# Construction of LL(1) Parsing Table

Prerequisite – Classification of top-down parsers, FIRST Set, FOLLOW Set

A top-down parser builds the parse tree from the top down, starting with the start non-terminal. There are two types of Top-Down Parsers:

- Top-Down Parser with Backtracking
- Top-Down Parsers without Backtracking

Top-Down Parsers without backtracking can further be divided into two parts:

In this article, we are going to discuss Non-Recursive Descent which is also known as LL(1) Parser.

**LL(1) Parsing:**

Here the 1st **L** represents that the scanning of the Input will be done from Left to Right manner and the second **L** shows that in this parsing technique we are going to use Left most Derivation Tree. And finally, the **1** represents the number of look-ahead, which means how many symbols are you going to see when you want to make a decision.

**Algorithm to construct LL(1) Parsing Table:**

**Step 1: **First check for left recursion in the grammar, if there is left recursion in the grammar remove that and go to step 2.

**Step 2: **Calculate First() and Follow() for all non-terminals.

**First****():**If there is a variable, and from that variable, if we try to drive all the strings then the beginning Terminal Symbol is called the First.- Follow(): What is the Terminal Symbol which follows a variable in the process of derivation.

**Step 3: **For each production A –> α. (A tends to alpha)

- Find First(α) and for each terminal in First(α), make entry A –> α in the table.
- If First(α) contains ε (epsilon) as terminal than, find the Follow(A) and for each terminal in Follow(A), make entry A –> α in the table.
- If the First(α) contains ε and Follow(A) contains $ as terminal, then make entry A –> α in the table for the $.

To construct the parsing table, we have two functions:

In the table, rows will contain the Non-Terminals and the column will contain the Terminal Symbols. All the **Null Productions** of the Grammars will go under the Follow elements and the remaining productions will lie under the elements of the First set.

Now, let’s understand with an example.

**Example-1:**

Consider the Grammar:

E --> TE' E' --> +TE' | ε T --> FT' T' --> *FT' | ε F --> id | (E) *ε denotes epsilon

Find their First and Follow sets:

First | Follow | |
---|---|---|

E –> TE’ | { id, ( } | { $, ) } |

E’ –> +TE’/ε | { +, ε } | { $, ) } |

T –> FT’ | { id, ( } | { +, $, ) } |

T’ –> *FT’/ε | { *, ε } | { +, $, ) } |

F –> id/(E) | { id, ( } | { *, +, $, ) } |

Now, the LL(1) Parsing Table is:

id | + | * | ( | ) | $ | |
---|---|---|---|---|---|---|

E | E –> TE’ | E –> TE’ | ||||

E’ | E’ –> +TE’ | E’ –> ε | E’ –> ε | |||

T | T –> FT’ | T –> FT’ | ||||

T’ | T’ –> ε | T’ –> *FT’ | T’ –> ε | T’ –> ε | ||

F | F –> id | F –> (E) |

As you can see that all the null productions are put under the Follow set of that symbol and all the remaining productions are lie under the First of that symbol.

**Note:** Every grammar is not feasible for LL(1) Parsing table. It may be possible that one cell may contain more than one production.

Let’s see with an example.

**Example-2:**

Consider the Grammar

S --> A | a A --> a

Find their First and Follow sets:

First | Follow | |
---|---|---|

S –> A/a | { a } | { $ } |

A –>a | { a } | { $ } |

Parsing Table:

a | $ | |
---|---|---|

S | S –> A, S –> a | |

A | A –> a |

Here, we can see that there are two productions into the same cell. Hence, this grammar is not feasible for LL(1) Parser.

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