LR(1) Item DFA - Computing Lookaheads

Question

I have trouble understanding how to compute the lookaheads for the LR(1)-items.

Lets say that I have this grammar:

S -> AB A -> aAb | a B -> d 

A LR(1)-item is an LR(0) item with a lookahead. So we will get the following LR(0)-item for state 0:

S -> .AB , {lookahead}  A -> .aAb,  {lookahead}  A -> .a,  {lookahead} 

State: 1

A ->  a.Ab, {lookahead}  A ->  a. ,{lookahead}  A -> .aAb ,{lookahead}  A ->.a ,{lookahead} 

Can somebody explain how to compute the lookaheads ? What is the general approach ?

Thank you in advance

Answer 1

The lookaheads used in an LR(1) parser are computed as follows. First, the start state has an item of the form

S -> .w  ($) 

for every production S -> w, where S is the start symbol. Here, the $ marker denotes the end of the input.

Next, for any state that contains an item of the form A -> x.By (t), where x is an arbitrary string of terminals and nonterminals and B is a nonterminal, you add an item of the form B -> .w (s) for every production B -> w and for every terminal in the set FIRST(yt). (Here, FIRST refers to FIRST sets, which are usually introduced when talking about LL parsers. If you haven't seen them before, I would take a few minutes to look over those lecture notes).

Let's try this out on your grammar. We start off by creating an item set containing

S -> .AB ($) 

Next, using our second rule, for every production of A, we add in a new item corresponding to that production and with lookaheads of every terminal in FIRST(B$). Since B always produces the string d, FIRST(B$) = d, so all of the productions we introduce will have lookahead d. This gives

S -> .AB ($) A -> .aAb (d) A -> .a (d) 

Now, let's build the state corresponding to seeing an 'a' in this initial state. We start by moving the dot over one step for each production that starts with a:

A -> a.Ab (d) A -> a. (d) 

Now, since the first item has a dot before a nonterminal, we use our rule to add one item for each production of A, giving those items lookahead FIRST(bd) = b. This gives

A -> a.Ab (d) A -> a. (d) A -> .aAb (b) A -> .a (b) 

Continuing this process will ultimately construct all the LR(1) states for this LR(1) parser. This is shown here:

[0] S -> .AB  ($) A -> .aAb (d) A -> .a   (d)  [1] A -> a.Ab (d) A -> a.   (d) A -> .aAb (b) A -> .a   (b)  [2] A -> a.Ab (b) A -> a.   (b) A -> .aAb (b) A -> .a   (b)  [3] A -> aA.b (d)  [4] A -> aAb. (d)  [5] S -> A.B  ($) B -> .d   ($)  [6] B -> d.   ($)  [7] S -> AB.  ($)  [8] A -> aA.b (b)  [9] A -> aAb. (b) 

In case it helps, I taught a compilers course last summer and have all the lecture slides available online. The slides on bottom-up parsing should cover all of the details of LR parsing and parse table construction, and I hope that you find them useful!

Hope this helps!

Answer 2

here is the LR(1) automaton for the grammar as the follow has been done above I think it's better for the understanding to trying draw the automaton and the flow will make the idea of the lookaheads clearer