Deterministic context-free language
In formal language theory, deterministic context-free languages (DCFL) are a proper subset of context-free languages. They are the context-free languages that can be accepted by a deterministic pushdown automaton.
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[hide]Description
The notion of the DCFL is closely related to the deterministic pushdown automaton (DPDA). It is where the language power of a pushdown automaton is reduced if we make it deterministic; the pushdown automaton becomes unable to choose between different state transition alternatives and as a consequence cannot recognize all context-free languages.[1] Unambiguous grammars do not always generate a DCFL. For example, the language of even-lengthpalindromes on the alphabet of 0 and 1 has the unambiguous context-free grammar S → 0S0 | 1S1 | ε. An arbitrary string of this language cannot be parsed without reading all its letters first which means that a pushdown automaton has to try alternative state transitions to accommodate for the different possible lengths of a semi-parsed string. [2]
Properties
Deterministic context-free languages can be recognized by a deterministic Turing machine in polynomial time and O(log2 n) space; as a corollary, DCFLis a subset of the complexity class SC.[3] The set of deterministic context-free languages is not closed under union but is closed under complement.
Importance
The languages of this class have great practical importance in computer science as they can be parsed much more efficienly than nondeterministic context-free languages. The complexity of the program and execution time of a deterministic pushdown automaton is vastly less than that of a nondeterministic one. In the naive implementation, the latter must make copies of the stack every time a nondeterministic step occurs. The best known algorithm to test membership in any context-free language is Valiant's algorithm, taking O(n2.378) time, where n is the length of the string. On the other hand, deterministic context-free languages can be accepted in O(n) time by a LR(k) parser.[4] This is very important for computer languagetranslation because many computer languages belong to this class of languages.
See also
References
- ^ Hopcroft, John; Jeffrey Ullman (1979). Introduction to automata theory, languages, and computation. Addison-Wesley. p. 233.
- ^ Hopcroft, John; Rajeev Motwani & Jeffrey Ullman (2001). Introduction to automata theory, languages, and computation 2nd edition. Addison-Wesley. pp. 249–253.
- ^ S. A. Cook. Deterministic CFL's are accepted simultaneously in polynomial time and log squared space. Proceedings of ACM STOC'79, pp. 338–345. 1979.
- ^ Knuth, Donald (July 1965). "On the Translation of Languages from Left to Right". Information and Control 8: 707 – 639. Retrieved 29 May 2011.
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