logics for weighted automata and...

Logics for Weighted Automata

and TransducersBenjamin Monmege

LIF, Aix-Marseille Université, France

Based on joint works with Paul Gastin, Benedikt Bollig and Marc Zeitoun

Software Verification

Critical Software• communication systems• e-commerce• health databases• energy production

Critical Software• communication systems• e-commerce• health databases• energy productionTO�BE��VERIFIED

Property to be verified

Is the property verified or not by the software?

• May an error state be reached?• Is there a book written by X, rented by Y?• Does this leader election protocol permit to elect the leader?

• What is the probability for an error state to be reached?• How many books, written by X, have been rented by Y?• What is the maximal delay ensuring that this leader election protocol permits the election?

From Boolean to Quantitative Verification

Formal Verification

Critical Software• communication systems• e-commerce• health databases• energy productionTO�

BE��VERIFIED

Formal Verification

BE��VERIFIED

Formal Model

ababcaabb

Is the property verified or not by the model?

Formal Verification

BE��VERIFIED

Formal Model

(a+ b)?c(ac)+

FG (pU q)

ababcaabb

Formal Specification

Is the property verified or not by the model?

Qualitative/QuantitativeQualitative, Boolean: [Büchi’60], [Elgot’61], [Trakhtenbrot’61]

Automata MSO logic

Quantitative, weights

Automata MSO logic

Weighted Automata

Automata MSO logic

Weighted Automata

Automata MSO logic

Weighted Automata

Find suitable weighted MSO logic

Automata MSO logic

Weighted Automata

Focus on definability / qualitative

Automata MSO logic

Weighted Automata

Focus on definability / qualitative

Focus on computation / quantitative

Weighted Automata

b, 0 b, 0

b,−1

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1)

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1 1

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

Semantics of aba: 1+(-1)+1 = 1

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

#a(w) - #b(w)

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1) (Z∪{−∞},max,+,−∞,0)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

#a(w) - #b(w)

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1) (Z∪{−∞},max,+,−∞,0)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

#a(w) - #b(w)

1 a,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1 b,0⎯ →⎯⎯ 2 a,0⎯ →⎯⎯ 2 2

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1) (Z∪{−∞},max,+,−∞,0)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

#a(w) - #b(w)

1 a,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1 b,0⎯ →⎯⎯ 2 a,0⎯ →⎯⎯ 2

0 a,0⎯ →⎯⎯ 0 a,0⎯ →⎯⎯ 0 b,0⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1) (Z∪{−∞},max,+,−∞,0)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

#a(w) - #b(w)

1 a,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1 b,0⎯ →⎯⎯ 2 a,0⎯ →⎯⎯ 2

0 a,0⎯ →⎯⎯ 0 a,0⎯ →⎯⎯ 0 b,0⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

Semantics of aaba: max(2,1) = 2

Weighted Automata

b, 0 b, 0

b,−1

(Z,+,×,0,1) (Z∪{−∞},max,+,−∞,0)

0 a,1⎯ →⎯⎯ 1 b,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,−1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

0 a,1⎯ →⎯⎯ 0 b,1⎯ →⎯⎯ 0 a,1⎯ →⎯⎯ 1

#a(w) - #b(w) max size of a’s blocks

1 a,1⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1 b,0⎯ →⎯⎯ 2 a,0⎯ →⎯⎯ 2

0 a,0⎯ →⎯⎯ 0 a,0⎯ →⎯⎯ 0 b,0⎯ →⎯⎯ 1 a,1⎯ →⎯⎯ 1

Semantics of aaba: max(2,1) = 2

How to Specify Quantitative Properties?

How to Specify Quantitative Properties?Weighted Monadic Second Order Logic [Droste&Gastin 05]generalized to trees [Droste&Vogler 06], infinite words [Droste&Rahonis 07], nested words [Mathissen 10] or pictures [Fichtner 11]

Weighted Regular Expressions over finite words [Kleene 56, Schützenberger 61]

Weighted Temporal Logics: PCTL [Hansson&Jonsson 94], WLTL [Mandrali 12]

• Core weighted logic for weighted automata

• Enhancing the logic to handle more properties: FO vs pebbles

• A special case: the transducers

Weighted MSOϕ ::= s | Pa(x) | x ! y | x ∈ X | ¬Pa(x) | ¬(x ! y) | ¬(x ∈ X)

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

[Droste-Gastin, 05]

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

[Droste-Gastin, 05]

Negation restricted to atomic formulae

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

[Droste-Gastin, 05]

Negation restricted to atomic formulae

Arbitrary constants from a semiring

Inspired from the boolean semiring

ϕ ::= s | Pa(x) | x ! y | x ∈ X | ¬Pa(x) | ¬(x ! y) | ¬(x ∈ X)

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

Semantics in a semiring S = (S,+,×, 0, 1)

Atomic formulae: 0,1

disjunction, existential quantifications: sum

conjunction, universal quantifications: product

B = ({0, 1},∨,∧, 0, 1)

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

Examples

ϕ1 = ∃xPa(x)

[[ϕ1]](w) = |w|a

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

Examples

ϕ1 = ∃xPa(x)

[[ϕ1]](w) = |w|a

ϕ2 = ∀x ∃y (y ! x ∧ Pa(y))

[[ϕ2]](abaab) = 1× 1× 2× 3× 3

[[ϕ2]](an) = n!

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

Examples

ϕ1 = ∃xPa(x)

[[ϕ1]](w) = |w|a

ϕ2 = ∀x ∃y (y ! x ∧ Pa(y))

[[ϕ2]](abaab) = 1× 1× 2× 3× 3

[[ϕ2]](an) = n!

Too big to be computed by a

weighted automaton[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

Examples

ϕ1 = ∃xPa(x)

[[ϕ1]](w) = |w|a

ϕ2 = ∀x ∃y (y ! x ∧ Pa(y))

[[ϕ2]](abaab) = 1× 1× 2× 3× 3

[[ϕ2]](an) = n!

Too big to be computed by a

weighted automaton

We need to restrict weighted MSO

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

Theorem: weighted automata = restricted wMSO

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

φ almost boolean

[Droste-Gastin, 05]

Weighted MSO

| ϕ ∨ ϕ | ϕ ∧ ϕ | ∃xϕ | ∀xϕ | ∃X ϕ | ∀X ϕ

φ almost booleancommutativity

[Droste-Gastin, 05]

Weighted MSO

Core weighted MSO logic

[Gastin-Monmege, 15]

Boolean fragment

ϕ ::= ⊤ | Pa(x) | x ! y | x ∈ X | ¬ϕ | ϕ ∧ ϕ | ∀xϕ | ∀X ϕ

Boolean fragment

Step formulae Ψ ::= s | ϕ ?Ψ : Ψ

Boolean fragment

if … then … else …

Boolean fragment

if … then … else …Pa(x) ? 1 : 0

Boolean fragment

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

Boolean fragment

x ∈ X1 ? s1 : (x ∈ X2 ? s2 : · · ·(x ∈ Xn−1 ? sn−1 : sn) · · · )

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

Boolean fragment

x ∈ X1 ? s1 : (x ∈ X2 ? s2 : · · ·(x ∈ Xn−1 ? sn−1 : sn) · · · )

[[Ψ ]](w,σ) = s some value occurring in Ψ

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

Boolean fragment

x ∈ X1 ? s1 : (x ∈ X2 ? s2 : · · ·(x ∈ Xn−1 ? sn−1 : sn) · · · )

[[Ψ ]](w,σ) = s some value occurring in Ψ

A step formula takes finitely many values

For each value, the pre-image is MSO-definable

Boolean fragment

core wMSO Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

Boolean fragment

core wMSO Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

no constants

Boolean fragment

core wMSO Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

if … then … else …

no constants

Boolean fragment

core wMSO Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

if … then … else …Assigns a value from Ψ

to each position

no constants

Boolean fragment

core wMSO Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

if … then … else …Assigns a value from Ψ

to each position

x Ψ |}(w,σ) = {{([[Ψ ]](w,σ[x !→ i]))i}} ∈ N⟨R⋆⟩

no constants

[Gastin-Monmege, 15] singleton multiset

Boolean fragment

core wMSO Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

x Ψ |}(w,σ) = {{([[Ψ ]](w,σ[x !→ i]))i}} ∈ N⟨R⋆⟩

Semantics

sums over multisets

{| 0 |}(w,σ)=∅

Multisets of weight structures

[Gastin-Monmege, 15][Droste-Perevoshchikov, 14]

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

multiset

A run generates a sequence of weights

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

{|A|} : Σ⋆ → N⟨R⋆⟩

weights of A

multiset

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

{|A|} : Σ⋆ → N⟨R⋆⟩

weights of A

Aggregation aggr : N⟨R⋆⟩ → S

multiset

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

{|A|} : Σ⋆ → N⟨R⋆⟩

weights of A

Semiring: sum-product aggrsp(A) =

r1···rn∈A r1 × · · ·× rn

multiset

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

{|A|} : Σ⋆ → N⟨R⋆⟩

weights of A

r1···rn∈A r1 × · · ·× rn

Valuation monoid: sum-valuation aggrsv(A) =

Val(A) =!

r1···rn∈A Val(r1 · · · rn)

multiset

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

{|A|} : Σ⋆ → N⟨R⋆⟩

weights of A

r1···rn∈A r1 × · · ·× rn

Valuation monoid: sum-valuation aggrsv(A) =

Val(A) =!

r1···rn∈A Val(r1 · · · rn)

Average value Discounted value…

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

{|A|} : Σ⋆ → N⟨R⋆⟩

weights of A

Concrete semantics [[A]] = aggr ◦ {|A|} : Σ⋆ → S

multiset

Ψ ::= s | ϕ ?Ψ : Ψ

Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

Theorem: weighted automata = core wMSO

Ψ ::= s | ϕ ?Ψ : Ψ

Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

{|− |} : Σ⋆ → N⟨R⋆⟩Abstract semantics

Ψ ::= s | ϕ ?Ψ : Ψ

Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

Concrete semantics [[−]] = aggr ◦ {|− |} : Σ⋆ → S

Ψ ::= s | ϕ ?Ψ : Ψ

Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

Concrete semantics [[−]] = aggr ◦ {|− |} : Σ⋆ → SEasy constructive proofs

preservation of the constants

no restriction on core wMSO

no hypotheses on weights

Extensions

More general models than words:

trees, nested words…

More powerful logics: deciding if a wMSO formula

is expressible in core wMSO?

More powerful automata: finding equivalent fragments of wMSO

Weighted FO logic

ϕ ::= ⊤ | Pa(x) | x ! y | x ∈ X | ¬ϕ | ϕ ∧ ϕ | ∀xϕ | ∀X ϕΨ ::= s | ϕ ?Ψ : ΨΦ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |

x Φ |!

X Φ |"

Weighted FO logic

Φ ::= s ϕ?Φ :Φ Φ+Φ Φ×Φ Φx∑ Φ

Reintroduction of the product Unconditional product

quantification

We can keep Boolean MSO or restrict to FO…

Weighted FO logic

quantification

ϕ2 = ∀x ∃y (y ! x ∧ Pa(y)) [[ϕ2]](an) = n!

Weighted FO logic

quantification

2y∏x∏!"# $%&(w)= 2

A = (Q,A, I, δ, T )

I ∈ SQ T ∈ SQ

δ : Q× Test×Move×Q→ S

Run as a finite sequence of configurations (W,σ, q, i,π)

σ : Peb→ pos(W )

π ∈ (Peb× pos(W ))∗with free pebbles

and a stack of currently dropped pebbles

Move = {→,←,↑}∪{↓xx ∈ Peb}

Pebble weighted automata

Weighted automata with pebbles

I Move = { ,!, "} [ {#x

| x 2 Peb}I Transition: q

weight test move��! q0

I Run ⇢ over word u 2 A+: sequence of transitions.

I weight(⇢): product of weights of transitions

I Non determinism: [[Ap,q

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

▶ a b b c a c b a ◀x

[Globerman and Harel, 96][Engelfriet and Hoogeboom,99]

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

▶ a b b c a c b a ◀

5 x1x7x7

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1x1x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

x2x2x2x2x2x1x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

x2x2x2x2x2x1x3x3x3x1x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

x2x2x2x2x2x1x3x3x3x1

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

Weight of

the run:

35 562 240

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

Weight of

the run:

35 562 240Non

determinism

Non determinism resolved by sum

I Move = { ,!, "} [ {#x

]](u) =X

⇢ run over u

weight(⇢)

1▹?5

a?→ 7b?→

c? ↓x

2¬x?→

3¬◃?←

I =�5 0 0 0

(a? + 7b?)! 0 c? #x

0! 0 0 00 0 2¬x?! x? 0 " 0 3¬.?

5 x1x7x7x1

x2x2x2x1x3x1x1

Weight of

the run:

35 562 240Non

determinism

Non determinism resolved by sum

Sum of the weights

of the runs:

5.7|w|b.∏wi=c2i-2(3i-1)

Translating a formula into an automaton

AΦ AΨlast?

first?

Sum by disjoint union of automata

Product:

Sum quantification:

Product quantification:

→ →

↓x last?, ↑

Challenging for the Boolean part: need unambiguous automata

Use deterministic automataof size non-elementary...

Take advantage of the pebblesto build a linear size automaton

Bϕokϕ

koϕBψ

koψlast?

←first?

Disjunction/conjunctionξ = φ∨ψ

Bϕokϕ

koϕBψ

koψlast?

←first?

Disjunction/conjunctionξ = φ∨ψ

Bϕokϕ

koϕ↓xlast?↑

last?↑

Existential/Universal quantificationsξ = ∃x φ

Logic equivalent to PWA?• Weighted FO misses a counting capability…

• Solution: weighted transitive closure operation

PebbleWA-ICALP.tex 1230 2010-07-07 13:59:03Z bm

Transitive closure logics! For ϕ with at least two first order free variables, define

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

! The transitive closure operator is defined by TCxyϕ =&

n≥1ϕ

! Bounded transitive closure : N-TCxyϕ = TCxy (x − N ≤ y ≤ x + N ∧ ϕ)

x yz1 z3 z2

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

13/17 PebbleWA-ICALP.tex 1230 2010-07-07 13:59:03Z bm

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

13/17 PebbleWA-ICALP.tex 1230 2010-07-07 13:59:03Z bm

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

ϕ1(x , y) = ϕ(x , y)

ϕn(x , y) = ∃z0 · · · ∃zn

!x = z0 ∧ zn = y ∧ diff(z0, . . . , zn) ∧

"#1≤ℓ≤n ϕ(zℓ−1, zℓ)

x yz2 z3 z1

n≥1ϕ

x yz1 z3 z2

Theorem: PWA = wFO + bounded-TC

Application to transductions

Tinput word output word

Pattern matching/replacement

Pattern matching/replacement Tree/Graph rewriting

Pattern matching/replacement

Update of XML databases

Tree/Graph rewriting

Existing models over words• Two-way Deterministic Finite-State Transducers • Functional One-way Finite-State Transducers • MSOT (à la Courcelle)

• Copyless Streaming String Transducers (Alur et al)

{Functions

{Relations

• Two-way Non-Deterministic Finite-State Transducers • Non-Deterministic Finite-State Transducers • Non-deterministic Copyless Streaming String Transducers

(Alur et Deshmukh) • NMSOT (with free second-order variables)

{Functions

{Relations

• Two-way Non-Deterministic Finite-State Transducers • Non-Deterministic Finite-State Transducers • Non-deterministic Copyless Streaming String Transducers

(Alur et Deshmukh) • NMSOT (with free second-order variables)

only finite valued relations…

Transduction as weights• Desire: weight transitions with words… Difficult to

equip A* with a semiring structure: how to combine several accepting runs?

• Works for deterministic or unambiguous automata: functional transducers

• For relations: semiring of languages

(2A*,∪,⋅,∅,{ε})

Examples(Px(a)?{aa} : (P

x(b)?{bb} :∅))

x∏ aba -> aabbaa

x(b)?{bb} :∅))

x∏ aba -> aabbaa

(Px(!)?{insert} :(P

x(a)?{a} :(P

x(b) : {b})))

x(b)?{bb} :∅))

x∏ aba -> aabbaa

(Px(!)?{insert} :(P

x(a)?{a} :(P

x(b) : {b})))

x∏ a*b -> ainsertb

x(b)?{bb} :∅))

x∏ aba -> aabbaa

(Px(!)?{insert} :(P

x(a)?{a} :(P

x(b) : {b})))

Py(!)?

y∑ (x = y)?{insert} : (Px(!)?{ε} : (P

x(a)?{a} : (P

x(b)?{b})))

x∏⎡⎣⎢ ⎤⎦⎥

x(b)?{bb} :∅))

x∏ aba -> aabbaa

(Px(!)?{insert} :(P

x(a)?{a} :(P

x(b) : {b})))

Py(!)?

y∑ (x = y)?{insert} : (Px(!)?{ε} : (P

x(a)?{a} : (P

x(b)?{b})))

x∏⎡⎣⎢ ⎤⎦⎥

a*b*a -> {ainsertba,abinserta}

Relation

ExamplesPx(a)?{a} : (P

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

x∏ababbaabb -> aaaaccccc

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Not comp. by 1-way Func Transducer

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Px(a)?{a} : (P

x(b)?{ε})

x∏ + Px(a)?{ε} : (P

x(b)×{c})

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Px(a)?{a} : (P

x(b)?{ε})

x∏ + Px(a)?{ε} : (P

x(b)×{c})

x∏ababbaabb -> {aaaa,ccccc}

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Px(a)?{a} : (P

x(b)?{ε})

x∏ + Px(a)?{ε} : (P

x(b)×{c})

Px(a)?{a,ε} : (P

x(b)?{b,ε})

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Px(a)?{a} : (P

x(b)?{ε})

x∏ + Px(a)?{ε} : (P

x(b)×{c})

Px(a)?{a,ε} : (P

x(b)?{b,ε})

x∏ aba -> {ε,a,b,ab,ba,aa,aba}

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Px(a)?{a} : (P

x(b)?{ε})

x∏ + Px(a)?{ε} : (P

x(b)×{c})

Px(a)?{a,ε} : (P

x(b)?{b,ε})

Px(a)?A*aA* : (P

x(b)?A*bA*)

x(b)?{ε})

x∏ × Px(a)?{ε} : (P

x(b) : {c})

Px(a)?{a} : (P

x(b)?{ε})

x∏ + Px(a)?{ε} : (P

x(b)×{c})

Px(a)?{a,ε} : (P

x(b)?{b,ε})

Px(a)?A*aA* : (P

x(b)?A*bA*)

x∏ aba -> A*aA*bA*aA*

Infinitely-valued relation

TransducersPx(a)?A*aA* : (P

x(b)?A*bA*)

a|A*aA* b|A*bA*

Infinite-valued, but deterministic

Reverse?

a|a, ←

A|ε, →

b|b, ←

first?

Reverse?

a|a, ←

A|ε, →

b|b, ←

first?

Impossible in FO… … because of order of

interpretation of product

Reverse?

a|a, ←

A|ε, →

b|b, ←

first?

Solution: in this non-commutative setting, add right-to-left products

Reverse?

a|a, ←

A|ε, →

b|b, ←

first?

Solution: in this non-commutative setting, add right-to-left products

Px(a)?{a} : (P

x(b)?{b})

Transitive closureϕ ::= ⊤ | Pa(x) | x ! y | x ∈ X | ¬ϕ | ϕ ∧ ϕ | ∀xϕ | ∀X ϕ

Φ ::= L ϕ?Φ :Φ Φ+Φ Φ×Φ Φx∑ Φ

x∏ N -TCx ,yΦ

Regular language

x∏ N -TCx ,yΦ

Regular languageAble to define right-

to-left product

Φ(x) := [1-TCx ,yx⨿ (y = x −1?Φ(x))](last, first)×Φ(first)

x∏ N -TCx ,yΦ

Regular languageAble to define right-

to-left product

Φ(x) := [1-TCx ,yx⨿ (y = x −1?Φ(x))](last, first)×Φ(first)

Theorem: Pebble Transducers = FO + bounded-TC

with regular language productions linear transformation from logic to transducers

Algorithmic questions

Algorithmic questionsTheorem: Evaluation of FO + bounded-TC with

complexity O(|formula|x|input|#variables)

• In functional setting: it is decidable if a 2-way transducer is recognisable by a 1-way transducer [Filiot,Gauwin,Reynier, Servais, 13]

• What about the general setting, with pebbles?

• Determinisability of functional non-det transducers is decidable in PTIME [Schützenberger, 75]

• Extension to the general setting?

Thank you!

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