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 ω -regular Borel classification of ω -regular languages Open problem det. co-Büchi recognizable det. Büchi recognizable Can ω -regular trace weakly recognizable languages also be classified in such a manner? 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 )) Rec ( R (Σ ω , I )) Distributed languages languages controllers REG trace- ω -REG Distributable * closed trace-closed sequential languages languages 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 )) Rec ( R (Σ ω , I )) Distributed languages languages controllers Distributed * Borel classification Church’s Syn- thesis Problem REG trace- ω -REG Distributable * closed trace-closed sequential languages languages controllers On ω -Regular Trace Languages 4 / 18
Preliminaries: Traces Let I ⊆ Σ 2 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 ⊆ Σ 2 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 c d b b On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 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 c d b b On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 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 c d · · · b b On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 be a symmetric, irreflexive independence relation , then (Σ , I ) is called a dependence alphabet . Assuming aIb and bIc Definition (Finite and infinite traces) a c c d A trace over (Σ , I ) is a labeled DAG: b b · · · a c d · · · b b On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 be a symmetric, irreflexive independence relation , then (Σ , I ) is called a dependence alphabet . Assuming aIb and bIc Definition (Finite and infinite traces) a c c d A trace over (Σ , I ) is a labeled DAG: b b · · · a c d t 1 ⊙ t 2 · · · b b On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 be a symmetric, irreflexive independence relation , then (Σ , I ) is called a dependence alphabet . Assuming aIb and bIc Definition (Finite and infinite traces) a c c d A trace over (Σ , I ) is a labeled DAG: b b · · · a c d t 1 ⊙ t 2 · · · b b a c d b b On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 be a symmetric, irreflexive independence relation , then (Σ , I ) is called a dependence alphabet . Assuming aIb and bIc Definition (Finite and infinite traces) a c c d A trace over (Σ , I ) is a labeled DAG: b b · · · a c d t 1 ⊙ t 2 · · · b b a c � d b b a c d b b t 1 ⊔ t 2 On ω -Regular Trace Languages 5 / 18
Preliminaries: Traces Let I ⊆ Σ 2 be a symmetric, irreflexive independence relation , then (Σ , I ) is called a dependence alphabet . Assuming aIb and bIc Definition (Finite and infinite traces) a c c d A trace over (Σ , I ) is a labeled DAG: b b · · · a c d t 1 ⊙ t 2 · · · b b a The set of all finite traces: M (Σ ∗ , I ) c � d The set of all infinite traces: R (Σ ω , I ) b b a c d b b t 1 ⊔ t 2 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 ) a c d Γ( abcdb ) = b b Mapping traces to trace-closed sets of words a c d Γ − 1 = { bacdb , abcdb , acbdb } b b 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 ) a c d Γ( abcdb ) = b b Mapping traces to trace-closed sets of words a c d Γ − 1 = { bacdb , abcdb , acbdb } b b 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 ) a c d Γ( abcdb ) = b b Mapping traces to trace-closed sets of words a c d Γ − 1 = { bacdb , abcdb , acbdb } b b 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 (Σ ω , 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 . 3 a b a 1 6 a b a b a b 0 4 8 b a a b a b 2 7 b a b 5 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 (Σ ω , I ) = � t ∈ T t ⊙ R (Σ ω , 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 (Σ ω , I ) = � t ∈ T t ⊙ R (Σ ω , 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 (Σ ω , I ) = � t ∈ T t ⊙ R (Σ ω , 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 (Σ ω , I ) = � t ∈ T t ⊙ R (Σ ω , 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
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