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

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Ähnlich wie Ambiguity Pilambda

Ähnlich wie Ambiguity Pilambda (20)

Mehr von Fatih Gökçe

Mehr von Fatih Gökçe (7)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

Ambiguity Pilambda