Introduction to Cocke-Younger-Kasami (CYK) algorithm

The Cocke-Younger-Kasami (CYK) algorithm is a common algorithm used in computer science for parsing context-free grammars. It is a bottom-up parsing algorithm that works by building a table of possible partial matches between the input string and the rules of the grammar.

How it works

The CYK algorithm works by taking a context-free grammar in Chomsky normal form and an input string, and then building a table of all possible ways the string can be parsed according to the grammar.

The first step in the algorithm is to initialize the table with the input string as the terminal symbols at the bottom of the table. The next step is to fill in the rest of the table, working up the rows and columns, by applying the rules of the grammar to the previously filled-in cells in the table.

Each cell in the table represents a non-terminal symbol in the grammar, and the value of the cell is a set of possible ways that the non-terminal can be derived from the input string so far. The CYK algorithm fills in the table by iterating over all possible ways of combining pairs of symbols in the table, and checking which rules in the grammar can derive those pairs of symbols.

Once the table is fully populated, the algorithm checks whether the top cell of the table contains the start symbol of the grammar. If it does, then the input string is fully parsed according to the grammar. If not, then the input string cannot be parsed by the grammar.

Pseudocode

The following is a pseudocode implementation of the CYK algorithm:

let G be a grammar in Chomsky normal form
let I be an input string
let T be a table of size |I| x |I| x |G|

// populate the bottom row of the table with the input string as terminal symbols
for i = 0 to |I| - 1
    for each non-terminal symbol A in G such that A -> I[i] is a rule in G
        T[i, i, A] = True

// fill in the rest of the table
for l = 1 to |I| - 1
    for i = 0 to |I| - l - 1
        j = i + l
        for each non-terminal symbol A in G
            for each rule of the form A -> BC in G
                for k = i to j - 1
                    if T[i, k, B] and T[k+1, j, C]
                        T[i, j, A] = True

// check if the input string can be parsed by the grammar
if T[0, |I|-1, S] is True, where S is the start symbol of G
    then the input string can be parsed by the grammar
otherwise it cannot

Applications

The CYK algorithm has a variety of applications in computer science, including natural language processing, code optimization, and finite-state machine analysis. It is commonly used in compilers and parsing tools to analyze and process source code.

Conclusion

The Cocke-Younger-Kasami (CYK) algorithm is a powerful bottom-up parsing algorithm that can be used to analyze and understand complex context-free grammars. By building a table of partial matches and applying the rules of the grammar in a systematic way, the CYK algorithm is able to parse even highly complex input strings quickly and effectively.