1. Regular Expressions, Backus-Naur Form and Reverse Polish Notation

2. Starter
What does the expression 3+4*5 evaluate to? The answer is 23 Why is the answer not 35?
The answer is 23

3. Objectives
To form simple regular expressions for string manipulation and matching
To be able to check language syntax by referring to BNF or syntax diagrams
To be able to convert simple infix notation to reverse Polish notation and vice versa

4. Natural Language
A natural language comprises a set of words written and spoken by humans
Governed by syntax rules that define the order in which words may be put together
“Work students young hard clever”  is an invalid construct
“Clever young students work hard” 
Governed by semantic rules which define the actual meaning when applied to real world concepts
“The peanut ate the monkey” is a valid construct but lacks meaning
“The monkey ate the peanut” 
Constructing rules for a natural language is difficult and this is why we use formal languages to communicate with computer systems

5. Formal Language
Used for the precise definition of syntactically correct programs for a programming language
Defined using an alphabet and syntax rules
The alphabet is defined as a finite set of symbols
The rules of syntax define how to construct strings of the language out of symbols
It is not feasible to list all the strings. Instead Regular expressions are used.

6. Regular Expressions
Is a way of expressing the language that a machine can use in a formal manner
They provide a means of identifying if a string of characters is allowed in a particular language
They can be represented using a finite state machine
Used extensively in operating systems for pattern matching in commands, searching for information using Google and search and replace in word processors

7. Regular Expressions
If the alphabet of some formal language is {a, b} then here are some regular expressions: a is a regular expression that matches a string consisting of the symbol a ab matches a string consisting of the symbol a followed by b a* matches a string consisting of zero or more a’s a+ matches a string consisting of one or more a’s abb? matches the strings ab or abb a|b matches a string consisting of the symbol a or the symbol b

8. Regular Expression Example
Define the syntax of a formal language with alphabet {a, b, c} in which valid strings consist of at least one a or at least one b, followed by two c’s. The language is L = (a+|b+)cc

9. Q1 A language L is defined by the alphabet {a, b, c} together with the regular expression (a|c)+bb. (a) Explain what represents a valid string in L. A non-empty string consisting of any combination of as and cs, terminated by two bs. (b) Give two examples of valid strings in L. aaccbb, cacccbb

10. Metacharacters | separates alternatives e.g. a |b can match a or b
? indicates there is zero or one of the preceding element
* indicates there is zero or more of the preceding element
+ indicates there is one or more of the preceding element
[ab] means a or b
[a-z] matches all 26 lower case letters
n[^t] means an n followed by a character that is not a t

11. Worked Examples
Write down the strings defined by the regular expression b[ea]d?
be, ba, bed, bad
Write down the strings defined by the regular expression 10*1
11, 101, 1001, 10001, ...
Write down the strings defined by the regular expression 10+1
101, 1001, 10001, ...

12. Backus-Naur Form (BNF)
Some languages cannot be defined by regular expressions
BNF allows representations of a wider range of languages
Expresses the rules for constructing valid strings in a regular language
Defines the terms of the language, characters, words and symbols

13. BNF Notation ::= means ‘is defined as’
| separates alternatives on the right-hand side of the rule
<digit> ::= 0|1|2|3|4|5|6|7|8|9
Recursive definition
<unsigned integer> ::= <digit>|<digit><unsigned integer>
Syntax of a programming language
<expression> ::= <term><arithmetic_operator><term>
<term> ::= <identifier>|<constant>|<expression>
<arithmetic_operator> ::= <add_op>|etc
<add_op> ::= + | -

14. Syntax Diagrams
Alternative way of defining the syntax rules of a language
See examples in book pg 65-66

15. Syntax Diagrams BNF of Select command
Syntax diagram showing Select command

16. Reverse Polish Notation (RPN)
We evaluate arithmetic expressions using infix notation e.g
An alternative is prefix notation where the operator occurs before the operands e.g
Postfix is where the operator occurs after the operands e.g
Postfix notation is also called RPN
RPN is used in scientific calculators

17. Examples of Infix and Postfix Expressions
x – y
xy-
(a+b)/(a-b)
ab+ab-/
x+(y^2)
xy2^+
5+((1+2)*4)-3
512+4*+3-
Infix means normal mathematical expressions and Postfix means RPN
expressions
Stacks are used in the evaluation of RPN expressions

18. Advantages of RPN
RPN expressions do not need brackets to show the order of evaluation. This makes them simpler to evaluate by a machine
If you try to use an infix (normal) calculator, there will be a limit on the length of expression that can be entered
Postfix expressions can be evaluated as they are entered, they are not limited to a certain number of operators in an expression

19. Evaluating a Postfix Expression
The infix expression 5+((1+2)*4)-3 can be written like this in RPN: * + 3 -
Input
Operation
Stack
Comment 5
Push operand 1
5, 1 2
5, 1, 2 +
Add
5, 3
Pop two values (1, 2) and push result (3) 4
5, 3, 4 *
Multiply
5, 12
Pop two values (3, 4) and push result (12) 17
Pop two values (5, 12) and push result (17) 3
17, 3 -
Subtract 14
Pop two values (17, 3) and push result (14)

20. BNF is used by compiler writers to express the syntax of a
programming language. The syntax for part of one such
language is written in BNF as follows:
<expression> ::= <integer> | <integer> <operator> <expression>
<integer> ::= 0|1|2|3|4|5|6|7|8|9
<operator> ::= +|-|*|/
(a) Do the following expressions conform to this grammar?
(b) (i) Express the infix expression 5+6*2 in RPN.
(ii) Give one advantage of RPN
Expression
Yes/No 1
4*9 2
8+6/2 3
-6*2 4
(4+5)*5

21. Regular Expressions Class work
Complete Q3 Regular expressions in the worksheet.
Download and install the regular expression tool Regex Coach
and try the regular expressions: a+ a*
(ac)*
a(a|b)*
1(1│0)*0(1│0)*1

22. Class work and Homework
June 2010 COMP3 Q4
June 2011 COMP3 Q9
Watch this You Tube video which demonstrates how to convert the infix expression to postfix expression using a Stack
June 2011 COMP3 Q5
Hand-in Monday 23rd April 2012