The document describes an algorithm for converting a left linear grammar to an equivalent right linear grammar in 4 steps: 1. If the left grammar has a rule with terminals on the right-hand side, add it to the right grammar. 2. If a rule has a non-terminal followed by terminals, make the non-terminal the start and add the terminals. 3. If a rule has a non-terminal preceding another non-terminal and terminals, make the first non-terminal produce the second and terminals. 4. If the start rule has a non-terminal and terminals, make the non-terminal produce the terminals. The algorithm is guaranteed to produce an equivalent right linear grammar

1. How to convert a left linear
grammar to a right linear grammar
Roger L. Costello
May 28, 2014

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

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

4. 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

5. 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:
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S → aA
A → aBb
B → Bb

6. 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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7. 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
7

8. 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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9. 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.
9

10. Convert left linear to right linear
Now we will see an algorithm for converting any
left linear grammar to its equivalent right linear
grammar.
S → Aa
A → ab
left linear
Both grammars generate this language: {aba}
S → abaA
A → ε
right linear
10

11. 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
11

12. 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
12

13. 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
13

14. Convert this left linear grammar
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S → Aa
A → ab
left linear

15. Right hand side has terminals
15
S → Aa
A → ab
left linear
2) If the left linear grammar has this rule A → p,
then add the following rule to the right linear
grammar: S → pA
S → abA
right linear

16. Right hand side of S has non-terminal
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S → Aa
A → ab
left linear
4) If the left linear grammar has S → Ap, then
add the following rule to the right linear
grammar: A → p
S → abA
A → a
right linear

17. Equivalent!
17
S → Aa
A → ab
left linear
Both grammars generate this language: {aba}
S → abA
A → a
right linear

18. Convert this left linear grammar
18
left linear
S0 → S
S → Ab
S → Sb
A → Aa
A → a
S → Ab
S → Sb
A → Aa
A → a
original grammar
make a new
start symbol
Convert this

19. Right hand side has terminals
19
S0 → S
S → Ab
S → Sb
A → Aa
A → a
left linear
S0 → aA
right linear
2) If the left linear grammar has this rule A → p,
then add the following rule to the right linear
grammar: S → pA

20. Right hand side has non-terminal
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S0 → S
S → Ab
S → Sb
A → Aa
A → a
left linear
S0 → aA
A → bS
A → aA
S → bS
right linear
3) If the left linear grammar has a rule B → Ap, add the
following rule to the right linear grammar: A → pB

21. Right hand side of start symbol has
non-terminal
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S0 → S
S → Ab
S → Sb
A → Aa
A → a
left linear
S0 → aA
A → bS
A → aA
S → bS
S → ε
right linear
4) If the left linear grammar has S → Ap, then
add the following rule to the right linear
grammar: A → p

22. Equivalent!
22
S0 → S
S → Ab
S → Sb
A → Aa
A → a
left linear
S0 → aA
A → bS
A → aA
S → bS
S → ε
right linear
Both grammars generate this language: {a+b+}

23. 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.
23

24. 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.
24

25. 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.
1) If the left linear grammar contains S → p, then put that rule in the right linear grammar.
2) If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
3) If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
4) If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
Algorithm:
25

26. Case 2: multiple rules needed
to produce p
Suppose p is produced by a sequence of n
production rules:
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1 p (p is composed of n symbols)
26

27. Case 2 (continued)
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ 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.
27

28. Algorithm inputs and outputs
algorithm
left linear rules
right linear rules
28

29. Case 2 (continued)
1) If the left linear grammar contains S → p, then put that rule in the right linear grammar.
2) If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
3) If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
4) If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
Algorithm:
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
S → A1p1
algorithm
A1 → p1 (see 4 below)
29

30. Case 2 (continued)
1) If the left linear grammar contains S → p, then put that rule in the right linear grammar.
2) If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
3) If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
4) If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
Algorithm:
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
A1 → A2p2
algorithm
A2 → p2A1 (see 3 below)
30

31. Case 2 (continued)
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
S → A1p1
A1 → A2p2
algorithm
A1 → p1
A2 → p2A1
31

32. Case 2 (continued)
S → A1p1
A1 → A2p2
algorithm
A1 → p1
A2 → p2A1
From A2 we obtain p2p1:
A2 → p2A1
→ p2p1
32

33. Case 2 (continued)
1) If the left linear grammar contains S → p, then put that rule in the right linear grammar.
2) If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
3) If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
4) If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
Algorithm:
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
A2 → A3p3
algorithm
A3 → p3A2 (see 3 below)
33

34. Case 2 (continued)
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
S → A1p1
A1 → A2p2
A2 → A3p3
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
34

35. Case 2 (continued)
S → A1p1
A1 → A2p2
A2 → A3p3
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
From A3 we obtain p3p2p1:
A3 → p3A2
→ p3p2A1
→ p3p2p1
35

36. Case 2 (continued)
1) If the left linear grammar contains S → p, then put that rule in the right linear grammar.
2) If the left linear grammar contains A → p, then put this rule in the right linear grammar: S → pA
3) If the left linear grammar contains B → Ap, then put this rule in the right linear grammar: A → pB
4) If the left linear grammar contains S → Ap, then put this rule in the right linear grammar: A → p
Algorithm:
S → A1p1
→ A2p2p1
→ A3p3p2p1
→ …
→ An-1pn-1…p3p2p1
→ pnpn-1…p3p2p1
An-1 → pn
algorithm
S → pnAn-1 (see 2 below)
36

37. Case 2 (concluded)
S → A1p1
A1 → A2p2
A2 → A3p3
…
An-1 → pn
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
…
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)
37

38. 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.
38

39. How we understand the algorithm
S → A1p1
A1 → A2p2
A2 → A3p3
…
An-1 → pn
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
…
An-1 → pn-1An-2
S → pnAn-1
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.
Make the rule with terminals on the RHS the starting rule
and output its terminals. Ascend through the other rules.
39

40. How we understand the algorithm
S → A1p1
A1 → A2p2
A2 → A3p3
…
An-1 → pn
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
…
An-1 → pn-1An-2
S → pnAn-1
If the left linear grammar contains A → p,
then put this rule in the right linear grammar:
S → pA
40

41. How we understand the algorithm
S → A1p1
A1 → A2p2
A2 → A3p3
…
An-1 → pn
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
…
An-1 → pn-1An-2
S → pnAn-1
If the left linear grammar contains B → Ap,
then put this rule in the right linear grammar:
A → pB
41

42. How we understand the algorithm
S → A1p1
A1 → A2p2
A2 → A3p3
…
An-1 → pn
algorithm
A1 → p1
A2 → p2A1
A3 → p3A2
…
An-1 → pn-1An-2
S → pnAn-1
If the left linear grammar contains S → Ap,
then put this rule in the right linear grammar:
A → p
42

43. 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.
43

44. Acknowledgement
The algorithm shown in these slides comes from
the wonderful book:
Introduction to Formal Languages by Gyorgy Revesz
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