Approximating ContextFree Grammar Ambiguity

1
Approximating Context-Free Grammar Ambiguity

• Claus Brabrand
• brabrand_at_brics.dk
• BRICS, Department of Computer Science
• University of Aarhus, Denmark

2
// Abstract
Approximating Context-Free Grammar Ambiguity
Context-free grammar ambiguity is
undecidable. However, just because its
undecidable, doesnt mean there arent (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, AMN, based
on the Mohri-Nederhof, 2000 regular
approximation of context-free grammars and show
how to boost the precision even further.
3
// Outline

• Introduction
• Vertical / Horizontal Ambiguity
• Characterization of Ambiguity
• (Over-)Approximation Framework
• Approximation (AMN)
• Assessment
• Related Work
• Conclusion

4
// Context-Free Grammar

• N finite set of nonterminals
• ? finite set of terminals
• s ? N start nonterminal
• ? N ? P(E) production function, E N ?
  ?

G ? N, ?, s, ? ?

• Assume
• All n?N reachable (from s)
• All n?N derive some (finite) string

L G ? P(?) language of G,
L(G)
5
// Relevant CFG Decision Problems

• Decidable
• Membership ? ? L(GCFG)
• Emptyness L(GCFG) ?
• Intersection (w/ REG) L(GCFG) ? L(RREG)
  L(CCFG)
• constructively
• Undecidable
• Intersection (w/ CFG) L(GCFG) ? L(GCFG) ?

• Ambiguity ???? 2 derivation trees ?

6
// Ambiguity Undecidable!

• Algorithms
• Undecidable!
• However

• Ambiguity ???? 2 derivation trees ?

s
s
?
T
T
?
?

?
ambiguous
unambiguous
7
// Side-Stepping Undecidability
However, just because its undecidable, doesnt
mean there arent (good) approximations! Indeed,
the whole area of static analysis works on
side-stepping undecidability.

• Unsafe approximation
• Safe approximation

ambiguous
unambiguous
unsafe approximation
ambiguous
ambiguous
unambiguous
unambiguous
safe (over-)approximation
safe (under-)approximation
8
// Motivation

• Use safe (over-)approximation
• Yes! ? G guaranteed unambiguous!!!
• Safely use any GLR parser on G
• Because never two parses at runtime!
• Hence
• dynamic parse ambiguity ? static parse ambiguity

ambiguous
unambiguous
.
Yes!
9
// Motivation (contd)

• Undecidability means therell always be a
  slack
• However, still useful!
• Possible interpretations of No?
• Treat as error (reject grammar)
• Please redesign your grammar (as in LALR(k))
• Treat as warning
• Here are some potential problems

ambiguous
.
unambiguous
.
No?
10
// Vertical Ambiguity

• Vertical ambiguity
• Example

G
?n ? N ??, ? ? ?(n) ? ? ? ? L(?) ?
L(?) ?
Z x A y x B y A a B a
?
Ambiguous string
xay
reduce/reduce conflict in Yacc
11
// Horizontal Ambiguity

• Horizontal ambiguity
• where
• Example

G
?n ? N ?? ? ?(n) ?i ? 1..?-1 L(?0 .. ?i-1)
L(?i .. ??-1 ) ?
P(?) ? P(?) ? P(?)
X Y xay x,y?? ? a?? ? x,xa?L(X) ?
y,ay?L(Y)
Z A B A x a x B a y y
?
Ambiguous string
xay
shift/reduce conflict in Yacc
12
// Characterization of Ambiguity

• Theorem 1
• Lemma 1a (?)
• Lemma 1b (?)

G ? G ? G unambiguous
G ? G ? G unambiguous
G ? G ? G unambiguous
13
// Proof (Lemma 1a) ?
G ? G ? G unambiguous

• or contrapositively
• Proof
• Assume G ambiguous (i.e. ? 2 der. trees for ?)
• Show
• by induction in max height of the 2 derivation
  trees

G ambiguous ? G ? G
G ? G
14
// Proof (Lemma 1a) ? (Base)

• Base case (height ? 1)
• The ambiguity means that (for p?p)
• Which means
• i.e., we have a vertical ambiguity

N
?
N
1
1
p
p
?
?
?
?

L(?) ? L(?) ? ? ? ?
G
15
// Proof (Lemma 1a) ? (I.H.)

• Induction step (height ? n)
• Assume induction hypothesis (for height ? n-1)
• The ambiguity means

N
N
1
1
p
p
?i
?i


?
n-1 ?
? n-1
..
..
..
..
?i
?i
??-1 ?0
??-1
?0
?
16
// Proof (Lemma 1a) ? (p?p)

• Case p q (different production)
• but then ?
• i.e., we have a vertical ambiguity

p ? p
L(?) ? L(?) ? ? ? ?
G
N
N
1
1
p
p
?i
?i


?
n-1 ?
? n-1
..
..
..
..
?i
?i
??-1 ?0
??-1
?0
?
17
// Proof (Lemma 1a) ? (pp,1)

• Case p ? q (same prod. ? )
• i.e. the top of the trees are the same
• Case
• ? ambiguity in subtreei ( deriving same ?i)
• Induction hypothesis (this subtree) ?

p p
?i ?i ?i
?i ?i ?i
?
G
G
N
N
1
1
p
p
?i
?i


?
n-1 ?
? n-1
..
..
..
..
?i
?i
??-1 ?0
??-1
?0
?
18
// Proof (Lemma 1a) ? (pp,2)
p p

• Case p ? q (same prod. ? )
• Case
• but then (assume WLOG
  )
• Now pick any k
• ...then

?i ?i ?i
?
?i ?i ? ?i
? ?i ?i ?i

• least such i
• 2nd least such j

?j?i ?j ? ?j
i ? k lt j
?
L(?0 .. ?k) L(?k1 .. ?? ) ? ?
G
N
N
1
1
p
p
?i
?j
?i
?j
. .
. .
?
n-1 ?
? n-1
?i

?j
?i
?j
k
k
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// Proof (Lemma 1b) ?
G ? G ? G unambiguous

• Contrapositively
• Assume (vertical conflict)
• Then for some N?N
• But then derive (using reachability
  derivability of N)

G ambiguous ? G ? G
N ? ? ? a, N ? ? ? a, L(?) ? L(?) ? a ? ?
s ? x N ? ? x ? ? ? x a ? ? x a y
s ? x N ? ? x ? ? ? x a ? ? x a y
20
// Proof (Lemma 1b) ? (contd)

• Assume (horizontal conflict)
• Then for some N?N
• But then derive (using reachability
  derivability of N)

N ? ? ? , L(?) L(?) ? ?
i.e.
?x,y ? ? ?a ? ? x,xa ? L(?) ? y,ay ? L(?)
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
21
// (Over-)Approximation (A)

• (Over-)Approximation A E ?
  P(?)
• A decidable ? and decidable on
  co-dom(A)
• Approximated vertical ambiguity
• Approximated horizontal ambiguity

?? ? E L(?) ? A(?)
?
G
A
?n ? N ??, ? ? ?(n) A(?) ? A(?) ?
G
A
?n ? N ?? ? ?(n) ?i ? 1..?-1 A(?0 .. ?i-1)
A(?i .. ??-1) ?
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// Ambiguity Approximation

• Theorem 2
• Proof
• Conflicts w/ smaller sets ? conflicts w/ larger
  sets

? ? G unambiguous
G
G
A
A
? ? ?
G
G
G
G
A
A
A(?) ? A(?) ? ? L(?) ? L(?) ?
A(?) A(?) ? ? L(?) L(?) ?
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// Compositionality (of As)

• Colloary 3
• Proof
• Follows from definition omited
• i.e. Approximations are compositional!

A, A decidable (over-)approximations
? A ? A decidable (over-)approximation
A
ambiguous
A ? A
unambiguous
ambiguous
unambiguous
?
ambiguous
unambiguous
A
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// Choice(s) of A?

• A?(?) ? (constant)
• Worst approximation
• but safe approximation!
• Useless
• Cannot determine that any grammars are
  unambiguous

ambiguous
unambiguous
worst approximation
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// Choice(s) of A? (contd)

• AMN(?) Mohri-Nederhof(?)
• CFG ? DFA (NFA) Approximation
• Properties of this Black-box
• Good (over-)approximation!
• Works on language, L(G)
• not on grammatical structure, G
• Approximation parameterizable
• E.g. unfold nonterminals n times

Regular Approximation of Context-Free Grammars
through Transformation Mohri-Nederhof, 2000
Black-box
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// Decidability (of AMN)

• ? decidable (using DFAs)
• O(XNFAYNFA)
• decidable (using DFAs)
• O(XNFAYNFA)
• AMN decidable
• With potential counterexamples (using DFAs)

X ? Y ?
X Y ?
? ? G unambiguous
AMN
AMN
27
// Decision Algorithm for (X Y)
?
?

• For X,Y regular languages
• All overlappings, xay, as DFAs variant of ?
  construction!

a
a
?
x
y
XNFA
YNFA
XNFA
YNFA
XYNFA
XYNFA
a
X
Y
? a ? path
X ? Y
?
?
x
a
y
a
X
Y
28
// Three Approximation Answers

• Y!
• G definitely not ambiguous!
• ?/D?
• ? Dont know?
• could not find any potential counterexamples.
• D? Dont know look at over-approx, D?
• and here are all potential counterexamples
• Note some strings do not even parse!
• Improve Parse S ?FIN D ? subset of real
  counterexamples

True answer
29
// Regaining Lost Precision!

• Now parse all counterexamples!
• i.e. parse DFA, DDFA
• 1) i.e. construct
• Decidable in O(DG)
• 2) Decide emptyness on C
• Decidable in O(C DG)
• Only potential counterexamples that parse!

L(CCFG) L(DDFA) ? L(GCFG)
L(CCFG) ?
30
// Three Approximation Answers

• Y!
• G definitely not ambiguous!
• ?/C?
• ? Dont know?
• could not find any counterexamples.
• C? Dont know look at over-approx, C?
• and here are all potential counterexamples
• Note all strings actually parse (maybe not
  ambiguously)!
• Improve extract finite under-approximation...?

True answer
31
// Asymptotic (Time) Complexity
h

• Mohri-Nederhof O(n2vh)
• Vertical Amb O(n3v4h4)
• Horizontal Amb O(n3v3h5)
• Total O(n3v3h4(vh)) ? O(g5)

N1 e1,1 ea,1 e1,p ea,p

• n N
• v max?(N), N?N
• h max?, ???(N), N?N
• g nvh G

v

n
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// Related Work (Dynamic)

• Dynamic disambiguation
• Disambiguation-by-convention
• Longest match, most specific match,
• Customizable
• Bison v. 1.5 dprec, merge
• ASFSDF disambiguation filters
• Dynamic ambiguity interception
• GLR (Tomita, Early, Bison, ASFSDF, )

33
// Related Work (Static)

• Static disambiguation
• Disambiguation-by-convention
• First match, most specific match,
• Customizable
• Yacc left, right, nonassoc, prec
• Static ambiguity interception
• LL(k), LA-LR(k),
• Our work goes here (but for GLR)!

34
// Implementation

• disamb (Java)

In progress!
35
// Assessment

• Quality of approximation
  Quantity of false-positives
• Precision
• Our \ LR(k) ?
• LR(k) \ Our ?
• False-positives ?
• Characterize ? / N?
• In terms of grammatical structure ?
• Efficiency (in practise)

In progress!
36
// Example Expression chains

• !?

E -gt E T -gt T T -gt T F -gt F F -gt
( E ) -gt x
37
// Example Balancing Structures

• Nasty
• Requires
• Unbounded memory ( xes)
• i.e. CFG structure
• Unbounded lookahead
• i.e. any finite k is insufficient
• ? False-positives!

S -gt A A A -gt x A x -gt y
Example string
xxyxxxyx
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// Future Work

• Permit
• With disambiguating conventions for
• Associativity
• Precedence
• Parsing optimization
• Exploit compile-time analysis information at
  runtime

E -gt E ? E
39
// Conclusion
Approximating Context-Free Grammar Ambiguity
Context-free grammar ambiguity is
undecidable. However, just because its
undecidable, doesnt mean there arent (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.
But wait, theres more
40
// Lessons Learned

• Framework
• Plug in your favorite (over-)approximation of
  L(?)
• Even take intersection of them A ?i Ai
• Approximation closed under intersection
• Methodology
• Just because its undecidable doesnt mean there
  arent (good) approximations
• Quantity of false-positives (practically
  motivated)
• What to do with false-positives (pratically
  motivated)
• Dont be scared of undecidability

41
bonus slides
42
// Membership Decidable!

• Membership (aka. parsing)
• Given ? ? ?
• Is the string, ?, in the language of G
• Algorithms
• LL(k) O(?)
• LA-LR(k) O(?)
• GLR O(?3)

? ? L(G)
43
// Parsing Greedily Left-to-Right

• The ambiguity problem for XY...
• In fact, already a problem if x goes too far
• Thus, we only have a problem if (X eats into
  Y)
• Essentially disambiguation by picking longest
  match

... may occur in 2 cases
x
y
- (too little) Not possible (due to
greediness)
x
y
- (too much) Only this is a problem!
x
y
x
y
X Y ?
? X ? X( prefix(Y) \ ? ) ? ? ?