DETERMINISTIC CONTEXT FREE LANGUAGES
DETERMINISTIC CONTEXT FREE
LANGUAGES
Ethakota Krishnamurthi Surendranath
Manoj Gujjari
• What is a CFL(Context Free Language)?
• Deterministic Context Free Language
– Description
– Properties
– Importance
CONTEXT FREE LANGUAGES
• A context-free language (CFL) is a language generated by
some context-free grammar (CFG).
• Different CF grammars can generate the same CF language, or
conversely, a given CF language can be generated by different
CF grammars.
• The class of context-free languages generalizes the class of
regular languages, i.e., every regular language is a contextfree language.
• The reverse of this is not true, i.e., every context-free
language is not necessarily regular.
DCFL
• Deterministic context-free languages are a proper
subset of context-free languages.
• Context free languages that can be accepted by Deterministic
Push-Down Automata.
• DCFLs are always unambiguous.
• Many programming languages can be described by means of
DCFLs.
DCFL Description
Consider $: end of string marker
Consider an example:
Let L = a* {anbn : n > 0}.
When it begins reading a’s, M must push them onto the stack.
In case there are going to be b’s it runs out of input without seeing b’s,
it needs a way to pop the a’s from the stack before it can accept.
The end-of-string marker allows the popping to happen, when all the
input has been read.
Add $ to make it easier to build DPDAs, it does not add power (to allow
building a PDA for L that was not context-free)
There exist CLFs that are not deterministic, e.g., L = {aibjck, i j or j k}.
A DETERMINISTIC PDA FOR L
L = a* {anbn : n > 0}.
A NDPDA FOR L( No end-marker)
L = a* {anbn : n > 0}.
Properties of DCFL
• DCFLS are not closed under union.
• DCFLS are not closed under intersection.
• DCFLS are closed under complement.
DCFLS are not closed under UNION
L1 = {aibjck, i, j, k 0 and i j}.
L2 = {aibjck, i, j, k 0 and j k}.
(a DCFL)
(a DCFL)
L = L1 L2.
= {aibjck, i, j, k 0 and (i j) or (j k)}.
L = L.
= {aibjck, i, j, k 0 and i = j = k}
{w {a, b, c}* : the letters are out of order}.
L = L a*b*c*.
= {anbncn, n 0}.
L is not even CF, much less DCF.
DCFLS are not closed under Intersection
L1 = {aibjck, i, j, k 0 and i = j}.
L2 = {aibjck, i, j, k 0 and j = k}.
L = L1 L2
=
L1 and L2 are deterministic context-free:
CFL HIERARCHY
References
• Automata, Computability and Complexity, Elaine Rich.
• http://en.wikipedia.org/wiki/Deterministic_contextfree_language
• http://en.wikipedia.org/wiki/Context-free_language
• http://en.wikipedia.org/wiki/Unambiguous_grammar
QUESTIONS?
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