1. Automata and Languages What do these have in common?
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2. Regular Expressions
The Finite State Machines that we have seen and regular expressions have equivalent power to express or recognize a language
What sort of languages can they accept? Or not accept?
How complicated may they be?
We now detour through formal languages
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3. Noam Chomsky Professor emeritus of linguistics at MIT
Developed a theory of generative grammars
This includes a language hierarchy
AKA Chomsky-Schützenberger Hierarchy
Includes recursively enumerable, context sensitive, context free and regular
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4. Recursively enumerable
Language Hierarchies
Type 3
Regular
Type 2
Context Free
Type 1
Context Sensitive
Type 0
Unrestricted or
Recursively enumerable
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5. Languages and Automata
Each of these languages corresponds to machine that can accept it
The weakest is a regular language, which can be accepted by a regular expression
Later machines correspond to stronger languages
Lets consider languages for a minute
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6. Formal Grammars
A grammar should be able to enumerate any legal sentence
Each grammar consists of four things
V – a finite set of non-terminals (aka variables)
T – a finite set of terminal symbols
Words made up from an alphabet
S – the start symbol
Must be an element of V
P – a set of productions
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7. C as an Example V – set of non-terminals T – set of terminals
Statement
Declaration
For-statement
T – set of terminals
Reserved words
Punctuation
Identifiers
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8. C example again S – Start symbol P – set of productions
Independently compilable part
Program
Function
Constant
P – set of productions
Rewrite rules
Start at the start symbol
End at terminals
Before we consider productions we must consider notation
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9. Copyright © 2003-2014 by Curt Hill
John Backus
Principle designer of FORTRAN
Substantial contributions to ALGOL 60
Designed Backus Normal Form
Eventually became a functional languages proponent
Turing award winner
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10. Copyright © 2003-2014 by Curt Hill
BNF
John Backus defined FORTRAN with a notation similar to Context Free languages independent of Chomsky in 1959
Peter Naur extended it slightly in describing ALGOL 60
Became known as BNF for Backus Normal Form or Backus Naur Form
A meta-language is any language that describes another language
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11. Copyright © 2003-2014 by Curt Hill
Simplest notation
Form of productions: LHS ::= RHS
Where:
LHS is a non-terminal (context free grammars)
RHS is any sequence of terminals and non-terminals, including empty
A common alternative to ::= is 
There can be many productions with exactly the same LHS, these are alternatives
If the RHS contains the LHS, the rule is recursive
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12. Copyright © 2003-2014 by Curt Hill
Notation
There is usually a simple way to distinguish terminals and non-terminals
Rosen and others enclose non-terminals in angle brackets
<if> ::= if ( <condition> ) <statement>
<if> ::= if ( <condition> ) <statement> else <statement>
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13. Copyright © 2003-2014 by Curt Hill
Simple extensions
Some times there is an alternation symbol that allows us to only need one production with the same LHS, often the vertical bar
<sign> ::= + | -
Some times things enclosed in [ and ] are optional, they may be present zero or one times
Some times things enclosed in { and } may be present 1 or more times
Thus [{x}] allows zero or more x items
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14. Copyright © 2003-2014 by Curt Hill
More
The extensions are often called EBNF
Syntax graphs are equivalent to EBNF
These tend to be more easy to read
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15. Syntax Graphs A circle represents a terminal
Reserved word or operator
No further definition
A rectangle represents a non-terminal
For statement or expression
Must be defined else where
An arrow represents the path between one item and another
The arrows may branch indicating alternatives
Recursion is also allowed
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16. Simple Expressions expression term + - term factor * / factor constant(
expression )
ident
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17. Productions
Productions may be represented as BNF, EBNF or syntax graphs
A production is a rewrite rule
We take a construction and find one way to rewrite it
In parsing we go from the distinguished symbol to any real program using application of these rewrite rules
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18. C For Production
For-statement ::= for ( expression; expression; expression) statement
This contains the terminals:
For ( ; )
Non-terminals
Expression
Statement
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19. Productions Again
Each non-terminal should have one or more productions that define it
Every non-terminal must have one or more productions
Multiple productions usually signify alternation
Recursion is allowed
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20. Recursion Productions may be recursive
Recall for-statement, here is Statement
Statement ::= expression ;
Statement ::= for-statement ;
Statement ::= if-statement ;
Statement ::= while-statement ;
Statement ::= compound-statement
Etc.
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21. Hierarchy Again Type Grammar Language Automata 3 Finite State Regular2
Context Free
Pushdown 1
Context Sensitive
Linear Bounded
Recursively enumerable
Unrestricted
Turing Machine
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22. How are these related?
Each of these grammars are related by how productions may be constructed
Regular are most restrictive
Unrestricted is the least restrictive
Lets compare
Upper case represent non-terminals
Lower case represent terminals
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23. Regular Grammars(3) A ::= b | A ::= bC | A ::= Cd
The production must have only one non-terminal on the left
The right-hand side must be:
A terminal
A terminal followed by a non-terminal
A non-terminal followed by a terminal
May not have a terminal non-terminal terminal on right
Terminal may lead or follow but not both
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24. Aside on Scanners
The first phase of a compiler is the lexical analyzer
AKA the scanner
It does the following:
Converts the source to a series of tokens
Removes comments and white space
The token stream is then used by the parser
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25. Scanners again A token could be: Parser inputs the stream of tokens
Any constant, usually typed
Any reserved word
Any punctuation mark
Any identifier
Parser inputs the stream of tokens
The scanner will often be just a finite state machine
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26. Context Free(2) A ::= aNy Single non-terminal on left
Any number or arrangement of non-terminals and terminals on the right
Most programming languages are largely context free
The optional else in C is not
These languages may be recognized by a pushdown machine
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27. Context Sensitive(1) x A y ::= x aNy y
Left hand side may have non-terminal surrounded by optional terminals
If terminals are present on left they must also be on right
Any number or arrangement of non-terminals and terminals on the right in between terminals
Recognized by linear bounded Turing machine
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28. Unrestricted(0) Anything on left and right
Terminals and non-terminals may be replaced by combinations of terminals and non-terminals in any combination
May be recognized by Turing machine
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29. Finally
It may seem strange that langauges and automata are related but they are
We find that most programming languages are context free
Sometimes with small exceptions
There are a number of table driven parsers for context free languages
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