Can we determine whether a given string is accepted by a context-free grammar? Is this problem decidable?
Determining whether a given string is accepted by a context-free grammar is a fundamental problem in computational complexity theory. This problem falls under the broader category of decidability, which deals with determining whether a particular property holds for a given input. In the case of context-free grammars, the problem of string acceptance is indeed decidable.
A context-free grammar is a formal system consisting of a set of production rules that describe how to generate strings in a language. It is defined by a tuple (V, Σ, R, S), where V is a set of non-terminal symbols, Σ is a set of terminal symbols, R is a set of production rules, and S is the start symbol. The language generated by a context-free grammar is the set of all strings that can be derived from the start symbol using the production rules.
To determine whether a given string is accepted by a context-free grammar, we can use various algorithms, such as the CYK algorithm or the Earley algorithm. These algorithms employ dynamic programming techniques to efficiently check if a string can be derived from the start symbol of the grammar.
The CYK algorithm, for example, constructs a table where each cell represents a substring of the input string and a set of non-terminals that can generate that substring. By iteratively filling the table based on the production rules of the grammar, the algorithm determines whether the start symbol can generate the entire input string. If the start symbol appears in the top-right cell of the table, then the string is accepted by the grammar; otherwise, it is not.
Consider the following example: Let's say we have a context-free grammar with the production rules:
S -> AB
A -> a
B -> b
If we want to determine whether the string "ab" is accepted by this grammar, we can apply the CYK algorithm. The algorithm constructs a table with two cells, one for each character in the input string. The table looks as follows:
| 1 | 2 |
—+—+—+
1 | A | S |
—+—+—+
2 | | B |
—+—+—+
Starting from the bottom row, we can see that the cell (2,2) contains the non-terminal B, which is generated by the production rule B -> b. Moving up to the top row, we find that the cell (1,2) contains the non-terminal S, which is generated by the production rule S -> AB. Finally, the cell (1,1) contains the non-terminal A, which is generated by the production rule A -> a. Since the start symbol S appears in the top-right cell, we can conclude that the string "ab" is accepted by the grammar.
The problem of determining whether a given string is accepted by a context-free grammar is decidable. Algorithms such as the CYK algorithm or the Earley algorithm can be used to efficiently check if a string can be derived from the start symbol of the grammar. These algorithms employ dynamic programming techniques to construct tables and determine the acceptance of the string.
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