On -Regular Trace Languages Classification and Synthesis Namit - - PowerPoint PPT Presentation

On ω-Regular Trace Languages

Classification and Synthesis Namit Chaturvedi and Marcus Gelderie GAMES 2012 September 09, 2012

On ω-Regular Trace Languages 1 / 18

Traces and trace languages

Traces as models of behaviors of distributed systems Traces model dependence/independence between actions Independent actions mean semantic equivalence of swapping in a sequence of actions

On ω-Regular Trace Languages 2 / 18

Traces and trace languages

Traces as models of behaviors of distributed systems Traces model dependence/independence between actions Independent actions mean semantic equivalence of swapping in a sequence of actions Trace languages correspond closely with word languages A single trace represents an equivalence class of such partially commutative sequences A language of traces is recognizable if and only if the corresponding “trace-closed” word language is recognizable

On ω-Regular Trace Languages 2 / 18

End of similarities

Borel classification of ω-regular languages

Open problem

Can ω-regular trace languages also be classified in such a manner?

ω-regular

• det. Büchi recognizable
• det. co-Büchi recognizable

weakly recognizable reachability safety

On ω-Regular Trace Languages 3 / 18

Motivation

Why Borel classification of ω-regular languages?

On ω-Regular Trace Languages 4 / 18

Motivation

Why Borel classification of ω-regular languages?

1 Classification is good 2 Natural descriptions of ω-regular languages with the help of regular

languages (using reachability, safety, and liveness conditions)

3 Easy construction of ω-automata from DFAs 4 Efficient algorithms for synthesis and verification for different classes 5 Effective classification procedure given an arbitrary ω-automaton On ω-Regular Trace Languages 4 / 18

Motivation

Why Borel classification of ω-regular languages?

1 Classification is good 2 Natural descriptions of ω-regular languages with the help of regular

languages (using reachability, safety, and liveness conditions)

3 Easy construction of ω-automata from DFAs 4 Efficient algorithms for synthesis and verification for different classes 5 Effective classification procedure given an arbitrary ω-automaton

Rec(M(Σ∗, I)) languages REG trace- closed languages Rec(R(Σω, I)) languages ω-REG trace-closed languages Distributed controllers Distributable* sequential controllers

On ω-Regular Trace Languages 4 / 18

Motivation

Why Borel classification of ω-regular languages?

1 Classification is good 2 Natural descriptions of ω-regular languages with the help of regular

languages (using reachability, safety, and liveness conditions)

3 Easy construction of ω-automata from DFAs 4 Efficient algorithms for synthesis and verification for different classes 5 Effective classification procedure given an arbitrary ω-automaton

Rec(M(Σ∗, I)) languages REG trace- closed languages Rec(R(Σω, I)) languages ω-REG trace-closed languages Distributed controllers Distributable* sequential controllers Borel classification Distributed* Church’s Syn- thesis Problem

On ω-Regular Trace Languages 4 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d · · · · · ·

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d · · · · · · Assuming aIb and bIc a b c d b c

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d · · · · · · Assuming aIb and bIc a b c d b c t1 ⊙ t2

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d · · · · · · Assuming aIb and bIc a b c d b c t1 ⊙ t2 a c d b b

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d · · · · · · Assuming aIb and bIc a b c d b c t1 ⊙ t2 a c d b b a b b c d

• t1 ⊔ t2

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces

Let I ⊆ Σ2 be a symmetric, irreflexive independence relation, then (Σ, I) is called a dependence alphabet.

Definition (Finite and infinite traces)

A trace over (Σ, I) is a labeled DAG: a b b c d · · · · · · The set of all finite traces: M(Σ∗, I) The set of all infinite traces: R(Σω, I) Assuming aIb and bIc a b c d b c t1 ⊙ t2 a c d b b a b b c d

• t1 ⊔ t2

On ω-Regular Trace Languages 5 / 18

Preliminaries: Traces, words, languages

There is a tight connection between traces and words Mapping words to traces, Γ: Σ∗ → M(Σ∗, I)

Γ(abcdb) = a b b c d

Mapping traces to trace-closed sets of words

Γ−1 = {bacdb, abcdb, acbdb} a b b c d

On ω-Regular Trace Languages 6 / 18

Preliminaries: Traces, words, languages

There is a tight connection between traces and words Mapping words to traces, Γ: Σ∗ → M(Σ∗, I)

Γ(abcdb) = a b b c d

Mapping traces to trace-closed sets of words

Γ−1 = {bacdb, abcdb, acbdb} a b b c d

Words u and v are trace equivalent if Γ(u) = Γ(v) A finite or infinite language L is trace closed if L = Γ−1(Γ(L)) or simply L = [L]∼I

On ω-Regular Trace Languages 6 / 18

Preliminaries: Traces, words, languages

There is a tight connection between traces and words Mapping words to traces, Γ: Σ∗ → M(Σ∗, I)

Γ(abcdb) = a b b c d

Mapping traces to trace-closed sets of words

Γ−1 = {bacdb, abcdb, acbdb} a b b c d

Words u and v are trace equivalent if Γ(u) = Γ(v) A finite or infinite language L is trace closed if L = Γ−1(Γ(L)) or simply L = [L]∼I

Definition

A trace language T ⊆ M(Σ∗, I) (resp. Θ ⊆ R(Σω, I)) is recognizable iff Γ−1(T) (resp. Γ−1(Θ)) is a recognizable word language.

On ω-Regular Trace Languages 6 / 18

Preliminaries: A final note on automata

Trace-closed languages are recognized by I-diamond automata Let Σ = {a, b}, aIb. Define K := [(aa)+(bb)+]∼I.

1 2 3 4 5 6 7 8 a b a b b a a b b a b a a b b a a b

On ω-Regular Trace Languages 7 / 18

Infinitary extensions of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)). Its infinitary extension is the ω-trace language given by ext(T) := T ⊙ R(Σω, I) =

t∈T t ⊙ R(Σω, I).

On ω-Regular Trace Languages 8 / 18

Infinitary extensions of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)). Its infinitary extension is the ω-trace language given by ext(T) := T ⊙ R(Σω, I) =

t∈T t ⊙ R(Σω, I).

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K).

On ω-Regular Trace Languages 8 / 18

Infinitary extensions of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)). Its infinitary extension is the ω-trace language given by ext(T) := T ⊙ R(Σω, I) =

t∈T t ⊙ R(Σω, I).

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). ext(K) ? = Γ−1(ext(T)), or equivalently, ext(K) ? = [ext(K)]∼I

On ω-Regular Trace Languages 8 / 18

Infinitary extensions of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)). Its infinitary extension is the ω-trace language given by ext(T) := T ⊙ R(Σω, I) =

t∈T t ⊙ R(Σω, I).

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). ext(K) = Γ−1(ext(T)), or equivalently, ext(K) = [ext(K)]∼I For trace-closed K ∈ REG, it may hold that ext(K) = [ext(K)]∼I.

On ω-Regular Trace Languages 8 / 18

Infinitary extensions of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)). Its infinitary extension is the ω-trace language given by ext(T) := T ⊙ R(Σω, I) =

t∈T t ⊙ R(Σω, I).

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). ext(K) = Γ−1(ext(T)), or equivalently, ext(K) = [ext(K)]∼I

abaabbaabb . . . / ∈ ext(K)

For trace-closed K ∈ REG, it may hold that ext(K) = [ext(K)]∼I.

On ω-Regular Trace Languages 8 / 18

Infinitary extensions of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)). Its infinitary extension is the ω-trace language given by ext(T) := T ⊙ R(Σω, I) =

t∈T t ⊙ R(Σω, I).

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). ext(K) = Γ−1(ext(T)), or equivalently, ext(K) = [ext(K)]∼I

abaabbaabb . . . / ∈ ext(K) Γ(ab(aabb)ω) = a b a b a b · · · · · · ∈ ext(T)

For trace-closed K ∈ REG, it may hold that ext(K) = [ext(K)]∼I.

On ω-Regular Trace Languages 8 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a a b b

K = [(aa)+(bb)+]∼I with aIb

On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a a b b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b} On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a a b b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b}

Kb−1b

On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a a b b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b}

Kb−1b K ∪ Kb−1b · I∗

b

On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a a b b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b}

Kb−1b K ∪ Kb−1b · I∗

b

K{b} = K ∪ [Kb−1b · I∗

b]∼I

On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a, b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b}

Kb−1b K ∪ Kb−1b · I∗

b

K{b} = K ∪ [Kb−1b · I∗

b]∼I

K{a,b} = K{b} ∪ [Ka−1a · I∗

a]∼I

On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a, b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b}

Kb−1b K ∪ Kb−1b · I∗

b

K{b} = K ∪ [Kb−1b · I∗

b]∼I

K{a,b} = K{b} ∪ [Ka−1a · I∗

a]∼I

KI = KΣ

On ω-Regular Trace Languages 9 / 18

Ensuring ext(K) is trace-closed

Definition (I-suffix extension)

For K ∈ REG, trace-closed, define KI := K ∪

a∈Σ[Ka−1a · I∗ a]∼I

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a, b

K = [(aa)+(bb)+]∼I with aIb

◮ Ib = {a}, Ia = {b}

Kb−1b K ∪ Kb−1b · I∗

b

K{b} = K ∪ [Kb−1b · I∗

b]∼I

K{a,b} = K{b} ∪ [Ka−1a · I∗

a]∼I

KI = KΣ

On ω-Regular Trace Languages 9 / 18

I-suffix extension suffices

Lemma

For any trace-closed K ∈ REG, it holds that ext(KI) = [ext(K)]∼I

On ω-Regular Trace Languages 10 / 18

I-suffix extension suffices

Lemma

For any trace-closed K ∈ REG, it holds that ext(KI) = [ext(K)]∼I For every T ∈ Rec(M(Σ∗, I)), the trace-closed language Γ−1(ext(T)) is I-diamond DWA recognizable.

On ω-Regular Trace Languages 10 / 18

I-suffix extension suffices

Lemma

For any trace-closed K ∈ REG, it holds that ext(KI) = [ext(K)]∼I For every T ∈ Rec(M(Σ∗, I)), the trace-closed language Γ−1(ext(T)) is I-diamond DWA recognizable.

Theorem (I-diamond DWA recognizability)

A trace-closed language L ⊆ Σω is recognized by an I-diamond DWA iff L ∈ BC(ext(K)) for a set K ⊆ REG of trace-closed regular languages.

On ω-Regular Trace Languages 10 / 18

I-suffix extension suffices

Lemma

For any trace-closed K ∈ REG, it holds that ext(KI) = [ext(K)]∼I For every T ∈ Rec(M(Σ∗, I)), the trace-closed language Γ−1(ext(T)) is I-diamond DWA recognizable.

Theorem (I-diamond DWA recognizability)

A trace-closed language L ⊆ Σω is recognized by an I-diamond DWA iff L ∈ BC(ext(K)) for a set K ⊆ REG of trace-closed regular languages.

Definition (Weakly recognizable trace languages)

A trace language Θ ⊆ R(Σω, I) is weakly recognizable if Γ−1(Θ) is I-diamond DWA recognizable.

On ω-Regular Trace Languages 10 / 18

Infinitary limits of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)), the infinitary limit lim(T) is the ω-trace language containing all θ ∈ R(Σω, I) such that there exists a sequence (ti)i∈◆, ti ∈ T satisfying ti ⊏ ti+1 and

i∈◆ ti = θ.

On ω-Regular Trace Languages 11 / 18

Infinitary limits of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)), the infinitary limit lim(T) is the ω-trace language containing all θ ∈ R(Σω, I) such that there exists a sequence (ti)i∈◆, ti ∈ T satisfying ti ⊏ ti+1 and

i∈◆ ti = θ.

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). lim(K) = Γ−1(lim(T))

On ω-Regular Trace Languages 11 / 18

Infinitary limits of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)), the infinitary limit lim(T) is the ω-trace language containing all θ ∈ R(Σω, I) such that there exists a sequence (ti)i∈◆, ti ∈ T satisfying ti ⊏ ti+1 and

i∈◆ ti = θ.

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). lim(K) = Γ−1(lim(T)) [lim(K)]∼I is not I-diamond DBA recognizable

On ω-Regular Trace Languages 11 / 18

Infinitary limits of regular trace languages

Definition

Let T ∈ Rec(M(Σ∗, I)), the infinitary limit lim(T) is the ω-trace language containing all θ ∈ R(Σω, I) such that there exists a sequence (ti)i∈◆, ti ∈ T satisfying ti ⊏ ti+1 and

i∈◆ ti = θ.

Let Σ = {a, b}, aIb, K = [(aa)+(bb)+]∼I, and T = Γ(K). lim(K) = Γ−1(lim(T)) [lim(K)]∼I is not I-diamond DBA recognizable

Definition (Limit-stability)

A trace-closed language K ∈ REG is limit-stable if lim(K) = [lim(K)]∼I. By extension, we say that T ⊆ M(Σ∗, I) is limit-stable if Γ−1(T) is.

On ω-Regular Trace Languages 11 / 18

Searching for limit-stable languages

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a b

K = [(aa)+(bb)+]∼I with aIb

On ω-Regular Trace Languages 12 / 18

Searching for limit-stable languages

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a b

K = [(aa)+(bb)+]∼I with aIb Γ(ab(aabb)ω) = a b a b a b · · · · · ·

On ω-Regular Trace Languages 12 / 18

Searching for limit-stable languages

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a b

K = [(aa)+(bb)+]∼I with aIb Γ(ab(aabb)ω) = a b a b a b · · · · · · Γ(ab(abba)ω) = a b a b a b · · · · · ·

On ω-Regular Trace Languages 12 / 18

Searching for limit-stable languages

Definition (F, I-cycle closure)

For a given (Σ, I), an I-diamond DFA is F, I-cycle closed if for all u ∼I v and all q ∈ Q, q u − →F q iff q v − →F q.

1 2 3 4 5 6 7 8 a b a b b a a b b a b a b a a b a b

K = [(aa)+(bb)+]∼I with aIb Γ(ab(aabb)ω) = a b a b a b · · · · · · Γ(ab(abba)ω) = a b a b a b · · · · · ·

On ω-Regular Trace Languages 12 / 18

F, I-cycle closure is necessary and sufficient

Theorem (I-diamond DBA recognizability)

A language K ∈ REG is limit-stable if and only if K is recognized by an F, I-cycle closed DFA.

On ω-Regular Trace Languages 13 / 18

F, I-cycle closure is necessary and sufficient

Theorem (I-diamond DBA recognizability)

A language K ∈ REG is limit-stable if and only if K is recognized by an F, I-cycle closed DFA. F, I-cycle closure is an efficiently decidable property An ω language recognized by a DBA is trace-closed if and only if the DBA is F, I-cycle closed

On ω-Regular Trace Languages 13 / 18

F, I-cycle closure is necessary and sufficient

Theorem (I-diamond DBA recognizability)

A language K ∈ REG is limit-stable if and only if K is recognized by an F, I-cycle closed DFA. F, I-cycle closure is an efficiently decidable property An ω language recognized by a DBA is trace-closed if and only if the DBA is F, I-cycle closed

Theorem (ω-regular trace languages)

A trace language Θ is ω-regular if and only if Γ−1(Θ) is a finite Boolean combination of I-diamond DBA recognizable trace-closed languages. That is, Rec(R(Σω, I)) = BC(lim(LS-M(Σ∗, I))).

On ω-Regular Trace Languages 13 / 18

Borel hierarchy of ω-regular trace languages

TLS KLS lim(TLS) lim(KLS) T K ext(T) KI ext(KI)

ω-regular (trace-closed) I-diamond DBA recognizable I-diamond co-DBA recognizable weakly recognizable (trace-closed) reachability safety

On ω-Regular Trace Languages 14 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t (a, (c, (a, c)

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t (a, ya) (c, yc) (a, c) ya, yc

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t (a, ya) (c, yc) (a, (b, (a, c) ya, yc (a, b)

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t (a, ya) (c, yc) (a, za) (b, yb) (a, c) ya, yc (a, b) za, yb

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t (a, ya) (c, yc) (a, za) (b, yb) (d, yd) (a, c) ya, yc (a, b) za, yb (d) yd

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Adversary’s choice of independence relation: I1 ⊆ Σ1 × Σ1 and infer I ⊆ Σ × Σ therefrom: (a, x)I(b, y) ⇔ aI1b A play of the distributed game over Σ ⊆ Σ1 × Σ2 is a trace θ ∈ R(Σω, I): θ = (a, xa) (b, xb) (c, xc) (d, xd) t (a, ya) (c, yc) (a, za) (b, yb) (d, yd) (a, c) ya, yc (a, b) za, yb (d) yd Player 1 moves (assuming Σ1 is ordered): CI := {(a1, . . . , ak) ⊆ 2Σ1 | a1 < · · · < ak and aiIaj, i = j} Player 2 moves: V := {x1, . . . , xk | xi ∈ Σ2, k ≤ |Σ1|}

On ω-Regular Trace Languages 15 / 18

Distributed Church’s Synthesis Problem

Game: an ω-trace language Θ ⊆ R(Σω, I) If θ ∈ Θ then Player 2 wins If θ / ∈ Θ then Player 1 wins

On ω-Regular Trace Languages 16 / 18

Distributed Church’s Synthesis Problem

Game: an ω-trace language Θ ⊆ R(Σω, I) If θ ∈ Θ then Player 2 wins If θ / ∈ Θ then Player 1 wins Given finite play prefixes, strategies guide players: Strategy for Player 1: λ1 : M(Σ∗, I) → CI Strategy for Player 2: λ2 : M(Σ∗, I) → VCI

On ω-Regular Trace Languages 16 / 18

Distributed Church’s Synthesis Problem

Game: an ω-trace language Θ ⊆ R(Σω, I) If θ ∈ Θ then Player 2 wins If θ / ∈ Θ then Player 1 wins Given finite play prefixes, strategies guide players: Strategy for Player 1: λ1 : M(Σ∗, I) → CI Strategy for Player 2: λ2 : M(Σ∗, I) → VCI To solve game Θ ∈ Rec(R(Σω, I)), solve game Γ−1(Θ).

On ω-Regular Trace Languages 16 / 18

Distributed Church’s Synthesis Problem

Game: an ω-trace language Θ ⊆ R(Σω, I) If θ ∈ Θ then Player 2 wins If θ / ∈ Θ then Player 1 wins Given finite play prefixes, strategies guide players: Strategy for Player 1: λ1 : M(Σ∗, I) → CI Strategy for Player 2: λ2 : M(Σ∗, I) → VCI To solve game Θ ∈ Rec(R(Σω, I)), solve game Γ−1(Θ). A sequential strategy of Player 2 is a transition system where each state q ∈ Q has exactly one outgoing transition (a, xa) ∈ Σ for each a ∈ Σ1.

On ω-Regular Trace Languages 16 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

(b, yb)

(b) yb

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

(b, yb)

(b) yb

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

(b, yb)

(b) yb

(d, yd) (d) yd

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

(b, yb)

(b) yb

(d, yd) (d) yd

(b, xb)(c, xc)(a, xa)(d, xd)(c, yc)(a, ya)

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

(b, yb)

(b) yb

(d, yd) (d) yd

(b, xb)(c, xc)(a, xa)(d, xd)(c, yc)(a, ya)

Theorem (Ştefănescu-Esparza-Muscholl, Mukund)

There exists an distributed strategy for Player 2 iff there exists a safe-branching* sequential strategy for Player 2.

On ω-Regular Trace Languages 17 / 18

Distributable sequential strategy for Player 2

θ = (a, xa) (b, xb) (c, xc) (d, xd) (a, ya) (c, yc)

(a, xa)(b, xb)(c, xc)(d, xd)(a, ya)(c, yc)

(a, za)

(a) za

(b, yb)

(b) yb

(d, yd) (d) yd

(b, xb)(c, xc)(a, xa)(d, xd)(c, yc)(a, ya)

Theorem (Ştefănescu-Esparza-Muscholl, Mukund)

There exists an distributed strategy for Player 2 iff there exists a safe-branching* sequential strategy for Player 2.

Ongoing work

Complete procedures for constructing safe-branching sequential strategies for various classes of trace-closed games.

On ω-Regular Trace Languages 17 / 18

Summary

Rec(M(Σ∗, I)) languages REG trace- closed languages Rec(R(Σω, I)) languages ω-REG trace-closed languages Distributed controllers Safe-branching sequential controllers

On ω-Regular Trace Languages 18 / 18

Summary

Rec(M(Σ∗, I)) languages REG trace- closed languages Rec(R(Σω, I)) languages ω-REG trace-closed languages Distributed controllers Safe-branching sequential controllers Borel classification

On ω-Regular Trace Languages 18 / 18

Summary

Rec(M(Σ∗, I)) languages REG trace- closed languages Rec(R(Σω, I)) languages ω-REG trace-closed languages Distributed controllers Safe-branching sequential controllers Borel classification Distributed Church’s Syn- thesis Problem

On ω-Regular Trace Languages 18 / 18