Parsing a context-free grammar involves analyzing a sequence of symbols according to a set of production rules defined by the grammar. This process is fundamental in various areas of computer science, including cybersecurity, as it allows us to understand and manipulate structured data. In this answer, we will describe the algorithm for parsing a context-free grammar and discuss its time complexity.

The most commonly used algorithm for parsing context-free grammars is the CYK algorithm, named after its inventors Cocke, Younger, and Kasami. This algorithm is based on dynamic programming and operates on the principle of bottom-up parsing. It builds a parse table that represents all possible parses for substrings of the input.

The CYK algorithm works as follows:

1. Initialize a parse table with dimensions n x n, where n is the length of the input string.

2. For each terminal symbol in the input string, fill in the corresponding cell in the parse table with the nonterminal symbols that produce it.

3. For each substring length l from 2 to n, and each starting position i from 1 to n-l+1, do the following:

a. For each partition point p from i to i+l-2, and each production rule A -> BC, check if the cells (i, p) and (p+1, i+l-1) contain nonterminal symbols B and C, respectively. If so, add A to the cell (i, i+l-1).

4. If the start symbol of the grammar is present in the cell (1, n), the input string can be derived from the grammar. Otherwise, it cannot.

The time complexity of the CYK algorithm is O(n^3 * |G|), where n is the length of the input string and |G| is the size of the grammar. This complexity arises from the nested loops used to fill in the parse table. The algorithm examines all possible partition points and production rules for each substring length, resulting in cubic time complexity.

To illustrate the algorithm, consider the following context-free grammar:

S -> AB | BC

A -> AA | a

B -> AB | b

C -> BC | c

And the input string "abc". The parse table for this example would look as follows:

| 1 | 2 | 3 |

——-|—–|—–|—–|

1 | A,S | B,C | S |

——-|—–|—–|—–|

2 | | B,C | A |

——-|—–|—–|—–|

3 | | | C |

——-|—–|—–|—–|

In this table, the cell (1, 3) contains the start symbol S, indicating that the input string "abc" can be derived from the given grammar.

The algorithm for parsing a context-free grammar, such as the CYK algorithm, allows us to analyze and understand structured data. It operates by building a parse table and checking for valid derivations according to the grammar's production rules. The time complexity of the CYK algorithm is O(n^3 * |G|), where n is the length of the input string and |G| is the size of the grammar.

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