Conjunctive grammars over a unary alphabet Artur Je z, Alexander - PowerPoint PPT Presentation

The case of a unary alphabet Σ = { a } . a n ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = { x + y | x ∈ X , y ∈ Y } ← → Regular ultimately periodic Theorem (Bar-Hillel et al., 1961) Every context-free language over { a } is regular. Problem The power of conjunctive grammars over { a } ? Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

The case of a unary alphabet Σ = { a } . a n ← → number n Language ← → set of numbers K · L ← → X ⊞ Y = { x + y | x ∈ X , y ∈ Y } ← → Regular ultimately periodic Theorem (Bar-Hillel et al., 1961) Every context-free language over { a } is regular. Problem The power of conjunctive grammars over { a } ? Can generate { a 4 n | n � 0 } (Je˙ z, 2007). Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 5 / 15

Using positional notation Our approach: using base- k notation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Σ k = { 0 , 1 , . . . , k − 1 } , strings in Σ ∗ k \ 0 Σ ∗ k . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Σ k = { 0 , 1 , . . . , k − 1 } , strings in Σ ∗ k \ 0 Σ ∗ k . Isomorphism Σ ∗ k \ 0 Σ ∗ k ↔ a ∗ . f k ( k-ary notation of n ) = a n Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Σ k = { 0 , 1 , . . . , k − 1 } , strings in Σ ∗ k \ 0 Σ ∗ k . Isomorphism Σ ∗ k \ 0 Σ ∗ k ↔ a ∗ . f k ( k-ary notation of n ) = a n Extends to languages: f k ( L ) = { f k ( w ) | w ∈ L } Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Σ k = { 0 , 1 , . . . , k − 1 } , strings in Σ ∗ k \ 0 Σ ∗ k . Isomorphism Σ ∗ k \ 0 Σ ∗ k ↔ a ∗ . f k ( k-ary notation of n ) = a n Extends to languages: f k ( L ) = { f k ( w ) | w ∈ L } Example f 4 ( 10 ∗ ) = { a 4 n | n � 0 } Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Σ k = { 0 , 1 , . . . , k − 1 } , strings in Σ ∗ k \ 0 Σ ∗ k . Isomorphism Σ ∗ k \ 0 Σ ∗ k ↔ a ∗ . f k ( k-ary notation of n ) = a n Extends to languages: f k ( L ) = { f k ( w ) | w ∈ L } Example f 4 ( 10 ∗ ) = { a 4 n | n � 0 } Equations over Σ ∗ k with ∩ , ∪ , ⊞ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Using positional notation Our approach: using base- k notation. a n ← → k -ary notation of n Σ k = { 0 , 1 , . . . , k − 1 } , strings in Σ ∗ k \ 0 Σ ∗ k . Isomorphism Σ ∗ k \ 0 Σ ∗ k ↔ a ∗ . f k ( k-ary notation of n ) = a n Extends to languages: f k ( L ) = { f k ( w ) | w ∈ L } Example f 4 ( 10 ∗ ) = { a 4 n | n � 0 } Equations over Σ ∗ k with ∩ , ∪ , ⊞ Isomorphism between language equations. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 6 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } Y 1 = ( Y 2 Y 2 ∩ Y 1 Y 3 ) ∪ a X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } Y 2 = ( Y 12 Y 2 ∩ Y 1 Y 1 ) ∪ aa X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } Y 3 = ( Y 12 Y 12 ∩ Y 1 Y 2 ) ∪ aaa X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Y 12 = ( Y 3 Y 3 ∩ Y 1 Y 2 ) Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } Y 1 = ( Y 2 Y 2 ∩ Y 1 Y 3 ) ∪ a X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } Y 2 = ( Y 12 Y 2 ∩ Y 1 Y 1 ) ∪ aa X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } Y 3 = ( Y 12 Y 12 ∩ Y 1 Y 2 ) ∪ aaa X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Y 12 = ( Y 3 Y 3 ∩ Y 1 Y 2 ) base 4: ( 10 ∗ , 20 ∗ , 30 ∗ , 120 ∗ ) Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } Y 1 = ( Y 2 Y 2 ∩ Y 1 Y 3 ) ∪ a X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } Y 2 = ( Y 12 Y 2 ∩ Y 1 Y 1 ) ∪ aa X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } Y 3 = ( Y 12 Y 12 ∩ Y 1 Y 2 ) ∪ aaa X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Y 12 = ( Y 3 Y 3 ∩ Y 1 Y 2 ) { a 4 n } , { a 2 · 4 n } , { a 3 · 4 n } , { a 6 · 4 n } � � base 4: ( 10 ∗ , 20 ∗ , 30 ∗ , 120 ∗ ) Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } Y 1 = ( Y 2 Y 2 ∩ Y 1 Y 3 ) ∪ a X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } Y 2 = ( Y 12 Y 2 ∩ Y 1 Y 1 ) ∪ aa X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } Y 3 = ( Y 12 Y 12 ∩ Y 1 Y 2 ) ∪ aaa X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Y 12 = ( Y 3 Y 3 ∩ Y 1 Y 2 ) { a 4 n } , { a 2 · 4 n } , { a 3 · 4 n } , { a 6 · 4 n } � � base 4: ( 10 ∗ , 20 ∗ , 30 ∗ , 120 ∗ ) X 2 ⊞ X 2 = 20 ∗ ⊞ 20 ∗ = 10 + ∪ 20 ∗ 20 ∗ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } Y 1 = ( Y 2 Y 2 ∩ Y 1 Y 3 ) ∪ a X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } Y 2 = ( Y 12 Y 2 ∩ Y 1 Y 1 ) ∪ aa X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } Y 3 = ( Y 12 Y 12 ∩ Y 1 Y 2 ) ∪ aaa X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Y 12 = ( Y 3 Y 3 ∩ Y 1 Y 2 ) { a 4 n } , { a 2 · 4 n } , { a 3 · 4 n } , { a 6 · 4 n } � � base 4: ( 10 ∗ , 20 ∗ , 30 ∗ , 120 ∗ ) X 2 ⊞ X 2 = 20 ∗ ⊞ 20 ∗ = 10 + ∪ 20 ∗ 20 ∗ X 1 ⊞ X 3 = 10 ∗ ⊞ 30 ∗ = 10 + ∪ 10 ∗ 30 ∗ ∪ 30 ∗ 10 ∗ , Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

Nonperiodic unary conjunctive languages Example (Je˙ z, DLT 2007) a ∗ N X 1 = ( X 2 ⊞ X 2 ∩ X 1 ⊞ X 3 ) ∪ { 1 } Y 1 = ( Y 2 Y 2 ∩ Y 1 Y 3 ) ∪ a X 2 = ( X 12 ⊞ X 2 ∩ X 1 ⊞ X 1 ) ∪ { 2 } Y 2 = ( Y 12 Y 2 ∩ Y 1 Y 1 ) ∪ aa X 3 = ( X 12 ⊞ X 12 ∩ X 1 ⊞ X 2 ) ∪ { 3 } Y 3 = ( Y 12 Y 12 ∩ Y 1 Y 2 ) ∪ aaa X 12 = X 3 ⊞ X 3 ∩ X 1 ⊞ X 2 Y 12 = ( Y 3 Y 3 ∩ Y 1 Y 2 ) { a 4 n } , { a 2 · 4 n } , { a 3 · 4 n } , { a 6 · 4 n } � � base 4: ( 10 ∗ , 20 ∗ , 30 ∗ , 120 ∗ ) X 2 ⊞ X 2 = 20 ∗ ⊞ 20 ∗ = 10 + ∪ 20 ∗ 20 ∗ X 1 ⊞ X 3 = 10 ∗ ⊞ 30 ∗ = 10 + ∪ 10 ∗ 30 ∗ ∪ 30 ∗ 10 ∗ , ( X 2 ⊞ X 2 ) ∩ ( X 1 ⊞ X 3 ) = 10 + . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 7 / 15

More unary conjunctive languages Theorem (Je˙ z, DLT 2007) For any regular language R ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a conjunctive grammar for f k ( R ) . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 8 / 15

More unary conjunctive languages Theorem (Je˙ z, DLT 2007) For any regular language R ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a conjunctive grammar for f k ( R ) . Note f k ( R ) has linear or exponential growth. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 8 / 15

More unary conjunctive languages Theorem (Je˙ z, DLT 2007) For any regular language R ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a conjunctive grammar for f k ( R ) . Note f k ( R ) has linear or exponential growth. * * * Theorem For every trellis automaton M over Σ k with L ( M ) ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a conjunctive grammar for f k ( L ( M )) . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 8 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; F ⊂ Q : final states. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; F ⊂ Q : final states. Equivalent to linear conjunctive grammars. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; F ⊂ Q : final states. Equivalent to linear conjunctive grammars. Closed under ∪ , ∩ , ∼ , not closed under concatenation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Trellis automata (one-way real-time cellular automata) Definition A trellis automaton is a M = ( Σ, Q , I , δ, F ) where: Σ : input alphabet; Q : finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q , transition function; F ⊂ Q : final states. Equivalent to linear conjunctive grammars. Closed under ∪ , ∩ , ∼ , not closed under concatenation. Can recognize { wcw } , { a n b n c n } , { a n b 2 n } , VALC. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 9 / 15

Main lemma Lemma For every trellis automaton M over Σ k with L ( M ) ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a system with ∪ , ∩ , ⊞ and regular constants, with least solution { 1 w 10 ∗ | w ⊞ 1 ∈ L ( M ) } , . . . , Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 10 / 15

Main lemma Lemma For every trellis automaton M over Σ k with L ( M ) ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a system with ∪ , ∩ , ⊞ and regular constants, with least solution { 1 w 10 ∗ | w ⊞ 1 ∈ L ( M ) } , . . . , 1 w 10 ∗ represents w . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 10 / 15

Main lemma Lemma For every trellis automaton M over Σ k with L ( M ) ⊆ Σ ∗ k \ 0 Σ ∗ k , there exists a system with ∪ , ∩ , ⊞ and regular constants, with least solution { 1 w 10 ∗ | w ⊞ 1 ∈ L ( M ) } , . . . , 1 w 10 ∗ represents w . Regular constants, can be changed to singleton. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 10 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) ⇔ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) ⇔ ∃ q ′ , q ′′ : δ ( q ′ , q ′′ ) = q , Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) ⇔ ∃ q ′ , q ′′ : δ ( q ′ , q ′′ ) = q , au ∈ L M ( q ′ ) , Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) ⇔ ∃ q ′ , q ′′ : δ ( q ′ , q ′′ ) = q , au ∈ L M ( q ′ ) , ub ∈ L M ( q ′′ ) . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) ⇔ ∃ q ′ , q ′′ : δ ( q ′ , q ′′ ) = q , au ∈ L M ( q ′ ) , ub ∈ L M ( q ′′ ) . Let 1 au 10 ∗ ⊆ X q ′ , 1 ub 10 ∗ ⊆ X q ′′ . � X q = ρ b ( X q ′ ) ∩ λ a ( X q ′′ ) q ′ , q ′′ : δ ( q ′ , q ′′ )= q a , b ∈ Σ k Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

The construction Set of variables { X q | q ∈ Q } , representing { L M ( q ) | q ∈ Q } . Actually, X q = { 1 w 10 ∗ | w ⊞ 1 ∈ L M ( q ) } aub ∈ L M ( q ) ⇔ ∃ q ′ , q ′′ : δ ( q ′ , q ′′ ) = q , au ∈ L M ( q ′ ) , ub ∈ L M ( q ′′ ) . Let 1 au 10 ∗ ⊆ X q ′ , 1 ub 10 ∗ ⊆ X q ′′ . � X q = ρ b ( X q ′ ) ∩ λ a ( X q ′′ ) q ′ , q ′′ : δ ( q ′ , q ′′ )= q a , b ∈ Σ k λ a ( 1 w 10 k ) = 1 aw 10 k ρ b ( 1 w 10 k ) = 1 wb 10 k − 1 Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 11 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word 1 w 10 ℓ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word 1 w 10 ℓ = 1 w ′ j ′ 10 ℓ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word 1 w 10 ℓ = 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 10 ℓ ∩ 1 Σ ∗ k j ′ 10 ∗ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word 1 w 10 ℓ = 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 10 ℓ ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ∩ 1 Σ ∗ 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 20 ℓ k j ′ 20 ∗ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word 1 w 10 ℓ = 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 10 ℓ ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ∩ 1 Σ ∗ 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 20 ℓ k j ′ 20 ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ 1 w ′ j ′ j 10 ℓ − 1 1 w ′ j ′ 20 ℓ k j 10 ∗ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

How does ρ look like The equations for ρ j : �� � k j ′ 20 ∗ � � ( X ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ) ∩ 1 Σ ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ k j 10 ∗ ρ j ( X ) = j ′ Word Operation New word 1 w 10 ℓ = 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 10 ℓ ∩ 1 Σ ∗ k j ′ 10 ∗ ⊞ 10 ∗ ∩ 1 Σ ∗ 1 w ′ j ′ 10 ℓ 1 w ′ j ′ 20 ℓ k j ′ 20 ∗ ⊞ ( j − 2 ) 10 ∗ ∩ 1 Σ ∗ 1 w ′ j ′ j 10 ℓ − 1 1 w ′ j ′ 20 ℓ k j 10 ∗ 1 w ′ j ′ j 10 ℓ − 1 1 wj 10 ℓ − 1 � j ′ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 12 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” — undecidable. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” — undecidable. Turing machine T recognizes X ; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” — undecidable. Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” — undecidable. Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Conjunctive grammar G for f k ( VALC ( T )) ; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” — undecidable. Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Conjunctive grammar G for f k ( VALC ( T )) ; L ( G ) = ∅ ⇔ L ( T ) = ∅ Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Undecidable properties Proposition “Given conjunctive grammar G over { a } , determine whether L ( G ) = ∅ ” — undecidable. Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Conjunctive grammar G for f k ( VALC ( T )) ; L ( G ) = ∅ ⇔ L ( T ) = ∅ Theorem For every fixed conjunctive L 0 ⊆ a ∗ , the problem “Given conjunctive grammar G over { a } , determine whether L ( G ) = L 0 ” — undecidable. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 13 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : g G ( n ) > g X ( n ) ( ∀ n � 1 ) Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : g G ( n ) > g X ( n ) ( ∀ n � 1 ) Turing machine T recognizes X ; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : g G ( n ) > g X ( n ) ( ∀ n � 1 ) Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : g G ( n ) > g X ( n ) ( ∀ n � 1 ) Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Conjunctive grammar G for f k ( VALC ( T )) ; Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : g G ( n ) > g X ( n ) ( ∀ n � 1 ) Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Conjunctive grammar G for f k ( VALC ( T )) ; g G ( n ) > g X ( n ) . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Unbounded growth Theorem ∀ r. e. set X = { g X ( n ) | n � 1 } , with g X ր ∃ conjunctive grammar G with L ( G ) = { a g G ( n ) | n � 1 } : g G ( n ) > g X ( n ) ( ∀ n � 1 ) Turing machine T recognizes X ; Trellis automaton M for VALC ( T ) ; Conjunctive grammar G for f k ( VALC ( T )) ; g G ( n ) > g X ( n ) . Remark Polynomial growth can be achieved. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 14 / 15

Conclusion Conjunctive grammar: CFG with intersection. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. ◮ Positional notation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. ◮ Positional notation. Unary notation of VALC ( T ) is conjunctive. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. ◮ Positional notation. Unary notation of VALC ( T ) is conjunctive. ◮ Undecidability. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. ◮ Positional notation. Unary notation of VALC ( T ) is conjunctive. ◮ Undecidability. ◮ Growth not recursively bounded. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. ◮ Positional notation. Unary notation of VALC ( T ) is conjunctive. ◮ Undecidability. ◮ Growth not recursively bounded. Research problems. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15

Conclusion Conjunctive grammar: CFG with intersection. Specifying unary notation of nontrivial languages. ◮ { a f k ( w ) | w ∈ L ( M ) } for any TA M . ◮ Equations over sets of integers. ◮ Positional notation. Unary notation of VALC ( T ) is conjunctive. ◮ Undecidability. ◮ Growth not recursively bounded. Research problems. ◮ Closure under complementation. Artur Je˙ z, Alexander Okhotin Conjunctive grammars over a unary alphabet September 7, 2007 15 / 15