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Module 29

Module 29. Parse/Derivation Trees Leftmost derivations, rightmost derivations Ambiguous Grammars Examples Arithmetic expressions If-then-else Statements Inherently ambiguous CFL’s. Context-Free Grammars. Parse Trees Leftmost/rightmost derivations Ambiguous grammars. Parse Tree.

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Module 29

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  1. Module 29 • Parse/Derivation Trees • Leftmost derivations, rightmost derivations • Ambiguous Grammars • Examples • Arithmetic expressions • If-then-else Statements • Inherently ambiguous CFL’s

  2. Context-Free Grammars Parse Trees Leftmost/rightmost derivations Ambiguous grammars

  3. Parse Tree • Parse/derivation trees are structured derivations • The structure graphically illustrates semantic information about the string • Formalization of concept we encountered in regular languages unit • Note, what we saw before were not exactly parse trees as we define them now, but they were close

  4. S S S ( S ) ( S ) l ( S ) l Parse Tree Example • Parse tree for string ( )(( )) and grammar BALG • BALG = (V, S, S, P) • V = {S}, S = {(, )}, S = S • P = S --> SS | (S) | l • One derivation of ( )(( )) • S ==> SS ==> (S)S ==> ( )S ==> ( )(S) ==> ( )((S)) ==> ( )(( )) • Parse tree

  5. Syntax: draw a unique arrow from each variable to each character that is a direct child of that variable A line instead of an arrow is ok The derived string can be read in a left to right traversal of the leaves Semantics The tree graphically illustrates the nesting structure of the string of parentheses S S S ( S ) ( S ) l ( S ) l Comments about Example *

  6. There is more than one derivation of the string ( )(( )). S ==> SS ==> (S)S ==>( )S ==> ( )(S) ==> ( )((S)) ==> ( )(( )) S ==> SS ==> (S)S ==> (S)(S) ==> ( )(S) ==> ( )((S)) ==> ( )(( )) S ==> SS ==> S(S) ==> S((S)) ==> S(( )) ==> (S)(( )) ==>( )(( )) Leftmost derivation Leftmost variable is always expanded Which one of the above is leftmost? Rightmost derivation Rightmost variable is always expanded Which one of the above is rightmost? S S S ( S ) ( S ) l ( S ) l Leftmost/Rightmost Derivations

  7. Fix a string and a grammar Any derivation corresponds to a unique parse tree Any parse tree can correspond to many different derivations Example The one parse tree corresponds to all three derivations Unique mappings For any parse tree, there is a unique leftmost/rightmost derivation that it corresponds to S S S ( S ) ( S ) l ( S ) l Comments • S ==> SS ==> (S)S ==>( )S ==> ( )(S) ==> ( )((S)) ==> ( )(( )) • S ==> SS ==> (S)S ==> (S)(S) ==> ( )(S) ==> ( )((S)) ==> ( )(( )) • S ==> SS ==> S(S) ==> S((S)) ==> S(( )) ==> (S)(( )) ==>( )(( ))

  8. Example * • S ==> SS ==> SSS ==> (S)SS ==> ( )SS ==> ( )S ==> ( ) • The above is a leftmost derivation of the string ( ) from the grammar BALG • Draw the corresponding parse tree • Draw the corresponding rightmost derivation • S ==> (S) ==> (SS) ==> (S(S)) ==> (S( )) ==> (( )) • The above is a rightmost derivation of the string (( )) from the grammar BALG • Draw the corresponding parse tree • Draw the corresponding leftmost derivation

  9. Ambiguous Grammars Examples: Arithmetic Expressions If-then-else statements Inherently ambiguous grammars

  10. Ambiguous Grammars • A grammar G is ambiguous if there exists a string x in L(G) with two or more distinct parse trees • (2 or more distinct leftmost/rightmost derivations) • Example • Grammar AG is ambiguous • String aaa in L(AG) has 2 rightmost derivations • S ==> SS ==> SSS ==> SSa ==> Saa ==> aaa • S ==> SS ==> Sa ==> SSa ==> Saa ==> aaa

  11. 2 Simple Examples • Grammar BALG is ambiguous • String ( ) in L(BALG) has >1 leftmost derivation • S ==> (S) ==> ( ) • S ==> (S) ==> (SS) ==>(S) ==>( ) • Give another leftmost derivation of ( ) from BALG • Grammar ABG is NOT ambiguous • Consider any string x in {aibi | i > 0} • There is a unique parse tree for x

  12. Legal Arithmetic Expressions • Develop a grammar MATHG = (V, S, S, P) for the language of legal arithmetic expressions • S = {0, 1, +, *, -, /, (, )} • Strings in the language include • 0 • 10 • 10*11111+100 • 10*(11111+100) • Strings not in the language include • 10+ • 11++101 • )(

  13. Grammar MATHG1 • V = {E, N} • S = {0, 1, +, *, -, /, (, )} • S = E • P: • E --> N | E+E | E*E | E/E | E-E | (E) • N --> N0 | N1 | 0 | 1

  14. E --> N | E+E | E*E | E/E | E-E | (E)N --> N0 | N1 | 0 | 1 MATHG1 is ambiguous • Come up with two distinct leftmost derivations of the string 11+0*11 • E ==> E+E ==> N+E ==> N1+E ==> 11+E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==> 11+0*11 • E ==> E*E ==> E+E*E ==> N+E*E ==> N1+E*E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==>11+0*11 • Draw the corresponding parse trees

  15. E ==> E+E ==> N+E ==> N1+E ==> 11+E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==> 11+0*11 E ==> E*E ==> E+E*E ==> N+E*E ==> N1+E*E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==>11+0*11 E E * E + E E E + N E N * E N 1 N N N 1 N N N 0 1 1 0 N 1 1 1 1 Corresponding Parse Trees E E

  16. E E E E * E + E E E + N E N * E N 1 N N N 1 N N N 0 1 1 0 N 1 1 1 1 Parse Tree Meanings Note how the parse trees captures the semantic meaning of string 11+0*11. More specifically, what number does the first parse tree represent? What number does the second parse tree represent?

  17. Implications • Two interpretations of string 11+0*11 • 11+(0*11) = 11 • (11+0)*11 = 1001 • What if a line in a program is • MSU_Tuition = 11+0*11; • What is MSU_Tuition? • Depends on how the expression 11+0*11 is parsed. • This is not good. • Ambiguity in grammars is undesirable, particularly if the grammar is used to develop a compiler for a programming language like C++. • In this case, there is an unambiguous grammar for the language of arithmetic expressions

  18. If-Then-Else Statements • A grammar ITEG = (V, S, S, P) for the language of legal If-Then-Else statements • V = (S, BOOL) • S = {D<85, D>50, grade=3.5, grade=3.0, if, then, else} • S = S • P: • S --> if BOOL then S else S | if BOOL then S |grade=3.5 | grade=3.0 • BOOL --> D<85 | D>50

  19. S --> if BOOL then S |grade=3.5 | grade=3.0 | if BOOL then S else S BOOL --> D<85 | D>50 ITEG is ambiguous • Come up with two distinct leftmost derivations of the string • if D<85 then if D>50 then grade=3.5 else grade=3.0 • S ==>if BOOL then S else S ==> if D<85 then S else S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0 • S ==>if BOOL then S ==> if D<85 then S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0 • Draw the corresponding parse trees

  20. S ==>if BOOL then S else S ==> if D<85 then S else S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0 S ==>if BOOL then S ==> if D<85 then S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0 if S if B then S else B then S S D<85 if grade=3.0 D<85 if else B then S B then S D>50 D>50 grade=3.5 grade=3.5 grade=3.0 Corresponding Parse Trees S S

  21. Parse Tree Meanings S S if B then S if S B then S else S D<85 if else B then S D<85 if grade=3.0 B then S D>50 grade=3.5 grade=3.0 D>50 grade=3.5 If you receive a 90 on type D points, what is your grade? By parse tree 1 By parse tree 2

  22. Implications • Two interpretations of string • if D<85 then if D>50 then grade=3.5 else grade=3.0 • Issue is which if-then does the last ELSE attach to? • This phenomenon is known as the “dangling else” • Answer: Typically, else binds to NEAREST if-then • In this case, there is an unambiguous grammar for handling if-then’s as well as if-then-else’s

  23. Inherently ambiguous CFL’s • A CFL L is inherently ambiguous iff for all CFG’s G such that L(G) = L, G is ambiguous • Examples so far • None of the CFL’s we’ve seen so far are inherently ambiguous • While the CFG’s we’ve seen ambiguous, there do exist unambiguous CFG’s for those CFL’s. • Later result • There exist inherently ambiguous CFL’s • Example: {aibjck | i=j or j=k or i=j=k} • Note i=j=k is unnecessary, but I added it here for clarity

  24. Summary • Parse trees illustrate “semantic” information about strings • Ambiguous grammars are undesirable • This means there are multiple parse trees for some string • These strings can be interpreted in multiple ways • There are some heuristics people use for taking an ambiguous grammar and making it unambiguous, but this is not the focus of this course • There are some inherently ambiguous CFL’s • Thus, the above heuristics do not always work

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