Deciding language inclusion problems using quasiorders

Deciding language inclusion problems
using quasiorders
Pierre Ganty IMDEA Software Institute
joint work with Pedro Valero (IMDEA) and Francesco Ranzato (Uni. Padova)

L2
ρ(L1)
• ρ is extensive: L ⊆ ρ(L)
• ρ is monotone: L ⊆ L implies ρ(L) ⊆ ρ(L )
• ρ is idempotent: ρ(ρ(L)) = ρ(L)

L2
ρ(L1)
L1 ⊆ L2 iff ρ(L1) ⊆ L2
Assume ρ(L2) = L2, then
ρ(L2) = L2

L1 as a ﬁxed point
q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq

q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq
Xq ∅
Xq ∅

q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq
Xq ∅
Xq ∅
{ε}
∅

q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq
Xq ∅
Xq ∅
{ε}
∅
{ε}
{a}

q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq
Lq (ba+
)∗
Lq a+
(ba+
)∗
Xq ∅
Xq ∅
{ε}
∅
{ε}
{a}

q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq
L1 = Lq = (ba+
)∗
= lfp λX. b ∪ Fn(X) q
Lq (ba+
)∗
Lq a+
(ba+
)∗
Xq ∅
Xq ∅
{ε}
∅
{ε}
{a}

q
q
b a
a
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq
L1 = Lq = (ba+
)∗
= lfp λX. b ∪ Fn(X) q
Lq (ba+
)∗
Lq a+
(ba+
)∗
Xq ∅
Xq ∅
{ε}
∅
{ε}
{a}
b Fn

ρ(L1) = ρ lfp λX. b ∪ Fn(X)
ρ(L1)
if ρ is complete for Fn (i.e. ρ Fn ρ = ρ Fn) then
ρ(L1) = lfp λX. ρ b ∪ Fn(X)

ρ(L1)
if ρ is complete for Fn (i.e. ρ Fn ρ = ρ Fn) then
ρ(L1) = lfp λX. ρ b ∪ Fn(X)
ρ is complete for Fn

Wanted
ρ(L2) = L2
ρ is complete for Fn

Quasiorder induced ρ
Given a quasiorder on words, deﬁne
ρ (X) = {y | ∃x ∈ X : x y}

ρ (X) = {y | ∃x ∈ X : x y}
∩ (L2 × L2) = ∅
x y implies a x a y for all a

ρ (X) = {y | ∃x ∈ X : x y}
∩ (L2 × L2) = ∅
ρ (L2) = L2
ρ is complete for λX. aX for all a

ρ (X) = {y | ∃x ∈ X : x y}
∩ (L2 × L2) = ∅
ρ (L2) = L2
ρ is complete for λX. aX for all a
is a left L2
consistent qo

Example of quasiorder
Nerode left quasiorder relative to L2 [Varricchio,deLuca’94]
x l
L2
y ⇔ L2 x−1
⊆ L2 y−1

x l
L2
y ⇔ L2 x−1
⊆ L2 y−1
Lx−1
= {w | wx ∈ L}

x l
L2
y ⇔ L2 x−1
⊆ L2 y−1
l
L2
is left L2 consistent

x l
L2
y ⇔ L2 x−1
⊆ L2 y−1
l
L2
l
L2
is a well-quasi order if L2 is regular

x l
L2
y ⇔ L2 x−1
⊆ L2 y−1
l
L2
l
L2
is a well-quasi order if L2 is regular
l
L2
is decidable if L2 is regular

Deciding L1 ⊆ L2 using Nerode’s left qo l
L2
L1 ⊆ L2 iﬀ ρ l
L2
(L1) ⊆ L2

L2
L1 ⊆ L2 iﬀ ρ l
L2
(L1) ⊆ L2
ρ l
L2
(L1)

L2
L1 ⊆ L2 iﬀ ρ l
L2
(L1) ⊆ L2
ρ l
L2
(L1)
L2

L2
L1 ⊆ L2 iﬀ ρ l
L2
(L1) ⊆ L2
iﬀ min l
L2
(ρ l
L2
(L1)) ⊆ L2
ρ l
L2
(L1)
L2

L2
L1 ⊆ L2 iff ρ l
L2
(L1) ⊆ L2
iff min l
L2
(ρ l
L2
(L1)) ⊆ L2
iff ∀w ∈ min l
L2
(ρ l
L2
(L1)): w ∈ L2
ρ l
L2
(L1)
L2

Deciding L1 ⊆ L2 using l
L2
(cont’d)
L1 ⊆ L2 iﬀ ∀w ∈ min l
L2
(ρ l
L2
(L1)): w ∈ L2

L2
(cont’d)
L2
(ρ l
L2
(L1)): w ∈ L2
iﬀ ∀w ∈ min l
L2
(lfp λX. ρ l
L2
b ∪ Fn(X) : w ∈ L2

L2
(cont’d)
L2
(ρ l
L2
(L1)): w ∈ L2
iﬀ ∀w ∈ lfp λX. min l
L2
b ∪ Fn(X) : w ∈ L2
iﬀ ∀w ∈ min l
L2
(lfp λX. ρ l
L2
b ∪ Fn(X) : w ∈ L2

Word-based antichain algorithm for L(A) ⊆ L2
Data: FA A = Q, δ, q0
, F
Data: L2 regular
1 Yq q∈Q := ∅;
2 repeat
3 Xq q∈Q := Yq q∈Q;
4 Yq q∈Q := min l
L2
b ∪ Fn( Yq q∈Q) ;
5 until ρ l
L2
( Yq q∈Q) ⊆ ρ l
L2
( Xq q∈Q);
6 forall u ∈ Yq0 do
7 if u /∈ L2 then return false;
8 return true;

What else?
x l
L2
y ⇔ L2 x−1
⊆ L2 y−1

What else?
State based quasiorder
Let A2 = Q2, δ2, q0
2, F2 be an automaton for L2
x l
A2
y ⇔ prex(F2) ⊆ prey(F2)
x l
L2
y ⇔ L2 x−1
⊆ L2 y−1

What else?
Let A2 = Q2, δ2, q0
x l
A2
x l
L2
y ⇔ L2 x−1
⊆ L2 y−1
1. l
A2
∩ (L2 × L2) = ∅
2. x l
A2
y implies a x l
A2
a y for all a
3. l
A2
is a well-quasiorder
4. l
A2
is decidable

What else?
Let A2 = Q2, δ2, q0
x l
A2
x l
L2
y ⇔ L2 x−1
⊆ L2 y−1
1. l
A2
∩ (L2 × L2) = ∅
2. x l
A2
y implies a x l
A2
a y for all a
3. l
A2
is a well-quasiorder
4. l
A2
is decidable
l
A2
decidable wqo

Word-based antichain algorithm for L(A1) ⊆ L(A2)
Data: FA A1 = Q1, δ1, q0
1, F1
2, F2
1 Yq q∈Q1
:= ∅;
2 repeat
3 Xq q∈Q1
:= Yq q∈Q1
;
4 Yq q∈Q1
:= min l
A2
b ∪ Fn( Yq q∈Q1
) ;
5 until ρ l
A2
( Yq q∈Q1
) ⊆ ρ l
A2
( Xq q∈Q1
);
6 forall u ∈ Yq0
1
do
7 if u /∈ L(A2) then return false;
8 return true;

Word-based antichain algorithm for L(A1) ⊆ L(A2)
1, F1
2, F2
1 Yq q∈Q1
:= ∅;
2 repeat
3 Xq q∈Q1
:= Yq q∈Q1
;
4 Yq q∈Q1
:= min l
A2
b ∪ Fn( Yq q∈Q1
) ;
5 until ρ l
A2
( Yq q∈Q1
) ⊆ ρ l
A2
( Xq q∈Q1
);
6 forall u ∈ Yq0
1
do
7 if u /∈ L(A2) then return false;
8 return true;
Let A2 = Q2, δ2, q0
x l
A2

Ditching words altogether
State-based antichain algorithm for L(A1) ⊆ L(A2)
1, F1
2, F2
1 Yq q∈Q1
:= ∅;
2 repeat
3 Xq q∈Q1
:= Yq q∈Q1
;
4 Yq q∈Q1 := [. . .];
5 until Yq ⊆∀∃
Xq, for all q ∈ Q1;
6 forall s ∈ Yq0
1
do
7 if q0
2 /∈ s then return false;
8 return true;
[. . .] = min⊆∀∃ {preA2
a (s) | ∃a ∈ Σ, q ∈ δ1(q, a), s ∈ Xq } ∪ F2

• Same as antichain algo
• Derived from general solution instantiated to l
A2
• Variants of the antichain algorithm as instantiations
• Simulation enhanced qo 2
• Nerode’s qo induce the coarsest qo based abstractions

• Same as antichain algo
• Derived from general solution instantiated to l
A2
• Variants of the antichain algorithm as instantiations
• Simulation enhanced qo 2
• Nerode’s qo induce the coarsest qo based abstractions
l
L2
and r
L2
are wqo’s if L2 is regular

L2 is a one-counter net trace set
q2
q1
q3a, 1
a, 0
a, −1
b, 1
The right Nerode qo for L2 ( r
L2
) is a wqo but it’s undecidable
u r
O2
v is a right L2 consistent decidable wqo
u r
O2
v iﬀ mu ≤ mv where mu, mv ∈ (N ∪ {⊥})3
L2 = L(O2) is the set of traces
from conﬁguration q2 0

Decision procedure for L(A1) ⊆ L(O2) using r
O2
Word-based antichain algorithm for L(A) ⊆ L(O2)
Data: FA A = Q, δ, q0
, F
Data: OCN O2
1 Yq q∈Q := ∅;
2 repeat
3 Xq q∈Q := Yq q∈Q;
4 Yq q∈Q := min r
O2
b ∪ Gn( Yq q∈Q) ;
5 until ρ r
O2
( Yq q∈Q) ⊆ ρ r
O2
( Xq q∈Q);
6 forall u ∈ Yq0 do
7 if u /∈ L2 then return false;
8 return true;

So far. . .
L(A1) ⊆ L(A2)
L(A1) ⊆ L(O2)

= lfp λX. ρ b ∪ Fn(X)
L2
ρ(L1)

L2
ρ(L1)
ρ Fn ρ = ρ Fn

L2
ρ(L1)
When L1 is regular, ρ is complete for λX. aX for all a
ρ Fn ρ = ρ Fn
lfp λ
Xq
Xq
.
{ } ∪ bXq
∅ ∪ aXq ∪ aXq

Quasiorders for the case L(G1) ⊆ L(A2)
FnG1
is more general: from aX, Xa, aXb, XX

FnG1
ρ(L1) = lfp λX. ρ b ∪ FnG1 (X)
if ρ is complete for λX. aX and λX. Xa for all a

FnG1
1. ∩ (L2 × L2) = ∅
2. x y implies a x a y and x a y a for all a

Myhill quasiorder relative to L2 [Varricchio, deLuca’94]
x L2
y ⇔ {(u, v) | u x v∈L2} ⊆ {(u, v) | u y v∈L2}
FnG1
1. ∩ (L2 × L2) = ∅

Myhill quasiorder relative to L2 [Varricchio, deLuca’94]
x L2
y ⇔ {(u, v) | u x v∈L2} ⊆ {(u, v) | u y v∈L2}
State based quasi order [Netys’15]
x A2
y ⇔ {(q, q ) | q
x
q } ⊆ {(q, q ) | q
y
q }
FnG1
1. ∩ (L2 × L2) = ∅

Greatest ﬁxpoint based approach
lfp λX. b ∪ Fn(X) ⊆ L2 iﬀ b ⊆ gfp λX. L2 ∩ Fn(X)
∪ becomes ∩
λX. aX becomes λX. a−1
X
ρ Fn ρ = ρ Fn becomes ρ Fn ρ = Fn ρ
gfp λX. ρ(L2 ∩ Fn(X)) stepwise equal gfp λX. L2 ∩ Fn(X)

∪ becomes ∩
X
Does not terminate in general

∪ becomes ∩
X
Does not terminate in general
Does always terminate 1

Conclusion
Promising framework for language inclusion

Conclusion
Extension to trees, ω-languages, timed languages,. . .

Conclusion
Extension to trees, ω-languages, timed languages,. . .
Technical Report 1904.01388 on arxiv.org

Deciding language inclusion problems using quasiorders

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