Support of Relational Algebra Knowledge ... - Hornad - TUKE

Support of Relational Algebra Knowledge Assessment
Henrieta Telepovska, Matus Toth *
Department of Computers and Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Kosice
Email: Henrieta.Telepovska@tuke.sk
Abstract.
In the paper a prototype of relational algebra performance
module is described. This module supports students
training and automated knowledge assessment of
relational algebra queries. The relational algebra is a basis
of SQL queries performance. Each select statement can be
written by operations of relational algebra and vice versa.
It is the main idea of our solution that is described in this
article. The relational algebra performance module uses a
module of select statements evaluation.
I. INTRODUCTION
Automated knowledge assessment becomes quite popular
in the last years ([7]). The main idea of advanced support
of relational algebra knowledge assessment is to automate
the process of student training and examination
([12][13][14]). It was developed prototype of relational
algebra performance module which uses module of SQL
statement processing especially SELECT statement
([10]). Both modules are plug-ins to LMS Moodle. On the
present LMS (Learning Management System) Moodle is
used as an electronic support in education on Department
of Computers and Informatics of Technical University in
Kosice. It provides 69 courses in all three degrees of
university study – undergraduate, graduate and
postgraduate. At first the Database Systems course was
developed to provide learning materials for students.
Later this course was extended about verification and
evaluation of students’ knowledge. It was used a quiz
module of LMS Moodle with its standard types of
questions.
A quiz module is a component of LMS Moodle ([11]).
This module enables to develop tests consisting of various
types questions. The questions and the answers are stored
in database. It is possible to generate a test from database
by defined requirements automatically.
LMS Moodle supports following question types -
embedded answers, matching, multiple choice, calculated,
random short-answer matching, short answer and
true/false. It is possible to set up various test parameters
or features. For example - shuffle questions, shuffle
answers, permit or deny reviewing a quiz, single or
multiple attempts at a quiz, review correct answers,
students’ responses, scores or general feedback and
different grading methods.
The quiz module enables to create the categories of
questions in which it is possible to distribute developed
questions. One category can contain one or more thematic
parts of a particular subject. The questions of Database
System course are organized into two basic parts – theory
and practice. The category of database system theory is a
collection of database system fundamental questions. This
category has 11 subcategories. The second category
named Database System Examples has 7 subcategories. It
contains practice examples that verify student knowledge
of database objects manipulation.
The examples are focused on mapping logical database
schema into physical schema. The next examples are
aimed at SQL (Structured Query Language) statements
for database objects creation – table, triggers, and stored
procedures. The very important questions are examples
that verify correctness of select statement writing. The
select statement examples are mainly focused on inner
and outer join, subquery and usage of aggregate functions.
The newest type of question is a relational algebra check
that verifies correctness of query writing by the relational
algebra operations.
II.RELATIONAL ALGEBRA
Relational algebra, together with the relational calculus
forms the mathematical basis for relational databases. E.F.
Codd described it as a basis for defining the relational
model ([2][3]). A relation is a fundamental concept of the
relational model. In the relational DBMS it is a data
structure that allows data storage in database. Relational
algebra is a set of formal operations of these relations,
resulting in a new relation. Relational algebra defines how
we can manipulate data in tables by relational operators.
Codd originally defined the eight relational operators,
which he called SELECT (or RESTRICT), PROJECT,
JOIN, PRODUCT, INTERSECT, UNION,
DIFFERENCE a DIVIDE. The most important operators
are SELECT, PROJECT and JOIN, which are mostly
used in the formulation of relational algebra expressions.
Relational operators are characterized by closeness
property, i.e. relational algebra operators are used over
existing relations (tables) and a result is new relation
again. There are two types of relational operators - unary
and binary. Unary operators such as SELECT and
PROJECT are used only at one relation, while an operator
such as JOIN is used over two relations ([1][3][4]).
There are two groups of relational algebra operations. The
first one includes set operations from mathematical set
theory, because each relation is defined as a set of tuples.
Set operations defined in the relational algebra: union,
intersection, difference and Cartesian product. The second
one consists of relational operations. This includes
operations such as: selection, projection and join
([1][3][4]). Chyba! Nenašiel sa žiaden zdroj odkazov.
Selection
Relational SELECT operator, also called RESTRICT,
retrieves all tuples that meet the specified criteria.
SELECT is the horizontal retrieve of relation ([1][3][4]).

Syntax of select operation:
σ ()
σ - symbol of select operation
condition - boolean expression specified over relation
attributes
relation_name – name of the relation which is processed
by select operation.
The result is a new relation that has the same attributes as
original relation defined by relation_name.
Projection
Projection is vertical retrieve from relation and it selects
table columns according to attribute list. Syntax of project
operation:
П ()
П - symbol of project operation
attribute_list – list of attribute (comma separated)
displayed in the result
relation_name – name of the relation which is processed
by projection
The result is a new relation with attributes that is specified
in the attribute list in the same order as listed. If the
attribute list contains only non-key attributes of the
relation, then duplicate tuples will not be displayed in the
final relation.
Union
Set operator UNION consolidates all tuples of two
relations, eliminating duplicate records. Both relations
must have the same relational scheme – degrees of
relations and domains of corresponding columns must be
identical then the relations are union-compatible. Syntax
R 3 = R 1 U R 2
U - symbol of union operation
Relations R 1 (a 1 , a 2 , … , a n ) and R 2 (b 1 , b 2 , … , b n ) must be
union-compatible. The degree of both relations is n.
The result is R 3 (c 1 , c 2 , … , c n ) relation with degree n and
each tuple of R 3 is either from R 1 or R 2.
If two relations are not union-compatible UNION
operator cannot be used. To overcome this problem, it can
be used the project operator, to narrow the number of
columns in each relation and the relations were unioncompatible.
Intersect
Intersection is a binary operation. The result tuples belong
to both relations. Syntax
R 3 = R 1 ∩ R 2
∩ - symbol of intersect
Relations R 1 (a 1 , a 2 , … , a n ) a R 2 (b 1 , b 2 , … , b n ) must be
union-compatible. The degree of both relations is n. The
result is R 3 (c 1 , c 2 , … , c n ) relation with degree n and each
tuple from R 3 is in R 1 and also R 2 .
Difference
Difference operator returns all records from a single
relation that are not in the second relation. Difference
operator requires that both relations were unioncompatible.
Syntax:
R 3 = R 1 - R 2
Relations R 1 (a 1 , a 2 , … , a n ) a R 2 (b 1 , b 2 , … , b m ) must be
union-compatible. The degree of both relations is n. The
result is R 3 (c 1 , c 2 , … , c n ) relation with degree n.
Cartesian product
Syntax
R 3 = R 1 X R 2 , where
R 1 (a 1 , a 2 , … , a n ) is a relation with degree n and the
number of tuples (the cardinality of relation) is i.
R 2 (b 1 , b 2 , … , b m ) is a relation with degree m and the
cardinality of relation is j.
The result R 3 (a 1 , a 2 , … , a n , b 1 , b 2 , … , b m ) is a relation
with degree n + m and the cardinality of result relation is
i*j.
Cartesian product combines tuples from two relations.
The result contains many tuples that have no relationship
between them.
Division
Syntax
R 3 = R 1 ÷ R 2
R 1 (a 1 , a 2 , … , a n ) is a relation with degree n and the
number of tuples (the cardinality of relation) is i.
R 2 (b 1 , b 2 , … , b m ) is a relation with degree m and the
cardinality of relation is j.
R 3 (c 1 , c 2 , … , c k ) is a relation with degree k=n-m and the
cardinality of relation is i ÷ j .
The result relation contains tuples from R 1 that
a combination with each tuple from R 2 is in R 1 . The result
tuples has only attributes from R 1 .
Join
Join operation is one of the basic operations of relational
algebra. It is a binary operation that combines two
relations in the prescribed manner. It is one of the most
important operations in relational databases, expressing
the relationship between the two relations. Join R1 and R 2
is a select operation applying over Cartesian product R 1
X R 2 with particular condition. Join operation is defined
over corresponding attributes both relations R 1 and R 2 .
These attributes have the same domain and semantics.
Syntax
R 3 = R1 ⋈ R 2
⋈ - symbol of join operation
R 1 (a 1 , a 2 , … , a n ) and R 2 (b 1 , b 2 , … , b m ) - relations
join_condition – boolean expression in the form
AND AND … AND
, where a condition has following form Ai Ө
Bj. Ai is an attribute of R 1 and Bj is a attribute R 2 . Symbol
Ө is one of next operators =, =, != .
R 3 (a 1 , a 2 , … , a n , b 1 , b 2 , … , b m ) – a result relation with
n+m attibutes
The result relation contains corresponding tuples that
satisfy a join condition.

Aggregate functions
Some database queries require an application of aggregate
functions to collection of data records. The aggregate
functions also called group functions are COUNT, SUM,
AVERAGE, MAXIMUM and MINIMUM functions.
Another type of database query involves grouping the
relation records according to the value of some relation
attributes and subsequent use of the aggregation function
independently of each group. Syntax
F ()
group_attributes – a list of relation attributes for grouping
records
F – function symbol
function_list – a list in form ,
a function is one of COUNT, SUM, AVERAGE,
MAXIMUM and MINIMUM functions and attribute is
one of relation attributes.
relation_name – a name of relation
The result relation contains a group attribute and one
attribute for each item from function_list. If no
group_attribute is defined the functions are applied to all
tuples of specified relation.
Sequences of operations
The usage of relational algebra operations in sequence can
be entered in two ways. The first way is one statement
created by immersion of operations. The second one is
implementation of operations in succession and storing
intermediate results.
The relations which store intermediate results must be
named. ([1][3][4])
III. MODULE OF RELATIONAL ALGEBRA
PERFORMANCE
A query written in a relational algebra can be expressed as
select statement of structured query language (SQL).
Execution of select statement on the database server
returns a relation. It is a way how to check the correctness
of written relational algebra statements. It was developed
the compiler of relational algebra statements into select
statements ([5]).
The check correctness of a student answer is realized by
execution of translated relational algebra statement on
database sever. The result relation is compared with
relation which is a result of correct teacher select
statement. Module of relational algebra performance
based on designed grammar transforms a query written by
relational algebra operations into select statement of SQL.
The select statement is then processed by module of SQL
statement evaluation ([6][10]).
Compiler
The lexical units of query are identified at the lexical
level. On the basis of designated formal grammar rules it
is checked syntax correctness of particular query ([8][9]).
It was designed a special language for writing relational
algebra queries for our solution. The special symbols (σ,
П, ⋈) which are not standard keyboard input were
replaced following keywords:
σ = SEL
П = PROJ
⋈
= JOIN
The symbol of aggregate functions 'F' was replaced FUN
keyword.
The intermediate results are written following:
MV1 = relational algebra operation
MV2 = relational algebra operation
...
V = relational algebra operation
MVi – symbol of intermediate result
V – symbol of result – resulting relation
Grammar
An alphabet is the foundation of every language. It is a
finite set of elements called symbols ([8][9]). Alphabet of
relational algebra:
• { a,b,...z } – roman letters
• { 0, 1, ...9 } - decimal digits
• { '.' , ',' , '(', ')', '+', '-', '*', '/', '=', '=', '!=' }
– special symbols
• {'SEL', 'PROJ', 'JOIN', 'FUN', 'SUM', 'AVERAGE',
'MAX', 'MIN', 'COUNT', 'AND', 'OR', 'NOT'} -
keywords
• {'MV1', 'MV2', ….'MVi', 'V' } - reserved words
Grammar of relational algebra language is
G = (N, T, P, S), where
N = {Program, Command, Select, Project, Join, JoinAF,
Agreg, Conditions, Condition, Relation, Function, Query,
Operation} – a finite set of nonterminal symbols
T = { SEL, PROJ, JOIN, FUN, AND, OR, NOT, SUM,
AVER, MAX, MIN, COUNT, RESULT, DOT, POINT,
LOZAT, NM, POZAT, PLUS, MINUS, MULL, DIV, LT,
LE, GT, GE, EQ, NE, PART, ID, KONST} - a finite set
of terminal symbols (Table 1)
P a set of rules
S = Program – start symbol
Table 1 Terminal symbols
Lexical Unit
Lex[[SEL]]
Lex[[PROJ]]
Lex[[JOIN]]
Lex[[FUN]]
Lex[[AND]]
Lex[[OR]]
Lex[[NOT]]
Lex[[SUM]]
Lex[[AVERAGE]]
Terminal Symbol
SEL
PROJ
JOIN
FUN
AND
OR
NOT
SUM
AVER

Lex[[MAX]]
Lex[[MIN]]
Lex[[COUNT]]
Lex[[ V ]]
Lex[[ . ]]
Lex[[ , ]]
Lex[[ ( ]]
Lex[[ ) ]]
Lex[[ + ]]
Lex[[ - ]]
Lex[[ * ]]
Lex[[ / ]]
Lex[[ < ]]
Lex[[ < ]]
Lex[[ > ]]
Lex[[ > ]]
Lex[[ = ]]
Lex[[ != ]]
Lexical Unit
MAX
MIN
COUNT
RESULT
DOT
POINT
LOZAT
POZAT
PLUS
MINUS
MULL
DIV
LT
LE
GT
GE
EQ
NE
Lex[[ MVi ]]
i → ( '0' | .. | '9' )
Lex[[ j 1 … j n ]]
j k → ( 'a' | .. | 'z' ) { ( 'a' | .. | 'z' ) }
Lex[[ c 1 … c n ]]
c k → ( '0' | .. | '9' ) { ( '0' | .. | '9' ) }
[( '.' | ',' ) ( '0' | .. | '9' ) {( '0' | .. | '9' )}]
Lex[[ ai 1 … i n a ]]
i k → znak , a → ” ' “
A set of rules:
Terminal
Symbol
PART
ID
NM
KONST
Relation → LOZAT ( ID | PART ) POZAT
Function → SUM | AVER | MAX | MIN | COUNT
Conditions → Condition {(AND | OR) Condition}
Condition → [NOT] (ID (EQ|NE|GT|GE|LT|LE)
Query | LOZAT Conditions POZAT)
Query
→ ID | NM | ST | LOZAT Query POZAT
| Query Operaction Query
Operation → PLUS | MINUS | MUL | DIV
It is a context-free grammar. The form of each grammar
rule is
A → α , where A ∈ Ν a α ∈ ( N U T).
On the left side of any rule it is no terminal symbol
The grammar is LL(1) grammar, because it satisfies
FIRST-FIRST a FIRST-FOLLOW conditions, i.e. for
A type rules are defined:
A → α1|α2 | ... | αn
FIRST-FIRST
If α i ⇒ * w, w ∈ T + , α j ⇒ * u, u ∈ T + ,
then FIRST 1 (α i ) ∩ FIRST 1 (α j ) = ∅ pre 1 ≤ i, j ≤ n, i ≠
j.
FIRST-FOLOW
If α i ⇒ * e ,t.j. e∈FIRST 1 (α i ) for i,
then FIRST 1 (α j )∩FOLLOW 1 (A )=∅ pre 1 ≤ j ≤ n, i ≠ j
Program → { PART Command } RESULT Command
Command → EQ (Select | Project | JoinAF | Agreg)
Select → SEL Conditions Relation
Project → PROJ ID { POINT ID } Relation
JoinAF → ID ( Join | AF )
Join → JOIN Condition { AND Condition }
( ID | PART )
Agreg → {POINT ID} FUN Function (ID|MUL)
{POINT Function (ID|MUL)} Relation
IV. IMPLEMENTATION
On Figure 1 the scenario of the relational algebra
performance is displayed. SQL_RA_Module controls the
whole process of RA_queries and select statements
treating. Compiler checks RA_query syntax and if any
errors occur in parsing SQL_RA_module finishes its
evaluation. RA_query is classified as incorrect.
Otherwise, a module returns select1_statement which
corresponds with student relational algebra query ([5]).
The select1_statement is executed on database server
([6][10]). The result relations of select1 and select2
statements are compared. If they are equal a student
RA_query is correct else it is incorrect. The module
displayed error message if an error occurs in any step of
performance.

SequenceDiagram_1
SQL_RA_Module Compiler Database
Student
RA_query
compile
select1_statement
errors?
errors
[errors = false]: execute select1 statement
result of select1
execute select2 statement
result of select2
result of comparison select1 with select2
[errors = true]: compiling error
Figure 1 The relational algebra performance scenario
V. CONCLUSION
Described solution enables to achieve accurate results. It
does not compare specified relation query as text but it
compares two relations. On the present the basic
operations of relational algebra (selection, projection,
join, and aggregate function) are implemented. The next
operations are developed. The dynamic evaluation of
select statement and relational algebra query is the first
step of understanding and correct writing of select
statement in SQL.
ACKNOWLEDGEMENT
This work is the result of the project implementation:
Knowledge-Based Software Life Cycle and Architectures
(Project VEGA No. 1/0350/08)
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* Matus Toth graduated on Department of Computers and
Informatics, Technical University of Kosice