LL Parsing: A piece of cake after LR


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1 LL Parsing: A piece of cake after LR Lecture 11 Dr. Sean Peisert ECS 142 Spring
2 LL Parsing Still specified using a CFG Still reads lefttoright (Lx) Now is leftmost derivation (xl) rather than rightmost (xr) Constructed from the top in the order that tokens appear in the token stream 2
3 Parsing Algorithms LL E E + E 2 + E 2 + E * E * E * 8 LR E 2 E 3 E E + E E 8 E E * E E E 3
4 LR vs. LL LR (bison/cup): harder to implement, but allows more grammars LL: easier to write, but less flexible in the grammars 4
5 Recursive Descent Parsing Consider the grammar: E T + E T T int int * T (E) Token stream is int5 * int2 Start with toplevel nonterminal E Try the rules E in order 5
6 Recursive Descent Parsing E0 T1 + E2 Then try a rule for T1 (E3) But ( does not match the input token int 5 Try T1 int. Token matches. But + after T 1 does not match the input token * Try T1 int * T2 This will match but + after T 1 will be unmatched Have exhausted the choices for T 1 Backtrack to choice for E 0 6
7 Recursive Descent Parsing E0 T1 Follow same steps as before for T 1 And succeed with T1 int * T2 and T2 int E0 T1 int5 * T2 7 int2
8 Recursive Descent Parsing Parsing: given a string of tokens t1, t2, t3, t4, t5, find its parse tree Recursive descent parsing. Try all of the productions exhaustively: At a given moment, the finge of the parse tree is: t1, t2,... tk, A Try all the productions for A: if A BC is a production, the new finge is t1, t2,... tk, BC Backtrack where the finge doesn t match the string. Stop when there are no more nonterminals. 8
9 Construction of Decision Table for each rule X α for each a in First(α) table (X, a) = X α so if we see the very first symbol, match the rule if ε is in First(alpha) for each b in Follows(X) table(x, b) = X α That s it! 9
10 NonLL Grammars There are a lot of nonll grammars. The patterns to look for are: LeftRecursive rules (goes into an infinite loop) which must be rewritten as rightrecursive Common prefix rules which must be rewritten to move the common prefix to one rule 10
11 Left Recursion A Aa A b For LL, First(A Aa) = b For LL, First(A b) = b this gives a conflict 11
12 Rewrite the Grammar A ba A aa A ε 12
13 Common Prefix X abc X abd This works for LR. For LL, the parser only looks at the First set of rules, not the entire rule. So the rules cause a conflict. 13
14 Rewritten Grammar X aby Y c Y d 14
15 Recursive Descent Simple and general, but unpopular (and slow) because of backtracking Often we can avoid backtracking (somewhat) 15
16 Predictive Parsing Like recursivedescent, but parser can predict which production to use By looking at the next few tokens No backtracking Predictive parsers accept LL(k) grammars 16
17 LL(1) Languages In recursivedescent, for each nonterminal and input token, there may be a choice of production LL(1) means that for each nonterminal and token, there is only one production that could lead to success. Can be specified as a 2D table 1 dim for curent nonterminal to expand 1 dim for next token A table entry contains one production 17
18 Example Grammar #1 Grammar: E T + E T T int int * T (E) Impossible to predict because T has two productions with int. For E it is not clear how to predict. Grammar must be leftfactored 18
19 Left Factoring Grammar E T + E T T int int * T (E) Factor out common prefixes of productions: E T X X + E ε T (E) int Y Y * T ε 19
20 LL(1) Parse Table Example LeftFactored Grammar E T X X + E ε T (E) int Y Y * T ε The LL(1) parsing table ($ is a special end marker): int * + ( ) $ T int Y (E) E T X T X X + E ε ε Y * T ε ε ε 20
21 LL(1) Parse Table Consider the [E, int] entry: When current nonterminal is E and next input is int, use production E TX This production can generate an int in the first place Consider the [Y, +] entry: When current nonterminal is y and current token is +, get rid of Y We ll see later why this is so 21
22 LL(1) Parse Table Blank entries indicate error situtations E.g., consider the [E, *] entry There is no way to derive a string starting with * from nonterminal E 22
23 Using Parse Tables Similar to recursive descent, except: For each nonterminal S We look at the next token a And choose the production shown at [S,a] Use a stack to keep track of pending nonterminals Reject when we encounter an error state Accept when we encounter an endtoinput 23
24 LL(1) Parsing Algorithm initialize stack = <S, $> and next (pointer to tokens) repeat case stack of <X, rest> if T[X, *next] = Y1...Yn then stack Y1...Yn rest> ; else error(); <t, rest> if T == *next ++ then stack <rest> ; else error(); until stack == < > 24
25 LL(1) Parsing Example Stack Input Action E$ int * int $ TX TX$ int * int $ int Y int YX$ int * int * terminal YX$ *int$ *T *TX$ *int$ terminal TX$ int$ int Y int YX$ int$ terminal YX$ $ ε X$ $ ε $ $ ACCEPT 25
26 Example Grammar #2 D v L c T L L m i L i T r v = Var, L = IdList c = colon, T = type m = comma, i = id, r = REAL 26
27 After fixing L (which was rightrecursive) D v L c T L i X X m i X X ε T r 27
28 Algorithm start with the start symbol on the stack repeat until done if top of stack == first char of input pop 1 of stack advance input else if top of stack == nonterminal T consult (T, input) if (T,input) == T α pop T push input in reverse order print T α reduce else if (T,input) == empty error  reject else if stack is empty and input is empty accept else if stack is empty and input is not empty reject 28
29 LL Decision Table v i m c r $ D D>vLcT L L>iX X X>miX X>e T T>r 29
30 stack input action D vimicr D, v=match D vlct, pop LHS(D), push RHS (vlct) in reverse order TcLv vimicr v, v = match so pop from stack, advance input TcL imicr L, i = match L ix pop L, push ix in reverse order TcXi imicr i, i = match, pop stack, advance TcX micr X, m = match X mix pop X, push mix in reverse order TcXim micr m, m = match pop stack, advance TcXi icr i, i = match pop stack, advance TcX cr X, c = match X ε, pop X, push in reverse order Tc cr c, c = match pop stack, advance T r T, r = match T r pop T, push r in reverse order r r r, r = match pop stack, advance 30
31 Done! Accept! 31
32 Status Project 2 Due Friday, Apr. 24, 11:55pm 32
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