Probabilistic -Regular Expressions Thomas Weidner Universitt - - PowerPoint PPT Presentation

▶
probabilistic regular expressions
SMART_READER_LITE
LIVE PREVIEW

Probabilistic -Regular Expressions Thomas Weidner Universitt - - PowerPoint PPT Presentation

Nov 17, 2022 328 likes •652 views

Probabilistic -Regular Expressions Thomas Weidner Universitt Leipzig LATA 2014 1. Probabilistic regular expressions on infinite words Motivation In this talk LATA 2014 Probabilistic -Regular Expressions Thomas Weidner (Universitt

slide-1
SLIDE 1

Probabilistic ω-Regular Expressions

Thomas Weidner

Universität Leipzig

LATA 2014

slide-2
SLIDE 2

Motivation

Background

▶ Classical regular expressions in almost every field

  • f theoretical computer science

(Kleene 1956) ▶ Probabilistic automata on finite words

well-studied with manifold applications

(Rabin 1963) ▶ Regular Expressions transferred to probabilistic setting

  • n finite words

(Bollig, Gastin, Monmege, Zeitoum 2012) ▶ Probabilistic automata extended to infinite words (Baier, Grösser 2005)

In this talk

  • 1. Probabilistic regular expressions on infinite words

expressively equivalent to probabilistic Muller-automata

  • 2. “Probabilistic star-free expressions”

with decidable emptiness and approximation problem

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 2 / 11

slide-3
SLIDE 3

Motivation

Background

▶ Classical regular expressions in almost every field

  • f theoretical computer science

(Kleene 1956) ▶ Probabilistic automata on finite words

well-studied with manifold applications

(Rabin 1963) ▶ Regular Expressions transferred to probabilistic setting

  • n finite words

(Bollig, Gastin, Monmege, Zeitoum 2012) ▶ Probabilistic automata extended to infinite words (Baier, Grösser 2005)

In this talk

  • 1. Probabilistic regular expressions on infinite words

expressively equivalent to probabilistic Muller-automata

  • 2. “Probabilistic star-free expressions”

with decidable emptiness and approximation problem

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 2 / 11

slide-4
SLIDE 4

Probabilistic ω-Regular Expressions

Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Add special syntax restrictions

Semantics

▶ ‖a‖(w)

= 󰊌

1 if w = a

  • therwise

and ‖p‖(w) = 󰊌 p if w = ε

  • therwise

▶ ‖E + F‖(w) = ‖E‖(w) + ‖F‖(w) ▶ ‖EF‖(w)

= ∑uv=w‖E‖(u)‖F‖(v)

▶ ‖E∗‖(w)

= ∑n≥0‖En‖(w)

(‖E∗‖(ε) = 1) ▶ 󰉞Eω󰉞(w)

= limn→∞‖EnΣω‖(w)

(‖Σω‖(w) = 1) ▶ Semantics well-defined

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 3 / 11

slide-5
SLIDE 5

Probabilistic ω-Regular Expressions

Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Add special syntax restrictions

Semantics

▶ ‖a‖(w)

= 󰊌

1 if w = a

  • therwise

and ‖p‖(w) = 󰊌 p if w = ε

  • therwise

▶ ‖E + F‖(w) = ‖E‖(w) + ‖F‖(w) ▶ ‖EF‖(w)

= ∑uv=w‖E‖(u)‖F‖(v)

▶ ‖E∗‖(w)

= ∑n≥0‖En‖(w)

(‖E∗‖(ε) = 1) ▶ 󰉞Eω󰉞(w)

= limn→∞‖EnΣω‖(w)

(‖Σω‖(w) = 1) ▶ Semantics well-defined

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 3 / 11

slide-6
SLIDE 6

Probabilistic ω-Regular Expressions

Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Add special syntax restrictions

Semantics

▶ ‖a‖(w)

= 󰊌

1 if w = a

  • therwise

and ‖p‖(w) = 󰊌 p if w = ε

  • therwise

▶ ‖E + F‖(w) = ‖E‖(w) + ‖F‖(w) ▶ ‖EF‖(w)

= ∑uv=w‖E‖(u)‖F‖(v)

▶ ‖E∗‖(w)

= ∑n≥0‖En‖(w)

(‖E∗‖(ε) = 1) ▶ 󰉞Eω󰉞(w)

= limn→∞‖EnΣω‖(w)

(‖Σω‖(w) = 1) ▶ Semantics well-defined

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 3 / 11

slide-7
SLIDE 7

Probabilistic ω-Regular Expressions

Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Add special syntax restrictions

Semantics

▶ ‖a‖(w)

= 󰊌

1 if w = a

  • therwise

and ‖p‖(w) = 󰊌 p if w = ε

  • therwise

▶ ‖E + F‖(w) = ‖E‖(w) + ‖F‖(w) ▶ ‖EF‖(w)

= ∑uv=w‖E‖(u)‖F‖(v)

▶ ‖E∗‖(w)

= ∑n≥0‖En‖(w)

(‖E∗‖(ε) = 1) ▶ 󰉞Eω󰉞(w)

= limn→∞‖EnΣω‖(w)

(‖Σω‖(w) = 1)

! Semantics not well-defined

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 3 / 11

slide-8
SLIDE 8

Probabilistic ω-Regular Expressions

Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Add special syntax restrictions

Semantics

▶ ‖a‖(w)

= 󰊌

1 if w = a

  • therwise

and ‖p‖(w) = 󰊌 p if w = ε

  • therwise

▶ ‖E + F‖(w) = ‖E‖(w) + ‖F‖(w) ▶ ‖EF‖(w)

= ∑uv=w‖E‖(u)‖F‖(v)

▶ ‖E∗‖(w)

= ∑n≥0‖En‖(w)

(‖E∗‖(ε) = 1) ▶ 󰉞Eω󰉞(w)

= limn→∞‖EnΣω‖(w)

(‖Σω‖(w) = 1) ▶ Semantics well-defined

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 3 / 11

slide-9
SLIDE 9

Probabilistic ω-Regular Expressions: Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Have to distinguish expressions on finite and infinite words ▶ Use Σω as placeholder to append other expressions

Definition

Set of probabilistic ω-regular expressions = smallest set ℛ such that

  • 1. Σω ∈ ℛ
  • 2. ∑a∈Σ aEa ∈ ℛ

if Ea ∈ ℛ for each a ∈ Σ

  • 3. pE + (1 − p)F ∈ ℛ

if E, F ∈ ℛ and p ∈ [0, 1]

  • 4. EF ∈ ℛ

if EΣω, F ∈ ℛ

  • 5. E∗F + Eω ∈ ℛ

if EΣω + F ∈ ℛ

  • 6. E ∈ ℛ

if E + F ∈ ℛ

  • 7. Close ℛ under usual distributivity, associativity, commutativity

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 4 / 11

slide-10
SLIDE 10

Probabilistic ω-Regular Expressions: Syntax

▶ Atomic expressions:

  • a (for a ∈ Σ)
  • p (for p ∈ [0, 1])

▶ Compound expressions:

  • E + F
  • E ⋅ F
  • E∗
  • Eω

▶ Have to distinguish expressions on finite and infinite words ▶ Use Σω as placeholder to append other expressions

Definition

Set of probabilistic ω-regular expressions = smallest set ℛ such that

  • 1. Σω ∈ ℛ
  • 2. ∑a∈Σ aEa ∈ ℛ

if Ea ∈ ℛ for each a ∈ Σ

  • 3. pE + (1 − p)F ∈ ℛ

if E, F ∈ ℛ and p ∈ [0, 1]

  • 4. EF ∈ ℛ

if EΣω, F ∈ ℛ

  • 5. E∗F + Eω ∈ ℛ

if EΣω + F ∈ ℛ

  • 6. E ∈ ℛ

if E + F ∈ ℛ

  • 7. Close ℛ under usual distributivity, associativity, commutativity

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 4 / 11

slide-11
SLIDE 11

Example: Ping Pong

some network device Ping Pong

▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ping request” or “nothing” Σ = {p, n} ▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression

E = 󰊅n∗p󰊅 1 10n󰊈

∗ 9

10n󰊈

ω

▶ ‖E‖(uvω) = 0 for all u, v ∈ Σ+ with v ∉ {n}+ ▶ ‖E‖(pnpn2pn3p …) > 0

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 5 / 11

slide-12
SLIDE 12

Example: Ping Pong

some network device Ping Pong/90%

▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ping request” or “nothing” Σ = {p, n} ▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression

E = 󰊅n∗p󰊅 1 10n󰊈

∗ 9

10n󰊈

ω

▶ ‖E‖(uvω) = 0 for all u, v ∈ Σ+ with v ∉ {n}+ ▶ ‖E‖(pnpn2pn3p …) > 0

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 5 / 11

slide-13
SLIDE 13

Example: Ping Pong

some network device Ping Pong/90%

▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ping request” or “nothing” Σ = {p, n} ▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression

E = 󰊅n∗p󰊅 1 10n󰊈

∗ 9

10n󰊈

ω

▶ ‖E‖(uvω) = 0 for all u, v ∈ Σ+ with v ∉ {n}+ ▶ ‖E‖(pnpn2pn3p …) > 0

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 5 / 11

slide-14
SLIDE 14

Example: Ping Pong

some network device Ping Pong/90%

▶ Network device, which responds to “Ping” messages ▶ Pong should be sent before next Ping ▶ Input = Sequence of “ping request” or “nothing” Σ = {p, n} ▶ Sending a Pong message successful 90% ▶ Probabilistic ω-Regular Expression

E = 󰊅n∗p󰊅 1 10n󰊈

∗ 9

10n󰊈

ω

▶ ‖E‖(uvω) = 0 for all u, v ∈ Σ+ with v ∉ {n}+ ▶ ‖E‖(pnpn2pn3p …) > 0

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 5 / 11

slide-15
SLIDE 15

Expressive Equivalence

Theorem

Let f ∶ Σω → [0, 1]. TFAE:

  • 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

slide-16
SLIDE 16

Idea of the Proof

Expression → Automaton

▶ Inductive construction on syntax of expression ▶ Based on ideas of the finite word case ▶ Uses automata with final states and Muller acceptance condition

Automaton → Expression

▶ Use induction on |X| for set X ⊆ Q to build expressions EX

p :

EX

p = 󰈭 q∉X

EX

p,qΣω + 󰈭 F⊆X

EX

p,inf=F

▶ In induction step:

Consider prefix-minimal runs visiting all states in X in fixed order

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 7 / 11

slide-17
SLIDE 17

Idea of the Proof

Expression → Automaton

▶ Inductive construction on syntax of expression ▶ Based on ideas of the finite word case ▶ Uses automata with final states and Muller acceptance condition

Automaton → Expression

▶ Use induction on |X| for set X ⊆ Q to build expressions EX

p :

EX

p = 󰈭 q∉X

EX

p,qΣω + 󰈭 F⊆X

EX

p,inf=F

▶ In induction step:

Consider prefix-minimal runs visiting all states in X in fixed order

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 7 / 11

slide-18
SLIDE 18

Decidability Results

▶ Effective transformations: automata ↔ expressions ▶ Automata-based decidability results transfer to expressions

Decidable:

▶ Given expression E, ∃u, v ∈ Σ+ ∶ ‖E‖(uvω) > 0 ?

Undecidable:

▶ Given expression E, ∃w ∈ Σω ∶ ‖E‖(w) > 0 (= 1)? ▶ Given expression E and ε > 0 such that either

  • 1. ∀w ∈ Σω ∶ ‖E‖(w) ≤ ε
  • 2. ∃w ∈ Σω ∶ ‖E‖(w) ≥ 1 − ε

Which is the case?

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 8 / 11

slide-19
SLIDE 19

Decidability Results

▶ Effective transformations: automata ↔ expressions ▶ Automata-based decidability results transfer to expressions

Decidable:

▶ Given expression E, ∃u, v ∈ Σ+ ∶ ‖E‖(uvω) > 0 ?

Undecidable:

▶ Given expression E, ∃w ∈ Σω ∶ ‖E‖(w) > 0 (= 1)? ▶ Given expression E and ε > 0 such that either

  • 1. ∀w ∈ Σω ∶ ‖E‖(w) ≤ ε
  • 2. ∃w ∈ Σω ∶ ‖E‖(w) ≥ 1 − ε

Which is the case?

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 8 / 11

slide-20
SLIDE 20

Decidability Results

▶ Effective transformations: automata ↔ expressions ▶ Automata-based decidability results transfer to expressions

Decidable:

▶ Given expression E, ∃u, v ∈ Σ+ ∶ ‖E‖(uvω) > 0 ?

Undecidable:

▶ Given expression E, ∃w ∈ Σω ∶ ‖E‖(w) > 0 (= 1)? ▶ Given expression E and ε > 0 such that either

  • 1. ∀w ∈ Σω ∶ ‖E‖(w) ≤ ε
  • 2. ∃w ∈ Σω ∶ ‖E‖(w) ≥ 1 − ε

Which is the case?

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 8 / 11

slide-21
SLIDE 21

A “Probabilistic Star-Free” Fragment of Expressions

▶ Restrict probabilistic iteration,

such that iteration almost surely terminates

▶ Allow only the following iteration constructs

1 E∗ and Eω for deterministic expression E 2 (pE)∗ for p < 1 and expression E

▶ Nesting 1 within 2 (2 within 1) not allowed

Almost ω-deterministic expressions

Example

▶ a∗b (1/3 ⋅ aa)∗ ⋅ 2/3 ⋅ abω ▶ 󰊄1/3 ⋅ (2/3 ⋅ a)∗a󰊇 ∗bω

Not an expression

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 9 / 11

slide-22
SLIDE 22

A “Probabilistic Star-Free” Fragment of Expressions

▶ Restrict probabilistic iteration,

such that iteration almost surely terminates

▶ Allow only the following iteration constructs

1 E∗ and Eω for deterministic expression E 2 (pE)∗ for p < 1 and expression E

▶ Nesting 1 within 2 (2 within 1) not allowed

Almost ω-deterministic expressions

Example

▶ a∗b (1/3 ⋅ aa)∗ ⋅ 2/3 ⋅ abω ▶ 󰊄1/3 ⋅ (2/3 ⋅ a)∗a󰊇 ∗bω

Not an expression

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 9 / 11

slide-23
SLIDE 23

A “Probabilistic Star-Free” Fragment of Expressions

▶ Restrict probabilistic iteration,

such that iteration almost surely terminates

▶ Allow only the following iteration constructs

1 E∗ and Eω for deterministic expression E 2 (pE)∗ for p < 1 and expression E

▶ Nesting 1 within 2 (2 within 1) not allowed

Almost ω-deterministic expressions

Example

▶ a∗b (1/3 ⋅ aa)∗ ⋅ 2/3 ⋅ abω ▶ 󰊄1/3 ⋅ (2/3 ⋅ a)∗a󰊇 ∗bω

Not an expression

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 9 / 11

slide-24
SLIDE 24

A “Probabilistic Star-Free” Fragment of Expressions

▶ Restrict probabilistic iteration,

such that iteration almost surely terminates

▶ Allow only the following iteration constructs

1 E∗ and Eω for deterministic expression E 2 (pE)∗ for p < 1 and expression E

▶ Nesting 1 within 2 (2 within 1) not allowed

Almost ω-deterministic expressions

Example

▶ a∗b (1/3 ⋅ aa)∗ ⋅ 2/3 ⋅ abω ▶ 󰊄1/3 ⋅ (2/3 ⋅ a)∗a󰊇 ∗bω

Not an expression

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 9 / 11

slide-25
SLIDE 25

Almost Limit-Deterministic Automata

Definition

A = (Q, δ, μ, Acc) almost limit-deterministic if for every SCC C ⊆ Q δ(p, a, q) ∈ {0, 1} for all p, q ∈ C, a ∈ Σ

  • r Prw

Aq(Cω) = 0 for all q ∈ C and w ∈ Σω

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 10 / 11

slide-26
SLIDE 26

Almost Limit-Deterministic Automata

Definition

A = (Q, δ, μ, Acc) almost limit-deterministic if for every SCC C ⊆ Q δ(p, a, q) ∈ {0, 1} for all p, q ∈ C, a ∈ Σ

  • r Prw

Aq(Cω) = 0 for all q ∈ C and w ∈ Σω

Example

a b

1 2a 1 3a 1 6a

a b b

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 10 / 11

slide-27
SLIDE 27

Almost Limit-Deterministic Automata

Definition

A = (Q, δ, μ, Acc) almost limit-deterministic if for every SCC C ⊆ Q δ(p, a, q) ∈ {0, 1} for all p, q ∈ C, a ∈ Σ

  • r Prw

Aq(Cω) = 0 for all q ∈ C and w ∈ Σω

Remark

▶ Only countable many runs in ALD automata

Decidable positive emptiness problem

▶ Class of ALD automata closed under

cross-product, complement, concatenation

▶ Cannot express infinitely many probabilistic choices

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 10 / 11

slide-28
SLIDE 28

Almost Limit-Deterministic Automata

Definition

A = (Q, δ, μ, Acc) almost limit-deterministic if for every SCC C ⊆ Q δ(p, a, q) ∈ {0, 1} for all p, q ∈ C, a ∈ Σ

  • r Prw

Aq(Cω) = 0 for all q ∈ C and w ∈ Σω

Theorem

Let A be almost limit deterministic and ε > 0. Then

∃ finite, computable V ⊆ [0, 1]∶ dH(‖A‖(Σω), V) ≤ ε,

where dH is Hausdorff-distance, i.e.

▶ ∀w ∈ Σω ∶ ∃x ∈ V ∶ |x − ‖A‖(w)| ≤ ε ▶ ∀x ∈ V ∶ ∃w ∈ Σω ∶ |x − ‖A‖(w)| ≤ ε

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 10 / 11

slide-29
SLIDE 29

Almost Limit-Deterministic Automata

Theorem

Let E almost ω-deterministic expression. There is almost limit deterministic automaton A s.t. ‖A‖ = ‖E‖.

Theorem

Let E be almost ω-deterministic and ε > 0. Then

∃ finite, computable V ⊆ [0, 1]∶ dH(‖E‖(Σω), V) ≤ ε,

where dH is Hausdorff-distance, i.e.

▶ ∀w ∈ Σω ∶ ∃x ∈ V ∶ |x − ‖E‖(w)| ≤ ε ▶ ∀x ∈ V ∶ ∃w ∈ Σω ∶ |x − ‖E‖(w)| ≤ ε

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 10 / 11

slide-30
SLIDE 30

Conclusion

Results

▶ We introduced probabilistic ω-regular expressions ▶ Expressively equivalent to probabilistic Muller-automata ▶ Almost ω-deterministic expressions

= “probabilistic star-free” fragment

▶ Decidable emptiness, approximation problems

Future research

▶ Applications for almost limit-deterministic automata ▶ Characterization of ALD automata by “probabilistic FO” logic ▶ Add weights to the expression framework

▶ Probabilistic weighted A. by Chattergee, Doyen, Henzinger ▶ Valuation monoids by Droste, et.al.

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 11 / 11

slide-31
SLIDE 31

Conclusion

Results

▶ We introduced probabilistic ω-regular expressions ▶ Expressively equivalent to probabilistic Muller-automata ▶ Almost ω-deterministic expressions

= “probabilistic star-free” fragment

▶ Decidable emptiness, approximation problems

Future research

▶ Applications for almost limit-deterministic automata ▶ Characterization of ALD automata by “probabilistic FO” logic ▶ Add weights to the expression framework

▶ Probabilistic weighted A. by Chattergee, Doyen, Henzinger ▶ Valuation monoids by Droste, et.al.

Thomas Weidner (Universität Leipzig) Probabilistic ω-Regular Expressions LATA 2014 11 / 11