Bottom-Up Parsing Techniques: LR(0) Parsing Explained

Compilers 4. Bottom-up Parsing Chih-Hung Wang References 1. C. N. Fischer and R. J. LeBlanc. Crafting a Compiler with C. Pearson Education Inc., 2009. 2. D. Grune, H. Bal, C. Jacobs, and K. Langendoen. Modern Compiler Design. John Wiley & Sons, 2000. 3. Alfred V. Aho, Ravi Sethi, and Jeffrey D. Ullman. Compilers: Principles, Techniques, and Tools. Addison-Wesley, 1986. (2nd Ed. 2006)

rest_expression expression term rest_expr IDENT IDENT IDENT  aap + ( noot + mies ) Creating a bottom-up parser automatically • Left-to-right parse, Rightmost-derivation • create a node when all children are present • handle: nodes representing the right-hand side of a production

LR(0) Parsing • Theoretically important but too weak to be useful. • running example: expression grammar input expression EOF expression  expression ‘+’ term | term term  IDENTIFIER | ‘(’ expression ‘)’ • short-hand notation Z  E $ E  E ‘+’ T | T T  i | ‘(’ E ‘)’

Z   E $ E   E ‘+’ T E   T T   i T   ‘(’ E ‘)’ LR(0) Parsing • keep track of progress inside potential handles when consuming input tokens • LR items: N   • initial set S0 Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’

 Closure algorithm for LR(0) The important part is the inference rule; it predicts new handle hypotheses from the hypothesis that we are looking for a certain non-terminal, and is sometimes called prediction rule; it corresponds to an  move, in that it allows the automation to move to another state without consuming input. Reduce item: an item with the dot at the end Shift item: the others

S0 Z   E $ E   E ‘+’ T E   T T   i T   ‘(’ E ‘)’ S3 Z  E  $ E  E  ‘+’ T S5 E  E + T  Transition Diagram S2 T E  T  i S1 T  i  E i S4 E  E ‘+’  T T   i T   ‘(’ E ‘)’ ‘+’ $ T S6 Z  E $ 

input i + i $ stack S0 LR(0) parsing example (1) Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ • shift input token (i) onto the stack • compute new state

LR(0) parsing example (2) Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 i S1 + i $ • reduce handle on top of the stack • compute new state

LR(0) parsing example (3) • reduce handle on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 T S2 + i $ i

LR(0) parsing example (4) • shift input token on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 E S3 + i $ T i

LR(0) parsing example (5) • shift input token on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 E S3 + S4 i $ T i

LR(0) parsing example (6) • reduce handle on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 E S3 + S4 i S1 $ T i

LR(0) parsing example (7) • reduce handle on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 E S3 + S4 T S5 $ T i i

LR(0) parsing example (8) • shift input token on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 E S3 $ E + T T i i

LR(0) parsing example (9) • reduce handle on top of the stack • compute new state Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 E S3 $ S6 E + T T i i

LR(0) parsing example (10) • accept! Z  E $ E  E ‘+’ T E  T T  i T  ‘(’ E ‘)’ stack input S0 Z E $ E + T T i i

Precomputing the item set (1) • Initial item set

Precomputing the item set (2) • Next item set

Complete transition diagram

The LR push-down automation • Two major moves and a minor move • Shift move • Remove the first token from the present input and pushes it onto the stack • Reduce move • N ->  •  are moved from the stack • N is then pushed onto the stack • Termination • The input has been parsed successfully when it has been reduced to the start symbol.

GOTO and ACTION tables

LR(0) parsing of the input i+i$

Another Example of LR(0) from Fischer (1)—closure0 G1 example

Another Example of LR(0) from Fischer (2) G2 example • S'S$ • SID|

Another Example of LR(0) from Fischer (3)

Another Example of LR(0) from Fischer (4)

Algorithm of LR(0) Construction (1)-goto Function

Algorithm of LR(0) Construction (2)- CFSM

Algorithm of LR(0) Construction (3)- goto Table

LR comments • The bottom-up parsing, unlike the top-down parsing, has no problems with left-recursion. • On the other hand, bottom-up parsing has a slight problem with right-recursion.

LR(0) conflicts (1) • shift-reduce conflict • Exist in a state when table construction cannot use the next k tokens to decide whether to shift the next input token or call for a reduction. • array indexing: T  i [ E ] T  i [ E ](shift) T  i(reduce) • -rule: RestExpr  Expr  Term  RestExpr (shift) RestExpr  (reduce)

LR(0) conflicts (2) • reduce-reduce conflict • Exist when table construction cannot use the next k tokens to distinguish between multiple reductions that cannot be applied in the inadequate state. • assignment statement: Z  V := E $ V  i (reduce) T  i (reduce) (Different reduce rules) • typical LR(0) table contains many conflicts

Handling LR(0) conflicts • Use a one-token look-ahead • Use a two-dimensional ACTION table • different construction of ACTION table • SLR(1) – Simple LR • LR(1) • LALR(1) – Look-Ahead LR

SLR(1) parsing • A handle should not be reduced to a non-terminal N if the look-ahead is a token that cannot follow N. • reduce N  iff token FOLLOW(N) • FOLLOW(N) • FOLLOW(Z) = { $ } • FOLLOW(E) = { ‘+’, ‘)’, $ } • FOLLOW(T) = { ‘+’, ‘)’, $ }

SLR(1) ACTION table shift

SLR(1) ACTION/GOTO table 1: Z  E $ 2: E  T 3: E  E ‘+’ T 4: T  i 5: T  ‘(’ E ‘)’ s7 sn – shift to state n rn – reduce rule n

Example of resolving conflicts (1) • A new rule T  i [E] 1: Z  E $ 2: E  T 3: E  E ‘+’ T 4: T  i 5: T  ‘(’ E ‘)’ 6: T  i ‘[‘ E ‘]’

Example of resolving conflicts (2) 1: Z  E $ 2: E  T 3: E  E ‘+’ T 4: T  i 5: T  ‘(’ E ‘)’ 6: T  i ‘[‘ E ‘]’ s5 T  i. T  i. [E]

Another Example of LR(0) Conflicts(1)

Another Example of LR(0) Conflicts(2)

Another Example of LR(0) Conflicts(3) num plus num times num $

Another Example of LR(0) Conflicts(4) Follow(E)= {plus, $}

Unfortunately … • SLR(1) leaves many shift-reduce conflicts unsolved • problem: FOLLOW(N) set is a union of all all look-aheads of all alternatives of N in all states • example S  A | x b A  a A b | B B  x Follow (S)={$} Follow(A) = {b, $} Follow(B) = {b, $}

SLR(1) automation

Another Example of SLR Problem Follow(A)={b, c, $}

Make the Grammar SLR(1) Follow(A1)={b, $}

Another Example 2 of SLR(1) (1) G3: SE$ EE+T|T TT*P|P PID|(E)

Another Example 2 of SLR(1) (2) G3: SE$ EE+T|T TT*P|P PID|(E) • SLR(1) Action Table for G3 Follow(E)={$+)} Follow(T)={$+)*}

SLR(1) Conflicts Ex2 Elem(List, Elem) ElemScalar ListList,Elem List Elem Scalar ID Scalar(Scalar) Fellow(Elem)={“)”,”,”,….}

LR(1) parsing • The LR(1) technique does not rely on FOLLOW sets, but rather keeps the specific look-ahead with each item • LR(1) item: N   {} •  - closure for LR(1) item sets: if set S contains an item P  N  {} then for each production rule N   S must contain the item N   {} where  = FIRST(  {} )

Bottom-Up Parsing Techniques: LR(0) Parsing Explained

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