Post on 03-Jan-2016
Copyright © 2003-2014 by Curt Hill
Grammar Types
The Chomsky HierarchyBNF and Derivation Trees
Introduction
• We are now familiar with the notion of a grammar and the language that it covers
• Next we wish to categorize grammars– This will be based on the forms that
the productions take
• We will start with the simplest and work up
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Chomsky Hierarchy
• Chomsky proposed an hierarchy of languages based on the strength of the rewriting rules
• There are four– Type 0 through Type 3
• The hierarchy is based on the strength of the rewriting rules
• Type 0 is strongest, 3 is weakest
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Type 3 - Regular Languages
• U n or U Wn• U and W are non-terminals and n
is a terminal• A non-terminal may only be
replaced by a terminal or non-terminal followed by a terminal
• Regular expressions are of this type– Do you know about regular expressions?
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Regular (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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Type 2 - Context Free• 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
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Type 1 - Context Sensitive
• xUy xvy• Where U is a non-terminal and v is
any sequence of terminals and/or non-terminals– x, y are terminals
• U may be rewritten to v only in the context of x and y before and after
• We may have another rule aUb aeb which is completely different replacement of U
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Type 0 - Unrestricted
• u v• Unrestricted both sides of the
production may have non-terminals or terminals, but u cannot be empty
• Unlike types 1-3 u could be a terminal
• Context is also important• Very powerful, very little work
done with it
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Language Hierarchies
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Type 3 Regular
Type 2 Context Free
Type 1 Context Sensitive
Type 0 Unrestricted
Languages and Automata
• Each of these languages corresponds to an automaton that can accept it
• The weakest is a regular language, which can be accepted by a regular expression or finite state automaton
• Later machines correspond to stronger languages
• We will consider these automatons later
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Hierarchy Again
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Type
Grammar Language Automata
3 Finite State Regular Finite
2 Context Free Context Free
Pushdown
1 Context Sensitive Context Sensitive
Linear Bounded
0 Recursively enumerable
Unrestricted Turing Machine
Again• We use regular (type 3) languages
are used for lexical analyzers– The lexical analyzer is typically the
front-end of a compiler
• Most programing languages have a context-free grammar (type 2) – With a few ambiguities
• Efficient algorithms exist to implement parsers for both of these – This cannot be said for type 0 and 1
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Derivation or parse trees
• A multi-way tree where:– Each interior node is a non-
terminal– Each leaf is a terminal– The start symbol is the root– Nested under each interior node
is the RHS of the production, with the LHS being the node itself
• This is a handy data structure for compilers and the like
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Example Parse Tree
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program
stmts
stmt
var expr =
term term = a
b
constvar
Example• Consider the following grammar• V= {a,b,c,S}• T = {a,b,c}• P = {
– S abS– S bcS– S bbS– S a– S cb
}
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bcbba
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S
bc
b
S
b
S
a
S bcS
S bbS
S a
Audience Participation
• Lets try on the board• bcabbbbbcb• Bbbcbba
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John Backus
• Principle designer of FORTRAN• Substantial contributions to
Algol60• Designed Backus Normal Form• Eventually became a functional
languages proponent• Turing award winner
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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
• Became known as BNF for Backus Normal Form or Backus Naur Form
• Meta-language is the language that describes another language
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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
• 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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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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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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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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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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Simple Expressions
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expressionterm
+
-term
factor*
/factor
constant ident ( )expression
Parse tree example
• Trees are recursive• Every sub-tree is a tree itself• Consider the parse of:
2 + 5 * ( 3 - 4 )– Using the previous syntax graph
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Expression: 2 + 5 * (3 – 4)
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term -
factor
3
term
factor
4
expression
*factor
5
termterm +
factor
2
expression
factor
( )
BNF is generative• A derivation is sentence generation• Leftmost derivation
– Only the leftmost non-terminal can be rewritten
– This is usually the kind of derivation used by compilers
– The previous derivation was leftmost
• There are also rightmost derivations
• The order of derivation does not affect the language defined
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Example BNF productions
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<program> ::= <stmts><stmts> ::= <stmt> | <stmt> ; <stmts><stmt> ::= <var> = <expr><var> ::= a | b | c | d<expr> ::= <term> + <term> | <term> - <term><term> ::= <var> | const
Example Derivation
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<program> => <stmts> => <stmt> => <var> = <expr> => a = <expr> => a = <term> + <term> => a = <var> + <term> => a = b + <term> => a = b + const
Exercises
• 13.1 b– 1, 5, 13, 19, 25, 35
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