DETERMINISTIC CONTEXT FREE LANGUAGES

Ethakota Krishnamurthi SurendranathManoj 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 context-free 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 DescriptionConsider $: end of string markerConsider 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}. (a DCFL)L2 = {aibjck, i, j, k 0 and j k}. (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 IntersectionL1 = {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_context-

free_language• http://en.wikipedia.org/wiki/Context-free_language• http://en.wikipedia.org/wiki/Unambiguous_grammar

QUESTIONS?