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On the complexity of the equational theory of generalized residuated boolean algerbas Zhe Lin and Minghui Ma Institute of Logic and Cognition, Sun Yat-Sen University TACL2017 Praha Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

R-algebras A residuated Boolean algebra , or r-algebra ,(B.J ´ o nsson and Tsinakis) is an algebra A = ( A , ∧ , ∨ , ′ , ⊤ , ⊥ , · , \ , / ) where ( A , ∧ , ∨ , ′ , ⊤ , ⊥ ) is a Boolean algebra, and · , \ and / are binary operators on A satisfying the following residuation property: for any a , b , c ∈ A , a · b ≤ c b ≤ a \ c a ≤ c / b iff iff The operators \ and / are called right and left residuals of · respectively. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

The left and right conjugates of · are binary operators on A defined by setting a ⊲ c = ( a \ c ′ ) ′ and c ⊲ b = ( c ′ / b ) ′ . The following conjugation property holds for any a , b , c ∈ A : a · b ≤ c ′ a ⊲ c ≤ b ′ c ⊳ b ≤ a ′ iff iff Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Let K be any class of algebras. The equational theory of K , denoted by Eq ( K ), is the set of all equations of the form s = t that are valid in K . The universal theory of K is the set of all first-order universal sentences that are valid in K denoted by Ueq ( K ), Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Eq ( NA ) is decidable (N´ emeti 1987) Eq ( UR ) is decidable. (Jipsen 1992) Ueq ( UR ) and Ueq ( RA ) are decidable (Buszkowski 2011) Eq ( ARA ) is undecidable (Kurucz, Nemeti, Sain and Simon 1993) Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Generalized residuated Boolean algebra Generalized residuated algebras admit a finite number of finitary operations o . With each n-ary operation ( o i ) (1 ≤ i ≤ m ) there are associated n residual operations ( o i / j ) (1 ≤ j ≤ n ) which satisfy the following generalized residuation law: ( o i )( α 1 , . . . , α n ) ≤ β iff α j ≤ ( o i / j )( α 1 , . . . , α j − 1 , β, α j +1 , . . . , α n ) A generalized residuated Boolean algebra is a Boolean algebra with generalized residual operations. A generalized residuated distributive lattice and lattice are defined naturally. The logics are denoted by RBL, RDLL, RLL respectively. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Figure: Outline of Proof Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Sequent Calculus ( Id ) A ⇒ A , ( D ) A ∧ ( B ∨ C ) ⇒ ( A ∧ B ) ∨ ( A ∧ C ) , ( ⊥ ) Γ[ ⊥ ] ⇒ A , ( ⊤ ) Γ ⇒ ⊤ , ( ¬ 1 ) A ∧ ¬ A ⇒ ⊥ , ( ¬ 2 ) ⊤ ⇒ A ∨ ¬ A , Γ[ A i ] ⇒ B ( ∧ R ) Γ ⇒ A Γ ⇒ B ( ∧ L ) Γ[ A 1 ∧ A 2 ] ⇒ B , , Γ ⇒ A ∧ B ( ∨ L ) Γ[ A 1 ] ⇒ B Γ[ A 2 ] ⇒ B Γ ⇒ A i ( ∨ R ) , . Γ[ A 1 ∨ A 2 ] ⇒ B Γ ⇒ A 1 ∨ A 2 ∆ ⇒ A ; Γ[ A ] ⇒ B ( Cut ) Γ[∆] ⇒ B Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Γ[( ϕ 1 , . . . , ϕ n ) o i ] ⇒ α Γ 1 ⇒ ϕ 1 ; . . . ; Γ n ⇒ ϕ n Γ[( o i )( ϕ 1 , . . . , ϕ n )] ⇒ α ( o i L ) ( o i R ) (Γ 1 , . . . , Γ n ) o i ⇒ α Γ[ ϕ j ] ⇒ α, ; Γ 1 ⇒ ϕ 1 ; . . . ; Γ n ⇒ ϕ n Γ[(Γ 1 , . . . , ( o i / j )( ϕ 1 , . . . , ϕ n ) , . . . , Γ n ) o i ] ⇒ α (( o i / j ) L ) ( ϕ 1 , . . . , Γ , . . . , ϕ ) o i ⇒ α Γ ⇒ ( o i / j )( ϕ 1 , . . . , Γ , . . . , ϕ )(( o i / j ) R ) Remark All above rules are invertible. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Frame semantics A frame is a pair F = ( W , R ) where W � = ∅ and R ⊆ W n +1 is an n + 1-ary relation on W . A model is a triple M = ( W , R , V ) where ( W , R ) is a frame and V : P → ℘ ( W ) is a valuation from the set of propositional variables P to the powerset of W . The satisfaction relation M , w | = ϕ between a model M with a point w and a formula ϕ is defined inductively as follows: Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

1 M , w | = p iff w ∈ V ( p ). 2 M , w �| = ⊥ . 3 M , w | = ϕ ⊃ ψ iff M , w �| = ϕ or M , w | = ψ . 4 M , w | = o ( ϕ 1 , . . . , ϕ n ) iff there are points u 1 , . . . , u n ∈ W such that Rwu 1 . . . u n and M , u i | = ϕ i for 1 ≤ i ≤ n . 5 M , w | = ( o / i )( ϕ 1 , . . . , ϕ n ) iff for all u 1 , . . . , u n ∈ W , if Ru i u 1 . . . w . . . u n and M , u j | = ϕ j for all 1 ≤ j ≤ n and j � = i , then M , u i | = ϕ i . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Unary case: 1 M , w | = ♦ A iff there exists u ∈ W with R ( w , u ) and M , u | = A . = � ↓ A iff for every u ∈ W , if R ( u , w ), then M , u | 2 M , w | = A . Binary case: 1 J , u | = A / B iff for all v , w ∈ W with S ( w , u , v ), if J , v | = B , then J , w | = A 2 J , u | = A \ B iff for all v , w ∈ W with S ( v , w , u ), if J , w | = A , then J , v | = B . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

From RBL to MRBNL The translation ( . ) # : L RBL (Prop) → L MRBNL (Prop) is defined as below: o i ( α 1 , . . . α n ) ‡ = ( . . . ( α 1 · i α 2 ) . . . ) · i α n ) . . . ) ( o i / j )( α 1 , . . . , α n ) = ( . . . ( α 1 · i α 2 ) . . . ) · i α j − 1 ) \ i ( . . . ( α j / i α n ) . . . / i α j +1 ) ((Γ 1 , . . . , Γ n ) o i ) ‡ = ( . . . (Γ 1 ◦ i Γ 2 ) . . . ) ◦ i Γ n ) . . . ) Theorem For any L RBL -sequent Γ ⇒ α , ⊢ RBL Γ ⇒ α if and only if ⊢ MRBNL ((Γ)) † ⊃ α † . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

From MRBNL to MK t The translation ( . ) # : L MBFNL (Prop) → L MK t (Prop) is defined as below: p # = p , ⊤ # = ⊤ , ⊥ # = ⊥ , ( ¬ α ) # = ¬ α # , ( α ∧ β ) # = α # ∧ β # , ( α ∨ β ) # = α # ∨ β # , ( α · i β ) # = ♦ i 1 ( ♦ i 1 α # ∧ ♦ i 2 β # ) , ( α \ i β ) # = � ↓ i 2 ( ♦ i 1 α # ⊃ � ↓ ( α/ i β ) # = � ↓ i 1 ( ♦ i 2 β # ⊃ � ↓ i 1 β # ) , i 1 α # ) . Theorem For any L MBFNL -sequent Γ ⇒ α , ⊢ MBFNL Γ ⇒ α if and only if ⊢ MK t ( f (Γ)) # ⊃ α # . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Figure: Translation # Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

From MK t to K t Let P ⊆ Prop and { x , q 1 , . . . , q n } �⊆ P be a distinguished propositional variable. Define a translation ( . ) ∗ : L K t 12 (P) → L K . t (P ∪ { x , q 1 , . . . , q n } ) recursively as follows: p ∗ = p , ⊥ ∗ = ⊥ , ( A ⊃ B ) ∗ = A ∗ ⊃ B ∗ . ( ♦ i A ) ∗ = ¬ x ∧ ♦ ( q i ∧ A ∗ ) , i A ) ∗ = ¬ x ⊃ � ↓ ( q i ⊃ A ∗ ) , ( � ↓ Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

Theorem For any L MK t -sequent Γ ⇒ α , ⊢ MK t Γ ⇒ α if and only if ⊢ K t ( f (Γ)) ∗ ⊃ α ∗ . Figure: Translation ∗ Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

( Id ) A ⇒ A , and inference rules Γ[ A ◦ B ] ⇒ C Γ ⇒ A ∆ ⇒ B ( · L ) Γ[ A · B ] ⇒ C , ( · R ) Γ ◦ ∆ ⇒ A · B , ∆ ⇒ A ; Γ[ A ] ⇒ B ( Cut ) Γ[∆] ⇒ B Γ[ A i ] ⇒ B ( ∧ R ) Γ ⇒ A Γ ⇒ B ( ∧ L ) Γ[ A 1 ∧ A 2 ] ⇒ B , , Γ ⇒ A ∧ B ( ∨ L ) Γ[ A 1 ] ⇒ B Γ[ A 2 ] ⇒ B Γ ⇒ A i ( ∨ R ) , . Γ[ A 1 ∨ A 2 ] ⇒ B Γ ⇒ A 1 ∨ A 2 ( · L), ( · R), ( ∧ R ) and ( ∨ L ) are invertible. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated

PSPACE-hard Lemma If ⊢ LG Γ[ A ∧ B ] ⇒ C and all formulae in Γ[ A ∧ B ] are ∨ -free and C is ∧ -free, then Γ[ A ] ⇒ C or Γ[ B ] ⇒ C. Lemma If ⊢ LG Γ ⇒ A ∨ B and all formulae in Γ are ∨ -free, then Γ ⇒ A or Γ[ B ] ⇒ B. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-Sen University On the complexity of the equational theory of generalized residuated