FABER F10

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CDT314 FABER Formal Languages, Automata and
Models of Computation Lecture 5 Mälardalen
University 2010
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ContentMidterm solutionsRegular vs. Nonregular
Languages Context-Free LanguagesContext-Free
GrammarsDerivation Trees. AmbiguityPush-Down
Automata, PDAApplications
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A Nonregular Language

DFA must have infinite number of states.
States
are distinct for each
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Non-regular languages
Context-Free Languages
Regular Languages
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Automata theory formal languages and formal
grammars
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Context-Free Languages

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Context-Free Languages
Context-Free Grammars
Pushdown Automata
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Context-Free Grammars

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Example
A context-free grammar
A derivation
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A context-free grammar
Another derivation
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Example
A context-free grammar
A derivation
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A context-free grammar
Another derivation
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Example
A context-free grammar
A derivation
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A context-free grammar
A derivation
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Definition Context-Free Grammars
Grammar
Variables
Terminal symbols
Start variables
Productions of the form
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Definition Context-Free Languages

• A language is context-free
• if and only if there is a grammar with

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Derivation Order
Leftmost derivation
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Derivation Order
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Leftmost derivation
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Rightmost derivation
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Derivation Trees

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Derivation Tree
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Derivation Tree
yield
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Partial Derivation Trees
Partial derivation tree
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Partial derivation tree
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sentential form

Partial derivation tree
yield
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Sometimes, derivation order doesnt matter
Leftmost
Rightmost
Same derivation tree
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Ambiguity

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derivation
( denotes multiplication)
leftmost derivation
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derivation
leftmost derivation
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Two derivation trees
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The grammar
is ambiguous!
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is ambiguous
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Definition
A context-free grammar is ambiguous if
some string has two or more
derivation trees (two or more leftmost/rightmost
derivations)
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Why do we care about ambiguity?
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Why do we care about ambiguity?
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Why do we care about ambiguity?
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Ambiguity is bad for programming languages We
want to remove ambiguity!
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We fix the ambiguous grammar
by introducing parentheses () to indicate
grouping, (precedence)
Non-ambiguous grammar
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Unique derivation tree
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The grammar
is non-ambiguous
Every string has a unique
derivation tree
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Inherent Ambiguity

• Some context free languages
• have only ambiguous grammars!

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The string
has two derivation trees
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Definition Context-Free Grammars
Grammar
Variables
Terminal symbols
Start variables
Productions of the form
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Definition Regular Grammars
Grammar
Variables
Terminal symbols
Start variables
Right or Left Linear Grammars. Productions of the
form
or
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Non-regular languages
Context-Free Languages
Regular Languages
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Context-Free Languages
Pushdown Automata
Context-Free Grammars
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Pushdown AutomataPDAs

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Pushdown Automaton - PDA
Input String
Stack
States
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The Stack
A PDA can write symbols on stack and read them
later on.
POP reading symbol PUSH writing symbol
All access to the stack only on the top! (Stack
top is written leftmost in the string, e.g. yxz)
A stack is valuable as it can hold an unlimited
amount of information.
The stack allows pushdown automata to recognize
some non-regular languages.
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The States
Pop oldreading stack symbol
Push new writing stack symbol
Input symbol
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input
stack
top
Replace
(An alternative is to start and finish with empty
stack)
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input
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input
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Example 3.7 Salling A PDA for simple nested
parenthesis strings
Time 0
Input
Stack
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Example 3.7
Time 1
Input
Stack
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Example 3.7
Time 2
Input
Stack
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Example 3.7
Time 3
Input
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Example 3.7
Time 4
Input
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Example 3.7
Time 5
Input
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Example 3.7
Time 6
Input
Stack
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Example 3.7
Time 7
Input
Stack
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ApplicationsCompilers

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Machine Code
Program
Add v,v,0 cmp v,5 jmplt ELSE THEN add x,
12,v ELSE WHILE cmp x,3 ...
v 5 if (vgt5) x 12 v while (x !3)
x x - 3 v 10 ......
Compiler
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Compiler
Lexical analyzer
parser
input
output
machine code
program
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A parser knows the grammar of the programming
language
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Parser
PROGRAM STMT_LIST STMT_LIST STMT
STMT_LIST STMT STMT EXPR IF_STMT
WHILE_STMT
STMT_LIST EXPR EXPR EXPR EXPR - EXPR
ID IF_STMT if (EXPR) then STMT
if (EXPR) then STMT else STMT WHILE_STMT
while (EXPR) do STMT
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The parser finds the derivation of a particular
input
derivation
Parser
input
E ? E E ? E E E ? 10 EE ? 10
2 E ? 10 2 5
E ? E E E E INT
10 2 5
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derivation
derivation tree
E ? E E ? E E E ? 10 EE ? 10
2 E ? 10 2 5
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derivation tree
machine code
mult a, 2, 5 add b, 10, a
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Parsing

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Parser
input string
derivation
grammar
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Example
Parser
derivation
input
?
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Exhaustive Search
Phase 1
Find derivation of
All possible derivations of length 1
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Phase 2
Phase 1
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Phase 2
Phase 3
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Final result of exhaustive search
(top-down parsing)
Parser
input
derivation
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Context Free Arthttp//www.contextfreeart.org/
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Context Free Arthttp//www.contextfreeart.org/