Items and Itemsets An itemset is merely a set of items In LR parsing terminology an item Looks like a production with a ‘.’ in it The ‘.’ indicates how far the parse has gone in recognizing a string that matches this production e.g. A -> aAb.BcC suggests that we’ve “seen” input that could replace aAb. If, by following the rules we get A -> aAbBcC. we can reduce by A -> aAbBcC

Building LR(0) Itemsets Start with an augmented grammar; if S is the grammar start symbol add S’ -> S The first set of items includes the closure of S’ -> S Itemset construction requires two functions Closure Goto

Closure of LR(0) Itemset If J is a set of items for Grammar G, then closure(J) is the set of items constructed from G by two rules 1) Each item in J is added to closure(J) 2) If A  α.Bβ is in closure(J) and B  φ is a production, add B  .φ to closure(J)

Closure Example Grammar: A  aBC A  aA B  bB B bC C cC C  λ Closure(J) A  a.BC A-> a.A A  .aBC A  .aA B  .bB B  .bC J A  a.BC A a.A

GoTo Goto(J,X) where J is a set of items and X is a grammar symbol – either terminal or non-terminal is defined to be closure of A αX.β for A  α.Xβ in J So, in English, Goto(J,X) is the closure of all items in J which have a ‘.’ immediately preceding X

Set of Items Construction Procedure items(G’) Begin C = {closure({[S’  .S]})} repeat for each set of items J in C and each grammar symbol X such that GoTo(J,X) is not empty and not in C do add GoTo(J,X) to C until no more sets of items can be added to C

Build LR(0) Itemsets for: {S  (S), S  λ} {S  (S), S  SS, S  λ}

Building LR(0) Table from Itemsets One row for each Itemset One column for each terminal or non-terminal symbol, and one for $ Table [J][X] is: Rn if J includes A  rhs., A  rhs is rule number n, and X is a terminal Sn if Goto(J,X) is itemset n

LR(0) Parse Table for: {S  (S), S  λ} {S  (S), S  SS, S  λ}

Building SLR Table from Itemsets One row for each Itemset One column for each terminal or non-terminal symbol, and one for $ Table [J][X] is: Rn if J includes A  rhs., A  rhs is rule number n, X is a terminal, AND X is in Follow(A) Sn if Goto(J,X) is itemset n

LR(0) and LR(1) Items LR(0) item “is” a production with a ‘.’ in it. LR(1) item has a “kernel” that looks like LR(0), but also has a “lookahead” – e.g. A  α.Xβ, {terminals} A  α.Xβ, a/b/c ≠ A  α.Xβ, a/b/d

Closure of LR(1) Itemset If J is a set of LR(1) items for Grammar G, then closure(J) includes 1) Each LR(1) item in J 2) If A  α.Bβ, a in closure(J) and B  φ is a production, add B  .φ, First(β,a) to closure(J)

LR(1) Itemset Construction Procedure items(G’) Begin C = {closure({[S’  .S, $]})} repeat for each set of items J in C and each grammar symbol X such that GoTo(J,X) is not empty and not in C do add GoTo(J,X) to C until no more sets of items can be added to C

Build LR(1) Itemsets for: {S  (S), S  SS, S  λ}

{S  CC, S  cC, C d} Is this grammar LR(0)? SLR? LR(1)? How can we tell?

LR(1) Table from LR(1) Itemsets One row for each Itemset One column for each terminal or non-terminal symbol, and one for $ Table [J][X] is: Rn if J includes A  rhs., a; A  rhs is rule number n; X = a Sn if Goto(J,X) in LR(1) itemset n