format(s, choose(s)))
s2 = { frozenset({1}), frozenset({2}), frozenset({3})}
print("CHOOSE {} = {}".format(s2, choose(s2)))
Unbounded universal quantifier
Notation: \A x: P
LaTeX notation:
Arguments: At least two arguments: a variable name and an expression.
Effect: This operator evaluates to a Boolean value. It evaluates to TRUE
,
when every element in the logical universe makes the expression P
evaluate to
TRUE
against the binding [x > e]
. More precisely, we have to consider
only the elements that produced a defined result when evaluating P
.
Neither TLC, nor Apalache support this operator. It is impossible to give operational semantics for this operator, unless we explicitly introduce the universe. It requires a firstorder logic solver. This operator may be useful when writing proofs with TLAPS.
Unbounded existential quantifier
Notation: \E x: P
LaTeX notation:
Arguments: At least two arguments: a variable name and an expression.
Effect: This operator evaluates to a Boolean value. It evaluates to TRUE
,
when at least one element in the logical universe makes the expression P
evaluate to TRUE
against the binding [x > e]
. More precisely, we have to
consider only the elements that produced a defined result when evaluating P
.
Neither TLC, nor Apalache support this operator. It is impossible to give operational semantics for this operator, unless we explicitly introduce the universe. It requires a firstorder logic solver. This operator may be useful when writing proofs with TLAPS.
Unbounded CHOOSE
Notation: CHOOSE x: P
LaTeX notation: CHOOSE x: P
Arguments: At least two arguments: a variable name and an expression.
Effect: This operator evaluates to some value v
in the logical universe
that evaluates P
to TRUE
against the binding [x > v]
.
Neither TLC, nor Apalache support this operator. It is impossible to give operational semantics for this operator, unless we explicitly introduce the universe and introduce a fixed rule for enumerating its elements.
Congratulations! You have reached the bottom of this page. If you want to learn
more about unbounded CHOOSE
, read Section 16.1.2 of Specifying Systems.