# Rel Language Reference

Reference Guide for the Rel language.

## Introduction

This is the *technical reference guide* for the Rel language. For an introduction to Rel see
the Rel Primer and our Rel Concept Guides.

## Values

Every value in Rel is a first-order relation. We write tuples as values separated by commas `(v1, v2, ...)`

, and
write relations as tuples separated by
semicolons (`;`

) and wrapped in curly braces (`{}`

). For example:

`{(1, 2); (3, 4); (5, 6)}`

is a relation with arity 2 and cardinality 3. It can be
visualized as the following table:

1 | 2 |

3 | 4 |

5 | 6 |

`{(1, 2, 3); (4, 5, 6)}`

is a relation with arity 3 and cardinality 2.
It can be visualized as the following table:

1 | 2 | 3 |

4 | 5 | 6 |

`{(1); (2); (3); (4)}`

is a relation with arity 1 (a unary relation) and cardinality 4.
It can be visualized as the following table:

1 |

2 |

3 |

4 |

We typically leave out the parentheses for unary relations, so this relation
can be written as `{1; 2; 3; 4}`

.

The purpose of this first section is mainly to explain what notation we use for values computed by Rel; but the notation is also supported by the Rel language itself.

In Rel, the parentheses of the tuple are in general optional because `,`

is an operator and
it binds stronger than the `;`

operator (see Comma
and Semicolon). Similar to other languages, tuples with
a trailing comma are allowed, as in `{(1,); (2,); (3,); (4,)}`

.

Values in Rel are restricted to first-order relations. This means that relations cannot contain relations. Relations can contain values such as integers, floating-point numbers, fixed-point decimals, characters, dates, datetimes, etc. Many of the types have some structure (such as dates), but are not variable-sized collections. The Rel system is designed to support arbitrary fixed-size Julia types as values.

### Terminology

Term | Explanation |
---|---|

Arity | Number of arguments in a relation. |

Cardinality | Number of tuples in a relation (size). |

Relation | A relation of any cardinality, arity, and with any properties. When we say `relation` , we do not mean that this relation is not also a function, a set, a singleton, etc. |

Function | Relation with a functional dependency from the initial arguments to the last argument. |

Empty relation | Relation of cardinality 0 (any arity). |

Tuple | Ordered list of values. We often identify a relation of cardinality 1 (any arity) with the tuple that it contains, and vice-versa. |

Set | Relation of arity 1 (unary relation). |

Nullary relation | Relation of arity 0. Nullary relations can only have cardinality 0 or 1. Nullary relations can be thought of as representing the boolean values true and false. See the Formulas section for a discussion of how this fits in the language design. |

Unary relation | Relation of arity 1 (a set). |

Binary relation | Relation of arity 2. |

Ternary relation | Relation of arity 3. |

N-ary relation | Relation of arity n. |

Singleton | Relation of cardinality 1 and arity 1. |

IDB relation | IDB is an abbreviation of intensional database. An IDB relation is a relation defined by a computation over IDB and EDB relations. IDB relations could be persisted or not, because they can always be recomputed from data that is already in the database. This corresponds to the definition of a view in SQL. |

EDB relation | EDB is an abbreviation of extensional database. An EDB relation has to be stored in the database because it is not defined by a computation. This corresponds to a table in SQL. |

## Lexical syntax

### Identifiers

An identifier is a sequence of any Unicode letter or underscore, followed by zero or more Unicode letters, underscores, or numbers.

```
Identifier := IdCharInit IdChar*
IdCharInit := [a-zA-Zα-ωΑ-Ω_]
IdChar := [a-zA-Zα-ωΑ-Ω_0-9]
```

For example, valid identifiers are: `x`

, `a1`

, `b1c`

, `θ`

, `φ`

, `Σ`

, `ß`

, `β1`

, `β2`

.

Due to missing features in the parser generator we use, the definition explicitly lists supported ranges of Unicode letters (hence the Greek ranges). This is currently incomplete, so we do not yet support other Unicode letters in identifiers.

### Comments

Rel uses `/* ... */`

for block (multi-line) comments and `// ...`

for end-of-line comments.

### Infix symbols

Infix symbols that are not defined in the Rel libraries are available to be user-defined:
`⊗`

, `⊙`

, `⊕`

, `⊔`

, `⊓`

, `∼`

, `≼`

, `≽`

, `≺`

, `≻`

, `→`

, `←`

, `≈`

.

For example:

```
@inline def (≈)(a, b) = a + 1 = b
def output = 1 ≈ 2
```

Note: the infix `∼`

used in alglib.rel (Unicode 8764) is not the same character as your keyboard’s `~`

(Unicode 126).

These symbols can also be useful as higher-order relational arguments, as this example from `alglib.rel`

shows:

```
@inline
def unary_relation(D, ∼) =
equal(arity[∼], 1)
```

## Declarations

### Definitions

Definitions define a named, computed relation.

```
Declaration :=
Attribute* "def" Id FormalParamsBracket* FormalParamsParen? "=" Expr
Attribute* "def" Id FormalParamsBracket* FormalParamsParen? "{" Expr "}"
```

Example | Description |
---|---|

`def five = 5` | `five` is a unary, singleton relation `{(5,)}` |

`def numbers = {1; 2; 3; 4}` | `numbers` is a unary relation |

`def grandparent = a, b: ∃(t: parent(a, t) and parent(t, b))` | `grandparent` is a binary relation |

`def grandparent(a, b) { ∃(t: parent(a, t) and parent(t, b)) }` | `grandparent` is a binary relation |

`def netsales[x, y, z] = sales[x, y, z] - returns[x, y, z]` | Assuming `sales` and `returns` have arity 4, `netsales` has arity 4. |

Note that in Rel any constant or variable can be used as a relation. For example, in
`def pi = 3.14`

the value of pi is the singleton relation {(3.14,)}.

Definitions are similar to views in SQL, though Rel has fewer restrictions on what computations are supported in a view.

Section Relational Application explains the square brackets syntax.

### Entities

Entity definitions in Rel have the form

```
Declaration :=
Attribute* "entity" entity_name:Id constructor_name:Id FormalParamsBracket* FormalParamsParen? "=" Expr
```

See the Entities concept Guide for an introduction to using entities in Rel.

We can optionally use one of `@hash`

and `@auto_number`

as the `Attribute`

to indicate how entities are constructed.
By default, `@hash`

is used.
`entity_name:Id`

is the name of the kind of entity, and `constructor_name:Id`

is the *constructor*,
used to create and reference the entities.
The relation `Expr`

describes the set of entities, each of which will be considered unique.
The tuples in this relation will be the keys from which the entities are constructed.

The declaration `@hash entity E c = r`

corresponds to `def c = hash[r]`

and `def E = last[c]`

.
Similarly, `@auto_number entity E c = r`

corresponds to `def c = auto_number[r]`

and `def E = last[c]`

.

`hash`

entities are 128-bit and `auto_number`

entities are 64-bit.
In both cases, *do not* make any assumption on the particular representation or ordering of the entities, other than their uniqueness.

## Expressions

### Formulas

Formulas are expressions that evaluate to either `true`

or `false`

. Formulas are the core of
the Rel language. This means that other constructs are compiled to formulas.

Formulas and relation-valued expressions may appear to be distinct types of expressions, but
in Rel `true`

is equivalent to `{()}`

, which is the zero arity relation containing its
only possible value: the empty tuple. Similarly, `false`

is the empty relation, `{}`

, of
arity 0. There are only two relations of arity 0.

`true` | `{()}` |

`false` | `{}` |

Formulas often have free variables (e.g. `x`

and `y`

in `p(x, y) and q(y)`

). Formulas with
free variables are not relations (they are just booleans, or equivalently, arity-0 relations).

#### Application (Atom)

```
Expr := Expr "(" {Expr ","}* ")"
```

The arity of the resulting expression must be zero. For example, in `p(x, y, z)`

, the
arity of `p`

must be three, and all three arguments are necessary. The formula `p(x, y)`

would evaluate to `false`

because there is no binary tuple in `p`

, which has arity 3.

Examples |
---|

`age("Martin", 39)` |

`parent(x, y)` |

`foo()` |

Section Relational Application introduces a syntax with
square brackets that supports partial application of the relation. For example,
`age["Martin"]`

has the value `{(39)}`

.

#### Conjunction

```
Expr := Expr "and" Expr
```

This construct is logical conjunction. Evaluates to `true`

if an only if the left and right
expression evaluates to `true`

.

The arguments of `and`

must be a formula (arity 0).

Examples |
---|

`flying(x) and mammal(x)` |

The comma operator (introduced later) can be used instead of `and`

, but
is more general: it does not require arity-0 arguments as `and`

does. Due to
this, the `and`

syntax can help catch mistakes and disambiguate overloaded
logic. The arity of expression `e1, e2`

entirely depends on the arity of `e1`

and `e2`

(it
can be any arity). On the other hand, the arity of `e1 and e2`

is always 0. If `e1`

and `e2`

are used with an incorrect assumption of their arity, then `and`

will give an error, while
`,`

will not.

The possible uses of `e1 and e2`

are:

`e1` | `e2` | `e1 and e2` |
---|---|---|

`false` | `false` | `false` |

`false` | `true` | `false` |

`true` | `false` | `false` |

`true` | `true` | `true` |

#### Disjunction

```
Expr := Expr "or" Expr
```

Examples |
---|

`flying(x) or floating(x)` |

The possible uses of `e1 or e2`

are:

`e1` | `e2` | `e1 or e2` |
---|---|---|

`false` | `false` | `false` |

`false` | `true` | `true` |

`true` | `false` | `true` |

`true` | `true` | `true` |

Similar to the role of the comma operator for `and`

, the semicolon operator is a
generalization of `or`

(see Section Semicolon (Union and Disjunction)). The semicolon operator is disjunction
(like `or`

) for arity 0, and union for relations of an arity bigger than 0. The section on
the semicolon operator explains how this matches the semantics of `or`

exactly.

#### Implication

```
Expr := Expr "implies" Expr
```

Implication `e1 implies e2`

is defined as `(not e1) or (e1 and e2)`

The possible uses of `e1 implies e2`

are:

`e1` | `e2` | `e1 implies e2` |
---|---|---|

`false` | `false` | `true` |

`false` | `true` | `true` |

`true` | `false` | `false` |

`true` | `true` | `true` |

#### Negation

```
Expr := "not" Expr
```

Examples |
---|

`not mammal(x)` |

Similar to the other logical operators, the argument of `not`

must be a formula (an arity-0 relation).

The possible uses of `not e`

are:

`e` | `not e` |
---|---|

`false` | `true` |

`true` | `false` |

#### Boolean Constant Relations

```
Expr := "true"
| "false"
```

The constants `true`

and `false`

are relations, not scalar values. The constant `true`

is
equivalent to `{()}`

. The constant `false`

is equivalent to the arity-0 relation `{}`

.

Because `true`

and `false`

are relations, there is never a relation persisted in the
database that has `true`

or `false`

occurring as a value in the tuples (otherwise that
relation would not be first-order). The Rel compiler statically eliminates any occurrences
of `true`

and `false`

according the laws of the language.

#### Comparison Operators

```
Expr := Expr "<" Expr
| Expr ">" Expr
| Expr "=" Expr
| Expr "!=" Expr
| Expr "≠" Expr
| Expr "<=" Expr
| Expr "≤" Expr
| Expr ">=" Expr
| Expr "≥" Expr
```

Comparison operators are translated into applications of infinite relations defined in the
standard library. For example, `e1 = e2`

is translated to `eq(e1, e2)`

.

Examples: `x < 5`

, `x > 5`

, `x != 0`

.

### Conditional Expression: If-Then-Else-End

```
Expr := "if" Expr "then" Expr "else" Expr "end"
```

The condition must be a formula (an arity-0 expression).

If the condition is `true`

, then the value of the expression is the `then`

branch. Otherwise
the value is the `else`

branch. The `then`

and `else`

branches can have arbitrary arity.

The semantics of conditional expressions is:

```
|[ if e1 then e2 else e3 end ]| => |[ (e1, e2); (not(e1), e3) ]|
```

Examples:

```
if 1 < 2 then 'a' else 'b' end
```

gives (‘a’,).

```
def p = 1
def q = 2,3
def r = 4,5
def output = if p(1) then q else r end
```

gives
`(2, 3)`

.

```
def output = if p > 1 then q else r end
```

gives
`(4, 5)`

.

```
def p = 1; 2; 3
def output = if x = 1 then 'a' else 'b' end for x in p
```

gives

```
(1, 'a')
(2, 'b')
(3, 'b')
```

### Constants

Constants in Rel are semantically equivalent to singleton relations (arity 1, cardinality 1).
In Rel, the value of *every* expression is a relation, and this includes constant
expressions. For example, the value of the constant `1`

is `{(1)}`

.

Note that `true`

and `false`

(see Section Boolean Constant Relations)
are arity-0 relations.

#### Integer Constants

Examples: `1`

, `2`

, `3`

#### Floating-point Constants

```
Expr := [0-9]+ "." [0-9]* exponent?
| [0-9]+ exponent
| "." [0-9]+ exponent?
decimalDigit := /[0-9]/
decimalDigits := decimalDigit+
exponent := ('e' | 'E') ('+' | '-')? decimalDigits
```

Examples:

`0.0`

`3.14`

`1.5e4`

`1e10`

`5e-4`

`5e+4`

`.5e4`

#### String Constants

Strings are enclosed by double quotes (`"`

). Multi-line strings are allowed, and enclosed by three double quotes (`"""`

).
A quote can be escaped as `\"`

, and the `\`

character can be specified as `"\\"`

.
There is no need to escape single quotes inside triple-quoted strings, unless they are next to the triple quotes.

Examples:

```
def f1 = "abc"
def f2 = """
a,b,c
1,2,"three"
"""
def f3 = "\""
def output = i, c : char[f3, i, c]
```

`f3`

will have length 1, and the output will be `{1, "}`

.

Characters are Unicode. `char["中文例子", 2] = '文'`

.

#### Character Constants

Enclosed by single quotes.

Examples: `'c'`

, `'文'`

.

#### Date Constants

Date constants use the ISO 8601 extended format.

```
Expr := [0-9][0-9][0-9][0-9] "-" [0-9][0-9] "-" [0-9][0-9]
```

Examples: `2021-01-01`

, `2100-04-03`

If a `Date`

does not exist (for example `2021-02-30`

or `2020-25-01`

) then a warning is
reported and the value of the constant is the unary empty relation.

#### DateTime Constants

DateTime constants use the ISO 8601 extended format. A timezone is required.

Example: `2021-01-01T21:00:00+00:00`

```
Expr := Date "T" Time Timezone
Date = [0-9][0-9][0-9][0-9] "-" [0-9][0-9] "-" [0-9][0-9]
Time = [0-9][0-9] ":" [0-9][0-9] ":" [0-9][0-9]
Timezone = "Z"
| "+" [0-9][0-9] ":" [0-9][0-9]
| "-" [0-9][0-9] ":" [0-9][0-9]
```

If a `DateTime`

is not valid then a warning is reported and the value of the constant is the
unary empty relation.

### Relational Abstractions

#### Bindings

Bindings are the mechanism by which variables are introduced in Rel. It is often useful to
immediately specify the expected domain of the new variables, and the binding syntax
supports that with `where`

and `in`

.

```
Bindings := FormalParams
Bindings := FormalParams "where" Expr
FormalParams := FormalParam
FormalParams := "(" FormalParams ")"
FormalParams := FormalParams "," FormalParams
FormalParam := FormalId
| FormalId "in" Expr
| FormalId "∈" Expr
| Constant
FormalId := Id
| Id "..."
```

Bindings are not expressions themselves, but are used in many language constructs that are expressions. Because the syntax for bindings is somewhat elaborate, it is useful to understand it separately from the language constructs in which it is used.

Examples of bindings:

`x`

- Introduces a single variable named
`x`

- Introduces a single variable named
`x, y, z`

- Introduces three variables:
`x`

,`y`

,`z`

- Introduces three variables:
`x in A`

- Introduces a single variable
`x`

whose domain is restricted to`A`

- Introduces a single variable
`x ∈ A`

- Equivalent to the previous example.

`x where A(x)`

- Equivalent to the previous example.

`x ∈ Int`

- Introduces a single variable
`x`

, with domain`Int`

. Domains do not have to be finite.

- Introduces a single variable
`z in {1; 2; 3; 4; 5}`

- Domains can also use literal relations

`z in 1`

- A domain can be a singleton (or even empty, but that’s not often useful)

`t where team(t) and soccer(t)`

- A domain can be a more complex logical formula.

`x in A, y in B, z in C`

or equivalently`x ∈ A, y ∈ B, z ∈ C`

- Introduces three variables, where
`x`

has domain`A`

,`y`

has`B`

,`z`

has`C`

.`A`

,`B`

, and`C`

need to have arity 1.

- Introduces three variables, where
`x ∈ Red, y ∈ Green where edge(x,y)`

`in`

and`where`

can be combined.

`x, y where edge(x,y) and Red(x) and Green(y)`

- Equivalent to the previous example.

`x, y where edge(x, y)`

- Introduces two variables
`x`

and`y`

in the`edge`

domain.

- Introduces two variables

If the domains of variables introduced are independent, then `in`

can be used. If the domain
involves multiple variables, then `where`

must be used.

For the `where`

case, the `Expr`

must be arity 0 (a formula). The formula can use all
variables in the binding.

For the `in`

/`∈`

case, the expression must have arity 1 if a normal variable is used.
Varargs are supported though, as in `x... in {(1,2,3)} : true`

. In the varargs case, the arity
of the expression (3, in our example) determines how many variables the vararg represents.

#### Relational Abstraction

Relational abstraction defines a relation without a name. This is by far the most important construct in the language. We will introduce some syntactic variations next, but they all translate into relational abstractions.

```
Expr := Bindings ":" Expr
```

The arity of a relational abstraction is the number of variables in the bindings + the arity of the expression. If the expression is a formula (arity 0), then only the variables from the bindings remain.

In the following examples, we use:

`def p = {1; 2; 3}`

`def q = {2; 3; 4}`

`def r = {(1, "a"); (2, "b"); (3, "c")}`

`def abc = {"a"; "b"; "c"}`

`def parent = {("John", "Mary"); ("Mary", "Felix"); ("Felix", "George")}`

It is common for a relational abstraction to have a formula as the expression. This is similar to a Datalog rule, but without a name for the relation that is defined. The following examples all use a formula.

`x: p(x) and q(x)`

- Unary relation (one variable
`x`

) - Value:
`{2; 3}`

- Unary relation (one variable
`x: p(x) or q(x)`

- Unary relation
- Value:
`{1; 2; 3; 4}`

`x, s: q(x) and r(x, s)`

- Binary relation
- Value:
`{(2, "b"); (3, "c")}`

`s: exists(x: q(x) and r(x, s))`

- Unary relation. The
`x`

is existentially quantified in the body of relational abstraction. - Value:
`{"b"; "c"}`

- Unary relation. The
`x, y: p(x) and q(y)`

- Binary relation
- This is the Cartesian product of
`p`

and`q`

, which is`{(1,2); (1,3); (1,4); (2,2); (2,3); (2,4), (3,2); (3,3); (3,4)}`

`x, t, y: parent(x, t) and parent(t, y)`

- Ternary relation
- Value:
`{("John", "Mary", "Felix"); ("Mary", "Felix", "George")}`

`x, y: parent(x, t) and parent(t, y)`

- This is an error because variable
`t`

is never introduced.

- This is an error because variable
`x, y: ∃(t: parent(x, t) and parent(t, y))`

- Value:
`{("John", "Felix"); ("Mary", "George")}`

- Value:
`x: Int(x) and x > 0`

- Value is the infinite relation
`{1; 2; 3; 4; ...}`

- Value is the infinite relation
`x, y: x = y`

- Binary relation of all values that are equal, e.g
`(1,1)`

and`('a', 'a')`

- Binary relation of all values that are equal, e.g
`x: x = 1`

- Singleton relation
`{(1,)}`

- Singleton relation
`x: true`

- Infinite unary relation (e.g. contains
`1`

, and`"abc"`

)

- Infinite unary relation (e.g. contains
`x: false`

- Empty, unary relation

`x...: p(x...) and q(x...)`

- Varargs are explained later, but hopefully they are intuitively clear. In this case,
`x...`

will only represent a single variable (because`p`

and`q`

have arity 1) and the value is`{1; 2}`

.

- Varargs are explained later, but hopefully they are intuitively clear. In this case,
`{x: p(x) and q(x)}`

- As discussed later, the only purpose of curly braces and parentheses is to indicate precedence. Sometimes a relational abstraction is more clear enclosed in curly braces. It does not change the semantics in any way.

The previous examples all use formulas for the body of the relational
abstraction. Expressions with arity bigger than 0 are allowed as well though. This makes it
a bit similar to a lambda, although it is important to keep in mind that the value is not
restricted to functions: the expression can compute more than one value. Using the same
sample data for `p`

, `q`

, `r`

and `parent`

as before:

`x, y: x + y`

- Ternary, infinite relation
- Example tuples:
`(1, 1, 2)`

,`(2, 3, 5)`

,`(3.5, 1.2, 4.7)`

`x: x + 1`

- Binary, infinite relation
- Example tuples:
`(1, 2)`

,`(1.0, 2.0)`

,`(3.14, 4.14)`

`x in Int: x + 1`

- Binary, infinite relation
- Example tuples:
`(1, 2)`

,`(2,3)`

,`(3,4)`

`x where Int(x): x + 1`

- Equivalent to the previous example.

A few slightly more advanced examples without sample data:

`p: today - date_of_birth[p]`

- Binary relation of person and their age. The arity is 1 (for
`p`

) plus 1 for the body.

- Binary relation of person and their age. The arity is 1 (for
`x: sum[v: scores(x, v)]`

- Sum scores
`v`

, grouped by`x`

- Note that there are two relational abstractions here: one outermost, and one as a parameter to sum.
- Aggregations are explained later in much more detail.

- Sum scores
`mean[x, y: (actual[x, y] - prediction[x, y]) ^ 2]`

- Compute the
`mse`

for a machine learning solution. Note that`mse`

is also defined in the standard library, which can be used as`mse[actual, prediction]`

.

- Compute the
`x: edge(x, x)`

- Unary relation of all nodes that have a self edge.

`∃(c ∈ course: ∀(s ∈ student: enrolled(c, s)))`

- The arguments to
`∃`

and`∀`

are relational abstractions.

- The arguments to
`sum[date store_nbr item_nbr: train_unit_sales[date, store_nbr, item_nbr]]`

- Sum up the sales for every unique
`(date, store_nbr, item_nbr)`

triple in the`train_unit_sales`

relation.

- Sum up the sales for every unique

Bindings can also use constants as an abbreviation for having to introduce a separate variable. Examples of constants in a binding of a relational abstraction:

`x, 1: abc(x)`

- Value:
`{("a", 1); ("b", 1); ("c", 1)}`

- Value:
`def count_abc = sum[x, 1: abc(x)]`

- Sums the last argument of all tuples, which all are
`1`

, so this defines the count of the number of tuples in`abc`

, which is`3`

. - The standard library defines
`count`

in this way generically, and it can be used as`count[abc]`

in this example.

- Sums the last argument of all tuples, which all are
`1: p(_)`

- Value:
`1`

(given that`p`

is not empty) - Note that there does not need to be any explicit variable at all in the left-hand-side of the relational abstraction.
- The underscore expression is syntactic sugar for an anonymous variable, see Section Underscore.

- Value:

#### Inverse Relational Abstraction

Relational abstractions are often used as a parameter to higher-order abstractions, such as
aggregations. In aggregations, typically the most important thing to see first is what value
is being aggregated. To make this easier to see immediately, the bindings and expression of
a relational abstraction can be inverted. The `:`

becomes a `|`

(or the keyword `for`

) in
this case. This syntax is designed to mimic the common mathematical notation.

```
Expr := Expr "|" Bindings
Expr := Expr "for" Bindings
```

This is purely a syntactic convenience: `e | b`

is equivalent to `e for b`

which is equivalent to `b: e`

.

Examples:

`x+1 | x in {1; 2; 3}`

- Value
`{(1,2); (2,3); (3,4)}`

- Equivalent to
`x in {1; 2; 3}: x+1`

- Value
`x+1 for x in {1; 2; 3}`

- Equivalent to the previous example.

`x^2 for x in {1; 2; 3}`

- Value
`{(1,1); (2,4); (3,9)}`

- Value
`{x^2 | x ∈ {1; 2; 3}}`

- Equivalent to the previous example.
- Similar to normal relational abstraction, curly braces can be used if that is more clear.

`x^2, x^3 for x in {1; 2; 3}`

- The comma operator is explained in more detail later, but hopefully it is clear that
the initial expression is now a tuple. The value of the entire expression is a ternary
relation
`{(1,1,1); (2,4,8); (3,9,27)}`

- The comma operator is explained in more detail later, but hopefully it is clear that
the initial expression is now a tuple. The value of the entire expression is a ternary
relation
`(x^2, x^3) for x in {1; 2; 3}`

- Equivalent to the previous example. Parentheses are used for precedence and have no other semantic here.

To users familiar with list comprehension in programming languages such as Python, Julia,
and Haskell it may appear as strange that the variables introduced in the comprehension
(e.g. `x`

in `x^2 for x in {1; 2; 3}`

) are part of the result. In these programming
languages, the value of the comprehension is a list of the value before the `for`

.

The distinction with Rel is that in Rel every value is a relation, and a relation is a
**set** of tuples. This means that duplicates are eliminated. For some computations this is
the intent, but often is not. Some examples to illustrate this:

`age[p] for p in employee`

- This is a binary relation of
`Person`

and`Int`

, not a unary relation of ages (which would contain every age value only once).

- This is a binary relation of
`max[age[p] for p in employee]`

- This is the previous example used as an argument to the
`max`

aggregation. The result is a singleton of the age of the oldest employee. Note that for this particular aggregation it does not matter whether age values are kept for all employees.

- This is the previous example used as an argument to the
`sum[age[p] for p in employee]`

- The same example, but now in a
`sum`

aggregation. Here it is very important that even if two different persons`p1`

and`p2`

have the same age, the age of both persons contributes to the sum. This is ensured by having`p`

be a part of the relation that is being aggregated, and we get the intended result.

- The same example, but now in a

Note that `max`

(resp. `sum`

) computes the maximum (resp. sum) of the last argument of the tuples in the relation,
which can have any arity.

The next section introduces the “from” language construct, which does **not** keep the variables
introduced, more akin to comprehension in programming languages.

#### “From” Expressions

From expressions are similar to relational abstractions, with one important distinction: the
variables introduced by the binding are **not** part of the resulting value. The arity of
the expression is entirely determined by the expression.

```
Expr := Expr "from" Bindings
```

The key difference between `for`

(introduced in the previous section) and `from`

(introduced
here) with some examples:

Example | Description |
---|---|

`x+1 for x in {1; 2; 3}` | Binary relation `{(1,2); (2,3); (3,4)}` |

`x+1 from x in {1; 2; 3}` | Unary relation `{2; 3; 4}` |

`x, x+1 from x in {1; 2; 3}` | Binary relation `{(1,2); (2,3); (3,4)}` |

`x, x+1 for x in {1; 2; 3}` | Ternary relation `{(1, 1,2); (2, 2,3); (3, 3,4)}` |

The third example shows that `from`

can be used in a similar way as the `for`

construct by
explicitly including the variables that introduced in the expression.

Expression | Value |
---|---|

`x+1 from x in {1; 2; 3}` | `{2; 3; 4}` |

`x^2 from x in {1; 2; 3}` | `{1; 4; 9}` |

`{x^2 from x in {1; 2; 3}}` | `{1; 4; 9}` |

`x^2, x^3 from x in {1; 2; 3}` | `{(1,1); (4,8); (9,27)}` |

`(x^2, x^3) from x in {1; 2; 3}` | `{(1,1); (4,8); (9,27)}` |

The `from`

construct is particularly useful if auxiliary variables are needed in some bigger
definition. Typically this comes up in the domain of a binding. For example, suppose that we
need to define a binary relation of a person `x`

and the name of the father of `x`

. The
problem is that we need to find the father of `x`

, but the father is not part of the
actual relation.

`def fathers_name = x where exists(f: father(x, f)): name[f]`

(wrong)- This definition is incorrect: the variable
`f`

is introduced with an existential quantifier in the domain of`x`

, but it cannot be used outside of that scope, so`name[f]`

will trigger a compilation error that`f`

is undefined.

- This definition is incorrect: the variable
`def fathers_name = name[f] | x: exists(f: father(x, f))`

(wrong)- The inverse relational abstraction syntax has the exact same problem. It will given an
error that
`f`

is not in scope in`name[f]`

.

- The inverse relational abstraction syntax has the exact same problem. It will given an
error that
`def fathers_name = x: {name[f] from f where father(x, f)}`

(correct)- This correct example introduces the variable
`f`

in a separate`from`

. The curly braces are optional, but may be helpful to understand the precedence and scope of`f`

.

- This correct example introduces the variable
`def fathers_name = x, name[f] from x, f where father(x, f)`

(correct)- If what is actually computed gets confusing, it can also help to just
explicitly list the desired result entirely, in this case
`x, name[f]`

. The variable`x`

must now be introduced in the`from`

binding (otherwise it would be undefined).

- If what is actually computed gets confusing, it can also help to just
explicitly list the desired result entirely, in this case
`def fathers_name[x] = name[father[x]]`

(correct)- In this example we actually did not need the variable
`f`

at all, so it could also have been written this way. The logic is not always this straightforward, though.

- In this example we actually did not need the variable

#### From-Where

To make Rel more accessible to users who are familiar with LINQ or SQL, Rel supports a similar syntax for queries:

```
Expr := Expr "from" Bindings
```

Although `where`

does not occur in this grammar rule, it is part of `Bindings`

and leads to a syntax very similar to LINQ (Language-Integrated Query).

s.name from s in students where s.age > 12 and s.age < 20 |

p.partkey from p in part where like[p.name, '@@1%'] |

This example uses composite relations (e.g., `s.age`

and `p.name`

). The semantics of relational
composition, also called navigation, are described in more detail below.

#### Overview of Relational Abstraction Alternatives

The previous sections introduced a few variants of relational abstractions, and it may be hard to remember the variants and when to use which one. When choosing how to write an abstraction, the first thing to consider is whether it is more readable to first introduce variables or state the value that is computed. This splits the options in two categories with only a few alternatives.

The following examples use `p = {1; 2; 3}`

Abstractions that have `Bindings`

first and `Expr`

second:

```
x in p: x+1 = {(1, 2); (2, 3); (3, 4)}
x in p where x < 3: x+1 = {(1, 2); (2, 3)}
```

Abstractions that have `Expr`

first and `Bindings`

second:

```
x+1 | x in p = {(1, 2); (2, 3); (3, 4)}
x+1 for x in p = {(1, 2); (2, 3); (3, 4)}
x+1 from x in p = {2; 3; 4}
x+1 from x in p where x < 3 = {2; 3}
```

The first and second are identical. The third variant is more SQL-like and lists the computed value separately.

### Relational Application

```
Expr := Expr "[" {Expr ","}* "]"
```

The square bracket syntax is used to project and select a slice of the relation. It is related to *currying* in functional languages in the sense that relations can be partially applied.

As opposed to application syntax with parentheses, the number of parameters can be less than the arity of the relation with relational application. For example, for a binary relation `p`

, only applications with two arguments are valid, e.g `p(x, y)`

. Normal application with parentheses is a formula, or equivalently a nullary relation. For binary `p`

, an example of a relational application is `p[x]`

, mimicking the syntax of arrays and matrix indexing from other languages. For a binary relation `p`

, the value of `p[x]`

is a unary relation.
By doing so the part of the expression that gets evaluated will not be returned.
This behavior is useful to access attributes/information that are stored in a key-value format.

We can use this syntax to define relations

```
def name[1] = "Noether", "Emmy"
def name[2] = "Hopper", "Grace"
def name[3] = "Curie", "Marie"
def age[1] = 53
def age[2] = 85
def age[3] = 66
```

and to query them

Example | Description |
---|---|

`name[2]` | `{"(Hopper", "Grace")}` |

`name[3, "Curie"]` | `{"Marie"}` |

`name[x-1 from x in {1;2}]` | `{("Noether", "Emmy")}` |

`(name[p], age[p] from p)[_]` | `{("Emmy", 53); ("Grace", 85); ("Marie", 66)}` |

`(age[p], name[p] for p)[(x : x%2 = 1), (x: x>60)]` | `{("Curie", "Marie")}` |

(In the last example, note that we have placed `age`

first, so we can filter it with `x : x > 60`

after filtering for `p`

.)

### Comma (Cartesian Product and Conjunction)

The comma operator is technically a Cartesian product operator, but when applied to singleton values, then it easier to understand as a tupling operator.

```
Expr := Expr "," Expr
```

The comma operator is used in the following examples:

Example | Description |
---|---|

`def p = 1, 2` | Singleton, binary relation (tuple) |

`1, 2, 3` | Ternary tuple |

`(1, 2, 3)` | Also a ternary tuple |

`{(1, 2, 3)}` | Also a ternary tuple |

`x^2, x^3 from x in {1; 2; 3}` | Binary relation (`x` is not in result) |

The comma operator can also be used in some perhaps surprising ways:

`def soccer_players[t] = players[t], soccer(t)`

- The comma serves here as an additional condition on the value of soccer players,
without having to write out the logic as the point-wise variant
`def soccer_players(t, p) = players(t, p) and soccer(t)`

.

- The comma serves here as an additional condition on the value of soccer players,
without having to write out the logic as the point-wise variant
`def order_price[pos] = order[pos, :price], not exists order[pos, :load_errors]`

- Similarly, the comma introduces a filter here, skipping rows from a CSV file with any errors.

`def count[R] = sum[{R, 1}]`

- Every tuple of R is extended with a 1, so that computing a
`sum`

aggregation actually computes the count.

- Every tuple of R is extended with a 1, so that computing a

Note that comma is really an operator and there is no dedicated syntax for tuples in the
language. For example, `(1, 2, 3)`

is an expression that uses parentheses (`Expr := "(" Expr ")"`

) and two applications of the comma operator. Tuples never occur as arguments of tuples
in a relation. For example, `((1, 2), 3)`

is the ternary relation `{(1, 2, 3)}`

, and `(1, 2), (3, 4)`

is the arity 4 relation `{(1, 2, 3, 4)}`

.

The comma operator is equivalent to `and`

when its arguments are restricted to relations of
arity 0. To illustrate this, here is the table of all possible arguments of arity-0 relations:

Left | Right | `Left , Right` |
---|---|---|

`{}` | `{}` | `{}` |

`{()}` | `{}` | `{}` |

`{}` | `{()}` | `{}` |

`{()}` | `{()}` | `{()}` |

Unfortunately the comma character is overloaded by the comma operator and the arguments of applications. In the context of an application, there is a semantic difference whether the comma is interpreted as a comma operator or separator of arguments.

`foo[x: y, z]`

In the absence of precedence rules, this example is ambiguous. It can be interpreted as either

`foo[{x: y}, z]`

or`foo[{x: y, z}]`

Independent of what the default interpretation is, it is advisable to be clear about the intent when any commas and relational abstractions are involved.

The comma operator binds stronger than relational abstraction, which means that

`foo[{x: y, z}]`

is the chosen representation in the absence of parentheses. If this is written without curly braces, then a warning will be reported [TODO].As a general guideline, unless there is only a single relational argument (as in

`sum`

,`count`

etc), it is best to always use curly braces or separate the arguments. For example, if the intent is to invoke`foo`

with two arguments, then use either`foo[x: y][z]`

or`foo[{x: y}, z]`

.

`foo[a, b: c, d]`

- A similar problem exists in the bindings of the abstraction, because a comma is also
used there to separate formal parameters. In this case, the possible interpretations
are
`foo[{a, b: c, d}]`

,`foo[{a, b: c}, d]`

,`foo[a, {b: c}, d]`

,`foo[a, {b: c, d}]`

. Similar to the previous example, precedence rules select`foo[{a, b: c, d}]`

. Similarly, the recommendation is to clarify the intent using curly braces.

- A similar problem exists in the bindings of the abstraction, because a comma is also
used there to separate formal parameters. In this case, the possible interpretations
are
`sum[R, 1]`

- In this incorrect example,
`sum`

is required to have two arguments (which is not the case). The first argument is the relation`R`

, and the second is the number`1`

. The intent of the logic is to count the number of tuples in`R`

by appending`1`

to all its tuples and then applying`sum`

to the relation. To achieve this, the relation argument must be put in parentheses or curly braces:`sum[{R, 1}]`

.

- In this incorrect example,

As a general rule, if a relation is used as an argument of an application, then it should be
wrapped in `{...}`

unless it is the only parameter and no commas are involved.

We are planning to introduce warnings for cases where the interpretation is ambiguous, which should hopefully address the confusion that arises from this. The warning would trigger for an expression that has a comma as the right-most operand (i.e. not written in curly braces or parentheses).

### Semicolon (Union and Disjunction)

The semicolon is a generalized union operator.

```
Expr := Expr ";" Expr
```

Example | Normalized |
---|---|

`p(x); q(x)` | `p(x) or q(x)` |

`p;q` | `x: p(x) or q(x)` |

`1;3` | `x: x=1 or x=3` |

`{};p` | `p` |

The semicolon operator is equivalent to `or`

when its arguments are restricted to relations
of arity 0. To illustrate this, here is the table of all possible arguments of arity-0 relations:

Left | Right | `Left ; Right` |
---|---|---|

`{}` | `{}` | `{}` |

`{()}` | `{}` | `{()}` |

`{}` | `{()}` | `{()}` |

`{()}` | `{()}` | `{()}` |

The semicolon operator does not require its arguments to have the same arity. The compilation of expressions without a single arity is not entirely implemented yet though.

### Precedence

Precedence can be indicated with either curly braces or parentheses. The two are exactly equivalent.

```
Expr := "{" Expr "}"
Expr := "(" Expr ")"
```

For readability, it can be helpful to always use `{...}`

for non-singleton relations, and
`(...)`

for singleton relations (scalars). For example, `(x + y) * z`

vs `{x, y, z: p(x, y, z)}`

.

All binary operators are left-associative unless otherwise stated below.

From highest precedence to lowest:

`_`

, variables, qualified names, literals,`()`

,`{}`

`.`

- relational application
`^`

(right assoc)- unary
`-`

`/`

,`%`

,`*`

,`÷`

,`×`

,`⊗`

,`⊙`

`-`

,`+`

,`⊕`

,`∩`

,`⊓`

`∪`

,`⊔`

,`=`

,`!=`

,`≈`

,`∼`

,`≠`

,`<`

,`>`

,`<=`

,`≤`

,`>=`

,`≥`

,`⊆`

,`⊇`

,`⊂`

,`⊃`

,`≼`

,`≽`

,`≺`

,`≻`

,`→`

,`←`

`¬`

,`not`

`forall`

,`∀`

,`exists`

,`∃`

`∧`

,`and`

`∨`

,`or`

`⇒`

(right assoc),`implies`

(right assoc),`⇐`

`≡`

,`⇔`

,`⊻`

,`≢`

,`⇎`

,`xor`

,`iff`

`<:`

,`:>`

`<++`

,`++>`

`,`

`;`

### Relation Literals

Relation literals do not have a dedicated syntax, but are by convention written using curly braces (outer), parentheses (around tuples), comma (in the tuples) and semicolon (between the tuples).

Example | Description |
---|---|

`{1}` | Singleton relation (arity 1, cardinality 1) |

`{1; 2; 3; 4}` | Unary relation (arity 1, cardinality 4) |

`{(1, 2); (3, 4); (5, 6)}` | Arity 2, cardinality 3 |

`{1; 2; 1; 2}` | Arity 1, cardinality 2 (duplicates are eliminated) |

`{()}` | Unit relation (cardinality 1, arity 0) |

`{}` | Empty relation (cardinality 0, any arity) |

Note that the arguments do not have to be constants, so the relation literal is not a relation constant. It can contain variables, and the tuples do also not have to be written explicitly with scalar variables.

For example, `{(friend, 1); (family, 2); (partner, 3)}`

for binary `friend`

, `family`

and
`partner`

has arity 3 and unknown cardinality. It can explicitly be defined as

```
x, y, v:
(friend(x, y) and v = 1) or
(family(x, y) and v = 2) or
(partner(x, y) and v = 3)
```

### Composition (Navigation)

```
Expr := Expr "." Expr
```

The semantics of `R.S`

is `x..., y... : exists(t: R(x..., t) and S(t, y...))`

. Note that the
arguments must have at least arity 1.

For specific cases, the semantics is simplified a little. In particular,`x.R`

is `R[x]`

.

Example | Normalized |
---|---|

`parent.parent` | `x, y: ∃(t: parent(x, t) and parent(t, y))` |

`t.players` | `p: players(t, p)` |

`t.players.age` | `v: ∃(p: players(t, p) and age(p, v))` |

`brand_name."Tesla"` | `b: brand_name(b, "Tesla")` |

### Restriction

```
Expr := Expr "<:" Expr
Expr := Expr ":>" Expr
```

Prefix join (or prefix restriction) `R <: S`

is syntax for `prefix_join[R, S]`

defined in
the standard library.

Suffix join (or suffix restriction) `R :> S`

is syntax for `suffix_join[R, S]`

defined in
the standard library.

### Override

```
Expr := Expr "<++" Expr
Expr := Expr "++>" Expr
```

Left override operator `R <++ S`

is syntax for `left_override[R, S]`

defined in the standard
library.

Right override operator `R ++> S`

is syntax for `right_override[R, S]`

defined in the
standard library.

### Arithmetic Operators

```
Expr := Expr "+" Expr
| Expr "-" Expr
| Expr "*" Expr
| Expr "/" Expr
| Expr "%" Expr
| Expr "÷" Expr
| Expr "^" Expr
```

### Underscore

```
Expr := "_"
```

Underscore is a variable without a name. It desugars into the relation `x: true`

, which is the infinite relation of all values. After further rewriting,
this typically becomes an existential quantification of a variable with a unique
name. Underscore is mostly useful if you do not care about certain arguments of
relation and want to avoid introducing a named variable and quantifier.

For example:

```
// Unary relation of all orders with a line item:
o: line_item(o, _)
// There does not exist an order for customer c:
not(order_customer(_, c))
// Count the number of distinct values of a function or relation `f`:
count[f[_]]
```

### Splat

```
Expr := Expr "..."
```

### Existential Quantification

```
Expr := "exists" Expr
| "∃" Expr
```

The argument of existential quantification can be an expression of any arity. This is because the expression includes the variables that are quantified over (they are not syntactically associated with the quantifier). For example:

```
def foo = 1;2;3
def output = exists(x, y : foo(x), foo(y), x > 2)
```

which evaluates to `true`

.

The value of an existential quantification is
`false`

if the argument is the empty relation, and `true`

otherwise.

Examples |
---|

`exists(p: person(p) and age[p] > 99)` |

`exists(centenarian)` |

`∃(centenarian)` |

`c: course(c) and not ∃(s: enrolled(c, s))` |

`c: course(c) and not ∃(enrolled[c])` |

`def grandparent(a, b) = exists(t: parent(a, t) and parent(t, b))` |

### Universal Quantification

```
Expr := "forall" "(" Bindings ":" Expr ")"
| "∀" "(" Bindings ":" Expr ")"
```

Universal quantification is `true`

if the range (the `Expr`

expression) is `true`

for the entire
domain (the domain specified in the bindings). For the empty domain,
universal quantification evaluates to `true`

.

The range (right-hand side) of universal quantification must be a formula (that is, an arity-0 relation).

Examples | Description |
---|---|

`forall(p in person: age[p] > 99)` | Are all persons older than 99? |

`∀(p in person: centenarian(p))` | Are all persons centenarians? Here we use a Unicode “forall” quantifier. |

`exists(c in course: forall(s in student: enrolled(c, s))` | Does there exist a course where all students are enrolled? |

`∃(c in course: ∀(s in student: enrolled(c, s))` | Same, with Unicode quantifiers. |

`c: course(c) and ∀(y in year: ∀(s where class[s] = y: enrolled(c, s)))` | Courses that have, for every year, all students from that year enrolled. |

`c: course(c) and ∃(y in year: ∀(s where class[s] = y: enrolled(c, s)))` | Courses that have all students from one year enrolled. |

Writing `forall x,y,... where E1(x,y,...) : E2`

is preferred to `forall x,y,... : E1(x,y,...) implies E2`

.

### Aggregation

There is no dedicated syntax for aggregations. Instead, there is one higher-order native
called `reduce`

that is used to define various aggregations in `stdlib.rel`

.

A few examples of aggregations:

Example | Description |
---|---|

`sum[salary]` | Sum all salaries |

`sum[p in employee: salary[p]]` | Sum employee salaries |

`Σ[salary[p] | p ∈ employee]` | Sum employee salaries |

`Σ[employee.salary]` | Wrong: duplicates eliminated |

`d in dept: Σ[salary[p] | p ∈ member[d]]` | Sum salaries by department |

`d in dept: Σ[d.member.salary]` | Wrong: duplicates eliminated |

`count[employee]` | Count number of employees |

`d in dept: count[d.member]` | Count number of employees by department |

`d, v: dept(d) and count[d.member](v)` | Aggregations can be used in formulas |

`max[age[p] | p ∈ person]` | Max age of all persons |

`max[age[p] for p in person]` | Max age of all persons |

`max[t.member.age] | t ∈ team` | Max age by team |

`max[Σ[salary[p] for p in member[t]]] for t in team` | Max sum of salaries by team |

`argmax[Σ[salary[p] for p in member[t]] for t in team]` | Team with max sum of salaries |

`argmin[Σ[salary[p] for p in member[t]] for t in team]` | Team with min sum of salaries |

`t: max[age[p] for p in member[t]]` | Max age by team (again) |

### Variable Scope

All variables have to be explicitly introduced to clarify their scope.

There is no implicit quantification as in Datalog.

All variables are introduced by `Bindings`

(although definitions support a subset of
`Bindings`

, `FormalParams`

).

A local scalar variable `x`

shadows relation definitions, schema, and natives with name `x`

.

For example, in the definition of `foo`

, the local variable `nation`

shadows the previous
definition of nation.

```
def nation = ...
def foo = nation: ...
```

Variable scope is static, not dynamic. This means that the local variable `nation`

only
shadows variables that are lexically in its scope.

## Other topics

### Types

To learn more about for types in Rel, see the Data Types Reference. The Rel standard library includes a number of predicates that test for a given type, such as the Number relation and the ones that follow.

### Mixing Arities and Types

The value of a Rel expression is a set of tuples that can have different arities and types. For example:

```
def myrel = 1; 2 ; "a" ; ("b", "c")
```

Technically, `myrel`

is three separate relations, sharing the same name:
`{1; 2}`

is unary and has integers; `{"a"}`

is unary and has strings,
and `{("b", "c")}`

is binary and has strings.
The language treats them as a single relation. For example, `first[x]`

is `{1; 2; "a"; "b"}`

.

### Integrity Constraints

Rel supports expressive Integrity Constraints, written using the Rel language itself. See the IC Concept Guide for an introduction to ICs, and the Integrity Constraint Reference for technical IC reference.

### Modules

Rel modules, which group together definitions and can be parameterized by relations, are relations as well. See the Rel Modules Guide.

### EDBs: `insert`

and `delete`

EDB relations can be persisted and updated with the transaction control relations `insert`

and `delete`

.
See the guide on Working with EDB Relations.

### Rel Libraries

When a database is created, a number of libraries are installed automatically as part of the database, including the Rel Standard Library (stdlib). See the Library Overview for more details.