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Overview Word Systems Regular expressiveness Linear temporal logic - - PowerPoint PPT Presentation

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ormal ethods roup Linear Temporal Logics and Grammars Joachim Baran The University of Manchester April 2006 Overview Word Systems Regular expressiveness Linear temporal logic B uchi-automata Right-linear grammars TL,

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SLIDE 1

Linear Temporal Logics and Grammars

Joachim Baran

The University of Manchester

April 2006

φormal µethods γ roup

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SLIDE 2

Overview – Word Systems

Regular expressiveness Linear temporal logic

νTL, QPTL, ETL, . . .

B¨ uchi-automata

  • ver infinite words

Right-linear grammars

  • ver infinite words
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SLIDE 3

Overview – Word Systems

Regular expressiveness Beyond context-free expressiveness Linear temporal logic

νTL, QPTL, ETL, . . .

Linear temporal logic

+

chop/concatenation LFLC B¨ uchi-automata

  • ver infinite words

Right-linear grammars

  • ver infinite words

Alternating context-free grammars

  • ver finite/infinite words
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SLIDE 4

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

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SLIDE 5

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

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SLIDE 6

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

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SLIDE 7

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p

p

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SLIDE 8

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p; q; p; q

q p q p

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SLIDE 9

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p; q; p; q

q p q p

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SLIDE 10

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p; q; p; q

q p q p

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SLIDE 11

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p; q; p; q

q p q p

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SLIDE 12

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p; q; p; q

q p q p

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SLIDE 13

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = νX.(p; q; X)

q

. . .

p q p q p q p q p

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SLIDE 14

Temporal Logic

Linear-time temporal logic with chop (LFLC):

  • propositional constants

p, q, . . .

  • special “empty” proposition

ε

  • connectives

∨, ∧

  • concatenation

;

  • fixed-point variables

X, Y, . . .

  • fixed-point operators

µ, ν

p ≡ {¬a, ¬b} q ≡ {¬a, b} r ≡ { a, ¬b} s ≡ { a, b}

Models:

M | = p ∨ p; q; p; q ∨ νX.(p; q; X)

p q p q p q

. . .

p q p q p q p q p

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SLIDE 15

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

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SLIDE 16

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

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SLIDE 17

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

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SLIDE 18

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

Languages: L

  • S → p

Ω(S) = 0

  • p
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SLIDE 19

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

Languages: L

  • S → pqpq

Ω(S) = 0

  • q

p q p

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SLIDE 20

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

Languages: L

  • S → pqS

Ω(S) = 0

  • q

. . .

p q p q p q p q p

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SLIDE 21

Grammars

Alternating Context-Free Grammar (ACFG):

  • terminals

p, q, . . . ∈ Σ

  • non-terminals

X, Y, . . . ∈ N

  • production rules

N → (N ∪ Σ)∗

  • designated initial symbol

S ∈ N

  • alternation function

λ : N → {∀, ∃}

  • parity function

Ω : N → N

X → pq X → pYq X → ε

Languages: L     S → p | pqpq | A A → pqA λ(S) = λ(A) = ∃ Ω(S) = Ω(A) = 0    

p q p q p q

. . .

p q p q p q p q p

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SLIDE 22

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 23

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 24

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 25

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 26

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 27

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 28

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 29

Relationship of LFLC and ACFGs

µX.(a; X ∨ b) L

  • S → aS | b

Ω(S) = 1

  • µX.(a; X ∨ b)

a; X ∨ b a X µX.(a; X ∨ b) a; X ∨ b a X b S aS a S | aS a S b

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SLIDE 30

Expressiveness of LFLC and ACFGs

  • beyond context-free expressiveness

L = {anbncn | n ≥ 0}

  • satisfiability is undecidable

L = L1 ∩ L2, where L1,L2 are context-free

due to

ϕ = ϕ1 ∧ ϕ2, ϕ1,ϕ2 are context-free

  • model-checking of finite words and

ultimately periodic infinite words is decidable

w | = ϕ or w ∈ L(G)

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SLIDE 31

ε is Obsolete

As long as ε |

= ϕ:

  • ϕ can be rewritten as an ε-free formula ϕ′
  • since ε

?

| = ϕ is decidable:

– ϕ becomes either ϕ′ = ϕ or ϕ′ = ϕ ∨ ε – new initial symbol S′, such that either

S′ → S or S′ → S | ε, λ(S) = ∃ ⇒ LFLC’s syntax/semantics is not concise

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SLIDE 32

Alternation Hierarchy

  • fixed-point alternation:

ϕ L(ϕ)

depth(ϕ)

ν X .(µY .(ε ∨ b; Y ) ∨ b; X ) (a∗b)ω

ϕ L(ϕ)

depth(ϕ)

νX.(µY.(ε ∨ b; Y) ∨ b; X) (a∗b)ω νX.µY.(a; Y ∨ b; X) (a∗b)ω

1

  • private communication with Martin Lange:

– Logic’s fixed-point alternation:

hierarchy collapses at level 0

– Grammar’s acceptance condition:

parity acceptance is equiv. to weak parity acc.

c.f. his submitted “Specifying Non-Regular Properties of Runs”

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SLIDE 33

Future Work

  • true expressiveness of LFLC/ACFGs

– proper inclusion in context-sensitive languages

  • model-checking decidability reconsidered

– non-periodic infinite words

  • decidable model-checking fragments

– extending w |

= ϕ to regular-L | = ϕ