Example CFG. Lectures 16 & 17 BottomUp Parsing. LL(1) Predictor Table Review. Stacks in LR Parsing 1. Sʹ " S. 2. S " AyB. 3. A " ab. 4.


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1 Example CFG Lectures 16 & 17 BottomUp Parsing CS 241: Foundations of Sequential Programs Fall Sʹ " S 2. S " AyB 3. A " ab 4. A " cd Matt Crane University of Waterloo 5. B " z 6. B " wz 2 LL(1) Predictor Table Review Stacks in LR Parsing Empty(α) = true if α * ε No α can satisfy this rule in our grammar First(α) = {x α * xβ} First(Sʹ) = { }; First(S) = {a, c}; First(A) = {a, c}; First(B) = {z, w} First(ab) = {a}; First(AyB) = {a, c}; First(yB) = {y} Follow(A) = {y Sʹ * αayβ} Follow(Sʹ) = {}; Follow(S) = { } Follow(A) = {y}; Follow(B) = { } α, β (N T)* x, y T A N Predict(A, x) = {A α x First(α)} {A β x Follow(A) and Empty(β)} Predict(A, a) = {A ab} {} Predict(A, w) = {} {} Predict(B, w) = {B wz} {} Predict(S, a) = {S A} {} Recall that a stack in LL/topdown parsing is used in the following way: input processed + stack = current derivation note that the stack here is read from top to bottom For LR/bottomup parsing, we have stack + input to be read = current derivation stack is read from bottom to top here 3 4
2 A Trace abywz Shift/Reduce Derivation Stack Input Read Unread Input Action abywz ε ε abywz Shift abywz abywz Shift a abywz a a bywz Shift b abywz ab ab ywz Reduce A " ab Aywz A ab ywz Shift y Aywz Ay aby wz Shift w Aywz Ayw abyw z Shift z Aywz Aywz abywz Reduce B " wz AyB AyB abywz Reduce S " AyB S S abywz Shift S S abywz ε Reduce Sʹ " S Sʹ Sʹ abywz ε ACCEPT Shift: moving a token from one place to another (push) Reduce: size of the stack may be reduced (pop RHS, push LHS) 5 Somehow we shifted at just the right time, and reduced just at the right time How did we know this? Recall that for LL(1) parsing, we had a predictor table For LR(1) parsing, we have an oracle, in the form of a DFA 6 Constructing DFA Oracle for LR(1) Grammars Building an LR(0) Automaton This is difficult to do Donald Knuth proved a theorem that we can construct a DFA (really, a transducer) for LR(1) grammars (1965) This transducer tells us when to shift or reduce We will build and use the transducer Definition: An item is a production with a dot ( ) somewhere on the RHS (which indicates a partially completed rule) How to construct the automaton: Make the start state the first rule, with the dot ( ) in front of the leftmost symbol of the RHS For each state, label an arc with the symbol that follows and advance the one position to the right in the next state If the precedes a nonterminal (e.g. A) add all productions with that nonterminal A on the LHS to the current state, with the in the leftmost position. 7 8
3 A Sample Construction of the DFA Small example CFG: 1. Sʹ " E 2. E " E + T 3. E " T 4. T " id Using the Automaton For each input token Start in the start state Read the stack (from the bottom up) and read the current input, and do the action indicated for the current input If there is a transition out of our current state on the current input, then shift (push) that input onto the stack We know we can reduce if the current state has only one item and is the rightmost symbol To reduce, pop the RHS off the stack, reread the stack (from bottomup), follow the transition for the LHS and push the LHS onto the stack Accept if Sʹ on the stack when all input is read 9 10 Using the Transducer Example: id+id+id Stack States Visited Input Read Unread Input Action ε 1 ε id+id+id Shift 1 2 id+id+id Shift id id id +id+id Reduce T " id T id +id+id Reduce E " T E id +id+id Shift + E id+ id+id Shift id E+id id+id +id Reduce T " id E+T id+id +id Reduce E " E+T E id+id +id Shift + E id+id+ id Shift id E+id id+id+id Reduce T " id E+T id+id+id Reduce E " E+T E id+id+id Shift E id+id+id ε Reduce Sʹ " E Sʹ 1 id+id+id ε ACCEPT What Can Go Wrong? A Problem What if the state looks like this? Do we try to shift the next character (as suggested by A α cβ) or do we reduce by B γ (as suggested by B γ )? A α cβ B γ This is known as a shiftreduce conflict 11 12
4 What Can Go Wrong? Another Problem What if the state looks like this? A α B β What Can Go Wrong? If any item A α occurs in a state in which it is not alone, then there is a shiftreduce or reducereduce conflict and the grammar is not LR(0) Do we reduce by A α, or by B β? This is known as a reducereduce conflict Example With Conflicts Consider rightassociative expressions. Modify our grammar slightly to allow (reverse RHS of second rule) 1. Sʹ " E Parsing With Conflicts Suppose we are parsing a string that looks like id Picture of the stack: 2. E " T+E 3. E " T 4. T " id Question: Should we reduce E T? Answer: It depends If input is id, then yes If input is id+, then no 15 16
5 Looking Ahead If we add a lookahead token to the automaton, we can fix the conflict For each A α, attach Follow(A) For our grammar: Follow(E) = Follow(T) = E T Consider our conflicting state: E T +E Interpretation: A reduce action A α F where F is the Follow(A) applies only if the next character is in F SLR(1) Parser When we add this one character of lookahead, we have an SLR(1) parser. SLR(1) = Simple LR with 1 character of lookahead SLR(1) resolves many, but not all, conflicts LR(1) parsing is more sophisticated than SLR(1) parsers LR(1) parsers strictly more grammars LR(1) automaton is more complex LR(1) and SLR(1) are identical as parsing algorithms, the only difference is in the respecting automaton they create There is also a parse called LALR(1) which falls between SLR(1) and LR(1). LALR(1) = Lookahead LR(1) This is what Yacc and Bison use Making This More Efficient Current running time of this algorithm: Outputting a Derivation In topdown parsing: output the rules used to expand by In bottomup parsing: Instead of scanning the stack each time Start the transducer in Easy: each time we do a reduction, output the rule But, this isn t quite right. Derivations should start with the start symbol. Bottomup parsing doesn t. Running time: 19 20
6 A Simple Observation Didn t we say that this was LR(1) parsing? Doesn t R mean rightmost derivation? Aren t we always reducing the leftmost nonterminal? Outputting the Parse Tree Create a tree stack Each time we reduce, pop the right hand side nodes from the tree stack Push the left hand side node and make its children the nodes we just popped Example: id+id+id But notice the direction we are creating the derivation. Write the derivation in reverse How The Tree is Actually Built LR Parsing How The Tree is Actually Built LL Parsing 23 24
7 Assignment Hints Note that the automaton in cfgr format specifies: state symbol shift/reduce next state / production rule terminal or nonterminal for shifts / for reductions P1, P2: Write a cfgr derivation by hand P3: Given a DFA, a state and one input, follow one transition in the DFA (i.e. did the DFA tell me to shift or reduce?) P4: Write a parser Read a CFG, the DFA and input, Output cfgr (derivation) P5: Write a parser for WLP4 Your parser will read tokens, build a parse tree and output a leftmost derivation Find a way to embed the WLP4 grammar and DFA table in your program Going Back Looking at: L = {a n b m n m 0} (nonll(1) language) 1. Sʹ S 2. S as 3. S T 4. T atb 5. T ε What do when you see the symbol: a b Final Fun Facts Theorem: For any augmented LR(1) grammar, there is an equivalent LR(0) grammar Theorem: The class of languages that can be parsed deterministically with a stack can be represented with an LR(1) grammar Comparing LL(1) vs. LR(1) 27
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