CSE 132A Discussion Week 2: Relational Calculus
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CSE 132A DiscussionWeek 2: Relational Calculus
Definition
● A nonprocedural query language, where each query is of the form:
{t | P(t)}
● Results: the set of all tuples t such that predicate P is true for t● t ∈ r denotes that tuple t is in relation r● t is a tuple variable, t(A) denotes the value of tuple t on attribute A● P is a formula similar to that of the predicate calculus
Predicate Calculus Formula● A set of attributes and constants● A set of comparison operators: (e.g. <, ≤, =, ≠, ≥, >)● A set of connectives: and (^) or (v), not (￢)● Implication (→): x → y, if x is true, then y is true.● A set of quantifiers:
○ ∃ t ∈ r (Q (t )) ≡ ”there exists” a tuple in t in relation r such that predicate Q (t) is true
○ ∀t ∈ r (Q (t )) ≡ Q is true “for all” tuples t in relation r
Logic Examples
Express “Each unicorn has a horn” in relational calculus
Logic Examples
Each unicorn has a horn.
∀a ∈ Animals,a is a unicorn → a has a horn
∀a ∈ Animals, a is a unicorn ∧ a has a horn
Logic Examples
Each unicorn has a horn.
∀a ∈ Animals,a is a unicorn → a has a horn
∀a ∈ Animals, a is a unicorn ∧ a has a horn
Some horses have horns.
Logic Examples
Each unicorn has a horn.
∀a ∈ Animals,a is a unicorn → a has a horn
∀a ∈ Animals, a is a unicorn ∧ a has a horn
Some horses have horns.
∃a ∈ Animals, a is a horse ∧ a has a horn
Our schemaConsider the following database schema for a BOOKSTORE database:
● Books (bookid, title, author, year) ● Customers (customerid, name, email) ● Purchases (customerid, bookid, year) ● Reviews (customerid, bookid, rating) ● Pricing (bookid, format, price)
Example 1 : from SQL to Relational CalculusFind books (show their titles) written by ’EDMUND MORGAN’ since year 1990.
SQL?
Example 1 : from SQL to Relational CalculusFind books (show their titles) written by ’EDMUND MORGAN’ since year 1990.
SELECT title
FROM Books
WHERE author = ‘EDMUND MORGAN’ AND year >= 1990
SELECT title
FROM Books
WHERE author = ‘EDMUND MORGAN’ AND year >= 1990
{r : title | …. }
SELECT title
FROM Books
WHERE author = ‘EDMUND MORGAN’ AND year >= 1990
{r : title | ∃b ∈ Books [....] }
SELECT title
FROM Books
WHERE author = ‘EDMUND MORGAN’ AND year >= 1990
{r : title | ∃b ∈ Books [ b(author)= ‘EDMUND MORGAN’ ^ b(year) ≥ 1990] }
Are we done here?
SELECT title
FROM Books
WHERE author = ‘EDMUND MORGAN’ AND year >= 1990
{r : title | ∃b ∈ Books [ b(author)= ‘EDMUND MORGAN’ ^ b(year) ≥ 1990 ^ b(title) = r(title)] }
Don’t forget about this!
Example 2What are the titles of the newest books?
Example 2What are the titles of the newest books?
{r : title | …. }
Example 2What are the titles of the newest books?
{r : title | ∃b ∈ Books [....] }
Example 2What are the titles of the newest books?
{r : title | ∃b ∈ Books [∀o ∈ Books[...]]}
Example 2What are the titles of the newest books?
{r : title | ∃b ∈ Books [∀o ∈ Books[b(years) ≥ o(years)]]}
Done?
Example 2What are the titles of the newest books?
{r : title | ∃b ∈ Books[∀o ∈ Books[b(years) ≥ o(years)]
^ b(title) = r(title)]}
Is this the only solution?
Example 2What are the titles of the newest books?
{r : title | ∃b ∈ Books[¬∃o ∈ Books[b(year)<o(year)]
∧b(title) = r(title)]}
Example 2What are the titles of the newest books?
{r : title | ∃b ∈ Books[¬∃o ∈ Books[b(year)<o(year)]
∧b(title) = r(title)]}
Translate back to SQL:
Example 2{r : title | ∃b ∈ Books[¬∃o ∈ Books[b(year)<o(year)] ∧b(title) = r(title)]}
SQL:
SELECT b.title
FROM Books b
WHERE NOT EXISTS (SELECT *
FROM Books o
WHERE b.year < o.year);All books that are newer than b
If there doesn’t exist book that are newer than b, then b is the newest book
Example 3
What are the titles of the Books which have been Purchased by every Customer?
Example 3
What are the titles of the Books which have been Purchased by every Customer?
{r :title | ∃b ∈ Books, ∀c ∈ Customers, ∃p ∈ Purchases[c(customerid) = p(customerid) ∧ p(bookid) = b(bookid)∧ b(title) = r(title) ]}
{r :title | ∃b ∈ Books, ¬∃c ∈ Customers, ¬ (∃p ∈ Purchases[ c(customerid) = p(customerid) ∧ p(bookid) = b(bookid) ∧ b(title) = r(title)) ]}
SELECT b.Title
FROM Books b
WHERE NOT EXISTS (
SELECT ∗
FROM Customers c
WHERE NOT EXISTS (
SELECT ∗
FROM Purchases p
WHERE c . customerid = p . customerid AND p. bookid = b. bookid ));
All purchases records of customer c brought the book b
Customer who didn’t buy book b
Book b that purchased by every customer
Example 4
Which book(s) are the cheapest?
Example 4
Which book(s) are the cheapest?
{r :title | ∃b ∈ Books, ∃p ∈ Pricing, ∀ c ∈ Pricing
[p(price) <= c(price) ∧
p(bookid) = b(bookid) ∧
b(title) = r(title)]}
Example 5
Which battleships launched before 1930 had 16-inch guns? List their names, their country, and the number of guns they carried?
Ships(name, yearLaunched, country, numGuns, gunSize, displacement)
Example 5
Which battleships launched before 1930 had 16-inch guns? List their names, their country, and the number of guns they carried?
{t :names, country, numGuns|∃s ∈ Ships[ t(name) = s(name) ∧ t(country) = s(country) ∧ t(numGuns) = s(numGuns)∧ s(yearLaunched) < 1930 ∧ s(gunSize) = 16]}
Ships(name, yearLaunched, country, numGuns, gunSize, displacement)
Any other questions?
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