More BottomUp Parsing


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1 More BottomUp Parsing Lecture 7 Dr. Sean Peisert ECS 142 Spring
2 Status Project 1 Back By Wednesday (ish) savior lexer in ~cs142/s09/bin Project 2 Due Friday, Apr. 24, 11:55pm My office hours 3pm today Discussion section 4:10pm today 2
3 The Stack Shift pushes a terminal on the stack Reduce pops 0 or more symbols off of the stack (production rhs) and pushes a nontermianl on the stack (production lhs) 3
4 When to Shift vs. Reduce? Decide based on the left string (the stack) Idea: use a finite automaton (DFA) to decide when to shift or reduce The DFA input is the stack The language consists of terminals and nonterminals We run the DFA on the stack and we examine the resulting state X and the token T after If X has a transition labeled T then shift If X is labeled with A ß on T then reduce 4
5 int + (int) + (int)$ int + (int) + (int)$ E + (int) + (int)$ E + (int ) + (int)$ E + (E ) + (int)$ E + (E) + (int)$ E + (int)$ E + (int )$ E + (E )$ E + (E) $ E $ shift E int shift(x3) E int shift E E+(E) shift(x3) E int shift E E+(E) accept 5
6 How is the DFA Constructed? The stack describes the context of the parse What nonterminal we are looking for What production rhs we are looking for What we have seen so far from the rhs Each DFA state describes several such contexts E.g., when we are looking for nonterminal E, we might be looking either for an int or an E+(E) rhs 6
7 LR(1) Items An LR(1) item is a pair, e.g.,: X α ß, a X αß is a production a is a terminal (the lookahead terminal) LR(1) means 1 lookahead terminal [X α ß, a] describes a context of the parser We are trying to find an X followed by an a, and We already have α on the top of the stack Thus we need to see a prefix derived from ßa 7
8 Note Symbol separates stack from rest of input: α γ where α is the stack and γ is the remaining string of terminals. In LR(1) items, is used to mark a prefix of a production RHS: X α ß, a Here, ß might contain nonterminals as well. In both cases, the stack is on the left. 8
9 Convention We add to our grammar a fresh new start symbol and a production S E Where E is the old start symbol The initial parsing context contains: S E, $ Trying to find an S as a string derived from E$ The stack is empty 9
10 LR(1) Items In context containing E E + (E), + If ( follows then we can perform a shift to context containing E E + ( E), + In context containing E E + (E), + We can perform a reduction with E E + (E) But only if a + follows 10
11 LR(1) Items Consider a context with the item E E + ( E), + We expect a string derived from E ) + There are two productions for E E E + (E) and E int We describe this by extending the context with two more items: E int, ) E E + (E), ) 11
12 First and Follows Consider the state S ß Aγ We re trying to match the string ßbδ Suppose that b is the next token. Either: A α, if b can start a string derived from α We say that b First(α) Or, the expansion of A is empty and b belongs to an expansion of γ (e.g., γ bω). b can appear after A in a derivation of the form S ßAbω We say that b Follow(A) in this case. 12
13 What productions can we use? Consider the state S ß Aγ We re trying to match the string ßbδ The expansion of A is empty and b belongs to an expansion of γ (e.g., γ bω). b can appear after A in a derivation of the form S ßAbω We say that b Follow(A) in this case. Can use as a production: A α can be used if α can expand to ε We say that ε First(A) in this case 13
14 Computing First Sets Definition: First(X) = {b X bα} U {ε X ε} First (b) = {b} For all productions X A1... An Add First (A1)  {ε} to First(X). Stop if ε First(A1)... Add First (An)  {ε} to First(X). Stop if ε First(An) Add ε to First(X) (ignore Ai if it is X) 14
15 First sets Example Grammar: E TX First sets T (E) int Y X + E ε Y * T ε First ( ( ) = { ( } First ( T ) = { int, ( } First ( ) ) = { ) } First ( E ) = { int, ( } First ( int ) = { int } First ( X ) = { +, ε } First ( + ) = { + } First ( Y ) = { *, ε } First ( * ) = { * } 15
16 Computing Follow Sets Definition: Follow(X) = {b S ß X b ω} Compute the First sets for all nonterminals first Add $ to Follow(S) (if S is the start nonterminal) For all productions Y X A1... An Add First (A1)  {ε} to Follow(X). Stop if ε First(A1)... Add First (An)  {ε} to Follow(X). Stop if ε First(An) Add Follow(Y) to Follow(X) 16
17 Follow sets Example Grammar: E TX First sets T (E) int Y X + E ε Y * T ε Follow ( + ) = { int, ( } Follow ( * ) = { int, ( } Follow ( ( ) = { int, ( } Follow ( E ) = { ), $ } Follow ( X ) = { $, ) } Follow ( T ) = { +, ), $ } Follow ( ) ) = { +, ), $ } Follow ( Y ) = { +, ), $ } Follow ( int ) = { *, +, ), $ } 17
18 LR(1) Items Consider a context with the item E E + ( E), + We expect a string derived from E ) + There are two productions for E E E + (E) and E int We describe this by extending the context with two more items: E int, ) E E + (E), ) 18
19 The Closure Operation The operation of extending the context with items is called the closure operation Closure(Items) = repeat for each [X α ß, a] in Items for each production Y γ for each b First(ßa) add [Y γ,b] to Items until Items is unchanged 19
20 Construct the Parsing DFA Construct the start context: Closure({S E, $)} S E, $ E E+(E), $ E int, $ E E+(E), + E int, + We abbreviate as: S E, $ E E+(E), $/+ E int, $, $/+ 20
21 Construct the Parsing DFA A DFA state is a closed set of LR(1) items This means that we performed Closure The start state contains [S E, $] A state that contains [X α, b] is labeled with reduce with X α on b And now the transitions... 21
22 DFA Transitions A state State that contains [X α γß, b] has a transition labeled y to a state that contains the items Transition(State, γ) γ can be a terminal or a nonterminal Transition(State, γ) Items for each [X α γß, b] State add [X αγ ß, b] to Items return Closure(Items) 22
23 Example Diagram 23
24 LR Parsing Tables Parsing tables (i.e., the DFA) can be constructed automatically for a CFG. But we still need to understand the construction to work with parser generators E.g., they report errors in terms of sets of items What kind of errors can we expect? 24
25 Shift/Reduce Conflicts If a DFA state contains both: [X α aß, b] and [Y γ, a] Then on input a we could either: Shift into state [X αa ß, b], or Reduce with [Y γ] This is called a shiftreduce conflict. 25
26 Shift/Reduce Conflicts Typically due to ambiguities in the grammar Classic example: the dangling else: S if E then S if E then S else S OTHER Will have a DFA state containing [S if E then S, else] [S if E then S else S, x ] If else follows then we can shift or reduce Default (bison, CUP, etc..) is to shift. Default behavior is as needed in this case. 26
27 Shift/Reduce Conflicts Consider the ambiguous grammar: E E + E E * E int Will have the DFA states containing [E E * E, +] [E E * E, +] [E E + E, +] E [E E + E, +] Again we have a shift/reduce input on + We need to reduce (* binds more tightly than +) Recall solution: declare precedence of + and * 27
28 Associativity Example: Stack contains 12 and the lookahead is  Left associative: (12)  5 Right associative: 1  (25) 28
29 Shift/Reduce Conflicts In bison declare precedence and associativity: %left + (left associative) %left * (left associative, higher precedence) Precedence of a rule = that of its last terminal See bison manual ( ConflictDependent Precedence ) for ways to override this default Bison resolves shift/reduce conflicts with a shift if: no precedence declared for either rule or terminal input terminal has higher precedence than the rule the precedences are the same and right associative 29
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