An Algebraic Query Language

Author

Ken Pu

An Algebraic Query Language

# 1
read: Section 2.4: An Algebraic Query Language (section)
model: flash
rules: presentation short bullet

Distinguish an abstract query language with an actual query lagnuage like SQL.

  • Relational Algebra: An Abstract Language

    Relational algebra constitutes a specialized algebra for data manipulation within the relational model. It provides a set of operations to construct new relations from given relations. When these given relations represent stored data, the newly constructed relations serve as answers to queries.

  • Its Role in Query Processing

    While not typically employed directly by users in commercial database management systems (DBMS) today, relational algebra holds a central position. Queries expressed in languages like SQL are often translated internally into relational algebra. This translation is fundamental for query optimizers, which seek to find the most efficient execution plan for a given query.

  • Distinction from Commercial Query Languages (SQL)

    Relational algebra is an abstract, foundational language, whereas SQL is the prominent commercial query language. SQL queries frequently correspond to “syntactically sugared” expressions of relational algebra. The constrained yet powerful nature of relational algebra, compared to general-purpose programming languages, offers significant advantages in terms of programming ease and the potential for compiler optimization.

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Why Do We Need a Special Query Language?

  • Discusses the need for a specialized database query language like relational algebra over conventional programming languages (e.g., C, Java).
  • Explains that relational algebra is intentionally less powerful than general-purpose languages.
  • Highlights the benefits of this limitation: ease of programming and the ability for the compiler to produce highly optimized code.
# 2
read: Section 2.4.1: Why Do We Need a Special Query Language? (section)
model: pro
rules: presentation short bullet

- Discusses the need for a specialized database query language like relational algebra over conventional programming languages (e.g., C, Java).
- Explains that relational algebra is intentionally less powerful than general-purpose languages.
- Highlights the benefits of this limitation: ease of programming and the ability for the compiler to produce highly optimized code.

  • The Sufficiency of Conventional Languages

    Conventional programming languages, such as C or Java, are Turing complete and can be used to ask any computable question about relations. One can represent a tuple as a struct or an object, and a relation as an array of these elements.

  • The Advantage of a Specialized Language

    The utility of relational algebra stems from the fact that it is intentionally less powerful than a general-purpose language like C or Java. For instance, one cannot write a relational algebra expression to determine if the number of tuples in a relation is even or odd. This limitation provides two significant advantages:

    • Ease of Programming: The language is declarative and focused solely on data retrieval and manipulation, simplifying the task for the programmer.
    • Optimization: The restricted nature of the language allows the database system’s compiler, or query optimizer, to produce highly efficient code for executing the query.
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What is an Algebra?

  • Defines a generic algebra as a system consisting of operators and atomic operands.
  • Uses the algebra of arithmetic as an analogy, where operands are variables and constants, and operators are addition, subtraction, etc.
  • Explains that an algebra allows for building complex expressions by applying operators to operands and other expressions.
# 3
read: Section 2.4.2: What is an Algebra? (section)
model: pro
rules: presentation definition short bullet

- Defines a generic algebra as a system consisting of operators and atomic operands.  Use formal mathematical notation.
- Uses the algebra of arithmetic as an analogy, where operands are variables and constants, and operators are addition, subtraction, etc.
- Explains that an algebra allows for building complex expressions by applying operators to operands and other expressions.

revise: add math notation.

An algebra is a formal system consisting of operands and operators.

Definition: Algebra

An algebra consists of a set of atomic operands, drawn from a domain \(S\), and a collection of operators, \(\mathbf{Op}\). Each operator \(f \in \mathbf{Op}\) is a n-ary function defined on \(S\):

\[ f : S \times S \times S \times S \dots \times S \to S \]

  • Atomic Operands

    These are the fundamental objects upon which operations are performed. In the algebra of arithmetic over the real numbers (\(\mathbb{R}\)), atomic operands include variables, such as \(x\), and constants, such as \(15\).

  • Operators

    An operator is a function that takes one or more operands as input and produces a result, typically in the same domain. A binary operator, such as addition (\(+\)), is a function that maps two operands to a result, e.g., \(+: \mathbb{R} \times \mathbb{R} \to \mathbb{R}\).

  • Expressions

    An algebra allows for the construction of complex expressions by applying operators to atomic operands and other expressions. Parentheses are used to group operators and operands to specify the order of evaluation. For instance, \((x + y) * z\) is a valid arithmetic expression.

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Overview of Relational Algebra

  • Introduces relational algebra’s atomic operands: variables standing for relations and constants which are finite relations.
  • Categorizes the operations of relational algebra into four broad classes:
    1. Usual set operations: union, intersection, and difference.
    1. Operations that remove parts of a relation: selection (rows) and projection (columns).
    1. Operations that combine tuples from two relations: Cartesian product and joins.
    1. Renaming operation: changes the schema (attribute names or relation name).
# 4.1
read: Section 2.4.3: Overview of Relational Algebra (section)
model: pro
rules: short presentation bullet

Focus on just the domain relational algebra.  Focus on the domain is relations. There are infinitely many objects in the domain: every possible instances of every possible relation schema.

The operations of relational algebra can be grouped into four primary classes. These operators take one or more relations as input and produce a new relation as output.

  • The usual set operations

    These include union (\(R \cup S\)), intersection (\(R \cap S\)), and difference (\(R - S\)). They are applied to relations as if the relations were sets of tuples.

  • Operations that remove parts of a relation

    Selection (\(\sigma\)) eliminates rows, or tuples, that do not satisfy a specified condition. Projection (\(\pi\)) eliminates columns, or attributes, that are not in a specified list.

  • Operations that combine tuples from two relations

    The Cartesian product (\(R \times S\)) combines tuples from two relations in all possible ways. Join operations, such as the natural join (\(R \bowtie S\)), selectively pair tuples from two relations based on a matching condition.

  • An operation for renaming

    The renaming operator (\(\rho\)) changes the schema of a relation. It can be used to alter the names of the attributes or the name of the relation itself. This is often necessary to resolve attribute name conflicts in other operations.

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Set Operations on Relations

  • Defines the three common set operations as applied to relations:
    • Union (\(R \cup S\)): set of elements in R, S, or both.
    • Intersection (\(R \cap S\)): set of elements in both R and S.
    • Difference (\(R - S\)): set of elements in R but not in S.
  • States the conditions for applying these operations: the two relations must have identical schemas (same set of attributes and domains) and the columns must be ordered identically.
# 5
read: Section 2.4.4: Set Operations on Relations (section)
model: pro
rules: short presentation bullet definition

- define the set operations (union, intersection, difference) using mathematical expressions.
- make some reminder that the operands are relations which are sets of tuples.

(don't bother to be too detailed.)

Since relations are, fundamentally, sets of tuples, the standard set-theoretic operations may be applied. A prerequisite for these operations is that the operand relations must be union-compatible, meaning they possess identical schemas: the same set of attributes with identical domains.

  • Union (\(\cup\))

    The union of two union-compatible relations, \(R\) and \(S\), is the set of tuples present in \(R\), in \(S\), or in both. Duplicates are eliminated as per set definition. \[ R \cup S = \{ t \mid t \in R \lor t \in S \} \]

  • Intersection (\(\cap\))

    The intersection of two union-compatible relations, \(R\) and \(S\), is the set of tuples present in both \(R\) and \(S\). \[ R \cap S = \{ t \mid t \in R \land t \in S \} \]

  • Difference (\(-\) )

    The difference of two union-compatible relations, \(R\) and \(S\), is the set of tuples present in \(R\) but not in \(S\). \[ R - S = \{ t \mid t \in R \land t \notin S \} \]

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# 5.1

rules: short presentation bullet

Discuss how database systems handle unions of two relations with different attribute names.  (Basically, the column names are ignored, only their domains matter.)

Then talk about how it chooses the output relation's attribute names.

Revise: keep the discussion very short and to the point.

  • Schema Requirements for Set Union

    For the union (and intersection and set difference) of relations \(R\) and \(S\) to proceed, their schemas must be positionally compatible. This necessitates that the relations possess an identical count of attributes, and that for each corresponding position, the attributes share the same data type, or domain. The specific attribute names, however, are largely ignored by the set operators.

  • Addressing Dissimilar Attribute Names

    If the attribute names in \(R\) and \(S\) differ, the resulting relation would assume the attributes of the first input relation.

For example:

\[ R = S(A:\text{INTEGER}, B:\text{TEXT}) \cup T(C:\text{INTEGER}, D:\text{TEXT}) \]

would have the schema \(R(A:\text{INTEGER}, B:\text{TEXT})\).

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Projection

  • Introduces the projection operator (\(\pi_{A_1, A_2, ..., A_n}(R)\)) to create a new relation with a subset of the columns from an existing relation R.
  • Specifies that the schema of the result contains only the listed attributes.
  • Emphasizes that duplicate tuples in the resulting relation are eliminated, as relational algebra operates on sets.
# 6
read: Section 2.4.5: Projection (section)
model: pro
rules: short presentation definition bullet

Use mathematical notations to express the logical formal definition of projection.  Follow:

- Introduces the projection operator ($\pi_{A_1, A_2, ..., A_n}(R)$) to create a new relation with a subset of the columns from an existing relation R.
- Specifies that the schema of the result contains only the listed attributes.
- Emphasizes that duplicate tuples in the resulting relation are eliminated, as relational algebra operates on sets.

Definition: Projection (\(\pi\))

The projection operator, denoted as \(\pi_{A_1, A_2, ..., A_n}(R)\), yields a new relation containing a subset of the columns of a relation \(R\). Formally, let \(R\) be a relation. The projection of \(R\) onto attributes \(A_1, A_2, ..., A_n\) is defined as: \[ \pi_{A_1, ..., A_n}(R) = \{ t[A_1, ..., A_n] \mid t \in R \} \] where \(t[A_1, ..., A_n]\) denotes a tuple formed by the components of tuple \(t\) corresponding to the attributes \(A_1, ..., A_n\).

  • The projection operator is a unary operation that selects specified attributes from a relation.

    It creates a “vertical” subset of the original relation, retaining only the columns listed in the subscript of the \(\pi\) operator. The schema of the resulting relation consists solely of these specified attributes, in the order they are listed.

  • The result of a projection is a set.

    As relational algebra operates on sets, any duplicate tuples that arise from projecting onto a smaller set of attributes are eliminated. The resulting relation will contain only unique tuples. For instance, if multiple tuples in the original relation share the same values for the projected attributes, only one such tuple will appear in the final result.

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# 6.1

Write a very short remark about how we are not focused on the number of operators.  We can have many projection operators (infinite) in order to handle all possible scenarios.

Note:

The projection operator, denoted by \(\pi\), allows for the selection of any specified subset of attributes from a given relation. There is no predefined limit on the number of such operators. Instead, a distinct projection operator is implicitly available for every possible combination of attributes one might wish to extract from a relation, addressing all operational needs.

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Selection

  • Describes the selection operator (\(\sigma_C(R)\)), which produces a new relation containing a subset of the tuples from R.
  • The tuples included in the result are those that satisfy a specified condition C.
  • The condition C involves attributes of R and constants, and the schema of the resulting relation is the same as R’s schema.
# 7
read: Section 2.4.6: Selection (section)
model: pro
rules: short presentation math definition bullet

Focus on:
- Describes the selection operator ($\sigma_C(R)$), which produces a new relation containing a subset of the tuples from R.
- The tuples included in the result are those that satisfy a specified condition C.
- The condition C involves attributes of R and constants, and the schema of the resulting relation is the same as R's schema.

Definition: Selection

The selection operator, denoted \(\sigma_C(R)\), yields a new relation containing a subset of the tuples from a relation \(R\). The tuples in the resulting relation are all those from \(R\) that satisfy a specified condition \(C\). The schema of the resulting relation is the same as the schema of \(R\).

  • The selection operation is denoted as \(\sigma_C(R)\).

    \(R\) is a relation, and \(C\) is a selection condition, which is a Boolean predicate. Formally, the result is the set of all tuples \(t\) in \(R\) for which the condition \(C\) holds true. \[ \sigma_C(R) = \{ t \mid t \in R \land C(t) \} \]

  • The condition \(C\) is a logical expression that involves the attributes of \(R\).

    The operands in the condition are attributes of \(R\) or constant values. The operators are the standard comparison operators (e.g., \(=, \ne, <, \le, >, \ge\)) and logical connectives (e.g., AND, OR, NOT).

  • The selection operator filters tuples.

    It is considered a “horizontal” operator because it selects a subset of the rows (tuples) of the original relation without modifying the columns (attributes).

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Cartesian Product

  • Defines the Cartesian product (\(R \times S\)) of two relations, which is the set of all possible concatenated tuples formed by pairing each tuple from R with each tuple from S.
  • The schema of the resulting relation is the union of the schemas of R and S.
  • Explains the need to disambiguate common attribute names, typically by prefixing the original relation name (e.g., R.A and S.A).
# 8
read: Section 2.4.7: Cartesian Product (section)
model: pro
rules: short presentation math definition bullet

Focus on:
- Defines the Cartesian product ($R \times S$) of two relations, which is the set of all possible concatenated tuples formed by pairing each tuple from R with each tuple from S.
- The schema of the resulting relation is the union of the schemas of R and S.
- Explains the need to disambiguate common attribute names, typically by prefixing the original relation name (e.g., R.A and S.A).

Give a fully worked out example.  If there is one in the section, just use the same example.

revise: add a formal definition of cartesian product using discrete math.  Keep everything else.

Definition: Cartesian Product

Let \(R\) be a relation with schema \(Sch(R)\) and \(S\) be a relation with schema \(Sch(S)\). The Cartesian product, denoted \(R \times S\), is the set of tuples formed by concatenating each tuple of \(R\) with each tuple of \(S\). \[ R \times S = \{ r \circ s \mid r \in R \land s \in S \} \] where \(\circ\) denotes tuple concatenation. The resulting relation contains all possible pairings of tuples from the two relations.

  • Schema

    The schema of the resulting relation is the union of the schemas for \(R\) and \(S\). If relations \(R\) and \(S\) have attributes with the same name, for instance attribute \(A\), then these attributes must be disambiguated in the resulting schema. A standard convention is to prefix the attribute with the relation name, yielding \(R.A\) and \(S.A\).

  • Cardinality

    Every tuple from \(R\) is paired with every tuple from \(S\). If \(R\) has \(n_R\) tuples and \(S\) has \(n_S\) tuples, the resulting relation \(R \times S\) will have \(n_R \times n_S\) tuples.

Example

Consider the relations \(R\) and \(S\) from Figure 2.14.

Relation \(R\):

A B
1 2
3 4

Relation \(S\):

B C D
2 5 6
4 7 8
9 10 11

The Cartesian product \(R \times S\) pairs each of the two tuples in \(R\) with each of the three tuples in \(S\). The attribute \(B\), which is common to both schemas, is disambiguated into \(R.B\) and \(S.B\). The result consists of \(2 \times 3 = 6\) tuples.

Result \(R \times S\):

A R.B S.B C D
1 2 2 5 6
1 2 4 7 8
1 2 9 10 11
3 4 2 5 6
3 4 4 7 8
3 4 9 10 11
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Natural Joins

  • Introduces the natural join (\(R \bowtie S\)), which pairs tuples from two relations that agree on the values of all attributes common to both schemas.
  • The resulting schema is the union of the attributes from both relations, with common attributes appearing only once.
  • Defines a ‘dangling tuple’ as a tuple in one relation that does not successfully pair with any tuple in the other relation.
# 9
read: Section 2.4.8: Natural Joins (section)
model: pro
rules: math short presentation bullet definition

Use formal definition in mathematical logic to define natural join..  Focus on:
- Introduces the natural join ($R \bowtie S$), which pairs tuples from two relations that agree on the values of all attributes common to both schemas.
- The resulting schema is the union of the attributes from both relations, with common attributes appearing only once.
- Defines a 'dangling tuple' as a tuple in one relation that does not successfully pair with any tuple in the other relation.

Definition: Natural Join

Let \(R\) and \(S\) be relations with schemas (sets of attributes) \(Sch(R)\) and \(Sch(S)\), respectively. The natural join of \(R\) and \(S\), denoted \(R \bowtie S\), is the set of tuples \(t\) defined over the schema \(Sch(R) \cup Sch(S)\) such that a tuple from \(R\) and a tuple from \(S\) can be recovered from \(t\) by projection. More formally: \[ R \bowtie S = \{ t \mid t[Sch(R)] \in R \text{ and } t[Sch(S)] \in S \} \] where \(t[X]\) denotes the projection of tuple \(t\) onto the set of attributes \(X\).

  • Pairing Condition

    The natural join operation pairs a tuple \(r \in R\) with a tuple \(s \in S\) if and only if \(r\) and \(s\) agree on the values of all attributes in the intersection of their schemas, \(Sch(R) \cap Sch(S)\). From each successfully paired pair \((r, s)\), a single result tuple \(t\) is formed.

  • Result Schema

    The schema of the resulting relation is the union of the schemas of the operand relations, \(Sch(R) \cup Sch(S)\). The attributes common to both \(R\) and \(S\) appear only once in the result.

  • Dangling Tuples

    A tuple in one of the operand relations is called a dangling tuple if it does not match any tuple in the other relation on the common attributes. Dangling tuples do not appear in the result of the natural join.

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# 9.1
model: pro

Write an example of natural join.  Make sure there is dangling tuples in both relations.  Identify the dangling tuples, and explain.

Keep everything short and succinct.

Consider the two relations \(R\) and \(S\) shown below. The schema for relation \(R\) is \((A, B, C)\) and the schema for relation \(S\) is \((B, C, D)\). The attributes common to both schemas are \(B\) and \(C\).

Relation \(R\):

A B C
a1 1 10
a2 1 20
a3 2 30

Relation \(S\):

B C D
1 10 d1
2 30 d2
2 40 d3

The natural join of \(R\) and \(S\), denoted \(R \bowtie S\), pairs tuples from \(R\) and \(S\) that agree on all common attributes. In this case, a tuple from \(R\) is paired with a tuple from \(S\) if and only if they have the same values for both attribute \(B\) and attribute \(C\). The resulting relation has a schema consisting of the union of the attributes from \(R\) and \(S\), which is \((A, B, C, D)\).

The result of the natural join \(R \bowtie S\) is:

A B C D
a1 1 10 d1
a3 2 30 d2

A tuple that fails to pair with any tuple of the other relation in a join is called a dangling tuple. Such tuples do not contribute to the result of the natural join.

In this example, both relations contain dangling tuples.

  • The tuple (a2, 1, 20) in relation \(R\) is a dangling tuple. There is no tuple in \(S\) where the values for attributes \(B\) and \(C\) are 1 and 20, respectively.

  • The tuple (2, 40, d3) in relation \(S\) is a dangling tuple. There is no tuple in \(R\) where the values for attributes \(B\) and \(C\) are 2 and 40, respectively.

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Theta-Joins

  • Describes the theta-join (\(R \bowtie_C S\)), which is a more general form of join where tuples are paired based on an arbitrary condition C.
  • Explains that a theta-join is equivalent to performing a Cartesian product followed by a selection based on the condition C.
  • Notes that unlike a natural join, common attributes are not merged in the result and must be disambiguated if necessary.
# 10

read: Section 2.4.9: Theta-Joins (section)
model: pro
rules: math short presentation bullet definition

Use formal definition with mathematical logic to define theta-join.  Focus on:
- Describes the theta-join ($R \bowtie_C S$), which is a more general form of join where tuples are paired based on an arbitrary condition C.
- Explains that a theta-join is equivalent to performing a Cartesian product followed by a selection based on the condition C.
- Notes that unlike a natural join, common attributes are not merged in the result and must be disambiguated if necessary.```

Then give an example of a theta-join.  Keep it short.

Definition: Theta-Join

Let \(R\) and \(S\) be relations. Let \(C\) be a predicate, or condition, that involves attributes from both \(R\) and \(S\). The theta-join of \(R\) and \(S\) with respect to condition \(C\), denoted as \(R \bowtie_C S\), is a relation containing all combinations of tuples from \(R\) and \(S\) that satisfy the condition \(C\). Formally, the theta-join is defined as a selection operation on the Cartesian product of the two relations:

\[ R \bowtie_C S = \sigma_C (R \times S) \]

  • The theta-join is a generalized join operation that allows for arbitrary comparison conditions, not just equality on common attributes.

    The condition \(C\) can involve any of the standard comparison operators: \(\{=, <, \le, >, \ge, \ne\}\). The attributes in the condition must be type-compatible for the comparison to be valid.

  • The schema of the resulting relation is the union of the schemas of the operand relations.

    Unlike a natural join, attributes common to both relations are not merged. If an attribute name exists in both relations, it must be disambiguated, typically by prefixing it with the relation name (e.g., \(R.A\) and \(S.A\)).

  • Example: Consider relations \(U(A, B, C)\) and \(V(B, C, D)\). The theta-join \(U \bowtie_{A < D} V\) pairs every tuple from \(U\) with every tuple from \(V\) and retains only those pairs where the value of attribute \(A\) from the \(U\)-tuple is less than the value of attribute \(D\) from the \(V\)-tuple.

    The resulting schema would be \((U.A, U.B, U.C, V.B, V.C, V.D)\). Note the disambiguation of common attributes \(B\) and \(C\).

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Combining Operations to Form Queries

  • Explains that relational algebra allows forming complex expressions by applying operations to the results of other operations.
  • Introduces two ways to represent these complex queries: using linear notation with parentheses, and as expression trees.
  • Notes that expression trees are evaluated bottom-up.
  • Points out that there can be multiple equivalent expressions for the same query, which is important for query optimization.
# 11
read: Section 2.4.10: Combining Operations to Form Queries (section)
model: pro
rules: math short presentation bullet

Summarize this section into a short succinct form.  Use mathematical expressions.
Use the following format:

- <topic>: short description

  > ... longer description, including example.

- <topic>: short description

  > longer description with example if possible.

  • Expression Composition

    Relational algebra allows operators to be applied to the results of other operations. This enables the construction of expressions of arbitrary complexity, forming sophisticated queries. Parentheses are used in the conventional manner to group operators and operands.

  • Expression Trees

    A complex expression can be represented as an expression tree. The leaves of the tree are the operand relations. The interior nodes are the algebraic operators, which are applied to the results of their children nodes. The tree is evaluated from the leaves to the root.

  • Equivalent Expressions

    A query can often be represented by more than one relational algebra expression. For example, to find the titles and years of movies made by Fox with a length of at least 100 minutes, one may write the expression:

    \[ \pi_{\text{title,year}} (\sigma_{\text{length} \ge 100}(\text{Movies}) \cap \sigma_{\text{studioName} = '\text{Fox'}}(\text{Movies})) \]

    An equivalent, and often more efficient, expression is:

    \[ \pi_{\text{title,year}} (\sigma_{\text{length} \ge 100 \text{ AND } \text{studioName} = '\text{Fox'}}(\text{Movies})) \]

    The existence of equivalent expressions is fundamental to query optimization.

    For the example above, count the number of scans through Movies relation.

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Naming and Renaming

  • Introduces the renaming operator (\(\rho_S(A_1, ..., A_n)(R)\)) to explicitly change the name of a relation and/or its attributes.
  • This is useful for controlling the schema of the result of an operation and is essential for tasks like comparing a relation with itself.
# 12
read: Section 2.4.11: Naming and Renaming (section)
model: pro
rules: math short presentation bullet definition

Provide a formal definition of the renaming operator.
Focus on:
- Introduces the renaming operator ($\rho_S(A_1, ..., A_n)(R)$) to explicitly change the name of a relation and/or its attributes.
- This is useful for controlling the schema of the result of an operation and is essential for tasks like comparing a relation with itself.

Then give an example of a simple relation, and apply a renaming operator on it.

Definition: Renaming (\(\rho\))

The renaming operator changes the schema of its operand relation without altering its instance (the set of tuples). Given a relation \(R\) with \(n\) attributes, the expression \(\rho_{S(A_1, A_2, \dots, A_n)}(R)\) yields a new relation named \(S\) with attributes \(A_1, A_2, \dots, A_n\). The tuples in \(S\) are identical to the tuples in \(R\).

  • The renaming operator, denoted by \(\rho\), provides a mechanism to alter the names of a relation and its attributes.

    The general form is \(\rho_{S(A_1, A_2, \dots, A_n)}(R)\).

    • \(R\) is the original relation.
    • \(S\) is the new name for the relation.
    • \(A_1, A_2, \dots, A_n\) are the new names for the attributes, listed in order.

    This operator is essential for queries that require comparing a relation with itself or for clarifying attribute origins in complex joins.

  • Consider a relation StarsIn(movieTitle, movieYear, starName).

    movieTitle movieYear starName
    Star Wars 1977 Carrie Fisher
    Star Wars 1977 Mark Hamill

    To rename this relation to Films with attributes title, year, and actor, we apply the expression:

    \[ \rho_{\text{Films(title, year, actor)}}(\text{StarsIn}) \]

    The resulting relation is Films(title, year, actor):

    title year actor
    Star Wars 1977 Carrie Fisher
    Star Wars 1977 Mark Hamill
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Relationships Among Operations

Demonstrates that some relational algebra operations can be expressed in terms of others.

  • Intersection can be defined using set difference: \(R \cap S = R - (R - S)\).
  • Theta-join can be defined using Cartesian product and selection: \(R \bowtie_C S = \sigma_C (R \times S)\).
  • Natural join can be defined using Cartesian product, selection, and projection.
# 13
read: Section 2.4.12: Relationships Among Operations (section)
model: pro
rules: math bullet presentation short

Demonstrates that some relational algebra operations can be expressed in terms of others.

- Intersection can be defined using set difference: $R \cap S = R - (R - S)$.
- Theta-join can be defined using Cartesian product and selection: $R \bowtie_C S = \sigma_C (R \times S)$.
- Natural join can be defined using Cartesian product, selection, and projection.

Certain relational algebra operations can be expressed in terms of other, more fundamental operations. This demonstrates a degree of redundancy within the set of operators.

  • The intersection of two relations, \(R\) and \(S\), can be defined using set difference.

    The expression is \(R \cap S = R - (R - S)\). This is computed by first finding all tuples in \(R\) that are not in \(S\), and then subtracting this set from the original relation \(R\). The result contains only those tuples that are in both \(R\) and \(S\).

  • The theta-join is expressible by means of the Cartesian product and the selection operators.

    The expression is \(R \bowtie_C S = \sigma_C (R \times S)\). This operation first computes all possible pairings of tuples from \(R\) and \(S\) via the Cartesian product, and then selects only those pairs that satisfy the condition \(C\).

  • The natural join can be expressed using Cartesian product, selection, and projection.

    The natural join \(R \bowtie S\) is equivalent to \(\pi_L (\sigma_C (R \times S))\). Here, the condition \(C\) equates all attributes common to both \(R\) and \(S\). The projection list \(L\) includes all attributes of \(R\) and all attributes of \(S\) that are not in \(R\), thereby eliminating the duplicate columns that result from the join condition.

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A Linear Notation for Algebraic Expressions

  • Presents an alternative to expression trees for representing complex queries: a sequence of assignment statements.
  • This notation invents names for temporary relations that correspond to the interior nodes of an expression tree.
  • The format used is RelationName(attributes) := expression.
# 14
read: Section 2.4.13: A Linear Notation for Algebraic Expressions (section)
model: pro
rules: presentation bullet math

- Presents an alternative to expression trees for representing complex queries: a sequence of assignment statements.
- This notation invents names for temporary relations that correspond to the interior nodes of an expression tree.
- The format used is `RelationName(attributes) := expression`.

An alternative to representing complex algebraic expressions with trees is a linear notation using a sequence of assignment statements.

  • Assignment Statements

    A sequence of assignments can be used to compute a complex query. Each assignment creates a temporary relation, corresponding to an interior node in an expression tree. The order of assignments must respect the dependencies of the query; the relations on the right side of an assignment must have been previously computed.

  • Notation

    The format for an assignment is as follows:

    \[ \text{RelationName}(\text{attribute}_1, \dots, \text{attribute}_n) := \text{Expression} \]

    The left side defines the name and schema of the temporary relation. The right side is any valid relational algebra expression. The final result is conventionally assigned to a relation named Answer.

  • Example

    Consider the query to find the titles and years of movies by ‘Fox’ that are at least 100 minutes long. The expression tree in Figure 2.18 can be represented by the following sequence of assignments:

    • \(R(t, y, l, i, s, p) := \sigma_{\text{length}\geq 100}(\text{Movies})\)
    • \(S(t, y, l, i, s, p) := \sigma_{\text{studioName}=\text{'Fox'}}(\text{Movies})\)
    • \(T(t, y, l, i, s, p) := R\cap S\)
    • \(\text{Answer}(\text{title}, \text{year}) := \pi_{(t, y)}(T)\)

  • Flexibility

    Several operations may be combined into a single assignment statement, allowing for more concise, though potentially less clear, expressions. The previous query could be written in a single line:

    \[ \text{Answer}(\text{title, year}) := \pi_{\text{title,year}}(\sigma_{\text{length} \ge 100 \text{ AND } \text{studioName} = \text{'Fox'}}(\text{Movies})) \]

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