Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf ·...

Deciding Emptiness of min-automata

Szymon Toruńczykjoint work with

Mikołaj Bojańczyk

LSV Cachan / University of Warsaw

Wednesday, November 18, 2009

“c tends to ∞”

liminf(c) = ∞

deterministic automata with counterstransitions invoke counter operations:

c:=min(d,e)

c:=c+1

acceptance condition is a boolean combination of:

Min-automata

Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)

Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞

a a a b a b a a b...

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...cdz

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

liminf(c) = ∞

c:=min(d,e)

c:=c+1

Min-automata

a a a b a b a a b...000

Max-automata

Extension of WMSO by the quanti$er

Max-automata

Extension of WMSO by the quanti$erUX φ(X)

„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”

which says

Max-automata

Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}

UX φ(X)

which says

Max-automata

UX “X is a block of a’s”

UX φ(X)

which says

Max-automata

UX φ(X)

which says

Max-automata Min-automata

UX φ(X)

which says

RX φ(X)

UX φ(X) which says

RX φ(X)

„there exist in"nitely many sets X of same size, satisfying φ(X)”

UX φ(X) which says

RX φ(X)

Language: {an1 b an2 b an3 b... : n1 n2 n3... converges to ∞}

Extension of WMSO by the quanti$erUX φ(X)

which says

RX φ(X)

Language: {an1 b an2 b an3 b... : n1 n2 n3... converges to ∞}

¬RX “X is a block of a’s”

max-automata min-automata

WMSO + U WMSO + R

eorem. WMSO+U has the same expressive power as deterministic max-automata.

WMSO + U WMSO + R

eorem. WMSO+R has the same expressive power as deterministic min-automata.

WMSO + U WMSO + R

WMSO + U + R

What if we allow both U and R?

WMSO + U + R

eorem. WMSO+U+R has the same expressive power as boolean combinations of min- and max-automata.

WMSO + U + R

Boolean combinations of min- & max-automata

WMSO + U + R

Boolean combinations of min- & max-automata

Equivalently: Nesting the quanti$ers U and R does not contribute anything to the expressive power of WMSO.

Emptiness of min-automata

eorem. Emptiness of min-automata is decidable.

1st proof. min-automata are a special case of ωBS-automata (Bojańczyk, Colcombet [06]), so emptiness is decidable. is gives bad complexity, however.

2nd proof. Reduction to the limitedness problem for distance-automata. Gives PSPACE algorithm, which is optimal.

eorem. Emptiness of max-automata is decidable.

eorem. Emptiness of a boolean combination of min- and max-automata is decidable.

Simplyfying the min-automata model

Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example

• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned

• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞

• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication

In Example 1, c:=c+1 can be written as:

!c d z

$1 ! !! 0 !! ! 0

!c d z

$1 ! !! 0 !! ! 0

c + 1 d z )

d:=min(c,c); c:=min(z,z) can be written as:

!c d z

$1 ! !! 0 !! ! 0

c + 1 d z )

!c d z

$1 ! !! 0 !! ! 0

!c d z

$! 0 !! ! !0 ! 0

c + 1 d z )

!c d z

$1 ! !! 0 !! ! 0

!c d z

$! 0 !! ! !0 ! 0

c + 1 d z )

z c z )

eorem. Min-automata are equivalent to min-automata in matrix form, with one state.

eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.

Example. Min-automaton which counts a’s on odd positions.

Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c.

Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:

-saw a in state q0 – go to q1; c:=c+1

-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0

-saw b in state q0 – go to q1

Min-automaton in matrix form with one state and two counters: c0, c1.

Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).

!c0 c1

"·#! 01 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

a a a b b b a a b...c0

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

a a a b b b a a b...0⊤

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

!c0 c1

"·#! 01 !

!c0 c1

"·#! 00 !

Example of a min-automaton A.

0 ⊤ ⊤

⊤ 1 ⊤

⊤ ⊤ 0

1 1 ⊤

⊤ ⊤ 0

⊤ ⊤ 0a: b:

0 ⊤ ⊤

⊤ 1 ⊤

⊤ ⊤ 0

1 1 ⊤

⊤ ⊤ 0

⊤ ⊤ 0a: b:

⊤ ⊤ 0

⊤ ⊤ 0abb:

0 ⊤ ⊤

⊤ 1 ⊤

⊤ ⊤ 0

1 1 ⊤

⊤ ⊤ 0

⊤ ⊤ 0a: b:

⊤ ⊤ 0

⊤ ⊤ 0abb:

0 ⊤ ⊤Initial valuation:

Acceptance condition: ¬ liminf(c3) = ∞

e tropical semiring

T = {0,1,2,..., ∞, ⊤}

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

with operations +, min

where ⊤+ x = x +⊤=⊤

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞

where ⊤+ x = x +⊤=⊤

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞

πn : T → Tn

maps n+1, n+2, ... to ∞

is a homomorphism of semirings

where ⊤+ x = x +⊤=⊤

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞

πn : T → Tn

maps n+1, n+2, ... to ∞

where ⊤+ x = x +⊤=⊤

πn : Tm → Tn for m>n

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞

πn : T → Tn

maps n+1, n+2, ... to ∞

where ⊤+ x = x +⊤=⊤

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞

πn : T → Tn

maps n+1, n+2, ... to ∞

where ⊤+ x = x +⊤=⊤

MkT – k by k matrices over Twith matrix multiplication

MkTn – k by k matrices over Tnwith matrix multiplication

πn : MkT → MkTn

maps n+1, n+2, ... to ∞

e tropical semiring

T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤

Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞

πn : T → Tn

maps n+1, n+2, ... to ∞

where ⊤+ x = x +⊤=⊤

MkT – k by k matrices over Twith matrix multiplication

MkTn – k by k matrices over Tnwith matrix multiplication

πn : MkT → MkTn

maps n+1, n+2, ... to ∞

3 ∞ 10 ∞ 12 ∞ ∞

3 32 10 11 12 7 ∞

∞ ∞ 10 ∞ 1∞ ∞ ∞

π6 π2

Pro$nite monoid

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

S0 S1 S2 S3 S7 S32 S1000

preserve multiplication

Pro$nite monoid

S0 S1 S2 S3 S7 S32 S1000

M3T0 M3T1 M3T2 M3T3

Pro$nite monoid

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

3 ∞ 10 ∞ 12 ∞ ∞

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

∞ ∞ 10 ∞ 12 ∞ ∞

3 ∞ 10 ∞ 12 ∞ ∞

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

∞ ∞ 10 ∞ 1∞ ∞ ∞

∞ ∞ 10 ∞ 12 ∞ ∞

3 ∞ 10 ∞ 12 ∞ ∞

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

∞ ∞ ∞0 ∞ ∞∞ ∞ ∞

∞ ∞ 10 ∞ 1∞ ∞ ∞

∞ ∞ 10 ∞ 12 ∞ ∞

3 ∞ 10 ∞ 12 ∞ ∞

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

S0 S1 S2 S3 S7 S32 S1000

Pro$nite monoid

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

Pro$nite monoid

-approximation:0

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

∞ ∞ ∞0 ∞ ∞∞ ∞ ∞

Pro$nite monoid

-approximation:1

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

∞ ∞ 10 ∞ 1∞ ∞ ∞

Pro$nite monoid

-approximation:2

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

∞ ∞ 10 ∞ 12 ∞ ∞

Pro$nite monoid

-approximation:3

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

3 ∞ 10 ∞ 12 ∞ ∞

Pro$nite monoid

-approximation:3 ∞ 10 ∞ 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

Pro$nite monoid

-approximation:3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

Metric Two elements are close if only an approx. with high threshold can distinguish them

S0 S1 S2 S3 S7 S32 S1000

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings

multiplication

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

multiplication

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

multiplication

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

...................

multiplication

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

multiplication

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

multiplication

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

multiplicationaddition

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

addition

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

addition

idempotent power (ω)

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

addition

ω-power

.... . . .........................

Pro$nite monoid

3 32 10 11 12 7 ∞

S0 S1 S2 S3 S7 S32 S1000

. . . . . . .........................

3 50 10 11 12 7 ∞

}d = 2-32

preserve

. . . . . . .........................

3 8 43 8 45 18 3

is continuous

.....................................

addition

ω-power

compact space.... . . .........................

0 ⊤ ⊤

⊤ 1 ⊤

⊤ ⊤ 0

1 1 ⊤

⊤ ⊤ 0

⊤ ⊤ 0a: b:

∞ ∞ 1⊤ ⊤ 0

⊤ ⊤ 0bω:

Ramsey eoremfor compact spaces

x x x x x x x x x x x x x x x x x x x x x x x x x

.x x x x x x x x x x x x x x x x x x x x x x x x x

colored by elements of a compact space

.x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .

..colored by elements of a compact space

x x x x x x x x x x x x x x x x x x x x x x x x x . ..

.x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x x x x . .. .

.x x .x

.x x .

x x x x x x x x x x x x x x x x x x x x x x x x x .

.x x .

x x x x x x x x x x x x x x x x x x x x x x x x x .

d( , )<1/Nlim

.x x .

x x x x x x x x x x x x x x x x x x x x x x x x x ..

d( , )<1/Nlim d( , )<1/N’lim

a b b a a b a a b a b b a a a b b a b a b a x x x

a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . . . . . ..

a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . . . . . ..induced transition matrix

a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .

a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lim

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lim

Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lim

Which limits lim are possible?

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lim

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+Which limits lim are possible?

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

∈Which limits lim are possible?

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

∈Which limits lim are possible?

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

eorem.

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

eorem. a b ( , )+__________

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

eorem. =a b ( , )+__________

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

eorem. a b=a b ( , )+__________

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

eorem. a b ( , )+,ω=a b ( , )+__________

d( , )<1/19)lim

.d( , )<1/10lim

d( , )<1/15lima b ( , )+__________

eorem. a b ( , )+,ω=if , are matrices over the (min,+)-semiring.a b

a b ( , )+__________

Simon’s Factorization eoremfor semigroups with stabilization

semigroup with stabilization

( S , · , # )

( S , · , # ){

• s # = (s n )# for n=1,2,3,...

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

• s# s# = s#

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

• s# s# = s#

• e# e = e# if e is idempotent

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

• s# s# = s#

e = e# if e is idempotent

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

• s# s# = s#

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

• s# s# = s#

Example 1 (in$nite) ({0,1,2,..., ∞}, +, ω ), 0ω =0, 1ω = 2ω = ...= ∞

( S , · , # ){

• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#

• s# s# = s#