Syntactical analysis Syntactical analysis Context-free grammars A - PowerPoint PPT Presentation

Syntactical analysis Syntactical analysis • Context-free grammars A context-free grammar is a 4-tuple G = ( N , Σ , P , S ) g ( ) • Derivations 1. N is a set of non terminals • Parse Trees 2. Σ is a set of terminals (disjoint from N ) • Left-recursive grammars P is a subset of ( N  Σ )  N P is a subset of ( N  Σ )*  N 3 3. • Top-down parsing An element ( α , A )  P is called a production non-recursive predictive parsers • A ::= α or α  A construction of parse tables • • Bottom-up parsing B i S  N is the start symbol 4. shift/reduce parsers • LR parsers • The sets N Σ P are finite The sets N , Σ , P are finite • Generalized parsing G li d i GLL parsers • GLR parsers • SGLR parsers SGLR parsers • • / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 0 / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 1 Syntactical analysis Syntactical analysis The language L ( G ) generated by the context-free grammar A context-free grammar can be consider as a simple rewrite G = ( N , Σ , P , S ) is: system:  A    if  A  P (  ,  ,   ( N  Σ )*, A  N ) L ( G ) = { w  Σ * | S  + w } Example p N = { E }, Σ = { + , * , ( , ) , - , a }, S = E , { }, { , , ( , ) , , }, , A sentence w  L ( G ) contains only terminals P = { E + E  E E * E  E A sentential form  is a string of terminals and non-terminals ( E )  E ( E )  E which can be derived from S : - E  E S  *  with   ( N  Σ ) * a  E } Derivation: E  -E  -(E)  -(E+E)  -(a+E)  -(a+a) A sentence in L ( G ) is a sentential form in which no non-terminals occur / Faculteit Wiskunde en Informatica / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 2 22-9-2011 PAGE 3

Syntactical analysis Syntactical analysis A parse tree for a context-free grammar is a G = ( N , Σ , P , S ) tree: Left/right derivations 1 1. The root is labeled with S (the start non terminal) The root is labeled with S (the start non-terminal) There are choices to be made for each derivation step: f • Each leaf is labeled with a terminal (  Σ )or ε 2. which non-terminal must be replaced? • 3. All other nodes are labeled with a non-terminal which alternative of the selected non-terminal must be applied? • If A is the label of a node and X 1 ,…, X n are the labels of the Always selecting the leftmost non-terminal in the sentential form gives • a leftmost derivation :  lm children (from left to right) then There exists also a rightmost derivation:  rm • X 1 ,…, X n  A must be a production rule in G (with X i is either a terminal or a Consider the context-free grammar for expressions: • non-terminal) Leftmost derivation for -(a+a) • E  -E  -(E)  -(E+E)  -(a+E)  -(a+a) ( ) ( ) ( ) ( ) Special case: ε  A with label A which has exactly one child with S i l A ith l b l A hi h h tl hild ith Rightmost derivation for -(a+a) • label ε E  -E  -(E)  -(E+E)  -(E+a)  -(a+a) Also called a nullable non-terminal • / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 4 / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 5 Syntactical analysis Syntactical analysis Example: Acceptor and parser E  -E  -(E)  -(E+E)  -(a+E)  -(a+a) For each grammar G there exists a decision procedure (acceptor) E  E  E  E  E  E  E AG for L ( G ): AG for L ( G ): AG : STRING  { true , false } - E - E - E - E - E - E such that AG ( w ) = true  w  L ( G ) AG ( ) t L ( G ) ( E ) ( E ) ( E ) ( E ) ( E ) A parser is an acceptor which constructs a parse tree as well. E + E E + E E + E A top-down parser constructs the tree starting from the root • A bottom-up parser constructs the tree starting from the leafs a a a • The parse tree abstracts from the derivation order p / Faculteit Wiskunde en Informatica / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 6 22-9-2011 PAGE 7

Syntactical analysis Syntactical analysis During parsing the following problems may occur: • A grammar is immediate left recursive if the g grammar contains a rule of the form A   A • The grammar is ambiguous • The grammar is left recursive • A grammar is left recursive if there exists a non- • The grammar contains cycles terminal A and a string   ( N  Σ )* such that A  A  terminal A and a string   ( N  Σ )* such that A  A  • This means that after one or more steps in a A grammar G is ambiguous if one word w  L ( G ) has at derivation an occurrence of A reduces again to an g least two parse trees least two parse trees occurrence of A without recognizing any terminal in • Expression grammar without associativities and priorities the input sentence. • Dangling else problem / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 8 / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 9 Syntactical analysis Syntactical analysis Examples of indirect left recursion Elimination of left recursion B   A A   A A   B   A where  =/> ε where  =/> ε or worse produce the sentential forms:   n B   A D A   B  A set of equivalent (non left recursive) rules are A set of equivalent (non left recursive) rules are ε  D  A’  A  G  D  A’  A’ ε  A’ A’ It is relatively easy to remove left recursion from a grammar / Faculteit Wiskunde en Informatica / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 10 22-9-2011 PAGE 11

Syntactical analysis Syntactical analysis Example: Example: (1) E + T  E (immediate left rec.) (1’) T E’  E (2) T  E (2’) + T E’  E’ (3) T F  T (immediate left rec.) (3) T * F  T (immediate left rec )  E  E’ (2 ) ε (2’’) ε (4) F  T (5) ( E )  F the same for: (6) a  F (6) F T’  T (3’) * F T’  T’ (4’) Applying the left recursion elimination transformation: Applying the left recursion elimination transformation:  T  T’ (4’’) ε (4 ) ε (1) E   E (with  = + T ) (2)   E (with  = T ) / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 12 / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 13 Syntactical analysis Syntactical analysis Indirect left recursion elimination This process is repeated until either • Suppose we have a rule of the form • t   A ; the process stops, or B   A • A  A ; the immediately left recursion elimination rule can be  1  B applied applied  2  B …  n  B • The rule B   A is now transformed into:  1   A    A  2   A …  n   A / Faculteit Wiskunde en Informatica / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 14 22-9-2011 PAGE 15

Syntactical analysis Syntactical analysis Left factorization Example • In general it is efficient to move the difference between the I l i i ffi i h diff b h if b then S else S  S alternatives of a non-terminal as far as possible to the left  S if b then S • Productions of the form Only at the occurrence of else it can be decided which y   1  A   1  A   2  A alternative should have been selected …   n  A An equivalent grammar is • Are equivalent with if b then S S’  S  A’  A  S’ else S  1  A   A’  S’ ε …  n  A’ / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 16 / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 17 Syntactical analysis Syntactical analysis Top-down parsing • Left recursion elimination and left factorization: • A top-down parser “guesses” the next alternative to be • introduce new (extra) non-terminals recognized, and verifies whether this alternative can be • change the structure of the derivation tree recognized in the input. If not, another alternative will be tried • may influence semantic actions connected to grammar rules • may influence semantic actions connected to grammar rules • Constructs the parse tree starting at the root C t t th t t ti t th t • Finds the leftmost derivation of the sentence • Alternative types of top-down parsers: Al i f d recursive descent parser with backtracking • recursive descent parser without backtracking (“predictive parser”) • non recursive predictive parser (uses push down automaton) non-recursive predictive parser (uses push-down automaton) • generalized parser • / Faculteit Wiskunde en Informatica / Faculteit Wiskunde en Informatica 22-9-2011 PAGE 18 22-9-2011 PAGE 19