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LR Parsing. LR Parsers. The most powerful shift-reduce parsing (yet efficient) is: LR(k) parsing . left to right right-most k lookhead scanning derivation (k is omitted it is 1) LR parsing is attractive because:
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LR Parsers • The most powerful shift-reduce parsing (yet efficient) is: LR(k) parsing. left to right right-most k lookhead scanning derivation (k is omitted it is 1) • LR parsing is attractive because: • LR parsing is most general non-backtracking shift-reduce parsing, yet it is still efficient. • The class of grammars that can be parsed using LR methods is a proper superset of the class of grammars that can be parsed with predictive parsers. LL(1)-Grammars LR(1)-Grammars • An LR-parser can detect a syntactic error as soon as it is possible to do so a left-to-right scan of the input. • Can recognize virtually all programming language constructs for which CFG can be written
LR Parsers • LR-Parsers • covers wide range of grammars. • SLR – simple LR parser • LR – most general LR parser • LALR – intermediate LR parser (look-ahead LR parser) • SLR, LR and LALR work same (they used the same algorithm), only their parsing tables are different.
LR Parsing Algorithm input stack output
A Configuration of LR Parsing Algorithm • A configuration of a LR parsing is: ( So X1 S1 ... XmSm, ai ai+1 ... an $ ) Stack Rest of Input • Sm and ai decides the parser action by consulting the parsing action table. (Initial Stack contains just So ) • A configuration of a LR parsing represents the right sentential form: X1 ... Xm ai ai+1 ... an $
Actions of A LR-Parser • shift s-- shifts the next input symbol and the state s onto the stack ( So X1 S1 ... Xm Sm, ai ai+1 ... an $ ) ( So X1 S1 ... Xm Sm ai s, ai+1 ... an $ ) • reduce A (or rn where n is a production number) • pop 2|| (=r) items from the stack; • then push A and s where s=goto[sm-r,A] ( So X1 S1 ... Xm Sm, ai ai+1 ... an $ ) ( So X1 S1 ... Xm-rSm-rA s, ai ... an $ ) • Output the reducing production reduce A • Accept – If action[Sm , ai ]= accept ,Parsing successfully completed • Error -- Parser detected an error (an empty entry in the action table) and calls an error recovery routine.
Reduce Action • pop 2|| (=r) items from the stack; let us assume that = Y1Y2...Yr • then push A and s where s=goto[sm-r,A] ( So X1 S1 ... Xm-rSm-rY1 Sm-r+1 ...Yr Sm, ai ai+1 ... an $ ) ( So X1 S1 ... Xm-rSm-rA s, ai ... an $ ) • In fact, Y1Y2...Yr is a handle. X1 ... Xm-rA ai ... an $ X1 ... XmY1...Yr ai ai+1 ... an $
LR Parsing Algorithm • set ip to point to the first symbol in ω$ • initialize stack to S0 • repeat forever • let ‘s’ be topmost state on stack & ‘a’ be symbol pointed to by ip • if action[s,a] = shift s’ • push a then s’ onto stack • advance ip to next input symbol • else if action[s,a] = reduce A • pop 2*| | symbols of stack • let s’ be state now on top of stack • push A then goto[s’,A] onto stack • output production A • else if action[s,a] == accept • return success • else • error()
SLR Parsing Tables for Expression Grammar Action Table Goto Table 1) E E+T 2) E T 3) T T*F 4) T F 5) F (E) 6) F id
Actions of SLR-Parser -- Example stackinputactionoutput 0 id*id+id$ shift 5 0id5 *id+id$ reduce by Fid Fid 0F3 *id+id$ reduce by TF TF 0T2 *id+id$ shift 7 0T2*7 id+id$ shift 5 0T2*7id5 +id$ reduce by Fid Fid 0T2*7F10 +id$ reduce by TT*F TT*F 0T2 +id$ reduce by ET ET 0E1 +id$ shift 6 0E1+6 id$ shift 5 0E1+6id5 $ reduce by Fid Fid 0E1+6F3 $ reduce by TF TF 0E1+6T9 $ reduce by EE+T EE+T 0E1 $ accept
LR Grammar • A grammar for which we can construct an LR parsing table in which every entry is uniquely defined is an LR grammar. • For an LR grammar, the shift reduce parser should be able to recognize handles when they appear on top of the stack. • Each state symbol summarizes the information contained in the stack below it. • The LR parser can determine from the state on top of the stack everything it needs to know about what is in the stack. • The next k input symbols can also help the LR parser to make shift reduce decisions. • Grammar that can be parsed by an LR parser by examining upto k input symbols on each move is called an LR(k) grammar.
Constructing SLR Parsing Tables – LR(0) Item • LR parser using SLR parsing table is called an SLR parser. • A grammar for which an SLR parser can be constructed is an SLR grammar. • An LR(0) item (item) of a grammar G is a production of G with a dot at the some position of the right side. • Ex: A aBb Possible LR(0) Items: A .aBb (four different possibility) A a.Bb A aB.b A aBb. • Sets of LR(0) items will be the states of action and goto table of the SLR parser. • A production rule of the form A yields only one item A . • Intuitively, an item shows how much of a production we have seen till the current point in the parsing procedure.
Constructing SLR Parsing Tables – LR(0) Item • A collection of sets of LR(0) items (the canonical LR(0) collection) is the basis for constructing SLR parsers. • To construct the of canonical LR(0) collection for a grammar we define an augmented grammar and two functions- closure and goto. • Augmented Grammar: G’ is grammar G with a new production rule S’S where S’ is the new starting symbol. i.e G {S’ S} where S is the start state of G. • The start state of G’ = S’. • This is done to signal to the parser when the parsing should stop to announce acceptance of input.
Constructing SLR Parsing Tables – LR(0) Item • Complete and Incomplete Items: An LR(0) item is complete if ‘.’ Is the last symbol in RHS else it is incomplete. For every rule A α, α≠ , there is only one complete item A α., but as many incomplete items as there are grammar symbols. Kernel and Non-Kernel items: Kernel items include the set of items that do not have the dot at leftmost end. S’ .S is an exception and is considered to be a kernel item. Non-kernel items are the items which have the dot at leftmost end. Sets of items are formed by taking the closure of a set of kernel items.
The Closure Operation • If I is a set of LR(0) items for a grammar G, then closure(I) is the set of LR(0) items constructed from I by the two rules: • Initially, every LR(0) item in I is added to closure(I). • If A .B is in closure(I) and B is a production rule of G;then B. will be in the closure(I). We will apply this rule until no more new LR(0) items can be added to closure(I).
The Closure Operation -- Example E’ E closure({E’ .E}) = E E+T { E’ .E kernel items E T E .E+T T T*F E .T T F T .T*F F (E) T .F F id F .(E) F .id }
Computation of Closure function closure ( I ) begin J := I; repeat for each item A .B in J and each production B of G such that B. is not in J do add B. to J until no more items can be added to J return J end
Goto Operation • If I is a set of LR(0) items and X is a grammar symbol (terminal or non-terminal), then goto(I,X) is defined as follows: • If A .X in I then every item in closure({A X.}) will be in goto(I,X). • If I is the set of items that are valid for some viable prefix , then goto(I,X) is the set of items that are valid for the viable prefix X. Example: I ={ E’ E., E E.+T} goto(I,+) = { E E+.T T .T*F T .F F .(E) F .id }
I ={ E’ .E, E .E+T, E .T, T .T*F, T .F, F .(E), F .id } goto (I, E) = { E’ E., E E.+T } goto (I, T) = { E T., T T.*F } goto (I, F) = {T F. } goto(I,id) = { F id. } goto (I, ( ) = { F (.E), E .E+T, E .T, T .T*F, T .F, F .(E), F .id } Example
Construction of The Canonical LR(0) Collection • To create the SLR parsing tables for a grammar G, we will create the canonical LR(0) collection of the grammar G’. • Algorithm: Procedure items( G’ ) begin C := { closure({S’.S}) } repeatfor each set of items I in C and each grammar symbol X if goto(I,X) is not empty and not in C add goto(I,X) to C until no more set of LR(0) items can be added to C. end • goto function is a DFA on the sets in C.
The Canonical LR(0) Collection -- Example I0: E’ .E . I4 : goto(I0, ( ) E .E+T F (.E)I7: goto(I2 , *) E .T E .E+T T T*.F T .T*F E .T F .(E) T .F T .T*F F .id F .(E) T .F F .id F .(E) I8: goto( I4 , E) F .id F (E.) I1: goto(I0, E) E E.+T E’ E.I5: goto(I0, id) I9:goto(I6, T) E E.+TF id.E E+T. I6: goto(I1, +) T T.*F I2: goto(I0, T) E E+.TI10: goto(I7, F) E T. T .T*FT T*F. T T.*F T .F I11 : goto(I8 , ) ) I3: goto(I0, F) F .(E) F (E). T F. F .id
Transition Diagram (DFA) of Goto Function E + T I0 I1 I2 I3 I4 I5 I6 I7 I8 to I2 to I3 to I4 I9 to I3 to I4 to I5 I10 to I4 to I5 I11 to I6 * to I7 F ( T id * F F ( id ( E ) id T id + F (
Constructing SLR Parsing Table (of an augumented grammar G’) • Construct the canonical collection of sets of LR(0) items for G’. C{I0,...,In} • State i is constructed from Ii . The parsing actions for state I are determined as follows: • If a is a terminal, A.a in Ii and goto(Ii,a)=Ij then action[i,a] is shift j. • If A. is in Ii , then action[i,a] is reduce A for all a in FOLLOW(A) where AS’. • If S’S. is in Ii , then action[i,$] is accept. • If any conflicting actions generated by these rules, the grammar is not SLR(1). • Create the parsing goto table • for all non-terminals A, if goto(Ii,A)=Ij then goto[i,A]=j • All entries not defined by (2) and (3) are errors. • Initial state of the parser is the one construcetd from the sets of items containing [S’.S]
Parsing Tables of Expression Grammar Action Table Goto Table
SLR(1) Grammar • An LR parser using SLR(1) parsing tables for a grammar G is called as the SLR(1) parser for G. • If a grammar G has an SLR(1) parsing table, it is called SLR(1) grammar (or SLR grammar in short). • Every SLR grammar is unambiguous, but every unambiguous grammar is not a SLR grammar.
shift/reduce and reduce/reduce conflicts • If a state does not know whether it will make a shift operation or reduction for a terminal, we say that there is a shift/reduce conflict. • If a state does not know whether it will make a reduction operation using the production rule i or j for a terminal, we say that there is a reduce/reduce conflict. • If the SLR parsing table of a grammar G has a conflict, we say that that grammar is not SLR grammar.
Conflict Example S L=R I0: S’ .S I1: S’ S. I6: S L=.R I9: S L=R. S R S .L=R R .L L *R S .R I2: S L.=R L .*R L id L .*R R L. L .id R L L .id R .L I3: S R. I4: L *.R I7: L *R. Problem R .L FOLLOW(R)={=,$} L .*R I8: R L. = shift 6 L .id reduce by R L shift/reduce conflict I5: L id. Action[2,=] = shift 6 Action[2,=] = reduce by R L [ S L=R *R=R] so follow(R) contains, =
Conflict Example2 S AaAb I0: S’ .S S BbBa S .AaAb A S .BbBa B A . B . Problem FOLLOW(A)={a,b} FOLLOW(B)={a,b} a reduce by A b reduce by A reduce by B reduce by B reduce/reduce conflict reduce/reduce conflict
Constructing Canonical LR(1) Parsing Tables • In SLR method, the state i makes a reduction by A when the current token is a: • if the A. in the Ii and a is FOLLOW(A) • In some situations, A cannot be followed by the terminal a in a right-sentential form when and the state i are on the top stack. This means that making reduction in this case is not correct. • Back to Slide no 22.
LR(1) Item • To avoid some of invalid reductions, the states need to carry more information. • Extra information is put into a state by including a terminal symbol as a second component in an item. • A LR(1) item is: A .,a where a is the look-head of the LR(1) item (a is a terminal or end-marker.) • Such an object is called LR(1) item. • 1 refers to the length of the second component • The lookahead has no effect in an item of the form [A .,a], where is not . • But an item of the form [A .,a] calls for a reduction by A only if the next input symbol is a. • The set of such a’s will be a subset of FOLLOW(A), but it could be a proper subset.
LR(1) Item (cont.) • When ( in the LR(1) item A .,a ) is not empty, the look-head does not have any affect. • When is empty (A .,a ), we do the reduction by A only if the next input symbol is a (not for any terminal in FOLLOW(A)). • A state will contain A .,a1 where {a1,...,an} FOLLOW(A) ... A .,an
Canonical Collection of Sets of LR(1) Items • The construction of the canonical collection of the sets of LR(1) items are similar to the construction of the canonical collection of the sets of LR(0) items, except that closure and goto operations work a little bit different. closure(I) is: ( where I is a set of LR(1) items) • every LR(1) item in I is in closure(I) • if A.B,a in closure(I) and B is a production rule of G;then B.,b will be in the closure(I) for each terminal b in FIRST(a) .
goto operation • If I is a set of LR(1) items and X is a grammar symbol (terminal or non-terminal), then goto(I,X) is defined as follows: • If A .X,a in I then every item in closure({A X.,a}) will be in goto(I,X).
Construction of The Canonical LR(1) Collection • Algorithm: C is { closure({S’.S,$}) } repeat the followings until no more set of LR(1) items can be added to C. for each I in C and each grammar symbol X if goto(I,X) is not empty and not in C add goto(I,X) to C • goto function is a DFA on the sets in C.
A Short Notation for The Sets of LR(1) Items • A set of LR(1) items containing the following items A .,a1 ... A .,an can be written as A .,a1/a2/.../an
Canonical LR(1) Collection -- Example S AaAb I0: S’ .S ,$ I1: S’ S. ,$ S BbBa S .AaAb ,$ A S .BbBa ,$ I2: S A.aAb ,$ B A . ,a B . ,b I3: S B.bBa ,$ I4: S Aa.Ab ,$ I6: S AaA.b ,$ I8: S AaAb. ,$ A . ,b I5: S Bb.Ba ,$ I7: S BbB.a ,$ I9: S BbBa. ,$ B . ,a S A a to I4 B b to I5 A a B b
An Example 1. S’ S 2. S C C 3. C c C 4. C d I0: closure({(S’ S, $)}) = (S’ S, $) (S C C, $) (C c C, c/d) (C d, c/d) I1: goto(I1, S) = (S’ S , $) I2: goto(I1, C) = (S C C, $) (C c C, $) (C d, $) I3: goto(I1, c) = (C c C, c/d) (C c C, c/d) (C d, c/d) I4: goto(I1, d) = (C d , c/d) I5: goto(I3, C) = (S C C , $)
S’ S, $ S C C, $ C c C, c/d C d, c/d S C c d (S’ S , $ C c d S C C, $ C c C, $ C d, $ S C C , $ c C c C, $ C c C, $ C d, $ C cC , $ C d , $ c C C c C, c/d C c C, c/d C d, c/d C c C , c/d I1 I5 I0 I2 I6 C I9 d I7 I8 I3 d I4 C d , c/d
An Example I6: goto(I3, c) = (C c C, $) (C c C, $) (C d, $) I7: goto(I3, d) = (C d , $) I8: goto(I4, C) = (C c C , c/d) : goto(I4, c) = I4 : goto(I4, d) = I5 I9: goto(I7, c) = (C c C , $) : goto(I7, c) = I7 : goto(I7, d) = I8
S C C c C c d d c d C d An Example I0 I1 I2 I5 I6 I9 I7 I3 I8 I4
c d $ S C 0 s3 s4 g1 g2 1 a 2 s6 s7 g5 3 s3 s4 g8 4 r3 r3 5 r1 6 s6 s7 g9 7 r3 8 r2 r2 9 r2 An Example
The Core of LR(1) Items • The core of a set of LR(1) Items is the set of their first components (i.e., LR(0) items) • The core of the set of LR(1) items { (C c C, c/d), (C c C, c/d), (C d, c/d) } is { C c C, C c C, C d }
Construction of LR(1) Parsing Tables • Construct the canonical collection of sets of LR(1) items for G’. C{I0,...,In} • Create the parsing action table as follows • If a is a terminal, A.a,b in Ii and goto(Ii,a)=Ij then action[i,a] is shift j. • If A.,a is in Ii , then action[i,a] is reduce A where AS’. • If S’S.,$ is in Ii , then action[i,$] is accept. • If any conflicting actions generated by these rules, the grammar is not LR(1). • Create the parsing goto table • for all non-terminals A, if goto(Ii,A)=Ij then goto[i,A]=j • All entries not defined by (2) and (3) are errors. • Initial state of the parser contains S’.S,$
LALR Parsing Tables • LALR stands for Lookahead LR. • LALR parsers are often used in practice because LALR parsing tables are smaller than LR(1) parsing tables. • The number of states in SLR and LALR parsing tables for a grammar G are equal. • But LALR parsers recognize more grammars than SLR parsers. • yacc creates a LALR parser for the given grammar. • A state of LALR parser will be again a set of LR(1) items.
Creating LALR Parsing Tables Canonical LR(1) Parser LALR Parser shrink # of states • This shrink process may introduce a reduce/reduce conflict in the resulting LALR parser (so the grammar is NOT LALR) • But, this shrik process does not produce a shift/reduce conflict.
The Core of A Set of LR(1) Items • The core of a set of LR(1) items is the set of its first component. Ex: S L.=R,$ S L.=R Core R L.,$ R L. • We will find the states (sets of LR(1) items) in a canonical LR(1) parser with same cores. Then we will merge them as a single state. I1:L id.,= A new state: I12: L id.,= L id.,$ I2:L id.,$ have same core, merge them • We will do this for all states of a canonical LR(1) parser to get the states of the LALR parser. • In fact, the number of the states of the LALR parser for a grammar will be equal to the number of states of the SLR parser for that grammar.
Creation of LALR Parsing Tables • Create the canonical LR(1) collection of the sets of LR(1) items for the given grammar. • For each core present; find all sets having that same core; replace those sets having same cores with a single set which is their union. C={I0,...,In} C’={J1,...,Jm} where m n • Create the parsing tables (action and goto tables) same as the construction of the parsing tables of LR(1) parser. • Note that: If J=I1 ... Ik since I1,...,Ik have same cores cores of goto(I1,X),...,goto(I2,X) must be same. • So, goto(J,X)=K where K is the union of all sets of items having same cores as goto(I1,X). • If no conflict is introduced, the grammar is LALR(1) grammar. (We may only introduce reduce/reduce conflicts; we cannot introduce a shift/reduce conflict)
S’ S, $ S C C, $ C c C, c/d C d, c/d I1 I5 I0 I2 I6 I9 I7 I8 I3 I4 S C c d (S’ S , $ C c d S C C, $ C c C, $ C d, $ S C C , $ c C C c C, $ C c C, $ C d, $ C cC , $ d C d , $ c C C c C, c/d C c C, c/d C d, c/d C c C , c/d d C d , c/d
S’ S, $ S C C, $ C c C, c/d C d, c/d I1 S C c d (S’ S , $ I5 I0 C c d S C C, $ C c C, $ C d, $ S C C , $ I2 c I6 C C c C, $ C c C, $ C d, $ d I7 C d , $ c C I89 C c C, c/d C c C, c/d C d, c/d C c C , c/d/$ I3 d I4 C d , c/d
S’ S, $ S C C, $ C c C, c/d C d, c/d I1 S C c (S’ S , $ I5 I0 C c d S C C, $ C c C, $ C d, $ S C C , $ I2 c I6 d C C c C, $ C c C, $ C d, $ d I47 C d , c/d/$ c d C I89 C c C, c/d C c C, c/d C d, c/d C c C , c/d/$ I3
