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For example, arbitrary palindromes and other languages with balanced symbols are not regular.Ĭontext-free languages are a more powerful class of languages, and do include languages with balanced constructs. Limits to the expressive power of context-free languages We saw that some interesting languages were not regular languages.
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An N-PDA can recognize any context-free language, and thus are more general. (Contrast this with DFAs and NFAs, which have exactly the same expressive power.) Deterministic context-free languages are the subset of context-free languages that can be recognized by a D-PDA. Interestingly, the power of D-PDAs and N-PDAs is not equivalent N-PDAs can recognize some languages that D-PDAs cannot. Note that nondeterministic pushdown automata (N-PDAs) are possible they may states that have multiple transitions on the same input symbol. The PDA shown above is a deterministic pushdown automaton (D-PDA) because each state has only one transition per input symbol. If the first symbol read is Δ, then the string is accepted (the empty string is a member of the language) For each a symbol read, an a symbol is pushed onto the tape When the first b symbol is encountered, the PDA attempts to pop a matching a symbol from the tape Subsequent b symbols must be matched with a symbols popped from the tape When the terminating Δ symbol is read, a matching Δ symbol must be popped, in which case the input string is accepted Any time an unexpected symbol is encountered (from the input string or the tape), the input string is rejected. The PDA works as follows: The special symbol Δ is pushed onto the tape It is appended to the input string, meaning that it will be the last symbol read from the input string Here is the state diagram for the PDA: It is pushed onto the tape as the automaton's first action 2. Let's describe a PDA that recognizes the language anbn (All strings of the form n a's followed by n b's, for n >= 0.) Note that the PDA a special symbol Δ, which is not part of the language's alphabet, is used in two ways 1.
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This is exactly equivalent to a stack data structure. In other words, the symbol popped from the tape by the automaton will be the one most recently pushed. Symbols are written to and read from the tape in last-in, first-out order. In any state, the automaton chooses to do one of three things: 1. The tape can serve as storage for data that the automaton needs to remember. However, an infinitely long "tape" is added to the automaton. Pushdown automata are similar to finite automata: they consist of states and transitions. Pushdown automata (PDAs) are a class of automaton that are powerful enough to serve as recognizers for context-free languages. You might wonder whether or not there is a kind of automaton that can serve as a recognizer for context-free languages. Various parsing algorithms exist to solve this problem. Parsing is thus a problem of recognizing whether or not an input string is a member of the language defined by a grammar. We also discussed the problem of parsing: given an input string and a context-free grammar, find a derivation (sequence of productions) that derive the input string from the grammar's start symbol. So every regular language is not N(P).Ĭontext-free languages and pushdown automata The strings in a context-free language can be generated by a context-free grammar. An example of a right regular grammar G with N = could be accepted by DPDA with final state, but not with empty stack, because strings of this language do not satisfy the prefix property. A → ε - where A is in N and ε is the empty string. A → Ba - where A and B are in N and a is in Σ 3. A → a - where A is a non-terminal in N and a is a terminal in Σ 2. In a left regular grammar (also called left linear grammar), all rules obey the forms 1. B → ε - where B is in N and ε denotes the empty string, i.e. B → aC - where B and C are in N and a is in Σ 3. B → a -where B is a non-terminal in N and a is a terminal in Σ 2. A right regular grammar (also called right linear grammar) is a formal grammar (N, Σ, P, S) such that all the production rules in P are of one of the following forms: N Non-Terminal Σ Input Set P Production Rules S Start Symbol 1.