CSc 453 Syntax Analysis Parsing Saumya Debray The

CSc 453 Syntax Analysis (Parsing) Saumya Debray The University of Arizona Tucson

Overview source program lexical analyzer (scanner) tokens syntax analyzer (parser) syntax tree symbol table manager Main Task: Take a token sequence from the scanner and verify that it is a syntactically correct program. Secondary Tasks: l l Process declarations and set up symbol table information accordingly, in preparation for semantic analysis. Construct a syntax tree in preparation for intermediate code CSc 453: Syntax Analysis 2 generation.

Context-free Grammars l l A context-free grammar for a language specifies the syntactic structure of programs in that language. Components of a grammar: l l a finite set of tokens (obtained from the scanner); a set of variables representing “related” sets of strings, e. g. , declarations, statements, expressions. a set of rules that show the structure of these strings. an indication of the “top-level” set of strings we care about. CSc 453: Syntax Analysis 3

Context-free Grammars: Definition Formally, a context-free grammar G is a 4 -tuple G = (V, T, P, S), where: l l l V is a finite set of variables (or nonterminals). These describe sets of “related” strings. T is a finite set of terminals (i. e. , tokens). P is a finite set of productions, each of the form A where A V is a variable, and (V T)* is a sequence of terminals and nonterminals. l S V is the start symbol. CSc 453: Syntax Analysis 4

Context-free Grammars: An Example A grammar for palindromic bit-strings: G = (V, T, P, S), where: l V = { S, B } l T = {0, 1} l P = { S B, S 0 S 0, S 1 S 1, B 0, B 1 } CSc 453: Syntax Analysis 5

Context-free Grammars: Terminology l Derivation: Suppose that l l and are strings of grammar symbols, and A is a production. Then, A (“ A derives ”). l : “derives in one step” * : “derives in 0 or more steps” * * if and * CSc 453: Syntax Analysis (0 steps) ( 1 steps) 6

Derivations: Example l Grammar for palindromes: G = (V, T, P, S), l l V = {S}, T = {0, 1}, P = { S 0 S 0 | 1 S 1 | 0 | 1 | }. A derivation of the string 10101: S 1 S 1 1 0 S 0 1 10101 (using S 1 S 1) (using S 0 S 0) (using S 1) CSc 453: Syntax Analysis 7

Leftmost and Rightmost Derivations l A leftmost derivation is one where, at each step, the leftmost nonterminal is replaced. (analogous for rightmost derivation) l Example: a grammar for arithmetic expressions: E E + E | E * E | id l Leftmost derivation: E E * E E + E * E id + id * id l Rightmost derivation: E E + E * id E + id * id CSc 453: Syntax Analysis 8

Context-free Grammars: Terminology l The language of a grammar G = (V, T, P, S) is L(G) = { w | w T* and S * w }. The language of a grammar contains only strings of terminal symbols. l Two grammars G 1 and G 2 are equivalent if L(G 1) = L(G 2). CSc 453: Syntax Analysis 9

Parse Trees l l A parse tree is a tree representation of a derivation. Constructing a parse tree: l The root is the start symbol S of the grammar. l Given a parse tree for X , if the next derivation step is X 1… n then the parse tree is obtained as: CSc 453: Syntax Analysis 10

Exercise: Derivations and Parse Trees Grammar: G = (V, T, P, S) l l l V = { A, B, C} T = {0, 1} P: l l l A→BC B→ 0 B 0 B→ 1 C→ 1 C C→ 0 Strings: l 00100110 Problem: 1. Leftmost derivation 2. Rightmost derivation 3. Parse trees S A CSc 453: Syntax Analysis 11

Approaches to Parsing l Top-down parsing: l l attempts to figure out the derivation for the input string, starting from the start symbol. Bottom-up parsing: l l starting with the input string, attempts to “derive in reverse” and end up with the start symbol; forms the basis for parsers obtained from parsergenerator tools such as yacc, bison. CSc 453: Syntax Analysis 12

Top-down Parsing l l l “top-down: ” starting with the start symbol of the grammar, try to derive the input string. Parsing process: use the current state of the parser, and the next input token, to guide the derivation process. Implementation: use a finite state automaton augmented with a runtime stack (“pushdown automaton”). CSc 453: Syntax Analysis 13

Bottom-up Parsing l l “bottom-up: ” work backwards from the input string to obtain a derivation for it. Parsing process: use the parser state to keep track of: l l l what has been so far, and given this, what the rest of the input might look like. Implementation: use a finite state automaton augmented with a runtime stack (“pushdown automaton”). CSc 453: Syntax Analysis 14

Parsing: Top-down vs. Bottom-up top-down parsing: S starting at the start symbol, infers a derivation that gives the input string bottom-up parsing: starting from the input string, infers a derivation-in-reverse that gives the start symbol parse tree input string CSc 453: Syntax Analysis 15

Parsing Problems: Ambiguity l l l A grammar G is ambiguous if some string in L(G) has more than one parse tree. Equivalently: if some string in L(G) has more than one leftmost (rightmost) derivation. Example: The grammar E E + E | E * E | id is ambiguous, since “id+id*id” has multiple parses: CSc 453: Syntax Analysis 16

Exercise: Ambiguity Which (if any) of these grammars are ambiguous? X→ XX X→(X) X→ X→AB A→ 0 A 1 | 1 B→ 1 B 0 | 0 CSc 453: Syntax Analysis 17

Dealing with Ambiguity Transform the grammar to an equivalent unambiguous grammar. Augment ambiguous grammar with disambiguating rules to specify which parse to use. 1. 2. l e. g. : ‘%left’, ‘%right’ annotations in yacc Comment: It is not possible to determine algorithmically whether: l l Two given CFGs are equivalent; A given CFG is ambiguous. CSc 453: Syntax Analysis 18

Bottom-up parsing: Approach (Implementation) Preprocess the grammar to compute some info about it. (FIRST and FOLLOW sets) Use this info to construct a pushdown automaton for the grammar: 1. 2. l l the automaton uses a table (“parsing table”) to guide its actions; constructing a parser amounts to constructing this table. CSc 453: Syntax Analysis 19

Bottom-up parsing: Approach Background 1. l l l general structure of bottom-up parsers: shift-reduce parsers LR parsers Construction of LR parsing automata Construction of LR parse tables 2. 3. l 4. (Discussion) FIRST and FOLLOW sets Putting it all together CSc 453: Syntax Analysis 20

GENERAL BACKGROUND CSc 453: Syntax Analysis 21

Shift-reduce Parsing An instance of bottom-up parsing Basic idea: repeat l l 1. 2. l in the string being processed, find a substring α such that A → α is a production; replace the substring α by A (i. e. , reverse a derivation step). until we get the start symbol. Technical issues: Figuring out 1. 2. which substring to replace; and which production to reduce with. CSc 453: Syntax Analysis 22

Shift-reduce Parsing: Example Grammar: S → a. ABe A → Abc | b B→d Input: abbcde a. Ade a. ABe S (using A → b) (using A → Abc) (using B → d) (using S → a. ABe) CSc 453: Syntax Analysis 23

Shift-Reduce Parsing: cont’d l Need to choose reductions carefully: abbcde a. AAc. Be … doesn’t work. l A handle of a string s is a substring s. t. : l l l matches the RHS of a rule A → ; and replacing by A (the LHS of the rule) represents a step in the reverse of a rightmost derivation of s. For shift-reduce parsing, reduce only handles. CSc 453: Syntax Analysis 24

Shift-reduce Parsing: Implementation Data Structures: l l l a stack, its bottom marked by ‘$’. Initially empty. the input string, its right end marked by ‘$’. Initially w. Actions: l repeat 1. Shift some ( 0) symbols from the input string onto the stack, until a handle appears on top of the stack. 2. Reduce to the LHS of the appropriate production. until ready to accept. l Acceptance: when input is empty and stack contains only the start symbol. CSc 453: Syntax Analysis 25

Example Stack (→) Input Action $ abbcde$ shift $a bbcde$ shift Grammar : $ab bcde$ reduce: A → b $a. A bcde$ shift S → a. ABe $a. Ab cde$ shift A → Abc | b $a. Abc de$ reduce: A → Abc B→d $a. A de$ shift $a. Ad e$ reduce: B → d $a. AB e$ shift $a. ABe $ reduce: S → a. ABe $S $ accept CSc 453: Syntax Analysis 26

Conflicts l Can’t decide whether to shift or to reduce ― both seem OK (“shift-reduce conflict”). Example: S → if E then S | if E then S else S | … l Can’t decide which production to reduce with ― several may fit (“reduce-reduce conflict”). Example: Stmt → id ( args ) | Expr → id ( args ) CSc 453: Syntax Analysis 27

LR Parsing l A kind of shift-reduce parsing. An LR(k) parser: l l Advantages: l l l scans the input L-to-R; produces a Rightmost derivation (in reverse); and uses k tokens of lookahead. very general and flexible, and handles a wide class of grammars; efficiently implementable. Disadvantages: l difficult to implement by hand (use tools such as yacc or bison). CSc 453: Syntax Analysis 28

LR PARSING: HOW THE PARSER WORKS CSc 453: Syntax Analysis 29

LR Parsing: Schematic l l The driver program is the same for all LR parsers (SLR(1), LALR(1), …). Only the parse table changes. Different LR parsing algorithms involve different tradeoffs between parsing power, parse table size. CSc 453: Syntax Analysis 30

LR Parsing: the parser stack l The parser stack holds strings of the form s 0 X 1 s 1 X 2 s 2 … Xmsm (sm is on top) where si are parser states and Xi are grammar symbols. (Note: the Xi and si always come in pairs, with the state component si on top. ) l A parser configuration is a pair stack contents, unexpended input CSc 453: Syntax Analysis 31

LR Parsing: Roadmap l LR parsing algorithm: l l l parse table structure parsing actions Parse table construction: l l l viable prefix automaton parse table construction from this automaton improving parsing power: different LR parsing algorithms CSc 453: Syntax Analysis 32

LR Parse Tables l The parse table has two parts: the action function and the goto function. l At each point, the parser’s next move is given by action[sm, ai], where: l l l sm is the state on top of the parser stack, and ai the next input token. The goto function is used only during reduce moves. CSc 453: Syntax Analysis 33

LR Parser Actions: shift Suppose: l l l Effects of shift move: l 1. 2. l the parser configuration is s 0 X 1 s 1 … Xmsm, ai … an , and action[sm, ai] = ‘shift sn’. push the next input symbol ai; and push the state sn New configuration: s 0 X 1 s 1 … Xmsm ai sn, CSc 453: Syntax Analysis ai+1 … an 34

LR Parser Actions: reduce Suppose: l l l Effects of reduce move: l 1. 2. 3. l the parser configuration is s 0 X 1 s 1 … Xmsm, ai … an , and action[sm, ai] = ‘reduce A → ’. pop n states and n grammar symbols off the stack (2 n symbols total), where n = | |. suppose the (newly uncovered) state on top of the stack is t, and goto[t, A] = u. push A, then u. New configuration: s 0 X 1 s 1 … Xm-nsm-n A u, ai … an CSc 453: Syntax Analysis 35

LR Parsing Algorithm set ip to the start of the input string w$. while TRUE do: 1. 2. 3. 4. 5. let s = state on top of parser stack, a = input symbol pointed at by ip. if action[s, a] == ‘shift t’ then: (i) push the input symbol a on the stack, then the state t; (ii) advance ip. if action[s, a] == ‘reduce A → ’ then: (i) pop 2*| | symbols off the stack; (ii) suppose t is the state that now gets uncovered on the stack; (iii) push the LHS grammar symbol A and the state u = goto[A, t]. if action[s, a] == ‘accept’ then accept; else signal a syntax error. CSc 453: Syntax Analysis 36

LR PARSING AUTOMATA CSc 453: Syntax Analysis 37

LR parsing: Viable Prefixes l l Goal: to be able to identify handles, and so produce a rightmost derivation in reverse. Given a configuration s 0 X 1 s 1 … Xmsm, ai … an : l X 1 X 2 … Xm ai … an is obtainable on a rightmost derivation. l l X 1 X 2 … Xm is called a viable prefix. The set of viable prefixes of a grammar are recognizable using a finite automaton. This automaton is used to recognize handles. CSc 453: Syntax Analysis 38

Viable Prefix Automata l An LR(0) item of a grammar G is a production of G with a dot “ ” somewhere in the RHS. l l Example: The rule A → a A b gives these LR(0) items: l A→ a. Ab l A→a Ab l A → a. A b l A→a. Ab Intuition: ‘A → ’ denotes that: l l we’ve seen something derivable from ; and it would be legal to see something derivable from at this point. CSc 453: Syntax Analysis 39

Overall Approach Given a grammar G with start symbol S: l Construct the augmented grammar by adding a new start symbol S ′ and a new production S ′ → S. l Construct a finite state automaton whose start state is labeled by the LR(0) item S ′ → S. l Use this automaton to construct the parsing table. CSc 453: Syntax Analysis 40

Viable Prefix NFA for LR(0) items l Each state is labeled by an LR(0) item. The initial state is labeled S′ → S. l Transitions: 1. where X is a terminal or nonterminal. 2. where X is a nonterminal, and X → is a production. CSc 453: Syntax Analysis 41

Viable Prefix NFA: Example Grammar : S→ 0 S 1 S→ CSc 453: Syntax Analysis 42

Viable Prefix NFA DFA l Given a set of LR(0) items I, the set closure(I) is constructed as follows: repeat 1. 2. add every item in I to closure(I); if A → B closure(I) and B is a nonterminal, then for each production B → , add the item B → to closure(I). until no new items can be added to closure(I). l Intuition: A → B closure(I) means something derivable from B is legal at this point. This means that something derivable from B 43 CSc 453: Syntax Analysis (and thus ) is also legal.

Viable Prefix NFA DFA (cont’d) l Given a set of LR(0) items I, the set goto(I, X) is defined as goto(I, X) = closure({ A → X | A → X I }) l Intuition: l l if A → X I then (a) we’ve seen something derivable from ; and (b) something derivable from X would be legal at this point. Suppose we now see something derivable from X. The parser should “go to” a state where (a) we’ve seen something derivable from X; and (b) something derivable from would be legal. CSc 453: Syntax Analysis 44

Example l l l Let I 0 = {S′ → S}. I 1 = closure(I 0) = { S′ → S, /* from I 0 */ S → 0 S 1, S → } goto (I 1, 0) = closure( { S → 0 S 1 } ) = {S → 0 S 1, S → 0 S 1, S → } CSc 453: Syntax Analysis 45

Viable Prefix DFA for LR(0) Items 1. 2. 3. 4. l Given a grammar G with start symbol S, construct the augmented grammar with new start symbol S′ and new production S ′ → S. C = { closure({ S ′ → S }) }; // C = a set of sets of items = set of parser states repeat : for each set of items I C : for each grammar symbol X : if ( goto(I, X) && goto(I, X) C ) : // new state add goto(I, X) to C; } until no change to C; return C. The set C gives the viable prefix automaton’s states; the goto() function gives the transitions between states. CSc 453: Syntax Analysis 46

PARSE TABLE CONSTRUCTION CSc 453: Syntax Analysis 47

SLR(1) Parse Table Construction I Given a grammar G with start symbol S: l l l Construct the augmented grammar G′ with start symbol S′. Construct the set of states {I 0, I 1, …, In} for the Viable Prefix DFA for the augmented grammar G′. Each DFA state Ii corresponds to a parser state si. The initial parser state s 0 coresponds to the DFA state I 0 obtained from the item S ′ → S. The parser actions in state si are defined by the items in the DFA state Ii. CSc 453: Syntax Analysis 48

Figuring out parsing actions I l Suppose a DFA state I contains an item A→ … a… where a is a terminal. l l In state I, we’re looking to see an ‘a’ in the input. l If we see an ‘a’ in the input, we should shift. A shift action needs a state s to be pushed l this state is computed from the DFA’s transitions: s = goto(I, a) CSc 453: Syntax Analysis 49

Figuring out parsing actions II l When should we reduce? l l Need to be careful about what terminals (lookahead tokens) we reduce on l l the parser state should have an item of the form ‘A → ’ too generous with reduce actions conflicts SLR(1) parsers: reduce with ‘A → ’ only if the next token can legitimately follow A. l need to figure out which tokens can legitimately follow A. l CSc 453: Syntax Analysis given by FIRST and FOLLOW sets 50

FIRST and FOLLOW sets S start symbol X 1 X 2 … Xn some sequence of grammar symbols FOLLOW: FIRST: what tokens can appear here? CSc 453: Syntax Analysis 51

FIRST Sets Defn: For any string of grammar symbols , l l l FIRST( ) = { a | a is a terminal and * a }. if * then is also in FIRST( ). Example: E T E′ E′ + T E′ | T F T′ T′ * F T′ | F ( E ) | id FIRST(E) = FIRST(T) = FIRST(F) = { (, id } FIRST(E′) = { +, } FIRST(T′) = { *, } CSc 453: Syntax Analysis 52

Computing FIRST Sets Given a sequence of grammar symbols A: l l if A is a terminal or A = then FIRST(A) = {A}. if A is a nonterminal with productions A 1 | … | n then: l l FIRST(A) = FIRST( 1) FIRST( n). if A is a sequence of symbols Y 1 … Yk then: l l for i = 1 to k do: § add each a FIRST(Yi), such that a , to FIRST(A). § if FIRST(Yi) then break; if is in each of FIRST(Y 1), …, FIRST(Yk) then add to FIRST(A). CSc 453: Syntax Analysis 53

Computing FIRST sets: cont’d l l For each nonterminal A in the grammar, initialize FIRST(A) = . repeat { for each nonterminal A in the grammar { compute FIRST(A); /* as described previously */ } } until there is no change to any FIRST set. CSc 453: Syntax Analysis 54

Example (FIRST Sets) X YZ | a Y b | Z c | l l X a, so add a to FIRST(X). X YZ, b FIRST(Y), so add b to FIRST(X). Y , i. e. FIRST(Y), so add non- symbols from FIRST(Z) to FIRST(X). ► add c to FIRST(X). FIRST(Y) and FIRST(Z), so add to FIRST(X). Final: FIRST(X) = { a, b, c, }. CSc 453: Syntax Analysis 55

FOLLOW Sets Definition: Given a grammar G = (V, T, P, S), for any nonterminal A V: l FOLLOW(A) = { a T | S * Aa for some , }. i. e. , FOLLOW(A) contains those terminals that can appear after A in something derivable from the start symbol S. l if S * A then $ is also in FOLLOW(A). ($ EOF, “end of input. ”) Example: E E + E | id FOLLOW(E) = { +, $ }. CSc 453: Syntax Analysis 56

Computing FOLLOW Sets Given a grammar G = (V, T, P, S): 1. add $ to FOLLOW(S); 2. repeat { l l l for each production A B in P, add every non - symbol in FIRST( ) to FOLLOW(B). for each production A B in P, where FIRST( ), add everything in FOLLOW(A) to FOLLOW(B). for each production A B in P, add everything in FOLLOW(A) to FOLLOW(B). } until no change to any FOLLOW set. CSc 453: Syntax Analysis 57

Example (FOLLOW Sets) X YZ | a Y b | Z c | l l X is start symbol: add $ to FOLLOW(X); X YZ, so add everything in FOLLOW(X) to FOLLOW(Z). ►add $ to FOLLOW(Z). X YZ, so add every non- symbol in FIRST(Z) to FOLLOW(Y). ►add c to FOLLOW(Y). X YZ and FIRST(Z), so add everything in FOLLOW(X) to FOLLOW(Y). ►add $ to FOLLOW(Y). CSc 453: Syntax Analysis 58

SLR(1) Parse Table Construction II Parsing action for parser state si: l action table entries: l l goto table entries: l l if DFA state Ii contains an item A → a where a is a terminal, and goto(Ii, a) = Ij : set action[i, a] = shift j. if DFA state Ii contains an item A → , where A S′: for each b FOLLOW(A), set action[i, b] = reduce A → . if state Ii contains the item S′ → S : set action[i, $] = accept. for each nonterminal A, if goto(Ii, A) = Ij, then goto[i, A] = j. any entry not defined by these steps is an error state. CSc 453: Syntax Analysis 59

PUTTING IT ALL TOGETHER CSc 453: Syntax Analysis 60

Constructing a SLR(1) Parser 1. 2. Compute FIRST and FOLLOW sets Compute the SLR(1) DFA augment the grammar compute the set of states of the DFA l l l 3. use closure(), goto() functions Construct the SLR(1) parse table the states of the parser are the states of the DFA parsing actions in a state determined by the items in that state l l l use FOLLOW sets to compute reduce actions CSc 453: Syntax Analysis 61

SLR(1) Shortcomings l l SLR(1) parsing uses reduce actions too liberally. Because of this it fails on many reasonable grammars. Example (simple pointer assignments): S→R | L=R L → *R | id R→L The SLR parse table has a state { S → L = R, R → L }, and FOLLOW(L) = { =, $ }. shift-reduce conflict. CSc 453: Syntax Analysis 62

Improving LR Parsing l SLR(1) parsing weaknesses can be addressed by incorporating lookahead into the LR items in parser states. The lookahead makes it possible to remove some “spurious” reduce actions in the parse table. The LALR(1) parsers produced by bison and yacc incorporate such lookahead items. l This improves parsing power, but at the cost of larger parse tables. CSc 453: Syntax Analysis 63

ERROR HANDLING CSc 453: Syntax Analysis 64

Normal parsing Error Handling Getting things back on track what does this mean?

Dealing with Errors: Issues l Reporting: What does the parser tell the user? l l Does the user need to know? What does the user need to know? What can the user do? Example of errors not reported: l web browsers l Handling: What does the parser do? l l l Unacceptable : ignore the error, crash, or quit continue to process the input. No code generation attempt to repair the error: transform an erroneous program into a similar but legal input

Error Reporting l Error messages should refer to the source program. prefer “line 11: X redefined” to “conflict in hash bucket 53” l Error messages should, as far as possible, indicate the location and nature of the error. avoid “syntax error” or “illegal character” l Error messages should be specific. prefer “x not declared in function foo” to “missing declaration” l They should not be redundant. CSc 453: Syntax Analysis 67

Error Recovery l Lexical errors: pass the illegal character to the parser and let it deal with the error. l Syntax errors: “panic mode error recovery” l l Essential idea: skip part of the input and pretend as though we saw something legal, then hope to be able to continue. Pop the stack until we find a state s such that goto[s, A] is defined for some nonterminal A. discard input tokens until we find some token a that can legitimately follow A (i. e. , a FOLLOW(A)). push the state goto[s, A] and continue parsing. CSc 453: Syntax Analysis 68

Error Handling in Yacc parsers A syntax error means the parser is out-of-sync with the input. It needs to get back in sync. 1. 2. 3. The “error token” is generated by the parser on encountering a syntax error. Stack entries are discarded until a state is found where the error token can be shifted. Input tokens are read and discarded until a token is found that can be shifted. CSc 453: Syntax Analysis 69