3.b V = {S, A, B, C, D}, E = {a,b,c}, S = S, R = S →A | € A →BC B→BD | bb C→CD cc D→B|C 4 CNF Step 4 Perform step four of converting the following CFG into CNF by removing remaining rules. V={S, A, B}, Σ = {a,b}, S = S, R=
Converting a Context-Free Grammar (CFG) to Chomsky Normal Form (CNF) involves transforming the CFG into a specific form where all productions are either of the form A -> BC (where A, B, and C are non-terminals) or A -> a (where A is a non-terminal, and 'a' is a terminal). CNF simplifies the structure of the grammar and is useful for various parsing algorithms and formal language analysis.
Here's a step-by-step explanation of how to convert a CFG to CNF:
1. Remove ε-productions (productions that derive the empty string):
If the CFG has ε-productions (productions of the form A -> ε), you need to remove them. This step ensures that the CNF doesn't allow generating the empty string.
2. Eliminate unit rules (productions of the form A -> B):
If your CFG contains unit rules, you should eliminate them. A unit rule is a production where a non-terminal A directly produces another non-terminal B. You can replace them with the set of productions that B generates.
3. Remove useless symbols (unreachable or non-generating symbols):
Identify and remove any non-terminals that are unreachable from the start symbol S or non-terminals that cannot derive any terminal string.
4. Replace terminals in productions with new non-terminals:
Ensure that each production's right-hand side contains only terminals or non-terminals. If a production contains a terminal 'a,' introduce a new non-terminal Aa, and add the production Aa -> a.
5. Break down productions into binary productions (A -> BC):
Ensure that each production has at most two symbols on the right-hand side. If a production has more than two symbols, break it down into a series of binary productions. For example, if you have a production A -> XYZ, introduce new non-terminals P, Q, and R and rewrite it as follows:
- A -> PQ
- P -> XY
- Q -> Z
6. Repeat step 5 until all productions are binary (A -> BC):
Continue the process of breaking down productions with more than two symbols on the right-hand side into binary productions until all productions in the grammar satisfy the CNF criteria.
7. Final result:
The final result is a CFG in Chomsky Normal Form. All productions are either of the form A -> BC or A -> a (where A, B, C are non-terminals, and 'a' is a terminal).
Chomsky Normal Form is a specific representation that simplifies the parsing process and analysis of context-free languages, making it a fundamental step in the study of formal language theory and parsing algorithms.
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