1. Context-free Grammars
cs466(Prasad)
L4CFG

2. Recursive definition
Grammar
Derivation
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3. What is the syntax of the description (meta-language)?
External form of the grammar
How do we associate a language with such a description?
Semantics or use of the grammar
Generators: Grammars specify languages by generating strings in the language.
Recognizers: Grammars can be viewed as a notation for describing a family of recognition algorithms.
cs466(Prasad)
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4. CFG = (V, ,P,S) V : Finite set of variables/non-terminals
: Alphabet/Finite set of terminals
P : Finite set of rules/productions
S : Start symbol
cs466(Prasad)
L4CFG

5. A-rule, null-rule, -rule.
a* represents a context-free language because we can write a CFG for it.
A-rule, null-rule, -rule.
Context-freeness: An A-rule can be applied whenever A occurs in a string, irrespective of the context (that is, non-terminals and terminals around A).
Cf. context-sensitive grammar (“declare-use”)
cs466(Prasad)
L4CFG

6. From CFG to Language One-step Derivation (Defn: 3.1.2)
w is derivable from v in CFG, if there is a finite sequence of rule applications such that:
cs466(Prasad)
L4CFG

7. Let G=(V, S, P, S) be a CFG. is a sentential form, if .
is a sentence, if
The language of G,
L(G) =
Normally, parenthesis is used in the meta-language for grouping. Parenthesis can be used as a terminal too: “(“ for a and “)” for b.
cs466(Prasad)
L4CFG

8. Derivation of aabb
Derivation tree S a b S a b l
cs466(Prasad)
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9. Factor left-hand side using syntax for alternatives |.
cs466(Prasad)
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10. Derivation Tree (abstracts derivation)l l
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11. Examples: CFGs and CFLs
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12. Constructive definitions : systematic buldup vs declarative definitions : generate and test
Same numeric variable = equality constraint; Different variables independence; TEMPLATE for sequencing of letters
cs466(Prasad)
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13. are part of the meta-language.
Parenthesis is a part of the meta-language here.
Note that parentheses
are part of the meta-language.
cs466(Prasad)
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14. CFG: Ease of understanding vs RG: Efficient left to right generation of string
cs466(Prasad)
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15. cs466(Prasad) L4CFG Pi is concatenation.
Same numeric variable = equality constraint; Different variables independence; TEMPLATE for sequencing of letters
cs466(Prasad)
L4CFG

16. Left to right generation of string.
CFG: Ease of understanding vs RG: Efficient left to right generation of string
Left to right generation of string.
cs466(Prasad)
L4CFG

17. Regular Grammar
A regular grammar is a CFG in which each of the rules has the form:
(Right Linear Grammar)
Left linear grammar A-> Ba
Cannot mix A->Ba and A->aB rules in regular grammars.
cs466(Prasad)
L4CFG

18. Equivalent Regular Grammar
A regular grammar is a CFG in which each of the rules has the form:
(Left Linear Grammar)
Right linear grammar A-> aB
Cannot mix A->Ba and A->aB rules in regular grammars.
cs466(Prasad)
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19. Extended regular grammar RHS begins with a string of terminals.
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20. Non-terminals are delimited by angular brackets
Non-terminals are delimited by angular brackets. (cf: complement language, DFA)
cs466(Prasad)
L4CFG

21. Derivations
Note that the above result does not hold for sentential forms.
S => B does not have a leftmost derivation
cs466(Prasad)
L4CFG