How to convert a left linear grammar to a right linear grammar

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How to convert a left linear
grammar to a right linear grammar
Roger L. Costello
May 28, 2014
Objective
This mini-tutorial will answer these questions:
1. What is a linear grammar? What is a left linear
grammar? What is a right linear grammar?
2
Objective
This mini-tutorial will answer these questions:
1. What is a linear grammar? What is a left linear
grammar? What is a right linear grammar?
2. Left linear grammars are evil – why?
3
Objective
This mini-tutorial will answer these questions:
1. What is a linear grammar? What is a left linear
grammar? What is a right linear grammar?
2. Left linear grammars are evil – why?
3. What algorithm can be used to convert a left
linear grammar to a right linear grammar?
4
Linear grammar
• A linear grammar is a context-free grammar
that has at most one non-terminal symbol on
the right hand side of each grammar rule.
– A rule may have just terminal symbols on the right
hand side (zero non-terminals).
• Here is a linear grammar:
S → aA
A → aBb
B → Bb
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Left linear grammar
• A left linear grammar is a linear grammar in
which the non-terminal symbol always occurs
on the left side.
• Here is a left linear grammar:
S → Aa
A → ab
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Right linear grammar
• A right linear grammar is a linear grammar in
which the non-terminal symbol always occurs
on the right side.
• Here is a right linear grammar:
S → abaA
A→ε
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Left linear grammars are evil
• Consider this rule from a left linear grammar:
A → Babc
• Can that rule be used to recognize this string:
abbabc
• We need to check the rule for B:
B → Cb | D
• Now we need to check the rules for C and D.
• This is very complicated.
• Left linear grammars require complex parsers.
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Right linear grammars are good
• Consider this rule from a right linear grammar:
A → abcB
• Can that rule be used to recognize this string:
abcabb
• We immediately see that the first part of the
string – abc – matches the first part of the rule.
Thus, the problem simplifies to this: can the rule
for B be used to recognize this string :
abb
• Parsers for right linear grammars are much
simpler.
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Convert left linear to right linear
Now we will see an algorithm for converting any
left linear grammar to its equivalent right linear
grammar.
left linear
right linear
S → Aa
A → ab
S → abaA
A→ε
Both grammars generate this language: {aba}
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May need to make a new start symbol
The algorithm on the following slides assume
that the left linear grammar doesn’t have any
rules with the start symbol on the right hand
side.
– If the left linear grammar has a rule with the start
symbol S on the right hand side, simply add this
rule:
S0 → S
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Symbols used by the algorithm
•
•
•
•
Let S denote the start symbol
Let A, B denote non-terminal symbols
Let p denote zero or more terminal symbols
Let ε denote the empty symbol
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Algorithm
1) If the left linear grammar has a rule S → p, then
make that a rule in the right linear grammar
2) If the left linear grammar has a rule A → p, then add
the following rule to the right linear grammar:
S → pA
3) If the left linear grammar has a rule B → Ap, add the
following rule to the right linear grammar:
A → pB
4) If the left linear grammar has a rule S → Ap, then
add the following rule to the right linear grammar:
A → p
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Convert this left linear grammar
left linear
S → Aa
A → ab
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Right hand side has terminals
left linear
right linear
S → Aa
A → ab
S → abA
2) If the left linear grammar has this rule A → p,
then add the following rule to the right linear
grammar: S → pA
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Right hand side of S has non-terminal
left linear
right linear
S → Aa
A → ab
S → abA
A→a
4) If the left linear grammar has S → Ap, then
add the following rule to the right linear
grammar: A → p
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Equivalent!
left linear
right linear
S → Aa
A → ab
S → abA
A→a
Both grammars generate this language: {aba}
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Convert this left linear grammar
original grammar
S → Ab
S → Sb
A → Aa
A→a
left linear
make a new
start symbol
S0 → S
S → Ab
S → Sb
A → Aa
A→a
Convert this
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Right hand side has terminals
left linear
right linear
S0 → S
S → Ab
S → Sb
A → Aa
A→a
S0 → aA
2) If the left linear grammar has this rule A → p,
then add the following rule to the right linear
grammar: S → pA
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Right hand side has non-terminal
left linear
right linear
S0 → S
S → Ab
S → Sb
A → Aa
A→a
S0 → aA
A → bS
A → aA
S → bS
3) If the left linear grammar has a rule B → Ap, add the
following rule to the right linear grammar: A → pB
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Right hand side of start symbol has
non-terminal
left linear
right linear
S0 → S
S → Ab
S → Sb
A → Aa
A→a
S0 → aA
A → bS
A → aA
S → bS
S→ε
4) If the left linear grammar has S → Ap, then
add the following rule to the right linear
grammar: A → p
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Equivalent!
left linear
right linear
S0 → S
S → Ab
S → Sb
A → Aa
A→a
S0 → aA
A → bS
A → aA
S → bS
S→ε
Both grammars generate this language: {a+b+}
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Will the algorithm always work?
• We have seen two examples where the
algorithm creates a right linear grammar that
is equivalent to the left linear grammar.
• But will the algorithm always produce an
equivalent grammar?
• Yes! The following slide shows why.
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Generate string p
• Let p = a string generated by the left linear
grammar.
• We will show that the grammar generated by
the algorithm also produces p.
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Case 1: the start symbol produces p
Suppose the left linear grammar has this rule:
S → p. Then the right linear grammar will
have the same rule (see 1 below). So the right
linear grammar will also produce p.
Algorithm:
1)
2)
3)
4)
If the left linear grammar contains S → p, then put that rule in the right linear grammar.
If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
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Case 2: multiple rules needed
to produce p
Suppose p is produced by a sequence of n
production rules:
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
p (p is composed of n symbols)
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
Let’s see what right linear rules will be generated
by the algorithm for the rules implied by this
production sequence.
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Algorithm inputs and outputs
left linear rules
algorithm
right linear rules
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
S → A1 p 1
algorithm
A1 → p1 (see 4 below)
Algorithm:
1)
2)
3)
4)
If the left linear grammar contains S → p, then put that rule in the right linear grammar.
If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
A1 → A2 p 2
algorithm
A2 → p2A1 (see 3 below)
Algorithm:
1)
2)
3)
4)
If the left linear grammar contains S → p, then put that rule in the right linear grammar.
If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
S → A1 p 1
A1 → A2 p 2
algorithm
A1 → p 1
A2 → p 2 A 1
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Case 2 (continued)
S → A1 p 1
A1 → A2 p 2
algorithm
A1 → p 1
A2 → p 2 A1
From A2 we obtain p2p1:
A2 → p 2 A1
→ p2 p1
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
A2 → A3 p 3
algorithm
A3 → p3A2 (see 3 below)
Algorithm:
1)
2)
3)
4)
If the left linear grammar contains S → p, then put that rule in the right linear grammar.
If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
S → A1 p 1
A1 → A2 p 2
A2 → A3 p 3
algorithm
A1 → p 1
A2 → p 2 A 1
A3 → p 3 A 2
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Case 2 (continued)
S → A1 p 1
A 1 → A2 p 2
A 2 → A3 p 3
algorithm
A 1 → p1
A 2 → p 2 A1
A 3 → p 3 A2
From A3 we obtain p3p2p1:
A3 → p 3 A2
→ p 3 p 2 A1
→ p3p2 p1
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Case 2 (continued)
S → A1 p 1
→ A2 p 2 p 1
→ A3 p 3 p 2 p 1
→…
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
An-1 → pn
algorithm
S → pnAn-1 (see 2 below)
Algorithm:
1)
2)
3)
4)
If the left linear grammar contains S → p, then put that rule in the right linear grammar.
If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
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Case 2 (concluded)
S → A1 p 1
A1 → A2 p 2
A2 → A3 p 3
…
An-1 → pn
algorithm
A1 → p 1
A2 → p 2 A1
A3 → p 3 A2
…
An-1 → pn-1An-2
S → pnAn-1
From S we obtain pn…p2p1:
S → pnAn-1
→ pnpn-1An-2
→…
→ pnpn-1…p3A2
→ pnpn-1…p3p2A1
→ pn…p3pn-1p2p1 (this is the desired string, p)
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We have shown that for any string p
generated by the left linear grammar,
the right linear grammar produced by
the algorithm will also generate p.
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How we understand the algorithm
S → A1 p 1
A1 → A2 p 2
A2 → A3 p 3
…
An-1 → pn
These rules descend through the non-terminals until
reaching a rule with terminals on the RHS, the terminals
are output, then we unwind from the descent and output
the terminals.
algorithm
A1 → p 1
A2 → p 2 A1
A3 → p 3 A2
…
An-1 → pn-1An-2
S → pnAn-1
Make the rule with terminals on the RHS the starting rule
and output its terminals. Ascend through the other rules.
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How we understand the algorithm
S → A1 p 1
A1 → A2 p 2
A2 → A3 p 3
…
An-1 → pn
algorithm
If the left linear grammar contains A → p,
then put this rule in the right linear grammar:
S → pA
A1 → p 1
A2 → p 2 A1
A3 → p 3 A2
…
An-1 → pn-1An-2
S → pnAn-1
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How we understand the algorithm
S → A1 p 1
A1 → A2 p 2
A2 → A3 p 3
…
An-1 → pn
algorithm
If the left linear grammar contains B → Ap,
then put this rule in the right linear grammar:
A → pB
A1 → p 1
A2 → p 2 A1
A3 → p 3 A2
…
An-1 → pn-1An-2
S → pnAn-1
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How we understand the algorithm
S → A1 p 1
A1 → A2 p 2
A2 → A3 p 3
…
An-1 → pn
If the left linear grammar contains S → Ap,
then put this rule in the right linear grammar:
A→p
algorithm
A1 → p 1
A2 → p 2 A1
A3 → p 3 A2
…
An-1 → pn-1An-2
S → pnAn-1
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Left-linear grammars generate
Type 3 languages
• Every left-linear grammar can be converted to
an equivalent right-linear grammar.
– “Equivalent right-linear grammar” means the
grammar generate the same language.
• Right-linear grammars generate Type 3
languages.
• Therefore, every left-linear grammar
generates a Type 3 language.
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Acknowledgement
The algorithm shown in these slides comes from
the wonderful book:
Introduction to Formal Languages by Gyorgy Revesz
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