Decomposition of Weighted Timed Automata Vitaly Perevo shchi kov - - PowerPoint PPT Presentation

Decomposition of Weighted Timed Automata

Vitaly Perevo shchi

• ≈she

kov (joint work with Manfred Droste)

Leipzig University, QuantLA

FFM 2015

Timed automata1

1 2

a, x ≥ 2, y ← 0 b, x ≤ 4 y ≥ 1 , x ← 0

A Nondeterministic finite automata with clocks Edges: ❵

a❀ ✣❀ Λ

• → ❵′ ∶

a ∈ Σ is a letter ✣ is a clock constraint Λ is a set of clocks to be reset.

1Alur, Dill ’92

Timed automata1

1 2

a, x ≥ 2, y ← 0 b, x ≤ 4 y ≥ 1 , x ← 0

A Nondeterministic finite automata with clocks Edges: ❵

a❀ ✣❀ Λ

• → ❵′ ∶

a ∈ Σ is a letter ✣ is a clock constraint Λ is a set of clocks to be reset.

Run ✚ ( 1 ❀ x=0

y=0) 2✿1

• →

delay ( 1 ❀ x=2✿1 y=2✿1) a

• →

switch ( 2 ❀ x=2✿1 y=0 ) 1✿1

• →

delay ( 2 ❀ x=3✿2 y=1✿1) b

• →

switch ( 1 ❀ x=0 y=1✿1)

Label(✚) ∶= (a❀2✿1)(b❀1✿1) ∈ (Σ × R≥0)+ =∶ TΣ+ is a timed word

1Alur, Dill ’92

Weighted Timed Automata (WTA)12

1 2

a, x ≥ 2, y ← 0 3 b, x ≤ 4 y ≥ 1 , x ← 0 2

A

5 6

Both edges and locations carry weights (costs):

discrete : costs of edges (for switches) continuous : cost rates of locations (for delays)

Run ✚ ( ❵0 ❀✗0)

t1

• →

e1

• → ( ❵1 ❀✗1)

t2

• →

e2

• → ✿✿✿

tn

• →

en

• → ( ❵n ❀✗n)

weight(✚) = ( wt(❵0) ⋅t1 + wt(e1) )+✿✿✿+( wt(❵n−1) ⋅tn + wt(en) ) Behavior: quantitative timed language: ∣∣A∣∣ ∶ TΣ+ → R ∪ {∞}: ∣∣A∣∣(w) = min{weight(✚) ∣ ✚ is a run with label w}

1Alur, La Torre, Pappas ’01 2Larsen, Behrmann, Brinksma, Fehnker, Hune, Pettersson, Romijn ’01

Average behavior1

1 2

a, x ≥ 2, y ← 0 3 b, x ≤ 4 y ≥ 1 , x ← 0 2

A

5 6

Both edges and locations carry weights (costs):

discrete : costs of edges (for switches) continuous : cost rates of locations (for delays)

Run ✚ ( ❵0 ❀✗0)

t1

• →

e1

• → ( ❵1 ❀✗1)

t2

• →

e2

• → ✿✿✿

tn

• →

en

• → ( ❵n ❀✗n)

weight(✚) = ( wt(❵0) ⋅ t1 + wt(e1) ) + ✿✿✿ + ( wt(❵n−1) ⋅ tn + wt(en) ) t1 + ✿✿✿ + tn

1Bouyer, Brinksma, Larsen ’04

Timed valuation monoids

A timed extension of valuation monoids of Droste and Meinecke. Definition A timed valuation monoid (M❀⊕❀val❀0): (M❀⊕❀0) is a commutative monoid; val ∶ T(M × M)+ → M is a timed valuation function

Timed valuation monoids

A timed extension of valuation monoids of Droste and Meinecke. Definition A timed valuation monoid (M❀⊕❀val❀0): (M❀⊕❀0) is a commutative monoid; val ∶ T(M × M)+ → M is a timed valuation function Runs in WTA: ✚ ∶ (❵0❀✗0)

t1

• →

e1

• → (❵1❀✗1)

t2

• →

e2

• → ✿✿✿

tn

• →

en

• → (❵n❀✗n)

Weight of ✚: val[ ⟨(wt(❵0)❀wt(e1))❀t1⟩ ❀✿✿✿❀ ⟨(wt(❵n−1)❀wt(en))❀tn⟩ ]

Timed valuation monoids

A timed extension of valuation monoids of Droste and Meinecke. Definition A timed valuation monoid (M❀⊕❀val❀0): (M❀⊕❀0) is a commutative monoid; val ∶ T(M × M)+ → M is a timed valuation function Runs in WTA: ✚ ∶ (❵0❀✗0)

t1

• →

e1

• → (❵1❀✗1)

t2

• →

e2

• → ✿✿✿

tn

• →

en

• → (❵n❀✗n)

Weight of ✚: val[ ⟨(wt(❵0)❀wt(e1))❀t1⟩ ❀✿✿✿❀ ⟨(wt(❵n−1)❀wt(en))❀tn⟩ ] The behavior of A: ∣∣A∣∣ ∶ TΣ+ → M w ↦ ⊕(weight(✚) ∣ ✚ is a run on w)

Unambiguous and deterministic TA

A timed automaton A = (L❀C❀I❀E❀F) over an alphabet Σ is: unambiguous if for each w ∈ TΣ+ there exists at most one accepting run. deterministic if for all e1 = ( ❵❀a ❀✣1❀Λ1❀❵1) ∈ E and e2 = ( ❵❀a ❀✣2❀Λ2❀❵2) ∈ E with e1 ≠ e2: ✣1 ∧ ✣2 is unsatisfiable.

Unambiguous and deterministic TA

A timed automaton A = (L❀C❀I❀E❀F) over an alphabet Σ is: unambiguous if for each w ∈ TΣ+ there exists at most one accepting run. deterministic if for all e1 = ( ❵❀a ❀✣1❀Λ1❀❵1) ∈ E and e2 = ( ❵❀a ❀✣2❀Λ2❀❵2) ∈ E with e1 ≠ e2: ✣1 ∧ ✣2 is unsatisfiable.

Operations for quantitative timed languages (QTL)

Let M = (M❀⊕❀val❀0) be a timed valuation monoid with val ∶ T(M × M)+ → M and Σ, Γ alphabets. Let h ∶ Γ → Σ be a renaming, v = (✌1❀t1)✿✿✿(✌n❀tn) ∈ TΓ+ and h(v) = (h(✌1)❀t1)✿✿✿(h(✌n)❀tn)

Operations for quantitative timed languages (QTL)

Let M = (M❀⊕❀val❀0) be a timed valuation monoid with val ∶ T(M × M)+ → M and Σ, Γ alphabets. Let h ∶ Γ → Σ be a renaming and r ∶ TΓ+ → M. Let h(r) ∶ TΣ+ → M w ↦ ⊕(r(v) ∣ v ∈ TΓ+ and h(v) = w)✿

Operations for quantitative timed languages (QTL)

Let M = (M❀⊕❀val❀0) be a timed valuation monoid with val ∶ T(M × M)+ → M and Σ, Γ alphabets. Let g ∶ Γ → M × M be a renaming and val○g ∶ TΓ+ → M v ↦ val(g(v))

Operations for quantitative timed languages (QTL)

Let M = (M❀⊕❀val❀0) be a timed valuation monoid with val ∶ T(M × M)+ → M and Σ, Γ alphabets. Let r ∶ TΓ+ → M and L ⊆ TΓ+. Let (r ∩ L) ∶ TΓ+ → M v ↦ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r(v)❀ if v ∈ L❀ 0❀

• therwise

A Nivat Decomposition Theorem for WTA

Let Σ be an alphabet and M = (M❀⊕❀val❀0) a timed valuation monoid. Let N(Σ❀M) be the class of QTL L ∶ TΣ+ → M with L = h((val○g) ∩ L ) where h ∶ Γ → Σ, g ∶ Γ → M × M are renamings, Γ an alphabet and L ⊆ TΓ+ is recognizable. N Det(Σ❀M) ⊆ N Unamb(Σ❀M) ⊆ N(Σ❀M). ❀

A Nivat Decomposition Theorem for WTA

Let Σ be an alphabet and M = (M❀⊕❀val❀0) a timed valuation monoid. Let N(Σ❀M) be the class of QTL L ∶ TΣ+ → M with L = h((val○g) ∩ L ) where h ∶ Γ → Σ, g ∶ Γ → M × M are renamings, Γ an alphabet and L ⊆ TΓ+ is recognizable. N Det(Σ❀M) ⊆ N Unamb(Σ❀M) ⊆ N(Σ❀M). Let Rec(Σ❀M) be the class of recognizable QTL.

A Nivat Decomposition Theorem for WTA

Let Σ be an alphabet and M = (M❀⊕❀val❀0) a timed valuation monoid. Let N(Σ❀M) be the class of QTL L ∶ TΣ+ → M with L = h((val○g) ∩ L ) where h ∶ Γ → Σ, g ∶ Γ → M × M are renamings, Γ an alphabet and L ⊆ TΓ+ is recognizable. N Det(Σ❀M) ⊆ N Unamb(Σ❀M) ⊆ N(Σ❀M). Let Rec(Σ❀M) be the class of recognizable QTL.

A Nivat Decomposition Theorem for WTA

Let Σ be an alphabet and M = (M❀⊕❀val❀0) a timed valuation monoid. Let N(Σ❀M) be the class of QTL L ∶ TΣ+ → M with L = h((val○g) ∩ L ) where h ∶ Γ → Σ, g ∶ Γ → M × M are renamings, Γ an alphabet and L ⊆ TΓ+ is recognizable. N Det(Σ❀M) ⊆ N Unamb(Σ❀M) ⊆ N(Σ❀M). Let Rec(Σ❀M) be the class of recognizable QTL. Theorem

1 Rec(Σ❀M) = N Det(Σ❀M) = N Unamb(Σ❀M). 2 If ⊕ is idempotent, then Rec(Σ❀M) = N(Σ❀M). 3 There exist an alphabet Σ0 and a timed valuation monoid M0

with Rec(Σ0❀M0) ≠ N(Σ0❀M0).

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

3 Multi-weighted timed setting (e.g., the reward-cost ratio)

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

3 Multi-weighted timed setting (e.g., the reward-cost ratio)

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

3 Multi-weighted timed setting (e.g., the reward-cost ratio)

Future work: Extension to infinite timed words, timed trees, etc. Decomposition of weighted register automata.

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

3 Multi-weighted timed setting (e.g., the reward-cost ratio)

Future work: Extension to infinite timed words, timed trees, etc. Decomposition of weighted register automata. Decomposition of weighted timed pushdown automata.

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

3 Multi-weighted timed setting (e.g., the reward-cost ratio)

Future work: Extension to infinite timed words, timed trees, etc. Decomposition of weighted register automata. Decomposition of weighted timed pushdown automata.

Conclusion

1 Decomposition of WTA. 2 Logical characterization of WTA (based on relative distance

logic of Wilke)

3 Multi-weighted timed setting (e.g., the reward-cost ratio)

Future work: Extension to infinite timed words, timed trees, etc. Decomposition of weighted register automata. Decomposition of weighted timed pushdown automata. THANK YOU!

A Nivat Theorem for WTA: Proof Idea

Lemma. N Unamb(Σ❀M) ⊆ Rec(Σ❀M).

A Nivat Theorem for WTA: Proof Idea

Lemma. N Unamb(Σ❀M) ⊆ Rec(Σ❀M). Proof idea. Let L ∶ TΣ+ → M with L = h((val○g) ∩ L) where g ∶ Γ → M × M, h ∶ Γ → Σ and L ⊆ TΓ+ is unambiguously recognizable.

A Nivat Theorem for WTA: Proof Idea

Lemma. N Unamb(Σ❀M) ⊆ Rec(Σ❀M). Proof idea. Let L ∶ TΣ+ → M with L = h((val○g) ∩ L) where g ∶ Γ → M × M, h ∶ Γ → Σ and L ⊆ TΓ+ is unambiguously recognizable. Closure properties for recognizable QTL: val○g ∶ TΓ+ → M is recognizable.

A Nivat Theorem for WTA: Proof Idea

Lemma. N Unamb(Σ❀M) ⊆ Rec(Σ❀M). Proof idea. Let L ∶ TΣ+ → M with L = h((val○g) ∩ L) where g ∶ Γ → M × M, h ∶ Γ → Σ and L ⊆ TΓ+ is unambiguously recognizable. Closure properties for recognizable QTL: val○g ∶ TΓ+ → M is recognizable. If r ∶ TΓ+ → M is recognizable, then r ∩ L is recognizable.

A Nivat Theorem for WTA: Proof Idea

Lemma. N Unamb(Σ❀M) ⊆ Rec(Σ❀M). Proof idea. Let L ∶ TΣ+ → M with L = h((val○g) ∩ L) where g ∶ Γ → M × M, h ∶ Γ → Σ and L ⊆ TΓ+ is unambiguously recognizable. Closure properties for recognizable QTL: val○g ∶ TΓ+ → M is recognizable. If r ∶ TΓ+ → M is recognizable, then r ∩ L is recognizable. If r ∶ TΓ+ → M is recognizable, then h(r) is recognizable.

A Nivat Theorem for WTA: Proof Idea

Lemma. N Unamb(Σ❀M) ⊆ Rec(Σ❀M). Proof idea. Let L ∶ TΣ+ → M with L = h((val○g) ∩ L) where g ∶ Γ → M × M, h ∶ Γ → Σ and L ⊆ TΓ+ is unambiguously recognizable. Closure properties for recognizable QTL: val○g ∶ TΓ+ → M is recognizable. If r ∶ TΓ+ → M is recognizable, then r ∩ L is recognizable. If r ∶ TΓ+ → M is recognizable, then h(r) is recognizable.

A Nivat Theorem for WTA: a Counterexample

Let M = (N❀+❀val❀0) with val[((m1❀m′

1)❀t1)❀✿✿✿❀((mn❀m′ n)❀tn)] = m′ 1 ⋅ ✿✿✿ ⋅ m′ n

Let Σ = {a}. Then, there exists a timed language L ⊆ TΣ+ which is recognizable but not unambiguously recognizable1. ❀ ❀

1Wilke ’94

A Nivat Theorem for WTA: a Counterexample

Let M = (N❀+❀val❀0) with val[((m1❀m′

1)❀t1)❀✿✿✿❀((mn❀m′ n)❀tn)] = m′ 1 ⋅ ✿✿✿ ⋅ m′ n

Let Σ = {a}. Then, there exists a timed language L ⊆ TΣ+ which is recognizable but not unambiguously recognizable1. Let L ∶ TΣ+ → N with L = char(L). ❀ ❀

1Wilke ’94

A Nivat Theorem for WTA: a Counterexample

Let M = (N❀+❀val❀0) with val[((m1❀m′

1)❀t1)❀✿✿✿❀((mn❀m′ n)❀tn)] = m′ 1 ⋅ ✿✿✿ ⋅ m′ n

Let Σ = {a}. Then, there exists a timed language L ⊆ TΣ+ which is recognizable but not unambiguously recognizable1. Let L ∶ TΣ+ → N with L = char(L). Let h ∶ Σ → Σ be the identity mapping, g ∶ Σ → N × N with g(a) = (1❀1). Then, L = h((val○g) ∩ L) ∈ N(Σ❀M)

1Wilke ’94

A Nivat Theorem for WTA: a Counterexample

Let M = (N❀+❀val❀0) with val[((m1❀m′

1)❀t1)❀✿✿✿❀((mn❀m′ n)❀tn)] = m′ 1 ⋅ ✿✿✿ ⋅ m′ n

Let Σ = {a}. Then, there exists a timed language L ⊆ TΣ+ which is recognizable but not unambiguously recognizable1. Let L ∶ TΣ+ → N with L = char(L). Let h ∶ Σ → Σ be the identity mapping, g ∶ Σ → N × N with g(a) = (1❀1). Then, L = h((val○g) ∩ L) ∈ N(Σ❀M) L is not recognizable, since: L is recognizable ⇔ L is unambiguously recognizable

1Wilke ’94

A Nivat Theorem for WTA: a Counterexample

Let M = (N❀+❀val❀0) with val[((m1❀m′

1)❀t1)❀✿✿✿❀((mn❀m′ n)❀tn)] = m′ 1 ⋅ ✿✿✿ ⋅ m′ n

Let Σ = {a}. Then, there exists a timed language L ⊆ TΣ+ which is recognizable but not unambiguously recognizable1. Let L ∶ TΣ+ → N with L = char(L). Let h ∶ Σ → Σ be the identity mapping, g ∶ Σ → N × N with g(a) = (1❀1). Then, L = h((val○g) ∩ L) ∈ N(Σ❀M) L is not recognizable, since: L is recognizable ⇔ L is unambiguously recognizable

1Wilke ’94