Language equations: the story of computational completeness - Users

Language equations: the story of computational completeness Alexander Okhotin Department of Mathematics, University of Turku; Academy of Finland

Dagstuhl, 17 December 2010 A. D.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

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Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn )

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

2 / 12

Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn ) Alphabet Σ.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

2 / 12

Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn ) Alphabet Σ. Xi : unknown formal languages.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

2 / 12

Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn ) Alphabet Σ. Xi : unknown formal languages. ϕi : variables, operations on languages, constant languages.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

2 / 12

Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn ) Alphabet Σ. Xi : unknown formal languages. ϕi : variables, operations on languages, constant languages. Solutions Xi = Li :

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

2 / 12

Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn ) Alphabet Σ. Xi : unknown formal languages. ϕi : variables, operations on languages, constant languages. Solutions Xi = Li : I

Unique solutions.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

2 / 12

Language equations System of equations:    ϕ1 (X1 , . . . , Xn ) = ψ1 (X1 , . . . , Xn ) .. .   ϕm (X1 , . . . , Xn ) = ψm (X1 , . . . , Xn ) Alphabet Σ. Xi : unknown formal languages. ϕi : variables, operations on languages, constant languages. Solutions Xi = Li : I I

Unique solutions. Least/greatest wrt. partial order: (L1 , . . . , Ln ) v (L01 , . . . , L0n )

Alexander Okhotin

Language equations

if Li ⊆ L0i for all i

Dagstuhl, 2010 A. D.

2 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}. I

Rules A → α1 | . . . | αn yield equation A = α1 ∪ . . . ∪ αn .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}. I I

Rules A → α1 | . . . | αn yield equation A = α1 ∪ . . . ∪ αn . Least solution.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}. I I I

Rules A → α1 | . . . | αn yield equation A = α1 ∪ . . . ∪ αn . Least solution. With operations {∪, ∩, ·}: conjunctive grammars (Okhotin, 2001).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}. I I I

2

Rules A → α1 | . . . | αn yield equation A = α1 ∪ . . . ∪ αn . Least solution. With operations {∪, ∩, ·}: conjunctive grammars (Okhotin, 2001).

One-sided concatenation: small fragment of MSO on infinite trees (Rabin, 1969).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}. I I I

2

Rules A → α1 | . . . | αn yield equation A = α1 ∪ . . . ∪ αn . Least solution. With operations {∪, ∩, ·}: conjunctive grammars (Okhotin, 2001).

One-sided concatenation: small fragment of MSO on infinite trees (Rabin, 1969). I

Solutions are regular. Everything is decidable.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Two simple well-known cases 1

Ginsburg, Rice (1962): context-free grammars represented by    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with operations {∪, ·}. I I I

2

Rules A → α1 | . . . | αn yield equation A = α1 ∪ . . . ∪ αn . Least solution. With operations {∪, ∩, ·}: conjunctive grammars (Okhotin, 2001).

One-sided concatenation: small fragment of MSO on infinite trees (Rabin, 1969). I I

Solutions are regular. Everything is decidable. Simpler proof of regularity. Everything is EXPTIME-complete. (Baader, Okhotin, 2006).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

3 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ...

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968). I

Turing machine T .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968). I

Turing machine T . VALC(T ) = {w \CT (w ) | w ∈ L(T )}

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968). I

Turing machine T . VALC(T ) = {w \CT (w ) | w ∈ L(T )}

I

Suitable CT : Σ∗ → Γ∗ encodes computations.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968). I

Turing machine T . VALC(T ) = {w \CT (w ) | w ∈ L(T )}

I I

Suitable CT : Σ∗ → Γ∗ encodes computations. Complement of VALC(T) is CF .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968). I

Turing machine T . VALC(T ) = {w \CT (w ) | w ∈ L(T )}

I I I

Suitable CT : Σ∗ → Γ∗ encodes computations. Complement of VALC(T) is CF . VALC(T ) = L(G1 ) ∩ L(G2 ) for context-free G1 , G2 .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Undecidable properties “Given CFG G = (Σ, N, P, S), is L(G ) equal to Σ∗ ?”: undecidable. Solution uniqueness for Xi = ϕi (X1 , . . . , Xn ) is undecidable. S0 = S0 ∪ S S = ... General method: computation histories (Hartmanis, 1968). I

Turing machine T . VALC(T ) = {w \CT (w ) | w ∈ L(T )}

I I I

Suitable CT : Σ∗ → Γ∗ encodes computations. Complement of VALC(T) is CF . VALC(T ) = L(G1 ) ∩ L(G2 ) for context-free G1 , G2 .

Solution existence for ϕi (X1 , . . . , Xn ) = ψi (X1 , . . . , Xn ) is undecidable (Parikh et al., 1985; Charatonik, 1994). Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

4 / 12

Computationally universal solutions Theorem (Okhotin, ICALP 2003) L ⊆ Σ∗ with |Σ| > 2 is given by a least solution of a system ϕ = ψ with {∪, ∩, ·} if and only if L is r.e.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

5 / 12

Computationally universal solutions Theorem (Okhotin, ICALP 2003) L ⊆ Σ∗ with |Σ| > 2 is given by a least solution of a system ϕ = ψ with {∪, ∩, ·} if and only if L is r.e. VALC(T ) = {w \CT (w ) | w ∈ L(T )}: accepting computations.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

5 / 12

Computationally universal solutions Theorem (Okhotin, ICALP 2003) L ⊆ Σ∗ with |Σ| > 2 is given by a least solution of a system ϕ = ψ with {∪, ∩, ·} if and only if L is r.e. VALC(T ) = {w \CT (w ) | w ∈ L(T )}: accepting computations. L(T ) is a least solution of VALC(T ) ⊆ X \Σ∗

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

5 / 12

Computationally universal solutions Theorem (Okhotin, ICALP 2003) L ⊆ Σ∗ with |Σ| > 2 is given by a least solution of a system ϕ = ψ with {∪, ∩, ·} if and only if L is r.e. VALC(T ) = {w \CT (w ) | w ∈ L(T )}: accepting computations. L(T ) is a least solution of VALC(T ) ⊆ X \Σ∗

I

VALC(T ) ∪ X \Σ∗ = X \Σ∗ .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

5 / 12

Computationally universal solutions Theorem (Okhotin, ICALP 2003) L ⊆ Σ∗ with |Σ| > 2 is given by a least (greatest, unique) solution of a system ϕ = ψ with {∪, ∩, ·} if and only if L is r.e. (co-r.e., recursive). VALC(T ) = {w \CT (w ) | w ∈ L(T )}: accepting computations. L(T ) is a least solution of VALC(T ) ⊆ X \Σ∗

I

VALC(T ) ∪ X \Σ∗ = X \Σ∗ .

Greatest and unique solutions.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

5 / 12

Computationally universal solutions Theorem (Okhotin, ICALP 2003) L ⊆ Σ∗ with |Σ| > 2 is given by a least (greatest, unique) solution of a system ϕ = ψ with {∪, ∩, ·} if and only if L is r.e. (co-r.e., recursive). VALC(T ) = {w \CT (w ) | w ∈ L(T )}: accepting computations. L(T ) is a least solution of VALC(T ) ⊆ X \Σ∗

I

VALC(T ) ∪ X \Σ∗ = X \Σ∗ .

Greatest and unique solutions. Matching upper bounds. Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

5 / 12

Fewer operations

Systems ϕ = ψ with {∪, ·} or {∩, ·} (Okhotin, 2005).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

6 / 12

Fewer operations

Systems ϕ = ψ with {∪, ·} or {∩, ·} (Okhotin, 2005). Systems ϕ = ψ with {∆, ·} (Okhotin, 2006).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

6 / 12

Fewer operations

Systems ϕ = ψ with {∪, ·} or {∩, ·} (Okhotin, 2005). Systems ϕ = ψ with {∆, ·} (Okhotin, 2006). LX = XL (Kunc, 2005). Finite constant L ⊆ {a, b}∗ :

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

6 / 12

Fewer operations

Systems ϕ = ψ with {∪, ·} or {∩, ·} (Okhotin, 2005). Systems ϕ = ψ with {∆, ·} (Okhotin, 2006). LX = XL (Kunc, 2005). Finite constant L ⊆ {a, b}∗ : greatest X is co-r.e.-complete.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

6 / 12

Fewer operations

Systems ϕ = ψ with {∪, ·} or {∩, ·} (Okhotin, 2005). Systems ϕ = ψ with {∆, ·} (Okhotin, 2006). LX = XL (Kunc, 2005). Finite constant L ⊆ {a, b}∗ : greatest X is co-r.e.-complete. Multiple-letter alphabet in all cases.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

6 / 12

The one-letter case: Σ = {a} Example (Leiss, 1995) X = aX

2

2

2

has a unique solution {n | ∃i > 0 : 23i 6 n < 23i+2 }.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

7 / 12

The one-letter case: Σ = {a} Example (Leiss, 1995) X = aX

2

2

2

has a unique solution {n | ∃i > 0 : 23i 6 n < 23i+2 }. = {n | base-8 notation of n begins with 1, 2 or 3}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

7 / 12

The one-letter case: Σ = {a} Example (Leiss, 1995) X = aX

2

2

2

has a unique solution {n | ∃i > 0 : 23i 6 n < 23i+2 }. = {n | base-8 notation of n begins with 1, 2 or 3}.

Example (Jez˙ , 2007) X1 = (X1 X3 ∩ X2 X2 ) ∪ {a} X2 = (X1 X1 ∩ X2 X6 ) ∪ {aa} X3 = (X1 X2 ∩ X6 X6 ) ∪ {aaa} X6 = (X1 X2 ∩ X3 X3 ) n

has a least solution Xi = {ai·4 | n > 0}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

7 / 12

The one-letter case: Σ = {a} Example (Leiss, 1995) X = aX

2

2

2

has a unique solution {n | ∃i > 0 : 23i 6 n < 23i+2 }. = {n | base-8 notation of n begins with 1, 2 or 3}.

Example (Jez˙ , 2007) X1 = (X1 X3 ∩ X2 X2 ) ∪ {a} X2 = (X1 X1 ∩ X2 X6 ) ∪ {aa} X3 = (X1 X2 ∩ X6 X6 ) ∪ {aaa} X6 = (X1 X2 ∩ X3 X3 ) n

has a least solution Xi = {ai·4 | n > 0}. In base-4: (10∗ )4 , (20∗ )4 , (30∗ )4 , (120∗ )4 . Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

7 / 12

Base-k notation Theorem (Jez˙ , Okhotin, 2007) Assume L ⊆ {0, 1, . . . , k − 1} is recognized by a one-way real-time cellular automaton. Then (L)k = {an | base-k notation of n is in L} is generated by a conjunctive grammar.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

8 / 12

Base-k notation Theorem (Jez˙ , Okhotin, 2007) Assume L ⊆ {0, 1, . . . , k − 1} is recognized by a one-way real-time cellular automaton. Then (L)k = {an | base-k notation of n is in L} is generated by a conjunctive grammar. That is, by unique solutions of systems    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with {∪, ∩, ·}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

8 / 12

Base-k notation Theorem (Jez˙ , Okhotin, 2007) Assume L ⊆ {0, 1, . . . , k − 1} is recognized by a one-way real-time cellular automaton. Then (L)k = {an | base-k notation of n is in L} is generated by a conjunctive grammar. That is, by unique solutions of systems    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with {∪, ∩, ·}. VALC(T ) recognized by such CA.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

8 / 12

Base-k notation Theorem (Jez˙ , Okhotin, 2007) Assume L ⊆ {0, 1, . . . , k − 1} is recognized by a one-way real-time cellular automaton. Then (L)k = {an | base-k notation of n is in L} is generated by a conjunctive grammar. That is, by unique solutions of systems    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with {∪, ∩, ·}. VALC(T ) recognized by such CA. Consider VALC(T ) over Σ = {0, 1, . . . , k − 1}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

8 / 12

Base-k notation Theorem (Jez˙ , Okhotin, 2007) Assume L ⊆ {0, 1, . . . , k − 1} is recognized by a one-way real-time cellular automaton. Then (L)k = {an | base-k notation of n is in L} is generated by a conjunctive grammar. That is, by unique solutions of systems    X1 = ϕ1 (X1 , . . . , Xn ) .. .   Xn = ϕn (X1 , . . . , Xn ) with {∪, ∩, ·}. VALC(T ) recognized by such CA. Consider VALC(T ) over Σ = {0, 1, . . . , k − 1}. (VALC(T ))k represented by language equations. Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

8 / 12

Extracting L(T ) from (VALC(T ))k Cf: VALC(T ) ⊆ X ]Σ∗ for larger alphabets.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

9 / 12

Extracting L(T ) from (VALC(T ))k Cf: VALC(T ) ⊆ X ]Σ∗ for larger alphabets. Here: same general idea, more difficult.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

9 / 12

Extracting L(T ) from (VALC(T ))k Cf: VALC(T ) ⊆ X ]Σ∗ for larger alphabets. Here: same general idea, more difficult. Extract number (w )k from number (CT (w )\w )k .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

9 / 12

Extracting L(T ) from (VALC(T ))k Cf: VALC(T ) ⊆ X ]Σ∗ for larger alphabets. Here: same general idea, more difficult. Extract number (w )k from number (CT (w )\w )k . A special encoding of VALC(T ) over base-6 alphabet.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

9 / 12

Extracting L(T ) from (VALC(T ))k Cf: VALC(T ) ⊆ X ]Σ∗ for larger alphabets. Here: same general idea, more difficult. Extract number (w )k from number (CT (w )\w )k . A special encoding of VALC(T ) over base-6 alphabet.

Theorem (Jez˙ , Okhotin, ICALP 2008) L ⊆ a∗ is given by unique (least, greatest) solution of a system ϕ = ψ with operations {∪, ∩, ·} if and only if L is recursive (r.e., co-r.e.).

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

9 / 12

Extracting L(T ) from (VALC(T ))k Cf: VALC(T ) ⊆ X ]Σ∗ for larger alphabets. Here: same general idea, more difficult. Extract number (w )k from number (CT (w )\w )k . A special encoding of VALC(T ) over base-6 alphabet.

Theorem (Jez˙ , Okhotin, ICALP 2008) L ⊆ a∗ is given by unique (least, greatest) solution of a system ϕ = ψ with operations {∪, ·} if and only if L is recursive (r.e., co-r.e.). Intersection not needed! (reconstruction from scratch)

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

9 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only?

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only? Denote τi (L) = {a16n+i | an ∈ L}: track i.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only? Denote τi (L) = {a16n+i | an ∈ L}: track i. Simulate equations with {∪, ·} using only {·}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only? Denote τi (L) = {a16n+i | an ∈ L}: track i. Simulate equations with {∪, ·} using only {·}.

Theorem (Jez˙ , Okhotin, STACS 2009) For every recursive (r.e., co-r.e.) language L ⊆ a∗ , the set σ(L) = {ε} ∪ τ6 (N) ∪ τ8 (N) ∪ τ9 (N) ∪ τ12 (N) ∪ τ13 (L). is representable by unique (least, greatest) solution of a system with {·}.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only? Denote τi (L) = {a16n+i | an ∈ L}: track i. Simulate equations with {∪, ·} using only {·}.

Theorem (Jez˙ , Okhotin, STACS 2009) For every recursive (r.e., co-r.e.) language L ⊆ a∗ , the set σ(L) = {ε} ∪ τ6 (N) ∪ τ8 (N) ∪ τ9 (N) ∪ τ12 (N) ∪ τ13 (L). is representable by unique (least, greatest) solution of a system with {·}.

Encoding checkedSby S X · {ε, a4 , a11 } = i∈{0,4,6,8,9, τi (a∗ ) ∪ i∈{1,3,7} τi (a+ ) ∪ {a11 }. 10,12,13}

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only? Denote τi (L) = {a16n+i | an ∈ L}: track i. Simulate equations with {∪, ·} using only {·}.

Theorem (Jez˙ , Okhotin, STACS 2009) For every recursive (r.e., co-r.e.) language L ⊆ a∗ , the set σ(L) = {ε} ∪ τ6 (N) ∪ τ8 (N) ∪ τ9 (N) ∪ τ12 (N) ∪ τ13 (L). is representable by unique (least, greatest) solution of a system with {·}.

Encoding checkedSby S X · {ε, a4 , a11 } = i∈{0,4,6,8,9, τi (a∗ ) ∪ i∈{1,3,7} τi (a+ ) ∪ {a11 }. 10,12,13}

KL = MN checked by σ(K )σ(L){ε, a} = σ(M)σ(N){ε, a}. Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

Unary alphabet, only concatenation Can anything be represented with concatenation only? Denote τi (L) = {a16n+i | an ∈ L}: track i. Simulate equations with {∪, ·} using only {·}.

Theorem (Jez˙ , Okhotin, STACS 2009) For every recursive (r.e., co-r.e.) language L ⊆ a∗ , the set σ(L) = {ε} ∪ τ6 (N) ∪ τ8 (N) ∪ τ9 (N) ∪ τ12 (N) ∪ τ13 (L). is representable by unique (least, greatest) solution of a system with {·}.

Encoding checkedSby S X · {ε, a4 , a11 } = i∈{0,4,6,8,9, τi (a∗ ) ∪ i∈{1,3,7} τi (a+ ) ∪ {a11 }. 10,12,13}

KL = MN checked by σ(K )σ(L){ε, a} = σ(M)σ(N){ε, a}. K ∪ L = M ∪ N checked by σ(K )σ(L){ε, a2 } = σ(M)σ(N){ε, a2 }. Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

10 / 12

One variable, two equations Encode all variables into one by π : 2a

Alexander Okhotin

Language equations

∗


n

∗

→ 2a .

Dagstuhl, 2010 A. D.

11 / 12

One variable, two equations Encode all variables into one by π : 2a

∗


n

∗

→ 2a .

Theorem (Lehtinen, Okhotin, 2010) For every recursive (r.e., co-r.e.) language L0 ⊆ a∗ there exists a system  XXK = XL XM = N with a unique (least, greatest, respectively) solution X = L00 , such that an ∈ L0 if and only if apn+d ∈ L00 .

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

11 / 12

One variable, two equations Encode all variables into one by π : 2a

∗


n

∗

→ 2a .

Theorem (Lehtinen, Okhotin, 2010) For every recursive (r.e., co-r.e.) language L0 ⊆ a∗ there exists a system  XXK = XL XM = N with a unique (least, greatest, respectively) solution X = L00 , such that an ∈ L0 if and only if apn+d ∈ L00 . Apparently the simplest form of language equations.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

11 / 12

One variable, two equations Encode all variables into one by π : 2a

∗


n

∗

→ 2a .

Theorem (Lehtinen, Okhotin, 2010) For every recursive (r.e., co-r.e.) language L0 ⊆ a∗ there exists a system  XXK = XL XM = N with a unique (least, greatest, respectively) solution X = L00 , such that an ∈ L0 if and only if apn+d ∈ L00 . Apparently the simplest form of language equations. Decision problems for such systems:

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

11 / 12

One variable, two equations Encode all variables into one by π : 2a

∗


n

∗

→ 2a .

Theorem (Lehtinen, Okhotin, 2010) For every recursive (r.e., co-r.e.) language L0 ⊆ a∗ there exists a system  XXK = XL XM = N with a unique (least, greatest, respectively) solution X = L00 , such that an ∈ L0 if and only if apn+d ∈ L00 . Apparently the simplest form of language equations. Decision problems for such systems: I

Existence of solution: Π01 -complete;

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

11 / 12

One variable, two equations Encode all variables into one by π : 2a

∗


n

∗

→ 2a .

Theorem (Lehtinen, Okhotin, 2010) For every recursive (r.e., co-r.e.) language L0 ⊆ a∗ there exists a system  XXK = XL XM = N with a unique (least, greatest, respectively) solution X = L00 , such that an ∈ L0 if and only if apn+d ∈ L00 . Apparently the simplest form of language equations. Decision problems for such systems: I I

Existence of solution: Π01 -complete; Uniqueness of solution: Π02 -complete;

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

11 / 12

One variable, two equations Encode all variables into one by π : 2a

∗


n

∗

→ 2a .

Theorem (Lehtinen, Okhotin, 2010) For every recursive (r.e., co-r.e.) language L0 ⊆ a∗ there exists a system  XXK = XL XM = N with a unique (least, greatest, respectively) solution X = L00 , such that an ∈ L0 if and only if apn+d ∈ L00 . Apparently the simplest form of language equations. Decision problems for such systems: I I I

Existence of solution: Π01 -complete; Uniqueness of solution: Π02 -complete; Having finitely many solutions: Σ03 -complete;

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

11 / 12

Any future work? Computational completeness in language equations:

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties?

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I

“Does a given system have countably many solutions?”

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Any more powerful models?

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Any more powerful models? I

Equations over sets of integers (Jez˙ , Okhotin, 2009–2010): reach up to Σ11 sets.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Any more powerful models? I

Equations over sets of integers (Jez˙ , Okhotin, 2009–2010): reach up to Σ11 sets.

Any weaker models?

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Any more powerful models? I

Equations over sets of integers (Jez˙ , Okhotin, 2009–2010): reach up to Σ11 sets.

Any weaker models? I

Yes, formal grammars.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Any more powerful models? I

Equations over sets of integers (Jez˙ , Okhotin, 2009–2010): reach up to Σ11 sets.

Any weaker models? I I

Yes, formal grammars. Problems for conjunctive and Boolean grammars.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12

Any future work? Computational completeness in language equations: I I I I

All or almost all rec/r.e./co-r.e. sets by unique/least/greatest sol. Upper bound: recursive constants, computable operations. Lower bound: XXK = XL, XM = N, all unary. No more room for improvement.

Any further properties? I I

“Does a given system have countably many solutions?” More to consider.

Any more powerful models? I

Equations over sets of integers (Jez˙ , Okhotin, 2009–2010): reach up to Σ11 sets.

Any weaker models? I I I

Yes, formal grammars. Problems for conjunctive and Boolean grammars. Perhaps, more general models.

Alexander Okhotin

Language equations

Dagstuhl, 2010 A. D.

12 / 12