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Relational Calculus
More “declarative” than relational algebra
– Foundation for query languages (such as SQL)
– Relational algebra used more for physical operators
Relational Calculus
Comes in two flavours: Tuple relational calculus (TRC)
and Domain relational calculus (DRC)
– TRC: Variables range over (i.e., get bound to) tuples
DRC: Variables range over domain elements (= field values)
– Both TRC and DRC are simple subsets of first-order logic
–
We will study DRC
Database Management Systems, R. Ramakrishnan
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Database Management Systems, R. Ramakrishnan
Domain Relational Calculus
DRC Formulas
Atomic formula:
Query has the form:
x1, x2,..., xn | p x1, x2,..., xn
x1, x2,..., xn ∈ Rname, or X op Y,
<, >, =, ≤,≥, ≠
–
Formula:
– an atomic formula, or … (more later)
Example of query with atomic formula
Formula is recursively defined
{< I , N , T , A > |< I , N , T , A >∈ Sailors }
Starting with simple atomic formulas
– Logical connectives
– Quantification
–
–
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DRC Formulas
{< I , N , T , A >|< I , N , T , A >∈ Sailors ∧ T > 7}
–
Equivalent relational algebra query?
Find sailors who are older than 18 or have a rating under 9, and
are called ‘Joe’
Database Management Systems, R. Ramakrishnan
Database Management Systems, R. Ramakrishnan
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Formula:
– an atomic formula, or
– ¬ p, p ∧ q, p ∨ q, where p and q are formulas, or
– ∃X ( p( X )) , where variable X is free in p(X), or
– ∀ X ( p( X )) , where variable X is free in p(X)
The use of quantifiers ∃ Xand ∀ Xis said to bind X.
Query using logical connectives
–
Equivalent relational algebra query?
DRC Formulas
Formula:
– an atomic formula, or
– ¬ p, p ∧ q, p ∨ q, where p and q are formulas, or
– (more later)
–
or X op constant
op is one of
–
Answer includes all tuples x1, x2,..., xn that
make the formula p x1, x2,..., xn be true.
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A variable that is not bound is free
For query:
–
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x 1, x 2 ,..., x n | p x 1, x 2 ,..., x n
The variables x1, ..., xn that appear to the left of `|’ must be the
only free variables in the formula p(...).
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Find sailors rated > 7 who’ve reserved boat # 103
Find sailors rated > 7 who’ve reserved a red boat
I, N,T, A | I, N,T, A ∈ Sailors ∧ T > 7 ∧
I, N,T , A | I, N,T , A ∈ Sailors ∧ T > 7 ∧
∃ Ir, Br, D Ir, Br, D ∈ Re serves ∧ Ir = I ∧ Br = 103
∃ Ir, Br, D Ir, Br, D ∈ Re serves ∧ Ir = I ∧
We have used ∃ Ir , Br , D ( . . .)
for ∃ Ir ∃ Br ( ∃ D (. . .) )
(
)
∃ B, BN,C B, BN, C ∈ Boats ∧ B = Br ∧ C = ' red '
as a shorthand
Observe how the parentheses control the scope of
each quantifier’s binding.
This may look cumbersome, but with a good user
interface, it is very intuitive.
Note the use of ∃ to find a tuple in Reserves that
j̀oins with’ the Sailors tuple under consideration.
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Database Management Systems, R. Ramakrishnan
Find sailors who’ve reserved all boats
Find sailors who’ve reserved all boats
I, N,T, A | I, N,T, A ∈ Sailors ∧
I, N,T, A | I, N,T, A ∈ Sailors ∧
∀B,BN ,C B, BN ,C ∈Boats
∀B,BN ,C B, BN ,C ∈Boats∧
∃ Ir, Br, D Ir, Br, D ∈ Re serves ∧ I = Ir ∧ Br = B
∃ Ir, Br, D Ir, Br, D ∈ Re serves ∧ I = Ir ∧ Br = B
Find all sailors I such that for each 3-tuple B, BN,C
there is a tuple in Reserves showing that sailor I has
reserved it.
Database Management Systems, R. Ramakrishnan
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What is wrong with this?
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Database Management Systems, R. Ramakrishnan
Find sailors who’ve reserved all boats
Find sailors who’ve reserved all boats
I, N,T, A | I, N,T, A ∈ Sailors ∧
I, N,T, A | I, N,T, A ∈ Sailors ∧
∀ B, BN, C ¬ B, BN, C ∈ Boats ∨
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¬∃B,BN ,C B, BN ,C ∈Boats ∧
∃ Ir, Br, D Ir, Br, D ∈ Re serves ∧ I = Ir ∧ Br = B
¬ ∃Ir,Br,D Ir, Br, D ∈Re serves∧I = Ir∧Br = B
Find all sailors I such that for each 3-tuple B, BN,C
either it is not a tuple in Boats or there is a tuple in
Reserves showing that sailor I has reserved it.
Find all sailors I such that there does not exist a 3tuple B, BN,C in Boats for which there does not exist
a tuple in Reserves showing that sailor I has
reserved it.
Database Management Systems, R. Ramakrishnan
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Database Management Systems, R. Ramakrishnan
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Find sailors who’ve reserved all red boats
Unsafe Queries, Expressive Power
Syntactically correct calculus queries that have an
infinite number of answers
I, N,T, A | I, N,T, A ∈ Sailors ∧
– Such queries are called unsafe.
– e.g., I , N ,T , A |¬ I , N ,T , A ∈Sailors
∀B,BN ,C B, BN ,C ∈Boats ∧C ='Red '
∃ Ir, Br, D Ir, Br, D ∈ Re serves ∧ I = Ir ∧ Br = B
Every relational algebra query can be expressed as
a safe query in DRC/TRC
Find all sailors I such that for each 3-tuple B, BN,C
there is a tuple in Reserves showing that sailor I has
reserved it.
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Summary
Relational calculus is “declarative” or nonoperational
– Users define queries in terms of what they want,
not in terms of how to compute it
Algebra and safe calculus have same
expressive power
– Relational “completeness”
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– And vice versa!
Relational Completeness: Query language (e.g.,
SQL) can express every query that is expressible
in relational algebra/calculus.
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