Cannot understand CYK Algorithm pseudo-code

Question

I was reading about the CYK algorithm, and there is one part of pseudo-code I cannot understand. The whole pseudo-code is:

let the input be a string S consisting of n characters: a1 ... an.
let the grammar contain r nonterminal symbols R1 ... Rr.
This grammar contains the subset Rs which is the set of start symbols.
let P[n,n,r] be an array of booleans. Initialize all elements of P to false.
for each i = 1 to n
  for each unit production Rj -> ai
    set P[i,1,j] = true
for each i = 2 to n -- Length of span
  for each j = 1 to n-i+1 -- Start of span
    for each k = 1 to i-1 -- Partition of span
      for each production RA -> RB RC
        if P[j,k,B] and P[j+k,i-k,C] then set P[j,i,A] = true
if any of P[1,n,x] is true (x is iterated over the set s, where s are all the indices for Rs) then
  S is member of language
else
  S is not member of language

These parts are which I am confused:

    for each production RA -> RB RC
      if P[j,k,B] and P[j+k,i-k,C] then set P[j,i,A] = true

Would someone give some hints about these pseudocode?

Answer 1

The pseudocode

For each production R_A → R_B R_C:

if P[j,k,B] and P[j+k,i-k,C] then set P[j,i,A] = true

Should be interpreted in the following way. Suppose that it's the case that P[j, k, B] is true. That means that the string formed from k characters starting at position j can derived from the nonterminal R_B. If it's also the case that P[j + k, i - k, C] is true, then the string formed from the i - k characters starting at position j + k can be derived from nonterminal R_C. Therefore, since R_A → R_B R_C is a production, it's the case that the string formed from the i characters starting at position j can be derived from R_A.

I think it might help to interpret that pseudocode as

For each production R_A → R_B R_C:

if P[j,k,B] == true and P[j+k,i-k,C] == true, then set P[j,i,A] = true

Hope this helps!