10 YEARS OF WEIGHTED LOGICS FOR WEIGHTED ...gastin/Talks/AutoMathA.pdf10 YEARS OF WEIGHTED LOGICS...

10 YEARS OF WEIGHTED LOGICS FOR WEIGHTED AUTOMATA

Paul Gastin,LSV, ENS Cachan

joint work with Benjamin MonmegeULB

Qualitative, 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

Find suitable weighted MSO logic

Automata MSO logic

Focus on definability / qualitative

Automata MSO logic

Focus on definability / qualitative

Focus on computation / quantitative

AUTOMATA: FOCUS ON QUALITATIVE

aa’s blocks are of even length

# a’s ≠ # b’s

# a’s ≠ # b’s 0 1

b,−1

Σ, 1Support

# a’s ≠ # b’s 0 1

b,−1

Σ, 1Support

probabilistic

Threshold

01 2 3

LOGIC: FOCUS ON QUALITATIVE

Probabilistic CTL Prob(p U q) ! 2

LOGIC: FOCUS ON QUALITATIVE

Probabilistic CTL Prob(p U q) ! 2

Timed CTL E p U<3 q

A p U<3 q p q

x < 1, {x}

AUTOMATA: FOCUS ON QUANTITATIVE

# a’s - # b’s 0 1

b,−1

Maximal length of a’s blocks 0 1 2

b, 0 b, 0

(max,+) Semiring

AUTOMATA: FOCUS ON QUANTITATIVE

# a’s - # b’s 0 1

b,−1

LOGIC: FOCUS ON QUANTITATIVE

Prob(p U q)

aababbaaaababaaab

xPa(x) ? 1 : 0

Prob(p U q)

aababbaaaababaaab

xPa(x) ? 1 : 0

Prob(p U q)

maxx,y!

∀z, x ! z ! y → Pa(z)"

z(x ! z ! y) ? 1 : 0

: −∞

AUTOMATA vs LOGIC

Qualitative: [Büchi’60], [Elgot’61], [Trakhtenbrot’61]

AUTOMATA = MSO

AUTOMATA vs LOGIC

Qualitative: [Büchi’60], [Elgot’61], [Trakhtenbrot’61]

Goal: Find a weighted logic expressively equivalent to

weighted automata

AUTOMATA = MSO

KEEP SYNTAX CHANGE SEMANTICSInspired by weighted automata on semirings

Qualitative: existence of an accepting path

Quantitative: number of accepting paths

µ(a) =

1 10 1

µ(b) =

1 00 1

Boolean semiring: ({0, 1},∨,∧, 0, 1)

µ(a) =

1 10 1

µ(b) =

1 00 1

µ(babaab) =

1 10 1

Boolean semiring: ({0, 1},∨,∧, 0, 1)

Natural semiring: (N,+,×, 0, 1)

µ(a) =

1 10 1

µ(b) =

1 00 1

µ(babaab) =

1 10 1

µ(babaab) =

1 30 1

WEIGHTED AUTOMATA ON SEMIRINGS

Probabilistic semiring: (R!0,+,×, 0, 1)

01 2 3

WEIGHTED AUTOMATA ON SEMIRINGS

(max,+) semiring:

Probabilistic semiring: (R!0,+,×, 0, 1)

(N ∪ {−∞},max,+,−∞, 0)

b, 0 b, 0

01 2 3

THE FIRST SOLUTION

KEEP SYNTAX CHANGE SEMANTICS

ϕ ::= 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, ICALP’05]

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

Negation restricted to atomic formulae

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

Negation restricted to atomic formulae

Arbitrary constants from a semiring

Inspired from the boolean semiring

| 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)

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

Examples

ϕ1 = ∃xPa(x)

[[ϕ1]](w) = |w|a

| 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!

| 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

| 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

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

Thm [Droste-Gastin] weighted automata = restricted wMSO

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

φ almost boolean

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

φ almost booleancommutativity

WEIGHTED AUTOMATA TO RESTRICTED WMSO

∃X = (X1, . . . , Xn) run(X) ∧ ∀xweight(X,x)

[[run(X)]](w,σ) =

1 if X is a run

0 otherwise.

[[run(X)]](w,σ) =

1 if X is a run

0 otherwise.

weight of the transition taken at x by run X

[[run(X)]](w,σ) =

1 if X is a run

0 otherwise.

weight of the run

[[run(X)]](w,σ) =

1 if X is a run

0 otherwise.

weight of the runsum over all runs

[[run(X)]](w,σ) =

1 if X is a run

0 otherwise.

weight of the runsum over all runs

We need UNAMBIGUOUS formulas

in the QUANTITATIVE semantics

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

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

∃x ∃y x < y ∧ Pa(x) ∧ Pb(y)

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

∃x ∃y x < y ∧ Pa(x) ∧ Pb(y)

∧ ∀z (x ! z ∨ (z < x ∧ ¬Pa(z)))

∧ ∀z (z ! y ∨ (y < z ∧ ¬Pb(z)))

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

ϕ !→ (ϕ+,ϕ−)

[[ϕ+]](w,σ) =

1 if w,σ |= ϕ

0 otherwise.[[ϕ−]](w,σ) =

0 if w,σ |= ϕ

1 otherwise.

∃x ∃y x < y ∧ Pa(x) ∧ Pb(y)

∧ ∀z (x ! z ∨ (z < x ∧ ¬Pa(z)))

∧ ∀z (z ! y ∨ (y < z ∧ ¬Pb(z)))

RESTRICTED WMSO TO WEIGHTED AUTOMATA

disjunction : closure under sum

existential quantification : closure under projections/renamings

universal quantification: main difficulty

φ almost boolean: finite image, pre-images are MSO-definable

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

φ almost boolean

MANY EXTENSIONS

Thm: weighted automata = restricted wMSO

• Finite words: [Droste-Gastin, ICALP’05] • Finite ranked trees: [Droste-Vogler, Theor. Comp. Sci.’06]

• Pictures: [Fichtner, STACS’06]

• Infinite words: [Droste-Rahonis, DLT’06]

• Finite unranked trees: [Droste-Vogler, Theor. of Computing Systems’09]

• Traces: [Fichtner-Kuske-Meinecke, Handbook of weighted automata ’09]

• Nested words: [Mathissen, LMCS’10]

• …

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

CHANGING THE SYNTAXTOWARDS EASIER FORMULAE

CHANGING THE SYNTAX

[Bollig-Gastin, DLT’09] [Droste-Meinecke’12] [Fülöp-Stüber-Vogler'12]

Boolean fragment

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

CHANGING THE SYNTAX

Boolean fragment

No need to write unambiguous formulae

∃x ∃y x < y ∧ Pa(x) ∧ Pb(y)

CHANGING THE SYNTAX

Boolean fragment

∃x ∃y x < y ∧ Pa(x) ∧ Pb(y)

∧ ∀z (x ! z ∨ (z < x ∧ ¬Pa(z)))

∧ ∀z (z ! y ∨ (y < z ∧ ¬Pb(z)))

CHANGING THE SYNTAXBoolean fragment

Quantitative fragment

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

Formulae are more readable!

x Pa(x)

∃xPa(x)

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Pa(x)

∃xPa(x)

y (y ! x ∧ Pa(y))

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Pa(x)

∃xPa(x)

y (y ! x ∧ Pa(y))

x ∀y (y ! x → Pa(y))

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Pa(x)

∃xPa(x)

y (y ! x ∧ Pa(y))

x ∀y (y ! x → Pa(y))We still need restrictions

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Pa(x)

∃xPa(x)

y (y ! x ∧ Pa(y))

x ∀y (y ! x → Pa(y))

Φ has no Σ or Π

We still need restrictions

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Pa(x)

∃xPa(x)

y (y ! x ∧ Pa(y))

x ∀y (y ! x → Pa(y))

Φ has no Σ or Π

We still need restrictions

commutative

Thm: weighted automata = restricted wMSO

CHANGING THE SYNTAX

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

BEYOND SEMIRINGSAVERAGE, DISCOUNTED SUMS, …

product

average:

discounted:

FROM PRODUCTS TO VALUATIONSA run generates a sequence of weights

We compute the weight of the run

Val(s1s2 · · · sn) =s1+s2+···+sn

Val(s1s2 · · · sn) = s1 + λs2 + · · ·+ λn−1sn

Val(s1s2 · · · sn) = s1 × s2 × · · ·× sn

[Droste-Meinecke, CIAA’10, IC’12]

Val(s1s2 · · · sn)

wgt(ρ) = s1s2 · · · sn

product

average:

discounted:

FROM PRODUCTS TO VALUATIONSA run generates a sequence of weights

Val(s1s2 · · · sn) =s1+s2+···+sn

Val(s1s2 · · · sn) = s1 + λs2 + · · ·+ λn−1sn

Val(s1s2 · · · sn) = s1 × s2 × · · ·× sn

Val: possibly non-associative,

non distributive

Val(s1s2 · · · sn)

wgt(ρ) = s1s2 · · · sn

A run generates a sequence of weights

Final semantics [[A]](w) =!

ρ run on w

Val(wgt(ρ))

Valuation monoid (S,+, 0,Val)

Val(s1s2 · · · sn)

wgt(ρ) = s1s2 · · · sn

FROM PRODUCTS TO VALUATIONS

Val : S+ → S

Semantics in a valuation monoid

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

(S,+, 0,Val)

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

+, Σ : ok

(S,+, 0,Val)

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Φ]](w,σ) = Val(([[Φ]](w,σ[x !→ i]))i)

+, Σ : ok

(S,+, 0,Val)

Π : Val

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Φ]](w,σ) = Val(([[Φ]](w,σ[x !→ i]))i)

+, Σ : ok

(S,+, 0,Val)

Π : Val

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Φ]](w,σ) = Val(([[Φ]](w,σ[x !→ i]))i)

+, Σ : ok

(S,+, 0,Val)

Π : Val

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Φ]](w,σ) = Val(([[Φ]](w,σ[x !→ i]))i)

+, Σ : ok

(S,+, 0,Val)

Π : Val

Φ ::= s | ϕ | Φ+ Φ | Φ× Φ |!

x Φ |!

X Φ |"

x Φ]](w,σ) = Val(([[Φ]](w,σ[x !→ i]))i)

+, Σ : ok

(S,+, 0,Val)

Thm [Droste-Meinecke]

weighted automata = restricted wMSO

Π : Val

EXTENDING THE SUMA run generates a sequence of weights

[Droste-Perevoshchikov, CSR’13]

Final semantics

Val(s1s2 · · · sn)

wgt(ρ) = s1s2 · · · sn

[[A]](w) = Average{{Val(wgt(ρ)) | ρ run on w}}

Final semantics

Val(s1s2 · · · sn)

wgt(ρ) = s1s2 · · · sn

not associative

Final semantics

Valuation structure

Val(s1s2 · · · sn)

wgt(ρ) = s1s2 · · · sn

not associative

(S, F,Val) F : N⟨S⟩ → S

ABSTRACT SEMANTICSMULTISETS OF WEIGHT-STRUCTURES

MULTISETS OF WEIGHT STRUCTURES

[Gastin-Monmege ’14][Droste-Perevoshchikov ’14]

Abstract semantics

wgt(ρ) = s1s2 · · · sn

{|A|}(w) = {{wgt(ρ) | ρ run on w}}

multiset

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

aggrsp(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

aggrsp(A) =!"

r1···rn∈A r1 × · · ·× rn

Valuation monoid: sum-valuation aggrsv(A) =

Val(A) =!

r1···rn∈A Val(r1 · · · rn)

Valuation structure: F-valuation aggrfv(A) = F ◦ Val(A) = F ({{Val(r1 · · · rn) | r1 · · · rn ∈ A}})

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

THE FINAL TOUCHCHANGING AGAIN THE SYNTAX

CHANGING AGAIN THE SYNTAX

[Gastin-Monmege, 14]

Boolean fragment

Step formulae Ψ ::= s | ϕ ?Ψ : Ψ

Boolean fragment

if … then … else … no sum, no product

Boolean fragment

Pa(x) ? 1 : 0

Boolean fragment

Pa(x) ? 1 : 0

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

Boolean fragment

Pa(x) ? 1 : 0

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

ϕ1(x) ? s1 : (ϕ2(x) ? s2 : · · ·(ϕn−1(x) ? sn−1 : sn) · · · )

Boolean fragment

Pa(x) ? 1 : 0

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

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

ϕ1(x) ? s1 : (ϕ2(x) ? s2 : · · ·(ϕn−1(x) ? sn−1 : sn) · · · )

Pa(x) ? 1 : (Pb(x) ?−1 : 0)

CHANGING AGAIN THE SYNTAXBoolean fragment

Pa(x) ? 1 : 0

[[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

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

x Φ |!

X Φ |"

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

x Φ |!

X Φ |"

no constants

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

x Φ |!

X Φ |"

if … then … else … No product

no constants

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

x Φ |!

X Φ |"

Assigns a value from Ψ to each position

no constants

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

x Φ |!

X Φ |"

Assigns a value from Ψ to each position

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

no constants

[Gastin-Monmege, 14] singleton multiset

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

x Φ |!

X Φ |"

Semantics

empty multiset

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

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

x Φ |!

X Φ |"

Semantics

sums over multisetempty multiset

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

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

Ψ ::= s | ϕ ?Ψ : Ψ

Φ ::= 0 | ϕ ?Φ : Φ | Φ+ Φ |!

x Φ |!

X Φ |"

Thm: 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

CONCLUDING REMARKSSimilar approaches

M-expressions for multi-operator monoids [Fülöp-Stüber-Vogler’12]

Assignment logic [Perevoshchikov PhD’15]

Other structures: ranked trees, unranked trees, …

multisets of weight-structures

core-wMSO: only the boolean fragment changes

Other structures: ranked trees, unranked trees, …

multisets of weight-structures

core-wMSO: only the boolean fragment changes

More operators in core-wMSO

any binary operation ♢ : S x S → S can be lifted to multisets of

weight-structures and added to core-wMSO

10 YEARS OF WEIGHTED LOGICS FOR WEIGHTED ...gastin/Talks/AutoMathA.pdf10 YEARS OF WEIGHTED LOGICS...

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