φ ormal µ ethods γ 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 ν TL, QPTL, ETL, . . . over infinite words over infinite words
Overview – Word Systems Regular expressiveness Linear temporal logic B¨ uchi-automata Right-linear grammars ν TL, QPTL, ETL, . . . over infinite words over infinite words Beyond context-free expressiveness Linear temporal logic Alternating + context-free grammars chop/concatenation over finite/infinite words LFLC
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p Models: p
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p ; q ; p ; q Models: p q p q
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p ; q ; p ; q Models: � p q p q
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p ; q ; p ; q Models: � p q p q
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p ; q ; p ; q Models: � p q p q
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p ; q ; p ; q Models: p � p q q
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = ν X . ( p ; q ; X ) Models: p q p q p q p q p q . . .
Temporal Logic Linear-time temporal logic with chop (LFLC): • propositional constants p , q , . . . p ≡ {¬ a , ¬ b } • special “empty” proposition ε q ≡ {¬ a , b } • connectives ∨ , ∧ ≡ { a , ¬ b } r • concatenation ; ≡ { a , b } s • fixed-point variables X , Y , . . . • fixed-point operators µ, ν M | = p ∨ p ; q ; p ; q ∨ ν X . ( p ; q ; X ) Models: p p q p q p q p q p q p q p q . . .
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ • production rules X → pYq • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ • production rules X → pYq • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ • production rules X → pYq • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ • production rules X → pYq • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N Languages: � � S → p L p Ω( S ) = 0
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ • production rules X → pYq • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N Languages: � � S → pqpq L Ω( S ) = 0 p q p q
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ • production rules X → pYq • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N Languages: � � S → pqS L Ω( S ) = 0 p q p q p q p q p q . . .
Grammars Alternating Context-Free Grammar (ACFG): • terminals p , q , . . . ∈ Σ X → pq • non-terminals X , Y , . . . ∈ N N → ( N ∪ Σ) ∗ X → pYq • production rules • designated initial symbol S ∈ N X → ε • alternation function λ : N → {∀ , ∃} • parity function Ω : N → N S → p | pqpq | A Languages: A → pqA p L λ ( S ) = λ ( A ) = ∃ p q p q Ω( S ) = Ω( A ) = 0 p q p q p q p q p q . . .
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Relationship of LFLC and ACFGs � � S → aS | b µ X . ( a ; X ∨ b ) L Ω( S ) = 1 µ X . ( a ; X ∨ b ) S a ; X ∨ b aS a a X S µ X . ( a ; X ∨ b ) | a ; X ∨ b aS a a X S b b
Expressiveness of LFLC and ACFGs • beyond context-free expressiveness L = { a n b n c n | n ≥ 0 } • satisfiability is undecidable L = L 1 ∩ L 2 , where L 1 , L 2 are context-free due to ϕ = ϕ 1 ∧ ϕ 2 , ϕ 1 , ϕ 2 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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