Iteration in Residuated Structures by Stepan Kuznetsov from Steklov - PowerPoint PPT Presentation

Iteration in Residuated Structures by Stepan Kuznetsov from Steklov Mathematical Institute (Moscow) October 20, 2017, The Wormshop

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤�

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit;

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit; ◮ \ and / are residuals of · : a ≤ c / b ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a \ c

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit; ◮ \ and / are residuals of · : a ≤ c / b ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a \ c ◮ ∨ and ∧ are for the lattice structure: a ∨ b = sup { a , b } , a ∧ b = inf { a , b } .

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit; ◮ \ and / are residuals of · : a ≤ c / b ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a \ c ◮ ∨ and ∧ are for the lattice structure: a ∨ b = sup { a , b } , a ∧ b = inf { a , b } . ◮ ∗ , the general case: 1 ∨ a ∨ ( a ∗ · a ∗ ) ≤ a ∗ , and if 1 ∨ a ∨ ( b · b ) ≤ b , then a ∗ ≤ b .

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit; ◮ \ and / are residuals of · : a ≤ c / b ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a \ c ◮ ∨ and ∧ are for the lattice structure: a ∨ b = sup { a , b } , a ∧ b = inf { a , b } . ◮ ∗ , the general case: 1 ∨ a ∨ ( a ∗ · a ∗ ) ≤ a ∗ , and if 1 ∨ a ∨ ( b · b ) ≤ b , then a ∗ ≤ b . ◮ ∗ , the ∗ -continuous case: p · q ∗ · r = sup { p · q n · r | n ≥ 0 }

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit; ◮ \ and / are residuals of · : a ≤ c / b ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a \ c ◮ ∨ and ∧ are for the lattice structure: a ∨ b = sup { a , b } , a ∧ b = inf { a , b } . ◮ ∗ , the general case: 1 ∨ a ∨ ( a ∗ · a ∗ ) ≤ a ∗ , and if 1 ∨ a ∨ ( b · b ) ≤ b , then a ∗ ≤ b . ◮ ∗ , the ∗ -continuous case: p · q ∗ · r = sup { p · q n · r | n ≥ 0 } References: Pratt 1990, Kozen 1994.

Residuated Kleene Algebras (Action Algebras) � A ; · , 1 , \ , /, ∨ , ∧ , ∗ , ≤� ◮ · gives a monoid structure, 1 is the unit; ◮ \ and / are residuals of · : a ≤ c / b ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a \ c ◮ ∨ and ∧ are for the lattice structure: a ∨ b = sup { a , b } , a ∧ b = inf { a , b } . ◮ ∗ , the general case: 1 ∨ a ∨ ( a ∗ · a ∗ ) ≤ a ∗ , and if 1 ∨ a ∨ ( b · b ) ≤ b , then a ∗ ≤ b . ◮ ∗ , the ∗ -continuous case: p · q ∗ · r = sup { p · q n · r | n ≥ 0 } References: Pratt 1990, Kozen 1994. Standard example: the algebra of languages over an alphabet, possibly with the empty word.

In This Talk... ... we consider the positive version of Kleene iteration ( + instead of ∗ ):

In This Talk... ... we consider the positive version of Kleene iteration ( + instead of ∗ ): ◮ a semigroup instead of a monoid;

In This Talk... ... we consider the positive version of Kleene iteration ( + instead of ∗ ): ◮ a semigroup instead of a monoid; ◮ a ∨ ( a + · a + ) ≤ a + , and if a ∨ ( b · b ) ≤ b , then a + ≤ b (for the general case);

In This Talk... ... we consider the positive version of Kleene iteration ( + instead of ∗ ): ◮ a semigroup instead of a monoid; ◮ a ∨ ( a + · a + ) ≤ a + , and if a ∨ ( b · b ) ≤ b , then a + ≤ b (for the general case); ◮ p · q + · r = sup { p · q n · r | n ≥ 1 } (for the ∗ -continuous case).

In This Talk... ... we consider the positive version of Kleene iteration ( + instead of ∗ ): ◮ a semigroup instead of a monoid; ◮ a ∨ ( a + · a + ) ≤ a + , and if a ∨ ( b · b ) ≤ b , then a + ≤ b (for the general case); ◮ p · q + · r = sup { p · q n · r | n ≥ 1 } (for the ∗ -continuous case). Standard example: the algebra of languages without the empty word.

Multiplicative-Only Fragment (the Lambek Calculus with Iteration, L + ω ) (for the ∗ -continuous case; cf. ACT ω by Buszkowski and Palka 2005–08) A → A A Π → B Π → A Γ B ∆ → C Π → A \ B , where Π is not empty Γ Π ( A \ B ) ∆ → C Π A → B Π → A Γ B ∆ → C Π → B / A , where Π is not empty Γ ( B / A ) Π ∆ → C Γ , A , B , ∆ → C Γ → A ∆ → B Γ , ∆ → A · B Γ , A · B , ∆ → C Γ , A n , ∆ → C for all n ≥ 1 Γ 1 → A . . . Γ n → A ( n ≥ 1) Γ 1 , . . . , Γ n → A + Γ , A + , ∆ → C Π → A Γ A ∆ → C ( cut ) Γ Π ∆ → C

Complexity Result Theorem L + ω is Π 0 1 -complete.

Complexity Result Theorem L + ω is Π 0 1 -complete. Proof idea: following Buszkowski & Palka for ACT ω , encode the totality problem for context-free grammars. The key trick that allows avoiding ∨ and ∧ is the usage of Lambek grammars with unique type assignment [Safiullin 2007].

Complexity Result Theorem L + ω is Π 0 1 -complete. Proof idea: following Buszkowski & Palka for ACT ω , encode the totality problem for context-free grammars. The key trick that allows avoiding ∨ and ∧ is the usage of Lambek grammars with unique type assignment [Safiullin 2007]. CFG → Lambek categorial grammar. a 1 ⊲ A 1 , a 2 ⊲ A 2 , C is the goal category. (Alphabet { a 1 , a 2 } ) a 1 ... a n ∈ L ⇐ ⇒ A 1 . . . A n → C is derivable. Checking derivability of ( A + · B + ) + → C is roughly equivalent to checking totality for the CFG.

Complexity Result Theorem L + ω is Π 0 1 -complete. Proof idea: following Buszkowski & Palka for ACT ω , encode the totality problem for context-free grammars. The key trick that allows avoiding ∨ and ∧ is the usage of Lambek grammars with unique type assignment [Safiullin 2007]. CFG → Lambek categorial grammar. a 1 ⊲ A 1 , a 2 ⊲ A 2 , C is the goal category. (Alphabet { a 1 , a 2 } ) a 1 ... a n ∈ L ⇐ ⇒ A 1 . . . A n → C is derivable. Checking derivability of ( A + · B + ) + → C is roughly equivalent to checking totality for the CFG. Open question: Safiullin’s result is not known for the case with empty word. Therefore, we cannot yet replace + with ∗ .

On The Other Side... Pratt’s axiomatisation for general (non necessarily ∗ -continuous) action algebras (a variant with positive iteration): A → A ( A · B ) · C → A · ( B · C ) A · ( B · C ) → ( A · B ) · C A → C / B B → A \ C A · B → C A · B → C A · B → C A → C / B A · B → C B → A \ C A → B i A 1 → B A 2 → B A → B B → C A → C A → B 1 ∨ B 2 A 1 ∨ A 2 → B A i → B A → B 1 A → B 2 A 1 ∧ A 2 → B A → B 1 ∧ B 2 A ∨ ( B · B ) → B A ∨ ( A + · A + ) → A + A + → B

On The Other Side... Pratt’s axiomatisation for general (non necessarily ∗ -continuous) action algebras (a variant with positive iteration): A → A ( A · B ) · C → A · ( B · C ) A · ( B · C ) → ( A · B ) · C A → C / B B → A \ C A · B → C A · B → C A · B → C A → C / B A · B → C B → A \ C A → B i A 1 → B A 2 → B A → B B → C A → C A → B 1 ∨ B 2 A 1 ∨ A 2 → B A i → B A → B 1 A → B 2 A 1 ∧ A 2 → B A → B 1 ∧ B 2 A ∨ ( B · B ) → B A ∨ ( A + · A + ) → A + A + → B NB: Pratt 1990 doesn’t cite Lambek 1958 (but cites Girard 1987).

Induction vs. *-continuity ACT ω is Π 0 1 -complete (Buszkowski & Palka); ACT Pratt is in Σ 0 1 (r.e.)

Induction vs. *-continuity ACT ω is Π 0 1 -complete (Buszkowski & Palka); ACT Pratt is in Σ 0 1 (r.e.) Therefore:

Induction vs. *-continuity ACT ω is Π 0 1 -complete (Buszkowski & Palka); ACT Pratt is in Σ 0 1 (r.e.) Therefore: ◮ there exists an action algebra that is not ∗ -continuous;

Induction vs. *-continuity ACT ω is Π 0 1 -complete (Buszkowski & Palka); ACT Pratt is in Σ 0 1 (r.e.) Therefore: ◮ there exists an action algebra that is not ∗ -continuous; ◮ the equational theories of all action algebras and ∗ -continuous action algebras differ, even in the fragment without ∨ and ∧ (for positive iteration).

Induction vs. *-continuity ACT ω is Π 0 1 -complete (Buszkowski & Palka); ACT Pratt is in Σ 0 1 (r.e.) Therefore: ◮ there exists an action algebra that is not ∗ -continuous; ◮ the equational theories of all action algebras and ∗ -continuous action algebras differ, even in the fragment without ∨ and ∧ (for positive iteration). Note that, as shown by Kozen, for the case without \ and / (but with ∨ ) the equational theories coincide.

Induction vs. *-continuity ACT ω is Π 0 1 -complete (Buszkowski & Palka); ACT Pratt is in Σ 0 1 (r.e.) Therefore: ◮ there exists an action algebra that is not ∗ -continuous; ◮ the equational theories of all action algebras and ∗ -continuous action algebras differ, even in the fragment without ∨ and ∧ (for positive iteration). Note that, as shown by Kozen, for the case without \ and / (but with ∨ ) the equational theories coincide. Open question 1: construct a concrete example of a formula valid in all *-continuous action algebras, but not in all action algebras.