Probabilistic -Regular Expressions Thomas Weidner Universitt - PowerPoint PPT Presentation

Probabilistic ω-Regular Expressions Thomas Weidner Universität Leipzig LATA 2014

1. Probabilistic regular expressions on infinite words Motivation In this talk LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) with decidable emptiness and approximation problem 2. “Probabilistic star-free expressions” expressively equivalent to probabilistic Muller-automata (Bollig, Gastin, Monmege, Zeitoum 2012) Background on finite words (Rabin 1963) well-studied with manifold applications (Kleene 1956) of theoretical computer science 2 / 11 ▶ Classical regular expressions in almost every field ▶ Probabilistic automata on finite words ▶ Regular Expressions transferred to probabilistic setting ▶ Probabilistic automata extended to infinite words (Baier, Grösser 2005)

Motivation Background LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) with decidable emptiness and approximation problem 2. “Probabilistic star-free expressions” expressively equivalent to probabilistic Muller-automata In this talk (Bollig, Gastin, Monmege, Zeitoum 2012) on finite words (Rabin 1963) well-studied with manifold applications (Kleene 1956) of theoretical computer science 2 / 11 ▶ Classical regular expressions in almost every field ▶ Probabilistic automata on finite words ▶ Regular Expressions transferred to probabilistic setting ▶ Probabilistic automata extended to infinite words (Baier, Grösser 2005) 1. Probabilistic regular expressions on infinite words

▶ Semantics well-defined ▶ Add special syntax restrictions ▶ ‖ a ‖( w ) = 󰊌 and ‖ p ‖( w ) = 󰊌 ▶ ‖ E + F ‖( w ) = ‖ E ‖( w ) + ‖ F ‖( w ) ▶ ‖ EF ‖( w ) = ∑ uv = w ‖ E ‖( u )‖ F ‖( v ) ▶ ‖ E ∗ ‖( w ) = ∑ n ≥ 0 ‖ E n ‖( w ) ▶ 󰉞 E ω 󰉞( w ) ( ‖ E ∗ ‖( ε ) = 1 ) 0 = lim n →∞ ‖ E n Σ ω ‖( w ) ( ‖ Σ ω ‖( w ) = 1 ) Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 otherwise Probabilistic ω-Regular Expressions if w = ε p • a (for a ∈ Σ ) • p (for p ∈ [ 0 , 1 ] ) • E + F • E ⋅ F • E ∗ • E ω Semantics 1 if w = a 0 otherwise Syntax 3 / 11 ▶ Atomic expressions: ▶ Compound expressions:

▶ Semantics well-defined ▶ Add special syntax restrictions ▶ 󰉞 E ω 󰉞( w ) Probabilistic ω-Regular Expressions otherwise LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ( ‖ Σ ω ‖( w ) = 1 ) = lim n →∞ ‖ E n Σ ω ‖( w ) ( ‖ E ∗ ‖( ε ) = 1 ) otherwise 0 if w = ε p Syntax 3 / 11 0 • E + F 1 Semantics • a (for a ∈ Σ ) • p (for p ∈ [ 0 , 1 ] ) if w = a • E ω • E ∗ • E ⋅ F ▶ Atomic expressions: ▶ Compound expressions: ▶ ‖ a ‖( w ) = 󰊌 and ‖ p ‖( w ) = 󰊌 ▶ ‖ E + F ‖( w ) = ‖ E ‖( w ) + ‖ F ‖( w ) ▶ ‖ EF ‖( w ) = ∑ uv = w ‖ E ‖( u )‖ F ‖( v ) ▶ ‖ E ∗ ‖( w ) = ∑ n ≥ 0 ‖ E n ‖( w )

▶ Semantics well-defined ▶ Add special syntax restrictions Probabilistic ω-Regular Expressions otherwise LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ( ‖ Σ ω ‖( w ) = 1 ) = lim n →∞ ‖ E n Σ ω ‖( w ) ( ‖ E ∗ ‖( ε ) = 1 ) otherwise 0 if w = ε p Syntax 3 / 11 0 • E ⋅ F 1 Semantics • E ω • a (for a ∈ Σ ) • p (for p ∈ [ 0 , 1 ] ) if w = a • E ∗ • E + F ▶ Atomic expressions: ▶ Compound expressions: ▶ ‖ a ‖( w ) = 󰊌 and ‖ p ‖( w ) = 󰊌 ▶ ‖ E + F ‖( w ) = ‖ E ‖( w ) + ‖ F ‖( w ) ▶ ‖ EF ‖( w ) = ∑ uv = w ‖ E ‖( u )‖ F ‖( v ) ▶ ‖ E ∗ ‖( w ) = ∑ n ≥ 0 ‖ E n ‖( w ) ▶ 󰉞 E ω 󰉞( w )

▶ Add special syntax restrictions Probabilistic ω-Regular Expressions otherwise LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ! Semantics not well-defined ( ‖ Σ ω ‖( w ) = 1 ) = lim n →∞ ‖ E n Σ ω ‖( w ) ( ‖ E ∗ ‖( ε ) = 1 ) otherwise 0 if w = ε p Syntax 3 / 11 0 • E ⋅ F 1 Semantics • E ω • a (for a ∈ Σ ) • p (for p ∈ [ 0 , 1 ] ) if w = a • E ∗ • E + F ▶ Atomic expressions: ▶ Compound expressions: ▶ ‖ a ‖( w ) = 󰊌 and ‖ p ‖( w ) = 󰊌 ▶ ‖ E + F ‖( w ) = ‖ E ‖( w ) + ‖ F ‖( w ) ▶ ‖ EF ‖( w ) = ∑ uv = w ‖ E ‖( u )‖ F ‖( v ) ▶ ‖ E ∗ ‖( w ) = ∑ n ≥ 0 ‖ E n ‖( w ) ▶ 󰉞 E ω 󰉞( w )

Probabilistic ω-Regular Expressions 0 LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ( ‖ Σ ω ‖( w ) = 1 ) = lim n →∞ ‖ E n Σ ω ‖( w ) ( ‖ E ∗ ‖( ε ) = 1 ) otherwise 0 if w = ε p Syntax otherwise 3 / 11 if w = a • E + F Semantics • E ω • E ∗ • a (for a ∈ Σ ) • p (for p ∈ [ 0 , 1 ] ) 1 • E ⋅ F ▶ Atomic expressions: ▶ Compound expressions: ▶ Add special syntax restrictions ▶ ‖ a ‖( w ) = 󰊌 and ‖ p ‖( w ) = 󰊌 ▶ ‖ E + F ‖( w ) = ‖ E ‖( w ) + ‖ F ‖( w ) ▶ ‖ EF ‖( w ) = ∑ uv = w ‖ E ‖( u )‖ F ‖( v ) ▶ ‖ E ∗ ‖( w ) = ∑ n ≥ 0 ‖ E n ‖( w ) ▶ 󰉞 E ω 󰉞( w ) ▶ Semantics well-defined

1. Σ ω ∈ ℛ 2. ∑ a ∈ Σ aE a ∈ ℛ if E a ∈ ℛ for each a ∈ Σ 5. E ∗ F + E ω ∈ ℛ if E Σ ω + F ∈ ℛ LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) 7. Close ℛ under usual distributivity, associativity, commutativity if E + F ∈ ℛ 6. E ∈ ℛ if E Σ ω , F ∈ ℛ 4. EF ∈ ℛ if E , F ∈ ℛ and p ∈ [ 0 , 1 ] 3. pE + ( 1 − p ) F ∈ ℛ Probabilistic ω-Regular Expressions: Syntax Set of probabilistic ω-regular expressions = smallest set ℛ such that Definition • E ω • E ∗ • E ⋅ F • E + F • p (for p ∈ [ 0 , 1 ] ) • a (for a ∈ Σ ) 4 / 11 ▶ Atomic expressions: ▶ Compound expressions: ▶ Have to distinguish expressions on finite and infinite words ▶ Use Σ ω as placeholder to append other expressions

Probabilistic ω-Regular Expressions: Syntax 3. pE + ( 1 − p ) F ∈ ℛ LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) 7. Close ℛ under usual distributivity, associativity, commutativity if E + F ∈ ℛ 6. E ∈ ℛ if E Σ ω , F ∈ ℛ 4. EF ∈ ℛ if E , F ∈ ℛ and p ∈ [ 0 , 1 ] 4 / 11 Set of probabilistic ω-regular expressions = smallest set ℛ such that • a (for a ∈ Σ ) Definition • E ω • p (for p ∈ [ 0 , 1 ] ) • E ∗ • E ⋅ F • E + F ▶ Atomic expressions: ▶ Compound expressions: ▶ Have to distinguish expressions on finite and infinite words ▶ Use Σ ω as placeholder to append other expressions 1. Σ ω ∈ ℛ 2. ∑ a ∈ Σ aE a ∈ ℛ if E a ∈ ℛ for each a ∈ Σ 5. E ∗ F + E ω ∈ ℛ if E Σ ω + F ∈ ℛ

▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression ∗ 9 E = 󰊅 n ∗ p 󰊅 1 ▶ ‖ E ‖( uv ω ) = 0 for all u , v ∈ Σ + with v ∉ { n } + ▶ ‖ E ‖( pnpn 2 pn 3 p …) > 0 LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ω 10n 󰊈 Example: Ping Pong 10n 󰊈 some Pong Ping device network 5 / 11 ▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ p ing request” or “ n othing” � Σ = { p , n }

▶ Probabilistic ω-Regular Expression ∗ 9 E = 󰊅 n ∗ p 󰊅 1 ▶ ‖ E ‖( uv ω ) = 0 for all u , v ∈ Σ + with v ∉ { n } + ▶ ‖ E ‖( pnpn 2 pn 3 p …) > 0 10n 󰊈 LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ω 10n 󰊈 Example: Ping Pong some Pong/90% Ping device network 5 / 11 ▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ p ing request” or “ n othing” � Σ = { p , n } ▶ Sending a Pong message successful 90%

▶ ‖ E ‖( uv ω ) = 0 for all u , v ∈ Σ + with v ∉ { n } + ▶ ‖ E ‖( pnpn 2 pn 3 p …) > 0 Example: Ping Pong some LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ω 10n 󰊈 10n 󰊈 5 / 11 Pong/90% Ping device network ▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ p ing request” or “ n othing” � Σ = { p , n } ▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression ∗ 9 E = 󰊅 n ∗ p 󰊅 1

Example: Ping Pong some LATA 2014 Probabilistic ω-Regular Expressions Thomas Weidner (Universität Leipzig) ω 10n 󰊈 10n 󰊈 5 / 11 Pong/90% Ping device network ▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ p ing request” or “ n othing” � Σ = { p , n } ▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression ∗ 9 E = 󰊅 n ∗ p 󰊅 1 ▶ ‖ E ‖( uv ω ) = 0 for all u , v ∈ Σ + with v ∉ { n } + ▶ ‖ E ‖( pnpn 2 pn 3 p …) > 0

Expressive Equivalence Theorem 1. f = ‖ A ‖ for some probabilistic Muller-automaton A 2. f = ‖ E ‖ for some probabilistic ω-regular expression E Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 6 / 11 Let f ∶ Σ ω → [ 0 , 1 ] . TFAE: