Simplifying CFL

1
Simplifying CFLs

• Reading 6.1

2
Grammar Simplification

• We want to change the production rule styles of
  CFLs to prove properties.
• Simplification removing undesirable types of
  production rules without changing the language
  that the grammar defines.
• Simplification does not always result in fewer
  rules!

3
Removing Lambda

• We want lambda-free grammars, so that we dont
  deal with special empty cases.
• We can write lambda-free grammars for any
  language that does not contain the string lambda.
• Otherwise, consider the language L ?

4
Removing Production Rules

• If you want to remove a rule of a CFL that has
  only one variable on the right,
• Exchange the rule with a set of rules that
  inserts all one-step derivations of the variable
  on the right.
• A -gt a aaA abBc
• B -gt abbA b

A -gt a aaA ababbAc abbc
5
Nullable Variables

• Nullable Variables are those that can be replaced
  with lambda.
• We often want to remove lambda production rules
• Doing so makes a grammar with no nullable
  variables

6
Algorithm for removing lambda and Nullable
Variables

• For any production rule in the grammar of the
  form
• A -gt ?
• Replace all right-hand occurrences of A with an
  option that has A replaced with ?.

7
Example Removing lambda

• S -gt aAb
• A -gt aAb ?
• Becomes
• S -gt aAb ab
• A -gt aAb ab

8
Example Removing Lambda
Iteration 1 (B) S -gt ABaC AaC A -gt BC C B -gt
b C -gt D ? D -gt d
Iteration 2 (C) S -gt ABaC AaC ABa Aa A -gt
BC C B ? B -gt b C -gt D D -gt d

• S -gt ABaC
• A -gt BC
• B -gt b ?
• C -gt D ?
• D -gt d

Iteration 3 (A) S -gt ABaC AaC ABa Aa BaC
aC Ba a A -gt BC C B B -gt b C
-gt D D -gt d
9
Useful Variables

• A variable in a grammar is said to be useful if
  there is at least one string w in the language
  such that
• S xAy w
• where x y are any sequences of variables and
  terminals.
• A variable is useful if it is used in the
  derivation of at least one string.
• Otherwise, it is useless.

10
Finding Useless Variables

• A variable may be useless if there is no way to
  get a terminal string from it
• S -gt aSb a A
• A -gt aA
• A is useless in this example, because it can
  never lead to terminals.

11
Finding Useless Variables

• A variable might be useless if there is no way to
  reach it from the start symbol.
• S -gt A
• A -gt aA a
• B -gt bA
• B is useless there is no other production rule
  that uses it.

12
Dependency Graph

• A way to tell if a variable is useless because it
  is unreachable from S.
• Make a node for every variable.
• Place arcs from node A to node B if B appears on
  the right of a production of A.
• A useless variable is any that cannot be reached
  in the graph from S.

13
Dependency Graph Example

• S -gt aaA ab
• A -gt bC
• B -gt Cab
• C - gt a bb
• B is useless because it is unreachable from S.

A
S
C
B
14
Algorithm for Removing All Useless Variables
(Part 1)

• Make a set V1, initialize to
• Loop
• Look at each variable V in turn
• Look at each of Vs production rules
• If everything on the right of the production rule
  is either a terminal or a variable in V1, add V
  to V1
• Continue Loop until the iteration adds no
  variables to V1.
• Keep all production rules whose variables are all
  in V1.

15
Algorithm for Removing All Useless Variables
(Part 2)

• Part 1 kept all variables that result in terminal
  strings.
• Part 2 will keep all variables reachable from the
  start symbol.
• Draw the dependency graph for the new grammar.
• Keep only the production rules in which all
  variables can be reached in the dependency graph.

16
Example

• Remove all useless symbols and productions from
  the following grammar
• S -gt aS A C
• A -gt a
• B -gt aa
• C -gt aCb

17
Step 1
Start with V1 Iteration 1 V1
A,B Iteration 2 V1 A,B,S Iteration 3 V1
A,B,S

• S -gt aS A C
• A -gt a
• B -gt aa
• C -gt aCb

18
Step 2

• S -gt aS A
• A -gt a
• B -gt aa

A
S
B
Final Grammar S -gt aS A A -gt a
19
Unit Productions

• Any production where one variable is replaced
  with another is a unit production.
• Any rule of the form
• A -gt B
• Unit production rules are sometimes undesirable.
  (Exhaustive Search Parsing)
• A -gt A can be removed with no effect.

20
Removing Unit Production Rules

• Part one Draw a unit-rule dependency graph.
  That is, a dependency graph based only on unit
  production rules.
• Find pairs A-gt B such that there is a walk in the
  graph from A to B.
• Part two For each pair A -gt B, add an option
  which substitutes non-unit resolutions of B to
  the right hand side of A.

21
Removing Unit Productions Example

• S -gt Aa B
• B -gt A bb
• A -gt a bc B

S
A
B
Pairs are S-gt BS -gt AA -gt BB -gt A
22
Removing Unit Rules

• Original S -gt Aa B Minus Units S-gt Aa
• B -gt A bb
  B -gt bb
• A -gt a bc B A -gt a
  bcFor each pair, add substitutions from the
  Minus Grammar
• (S -gt A)
• S -gt Aa a bc
• (S -gt B)
• S -gt Aa a bc bb

(B -gt A)B -gt bb a bc (A -gt B)A -gt a bc
bb
23
Removing Units

• Final Grammar is
• S -gt Aa a bc bb
• B -gt bb a bc
• A -gt a bc bb

24
Complete Simplification

1. Remove lambda productions
2. Remove unit productions
3. Remove useless productions