1、Bottom-up parsing,Goal of parser : build a derivation Top-down parser : build a derivation by working from the start symbol towards the input. Builds parse tree from root to leaves Builds leftmost derivation Bottom-up parser : build a derivation by working from the input back toward the start symbol
2、 Builds parse tree from leaves to root Builds reverse rightmost derivation,Bottom-up parsing,The parser looks for a substring of the parse trees frontier. .that matches the rhs of a production and .whose reduction to the non-terminal on the lhs represents on step along the reverse of a rightmost der
3、ivation Such a substring is called a handle. Important: Not all substrings that match a rhs are handles.,Bottom-up parsing techniques,Shift-reduce parsing Shift input symbols until a handle is found. Then, reduce the substring to the non-terminal on the lhs of the corresponding production. Operator-
4、precedence parsing Based on shift-reduce parsing. Identifies handles based on precedence rules.,Example: Shift-reduce parsing,1. S E 2. E E + E 3. E E * E 4. E num 5. E id,Input to parse: id1 + num * id2,Accept,$ S,Reduce (rule 1),$ E,Reduce (rule 2),$ E + E,Reduce (rule 3),$ E + E * E,Reduce (rule
5、5),$ E + E * id2,Shift,$ E + E *,Shift,$ E + E,Reduce (rule 4),$ E + num,Shift,$ E +,Shift,$ E,Reduce (rule 5),$ id1,Shift,$,ACTION,Grammar:,Handles: underlined,STACK,Shift-Reduce parsing,A shift-reduce parser has 4 actions: Shift - next input symbol is shifted onto the stack Reduce - handle is at t
6、op of stack pop handle push appropriate lhs Accept - stop parsing & report success Error - call error reporting/recovery routine,Shift-Reduce parsing,How can we know when we have found a handle? Analyze the grammar beforehand. Build tables Look ahead in the input LR(1) parsers recognize precisely th
7、ose languages in which one symbol of look-ahead is enough to determine whether to reduce or shift. L : for left-to-right parse of the input R : for reverse rightmost derivation 1: for one symbol of lookahead,How does it work?,Read input, one token at a time Use stack to keep track of current state T
8、he state at the top of the stack summarizes the information below. The stack contains information about what has been parsed so far. Use parsing table to determine action based on current state and look-ahead symbol. How do we build a parsing table?,LR parsing techniques,SLR (not in the book) Simple
9、 LR parsing Easy to implement, not strong enough Uses LR(0) items Canonical LR Larger parser but powerful Uses LR(1) items LALR (not in the book) Condensed version of canonical LR May introduce conflicts Uses LR(1) items,Class examples,L,R,id,L,* R,L,R,S,L = R,S,S,S,Finding handles,As a shift/reduce
10、 parser processes the input, it must keep track of all potential handles. For example, consider the usual expression grammar and the input string x+y. Suppose the parser has processed x and reduced it to E. Then, the current state can be represented by E +E where means that an E has already been par
11、sed and that +E is a potential suffix, which, if found, will result in a successful parse. Our goal is to eventually reach state E+E, which represents an actual handle and should result in the reduction EE+E,LR parsing,Typically, LR parsing works by building an automaton where each state represents
12、what has been parsed so far and what we hope to parse in the future. In other words, states contain productions with dots, as described earlier. Such productions are called items States containing handles (meaning the dot is all the way to the right end of the production) lead to actual reductions d
13、epending on the lookahead.,SLR parsing,SLR parsers build automata where states contain items (a.k.a. LR(0) items) and reductions are decided based on FOLLOW set information. We will build an SLR table for the augmented grammar,SS S L=R S R L *R L id R L,SLR parsing,When parsing begins, we have not p
14、arsed any input at all and we hope to parse an S. This is represented by SS. Note that in order to parse that S, we must either parse an L=R or an R. This is represented by SL=R and SR closure of a state: if AaBb represents the current state and B is a production, then add B to the state. Justificat
15、ion: aBb means that we hope to see a B next. But parsing a B is equivalent to parsing a , so we can say that we hope to see a next,SLR parsing,Use the closure operation to define states containing LR(0) items. The first state will be:From this state, if we parse, say, an id, then we go to state If,
16、after some steps we parse input that reduces to an L, then we go to state,S S S L=R S R L *R L id R L,L id ,S L =R R L ,id,SLR parsing,Continuing the same way, we define all LR(0) item states:,S S S L=R S R L *R L id R L,L id ,S L =R R L ,S S ,I0,I1,I2,I3,S R ,I4,L * R R L L id L * R,I5,S,L,*,id,R,S
17、 L= R R L L *R L id,I6,=,R,S L=R ,R L ,L,L,I7,id,I3,*,*,L *R ,R,I8,I9,SLR parsing,The automaton and the FOLLOW sets tell us how to build the parsing table: Shift actions If from state i, you can go to state j when parsing a token t, then slot i,t of the table should contain action “shift and go to s
18、tate j“, written sj Reduce actions If a state i contains a handle A, then slot i, t of the table should contain action “reduce using A“, for all tokens t that are in FOLLOW (A). This is written r(A) The reasoning is that if the lookahead is a symbol that may follow A, then a reduction A should lead
19、closer to a successful parse. continued on next slide,SLR parsing,The automaton and the FOLLOW sets tell us how to build the parsing table: Reduce actions, continued Transitions on non-terminals represent several steps together that have resulted in a reduction. For example, if we are in state 0 and
20、 parse a bit of input that ends up being reduced to an L, then we should go to state 2. Such actions are recorded in a separate part of the parsing table, called the GOTO part.,SLR parsing,Before we can build the parsing table, we need to compute the FOLLOW sets:,S S S L=R S R L *R L id R L,FOLLOW(S
21、) = $ FOLLOW(S) = $ FOLLOW(L) = $, = FOLLOW(R) = $, =,SLR parsing,state action gotoid = * $ S L R0 s3 s5 1 2 41 accept2 s6/r(RL)3 r(Lid) r(Lid)4 r(SR)5 s3 s5 7 86 s3 s5 7 97 r(RL) r(RL)8 r(L*R) r(L*R)9 r(SL=R),Note the shift/reduce conflict on state 2 when the lookahead is an =,Conflicts in LR parsi
22、ng,There are two types of conflicts in LR parsing: shift/reduce On some particular lookahead it is possible to shift or reduce The if/else ambiguity would give rise to a shift/reduce conflict reduce/reduce This occurs when a state contains more than one handle that may be reduced on the same lookahe
23、ad.,Conflicts in SLR parsing,The parser we built has a shift/reduce conflict. Does that mean that the original grammar was ambiguous? Not necessarily. Lets examine the conflict: it seems to occur when we have parsed an L and are seeing an =. A reduce at that point would turn the L into an R. However
24、, note that a reduction at that point would never actually lead to a successful parse. In practice, L should only be reduced to an R when the lookahead is EOF ($). An easy way to understand this is by considering that L represents l-values while R represents r-values.,Conflicts in SLR parsing,The co
25、nflict occurred because we made a decision about when to reduce based on what token may follow a non-terminal at any time. However, the fact that a token t may follow a non-terminal N in some derivation does not necessarily imply that t will follow N in some other derivation. SLR parsing does not ma
26、ke a distinction.,Conflicts in SLR parsing,SLR parsing is weak. Solution : instead of using general FOLLOW information, try to keep track of exactly what tokens many follow a non-terminal in each possible derivation and perform reductions based on that knowledge. Save this information in the states.
27、 This gives rise to LR(1) items: items where we also save the possible lookaheads.,Canonical LR(1) parsing,In the beginning, all we know is that we have not read any input (SS), we hope to parse an S and after that we should expect to see a $ as lookahead. We write this as: SS, $ Now, consider a gen
28、eral item A, x. It means that we have parsed an , we hope to parse and after those we should expect an x. Recall that if there is a production , we should add to the state. What kind of lookahead should we expect to see after we have parsed ? We should expect to see whatever starts a . If is empty o
29、r can vanish, then we should expect to see an x after we have parsed (and reduced it to B),Canonical LR(1) parsing,The closure function for LR(1) items is then defined as follows: For each item A, x in state I, each production in the grammar, and each terminal b in FIRST(x), add , b to I If a state
30、contains core item with multiple possible lookaheads b1, b2,., we write , b1/b2 as shorthand for , b1 and , b2,id,Canonical LR(1) parsing,L id , =/$,S L =R, $ R L , $,S S , $,I0,I1,I2,I3,S R, =/$,I4,I5,S,L,*,id,R,I6,=,R,SL=R, $,R L, =/$,L,L,I7,id,*,*,L *R , =/$,R,I8,I9,S S, $ S L=R, $ S R, $ L *R, =
31、/$ L id, =/$ R L, $,L *R, =/$ R L, =/$ L id, =/$ L *R, =/$,Lid, $,I3,R L, $,I7,S L= R, $ R L, $ L *R, $ L id, $,I5,*,L *R , $,I8,L *R, $ R L, $ L id, $ L *R, $,L,R,id,I3,Canonical LR(1) parsing,The table is created in the same way as SLR, except we now use the possible lookahead tokens saved in each
32、 state, instead of the FOLLOW sets. Note that the conflict that had appeared in the SLR parser is now gone. However, the LR(1) parser has many more states. This is not very practical. It may be possible to merge states!,LALR(1) parsing,This is the result of an effort to reduce the number of states i
33、n an LR(1) parser. We notice that some states in our LR(1) automaton have the same core items and differ only in the possible lookahead information. Furthermore, their transitions are similar. States I3 and I3, I5 and I5, I7 and I7, I8 and I8 We shrink our parser by merging such states. SLR : 10 sta
34、tes, LR(1): 14 states, LALR(1) : 10 states,id,LALR(1) parsing,L id , =/$,S L =R, $ R L , $,S S , $,I0,I1,I2,I3,S R, =/$,I4,I5,S,L,*,id,R,I6,=,R,SL=R, $,R L, =/$,L,L,I7,id,*,*,L *R , =/$,R,I8,I9,S S, $ S L=R, $ S R, $ L *R, =/$ L id, =/$ R L, $,L *R, =/$ R L, =/$ L id, =/$ L *R, =/$,S L= R, $ R L, $
35、L *R, $ L id, $,I3,Conflicts in LALR(1) parsing,Note that the conflict that had vanished when we created the LR(1) parser has not reappeared. Can LALR(1) parsers introduce conflicts that did not exist in the LR(1) parser? Unfortunately YES. BUT, only reduce/reduce conflicts.,Conflicts in LALR(1) par
36、sing,LALR(1) parsers cannot introduce shift/reduce conflicts. Such conflicts are caused when a lookahead is the same as a token on which we can shift. They depend on the core of the item. But we only merge states that had the same core to begin with. The only way for an LALR(1) parser to have a shift/reduce conflict is if one existed already in the LR(1) parser. LALR(1) parsers can introduce reduce/reduce conflicts. Heres a situation when this might happen:,A B , x A C , y,A B , y A C , x,merges with,to give:,A B , x/y A C , x/y,
