# LL(1) Parsing Algorithm

Prerequisite — construction of LL(1) parsing table.

LL(1) parsing is a top-down parsing method in the syntax analysis phase of compiler design. Required components for LL(1) parsing are input string, a stack, parsing table for given grammar, and parser. Here, we discuss a parser that determines that given string can be generated from a given grammar(or parsing table) or not.

Let given grammar is G = (V, T, S, P)

where V-variable symbol set, T-terminal symbol set, S- start symbol, P- production set.

**LL(1) Parser algorithm:****Input-** 1. stack = S //stack initially contains only S.

2. input string = w$

where S is the start symbol of grammar, w is given string, and $ is used for the end of string.

3. PT is a parsing table of given grammar in the form of a matrix or 2D array.

**Output-** determines that given string can be produced by given grammar(parsing table) or not, if not then it produces an error.

**Steps:**

1. while(stack is not empty) { // initially it is S 2. A = top symbol of stack; //initially it is first symbol in string, it can be $ also 3. r = next input symbol of given string; 4. if (A∈T or A==$) { 5. if(A==r){ 6. pop A from stack; 7. remove r from input; 8. } 9. else 10. ERROR(); 11. } 12. else if (A∈V) { 13. if(PT[A,r]= A⇢B1B2....Bk) { 14. pop A from stack; // B1 on top of stack at final of this step 15. push Bk,Bk-1......B1 on stack 16. } 17. else if (PT[A,r] = error()) 18. error(); 19. } 20. } // if parser terminate without error() // then given string can generated by given parsing table.

**Time complexity**

As we know that size of a grammar for a language is finite. Like, a finite number of variables, terminals, and productions. If grammar is finite then its LL(1) parsing table is also finite of size O(V*T). Let

*p*is the maximum of lengths of strings in RHS of all productions and*l*is the length of given string and

*l* is very larger than *p*. *if* block at line 4 of algorithm always runs for O(1) time. *else if* block at line 12 in algorithm takes *O(|P|*p)* as upper bound for a single next input symbol. And *while loop* can run for more than *l* times, but we have considered the repeated *while loop* for a single next input symbol in *O(|P|*p)*. So, the total time complexity is

T(l) = O(l)*O(|P|*p) = O(l*|P|*p) = O(l) { as l >>>|P|*p }

The time complexity of this algorithm is the order of length of the input string.

**Comparison with Context-free language (CFL) :**

Languages in LL(1) grammar is a proper subset of CFL. Using the CYK algorithm we can find membership of a string for a given Context-free grammar(CFG). CYK takes *O(l ^{3})* time for the membership test for CFG. But for LL(1) grammar we can do a membership test in

*O(l)*time which is linear using the above algorithm. If we know that given CFG is LL(1) grammar then use LL(1) parser for parsing rather than CYK algorithm.

**Example –**

Let the grammar *G = (V, T, S’, P*) is

S'→ S$S → xYzS | aY → xYz | y

Parsing table(*PT*) for this grammar

a | x | y | z | $ | |

S’ | S’ → S$ | S’ → S$ | error | error | error |

S | S → a | S → xYzS | error | error | error |

Y | error | Y → xYz | Y → y | error | error |

Letstring1 = xxyzza,

We have to add *$* with this string,

We will use the above parsing algorithm, diagram for the process :

For *string1* we got an empty stack, and while loop or algorithm terminated without error. So, *string1* belongs to language for given grammar *G*.

Letstring2 = xxyzzz,

Same way as above we will use the algorithm to parse the string2, here is the diagram

For *string2*, at the last stage as in the above diagram when the top of the stack is *S* and the next input symbol of the string is *z*, but in *PT[S,z] = error*. The algorithm terminated with an error. So, *string2* is not in the language of grammar *G*.

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