# Bottom-Up Parsing. Parser Generation. LR Parsing. Constructing LR Parser

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1 Parser Generation Main Problem: given a grammar G, how to build a top-down parser or a bottom-up parser for it? parser : a program that, given a sentence, reconstructs a derivation for that sentence ---- if done sucessfully, it recognize the sentence all parsers read their input left-to-right, but construct parse tree differently. bottom-up parsers --- construct the tree from leaves to root Bottom-Up Parsing Construct parse tree bottom-up --- from leaves to the root Bottom-up parsing always constructs right-most derivation Important parsing algorithms: shift-reduce, LR parsing LR parser components: input, stack (strings of grammar symbols and states), driver routine, parsing tables. input: a 1 a 2 a 3 a 4... a n \$ shift-reduce, LR, LR, LLR, operator precedence top-down parsers --- construct the tree from root to leaves recursive descent, predictive parsing, LL(1) stack s m X m.. s 1 X 1 s 0 LR Parsing parsing output Copyright Zhong hao, Yale University Parser Generation: Page 1 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 2 of 27 LR Parsing sequence of new state symbols s 0, s 1, s 2,..., s m each state sumarize the information contained in the stack below it. Parsing configurations: (stack, remaining input) written as (s 0 X 1 s 1 X 2 s 2...X m s m, a i a i+1 a i+2...a n \$) next move is determined by s m and a i Parsing tables: CTION[s,a] and GOTO[s,X] Table CTION[s,a] --- s : state, a : terminal its entries (1) shift s k (2) reduce -> (3) accept (4) error Table G GOTO[s,X] --- s : state, X : non-terminal its entries are states Constructing LR Parser How to construct the parsing table CTION and GOTO? basic idea: first construct DF to recognize handles, then use DF to construct the parsing tables! different parsing table yield different LR parsers LR(1), LR(1), or LLR(1) augmented grammar for context-free grammar G = G(T,N,P,) is defined as G = G (T, N { }, P { -> }, ) adding non-terminal and the production ->, and is the new start symbol. When -> is reduced, parser accepts. LR(0) item for productions of a context-free grammar G is a production with dot at some position in the r.h.s. For -> XYZ, its items are ->.XYZ -> X.YZ -> XY.Z -> XYZ. For ->, its items are just ->. Copyright Zhong hao, Yale University Parser Generation: Page 3 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 4 of 27

2 LR(0) items and LR(0) DF Informally, item -> X.YZ means a string derivable from X has been seen, and one from YZ is expected. LR(0) items are used as state names for LR(0) DF or LR(0) NF that recognizes viable prefixes. Viable prefixes of a CFG are prefixes of right-sentential forms with no symbols to right of the handle; we can always add terminals on right to form a rightsentential form. Two way to construct the LR(0) DF: 1. first construct LR(0) NF and then convert it to a DF! 2. construct the LR(0) DF directly! From LR(0) DF to the Parsing Table transition table for the DF is the GOTO table; the states of DF are states of the parser. Example: LR(0) Items CFG Grammar: E -> E + T T T -> T * F F ugmented Grammar: LR(0) terms: E -> E E -> E + T T T -> T * F F E ->. E T ->. T * F F -> ( E. ) E -> E. T -> T. * F F -> ( E ). E ->. E + T T -> T *. F F ->. id E -> E. + T T -> T * F. F -> id. E -> E +. T T ->. F E -> E + T. T -> F. E ->. T F ->. ( E ) E -> T. F -> (. E ) Copyright Zhong hao, Yale University Parser Generation: Page 5 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 6 of 27 From LR(0) NF to LR(0) DF Construct LR(0) NF with all LR(0) items of G as states, connect states by moving the dot; final states are those with dots at the end. 1. for each item ->.X ->.X -> X. 2. for each pair ->.B, B ->. (expect to see a string derivable from ) ->.B B ->. Convert NF to DF using subset construction algorithm. The states of the resulting LR(0) DF --- C = {I 1,I 2,..., I n } are called canonical LR(0) collection for grammar G Disadvantage: the NF is often huge, and converting from NF to DF is tedious and time-consuming. X Building LR(0) DF Directly Instead of building DF from NF, we can build the LR(0) DF directly. Given a set of LR(0) items I, CLOURE(I) is defined as for each item ->.B in I and each production B -> add B ->. to I, if it s not in I until I does not change GOTO(I,X) is defined as CLOURE(all items -> X. for each ->.X in I) Canonical LR(0) collection is computed by the following procedure: I 0 = CLOURE({ ->.}) and C = {I 0 } for each I C and grammar symbol X T = GOTO(I,X); if T and T C then C = C { T }; until C does not change Resulting LR(0) DF: C is the set of states; GOTO is the transition table Copyright Zhong hao, Yale University Parser Generation: Page 7 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 8 of 27

3 Constructing LR(1) Parsing Table From the LR(0) DF, we can construct the parsing table LR(1) parsing table. The parser based on LR(1) parsing table is called LR(1) parser. The LR(1) grammars are those whose LR(1) parsing table does not contain any conflicts. lgorithm --- use C = {I 0,...,I n }, GOTO, FOLLOW: 1. If -> a.a is in I i and GOTO(I i,a) = I j where a is a terminal, set CTION[i,a] to shift j. 2. If ->. is in I i, set CTION[i,a] to reduce -> for all terminal a in FOLLOW(). 3. If ->. is in I i, set CTION[ i,\$] to accept 4. If GOTO(I i,) = I j, set GOTO[i,] = j 5. set all other entries to error 6. set initial state to be I i with ->. Limitation of LR(1) Parser Unfortunately, many unambiguous grammars are not LR(1) gammars -> L = R R L -> *R id R -> L Canonical LR(0) collection --- L means l-value R means r-value * means contents of I0 : ->. I3: -> R. I6: -> L=.R ->.L=R R ->.L ->.R I4: L -> *.R L ->.*R L ->.*R R ->.L L ->.id L ->.id L ->.*R R ->.L L ->.id I7: L -> *R. I1 : ->. I5: L -> id. I8: R -> L. I2 : -> L.=R I9: -> L=R. R -> L. FOLLOW(R) = {=,...} state 2 has a shift/reduce conflict on = : shift 6 or reduce R -> L Copyright Zhong hao, Yale University Parser Generation: Page 9 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 10 of 27 LR(1) Parsing Conflict arises because LR(0) states do not encode enough left context --- in the previous example, reduction R -> L is wrong upon input = because R =... never appears in right-sentential form. olution: split LR(0) states by adding terminals to states, for example, [ - >., a] results in reduction only if next symbol is a. n LR(1) term is in the form of [ ->., a ] where -> is a production and a is a terminal or \$ To build LR(1) parsing table --- we first build LR(1) DF --- then construct the parsing table using the same LR(1) algorithm except 2. only if [ ->., a] is in I i, then set CTION[i,a] to reduce -> To way to build LR(1) DF ---- from NF -> DF or build DF directly Building LR(1) DF Construct LR(1) NF with all LR(1) items of G as states, connect states by moving the dot; then convert the NF to DF. 1. for each item [ ->.X,a] Construct the LR(1) DF directly (see the Dragon book) Given a set of LR(1) items I, ->.X a 2. for each pair [ ->.B, a] B ->. and b in FIRT( a ). ->.B a CLOURE(I) is now defined as for each item [ ->.B, a] in I and each production B -> and each terminal b in FIRT( a) add [B ->. a] to I, if it s not in I until I does not change X -> X. a B ->. b Copyright Zhong hao, Yale University Parser Generation: Page 11 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 12 of 27

4 Constructing LR(1) Parser Canonical LR(1) collection is computed by the following procedure: I 0 = CLOURE([ ->., \$]) and C = {I 0 } for each I C and grammar symbol X T = GOTO(I,X); if T and T C then C = C { T }; until C does not change Resulting LR(1) DF: C is the set of states; GOTO is the transition table From the LR(1) DF, we can construct the parsing table LR(1) parsing table. The parser based on LR(1) parsing table is called LR(1) parser. The LR(1) grammars are those whose LR(1) parsing table does not contain any conflicts (no duplicate entries). Example: -> -> C C C -> c C d LLR(1) Parsing Bad News: LR(1) parsing tables are too big; for PCL, LR tables has about 100 states, LR table has about 1000 states. LLR (Lookhead-LR) parsing tables have same number of states as LR, but use lookahead for reductions. The LLR(1) DF can be constructed from the LR(1) DF. LLR(1) states can be constructed from LR(1) states by merging states with same core, or same LR(0) items, and union their lookahead sets. Merging I8: C -> cc., c/d I9: C -> cc., \$ into a new state I89: C -> cc., c/d/\$ Merging I3: C -> c.c, c/d I6: C -> c.c, \$ C ->.cc, c/d C ->.cc, \$ C ->.d, c/d C ->.d, \$ into a new state I36: C -> c.c, c/d/\$ C ->.cc, c/d/\$ C ->.d, c/d/\$i Copyright Zhong hao, Yale University Parser Generation: Page 13 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 14 of 27 LLR(1) Parsing (cont d) From the LLR(1) DF, we can construct the parsing table LLR(1) parsing table. The parser based on LLR(1) parsing table is called LLR(1) parser. The LLR(1) grammars are those whose LLR(1) parsing table does not contain any conflicts (no duplicate entries). LLR(1) DF and LLR(1) parsing table can be constructed without creating LR(1) DF --- see Dragon book for detailed algorithm. LLR parser makes same number of moves as LR parser on correct input. On incorrect input, LLR parser may make erroneous reductions, but will signal error before shifting input, i.e., merging states makes reduce determination less accurate, but has no effect on shift actions. ummary: LR Parser Relation of three LR parsers: LR(1) > LLR(1) > LR(1) Most programming language constructs are LLR(1). The LR(1) is unnecessary in practice, but the LR(1) is not enough. YCC is an LLR(1) Parser Generator. When parsing ambiguious grammars using LR parsers, the parsing table will contain multiple entries. We can specify the precedence and associativity for terminals and productions to resolve the conflicts. YCC uses this trick. Other Issues in parser implementation: 1. compact representation of parsing table 2. error recovery and diagnosis. Copyright Zhong hao, Yale University Parser Generation: Page 15 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 16 of 27

6 Left Recursion Elimination Elimination of Left Recursion (useful for top-down parsing only) replace productions of the form with Important: read ppel pp for details -> -> -> (yields different parse trees but same language) example: E -> E + T T T -> T * F F become E -> T E E -> + T E T -> F T T -> * F T Left Factoring ome grammars are unsuitable for recursive descent, even if there is no left recursion dangling-else stmt -> if expr then stmt if expr then stmt else stmt... input symbol if does not uniquely determine alternative. Left Factoring --- factor out the common prefixes (see HU pp 178) change the production -> x y x z to -> x -> y z thus stmt -> if expr then stmt -> else stmt Copyright Zhong hao, Yale University Parser Generation: Page 21 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 22 of 27 Predictive Parsing Predictive parsing is just table-driven recursive descent; it contains: parsing stack --- contains terminals and non-terminals parsing table : a 2-dimensional table M[X,a] where X is non-terminal, a is terminal, and table entries are grammar productions or error indicators. algorithm \$ is end-of-file, is start symbol push(\$); push(); while top <> \$ do ( a <- the input char if top is a terminal or \$ then (if top == a then pop(); advance() else err()) else if M[top,a] is X->Y 1 Y 2...Y k then (pop(); push(y k );...; push(y 1 )) else err() ) input: X... Y Z \$ stack a 1 a 2 a 3 a 4... a n \$ Predictive Parser Parsing Table M output Constructing Predictive Parser The key is to build the parse table M[,a] for each production -> do for each a FIRT( ) do add -> to M[,a] if FIRT( ) then for each b FOLLOW() do add -> to M[,b] rest of M is error FIRT( ) is a set of terminals (plus ) that begin strings derived from, where is any string of non-terminals and terminals. FOLLOW() is a set of terminals that can follow in a sentential form, where is any non-terminal Copyright Zhong hao, Yale University Parser Generation: Page 23 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 24 of 27

7 First & Follow To compute FIRT(X) for any grammar symbol X : FIRT(X) = {X}, if X is a terminal; FIRT(X) = FIRT(X) {a}, if X -> a ; FIRT(X) = FIRT(X) { }, if X -> ; and FIRT(X) = FIRT(X) FIRT(Y 1 Y 2...Y k ), if X -> Y 1 Y 2...Y k. until nothing new is added to any FIRT FIRT(Y 1 Y 2...Y k ) = FIRT(Y 1 )-{ } FIRT(Y 2 )-{ } if FIRT(Y 1 ) FIRT(Y 3 )-{ } if FIRT(Y 1 Y 2 )... FIRT(Y k )-{ } if FIRT(Y 1...Y k-1 ) { } if all FIRT(Y i ) (i=1,...,k) contain First & Follow (cont d) To compute FOLLOW(X) for any non-terminal X : FOLLOW() = FOLLOW() {\$}, if is start symbol; FOLLOW(B) = FOLLOW(B) (FIRT( ) - { }), if -> B and FOLLOW(B) = FOLLOW(B) FOLLOW() if -> B or -> B and FIRT( ) Example: reason FOLLOW(E) = {\$}, E is start symbol = {\$ )}, F -> ( E ) FOLLOW(E )= FOLLOW(E) E -> T E = {\$ )}, (Read ppel pp for detailed examples) Copyright Zhong hao, Yale University Parser Generation: Page 25 of 27 Copyright Zhong hao, Yale University Parser Generation: Page 26 of 27 ummary: LL(1) Grammars grammar is LL(1) if parsing table M[,a] has no duplicate entries, which is equivalent to specifying that for each production 1. ll FIRT( i ) are disjoint. -> n 2. t most one i can derive ; in that case, FOLLOW() must be disjoint from FIRT( 1 ) FIRT( 2 ) FIRT( n ) Left-recursion and ambiguity grammar lead to multiple entries in the parsing table. (try the dangling-else example) The main difficulty in using (top-down) predicative parsing is in rewriting a grammar into an LL(1) grammar. There is no general rule on how to resolve multiple entries in the parsing table. Copyright Zhong hao, Yale University Parser Generation: Page 27 of 27

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