Ambiguity Pilambda

Approximating Context-Free Grammar Ambiguity Claus Brabrand [email_address] BRICS, Department of Computer Science University of Aarhus, Denmark

// Abstract “ Approximating Context-Free Grammar Ambiguity” Context-free grammar ambiguity is undecidable. However, just because it’s undecidable, doesn’t mean there aren’t (good) approximations! Indeed, the whole area of static analysis works on “side-stepping undecidability” . We exhibit a characterization of context-free ambiguity which induces a whole framework for approximating the problem. In particular, we give an approximation, A MN , based on the [Mohri-Nederhof, 2000] regular approximation of context-free grammars and show how to boost the precision even further.

// Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

// Context-Free Grammar ,[object Object],[object Object],[object Object],[object Object],G =  N,  , s,   ,[object Object],[object Object],[object Object],L : G  P (  *) language of G, L (G)

// Relevant CFG Decision Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

// Ambiguity: Undecidable! ,[object Object],[object Object],[object Object], T s  T’ s  = unambiguous ambiguous ,[object Object],?

// “Side-Stepping Undecidability” ,[object Object],[object Object],However, just because it’s undecidable, doesn’t mean there aren’t (good) approximations ! Indeed, the whole area of static analysis works on “ side-stepping undecidability ” . unambiguous ambiguous safe (over-) approximation unambiguous ambiguous safe (under-) approximation unambiguous ambiguous unsafe approximation

// Motivation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],unambiguous ambiguous Yes! .

// Motivation (cont’d) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],unambiguous ambiguous No? . .

// Vertical Ambiguity ,[object Object],[object Object], n  N :  ,  ’   (n) :    ’  L (  )  L (  ’) =  x a y Z : x A y : x B y A : a B : a Ambiguous string: ~ “ reduce/reduce conflict ” in [Yacc]  G

// Horizontal Ambiguity ,[object Object],[object Object],[object Object], n  N :    (n):  i  [1..|  |-1]: L (  0 ..  i-1 ) L (  i ..  |  |-1 ) =  : P (  *)  P (  *)  P (  *) X Y = { x a y | x,y  *  a  +  x,x a  L (X)  y, a y  L (Y) } x a y Z : A B A : x a : x B : a y : y Ambiguous string: ~ “ shift/reduce conflict ” in [Yacc]  G      

// Characterization of Ambiguity ,[object Object],[object Object],[object Object],G  G  G unambiguous G  G  G unambiguous G  G  G unambiguous

// Proof (Lemma 1a): “  ” ,[object Object],[object Object],[object Object],[object Object],[object Object],G  G  G unambiguous G ambiguous  G  G G  G

// Proof (Lemma 1a): “  ” (Base) ,[object Object],[object Object],[object Object],[object Object],N  ’ 1  N  1   L (  )  L (  ’)  {  }   p p’ = G

// Proof (Lemma 1a): “  ” (I.H.) ,[object Object],[object Object],[object Object],N n-1  N  n-1  i  ’ i’ … …  i … …  ’ i’ p p’ 1 1  |  -1| =  ’ 0  ’ |  ’-1|  0 .. .. .. ..  = 

// Proof (Lemma 1a): “  ” (p  p’) ,[object Object],[object Object],[object Object],L (  )  L (  ’)  {  }   p  p’ G N n-1  N  n-1  i  ’ i’ … …  i … …  ’ i’ p p’ 1 1  |  -1| =  ’ 0  ’ |  ’-1|  0 .. .. .. ..  = 

// Proof (Lemma 1a): “  ” (p=p’,1) ,[object Object],[object Object],[object Object],[object Object],[object Object], i :  i =  ’ i p = p’  i :  i =  ’ i  N n-1  N  n-1  i  i … …  i … …  i’ p p’ 1 1  |  -1| =  0  |  -1|  0 .. .. .. ..  =  G G

// Proof (Lemma 1a): “  ” (p=p’,2) ,[object Object],[object Object],[object Object],[object Object],[object Object],N n-1  N  n-1  i . … .  i p  i :  i   ’ i p = p’ p 1 1  i :  i =  ’ i   i :  i =  ’ i   j  i:  j   ’ j =   j  j  ’ i . … .  i  j  ’ j i  k < j L (  0 ..  k ) L (  k+1 ..  |  | )    k k ,[object Object],[object Object],G  

// Proof (Lemma 1b): “  ” ,[object Object],[object Object],[object Object],[object Object],s  * x N   x    * x a   * x a y s  * x N   x  ’   * x a   * x a y N    * a , N   ’  * a , L (  )  L (  ’ )  { a }   G  G  G unambiguous G ambiguous  G  G

// Proof (Lemma 1b): “  ” (cont’d) ,[object Object],[object Object],[object Object],s  * v N   v     * v x    * v x a y   * v x a y w s  * v N   v     * v x a    * v x a y   * v x a y w N    , L (  ) L (  )    x,y   * :  a   + : x,x a  L (  )  y, a y  L (  ) i.e.  

// (Over-)Approximation (A) ,[object Object],[object Object],[object Object],[object Object],  E* : L (  )  A (  )  n  N :  ,  ’   (n) : A (  )  A (  ’) =  A A  n  N:    (n):  i  [1..|  |-1]: A (  0 ..  i-1 ) A (  i ..  |  |-1 ) =   G G    

// Ambiguity Approximation ,[object Object],[object Object],[object Object],  G unambiguous A (  )  A (  ) =   L (  )  L (  ) =  A (  ) A (  ) =   L (  ) L (  ) =  A A    A A G G G G G G    

// Compositionality (of A’s) ,[object Object],[object Object],[object Object],[object Object],A , A’ decidable (over-)approximations  A  A’ decidable (over-)approximation unambiguous ambiguous unambiguous ambiguous unambiguous ambiguous A A’ A  A’ 

// Choice(s) of A? ,[object Object],[object Object],[object Object],[object Object],[object Object],unambiguous ambiguous worst approximation

// Choice(s) of A? (cont’d) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Black-box

// Decidability (of A MN ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],X  Y =  X Y =    G unambiguous A MN A MN    

[object Object],[object Object],// Decision Algorithm for (X Y)  X NFA Y NFA [X;Y] NFA   a  path : X NFA Y NFA [X;Y] NFA a a x y x a y a a X Y Y X X  Y   

// Three Approximation Answers ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],True answer

// Regaining Lost Precision! ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],L (C CFG ) = L (D DFA )  L (G CFG ) L (C CFG ) = 

[object Object],[object Object],[object Object],[object Object],// Asymptotic (Time) Complexity N 1 : e 1,1 … e a,1 : … : e 1,p … e a,p h n v ,[object Object],[object Object],[object Object],[object Object]

// Related Work (Dynamic) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

// Related Work (Static) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

// Implementation ,[object Object],In progress…!

// Assessment ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],In progress…!

// Example: Expression chains ,[object Object],E -> E + T -> T T -> T * F -> F F -> ( E ) -> x

// Example: Balancing Structures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],S -> A A A -> x A x -> y xxyxx xyx Example string:

// Future Work ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],E -> E  E

// Conclusion But wait, there’s more… “ Approximating Context-Free Grammar Ambiguity” Context-free grammar ambiguity is undecidable. However, just because it’s undecidable, doesn’t mean there aren’t (good) approximations! Indeed, the whole area of static analysis works on “side-stepping undecidability” . We exhibit a characterization of context-free ambiguity which induces a whole framework for (over-)approximation. In particular, we give an approximation based on the [Mohri-Nederhof, 2000] regular approximation of context-free grammars and show how to boost the precision even further.

// Lessons Learned ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

// Membership: Decidable! ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],  L (G)

[object Object],[object Object],[object Object],[object Object],// Parsing Greedily Left-to-Right x y x’ y’ x y - (“too little”): Not possible (due to greediness) ... may occur in 2 cases: - (“too much”): Only this is a problem!  X  X;( prefix(Y) {  } )    X Y  x’ y’

Ambiguity Pilambda

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Ambiguity Pilambda