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-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas R-algebras A residuated Boolean algebra, or r-algebra,(B.J´onsson and Tsinakis) is an algebra A = (A; ^; _;0 ; >; ?; ·; n; =) where (A; ^; _;0 ; >; ?) is a Boolean algebra, and ·; n and = are binary operators on A satisfying the following residuation property: for any a; b; c 2 A, a · b ≤ c iff b ≤ anc iff a ≤ c=b The operators n and = are called right and left residuals of · respectively. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas The left and right conjugates of · are binary operators on A defined by setting 0 0 0 0 a B c = (anc ) and c B b = (c =b) . The following conjugation property holds for any a; b; c 2 A: 0 0 0 a · b ≤ c iff a B c ≤ b iff c C b ≤ a Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas 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-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Eq(NA) is decidable (N´emeti1987) 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-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Generalized residuated Boolean algebra Generalized residuated algebras admit a finite number of finitary operations o. With each n-ary operation (oi ) (1 ≤ i ≤ m) there are associated n residual operations (oi =j) (1 ≤ j ≤ n) which satisfy the following generalized residuation law: (oi )(α1; : : : ; αn) ≤ β iff αj ≤ (oi =j)(α1; : : : ; αj−1; β; αj+1; : : : ; αn) 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-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Figure: Outline of Proof Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Sequent Calculus (Id) A ) A; (D) A ^ (B _ C) ) (A ^ B) _ (A ^ C); (?) Γ[?] ) A; (>)Γ ) >; (:1) A ^ :A )?; (:2) > ) A _:A; Γ[A ] ) B Γ ) A Γ ) B (^L) i ; (^R) ; Γ[A1 ^ A2] ) B Γ ) A ^ B Γ[A ] ) B Γ[A ] ) B Γ ) A (_L) 1 2 ; (_R) i : Γ[A1 _ A2] ) B Γ ) A1 _ A2 ∆ ) A; Γ[A] ) B (Cut) Γ[∆] ) B Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Γ[('1;:::;'n)oi ] ) α Γ1 ) '1; ::: ;Γn ) 'n (oi L) (oi R) Γ[(oi )('1;:::;'n)] ) α (Γ1;:::; Γn)oi ) α Γ['j ] ) α; ;Γ1 ) '1; ::: ;Γn ) 'n ((oi =j)L) Γ[(Γ1;:::; (oi =j)('1;:::;'n);:::; Γn)oi ] ) α ('1;:::; Γ;:::;')oi ) α ((oi =j)R) Γ ) (oi =j)('1;:::; Γ;:::;') Remark All above rules are invertible. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Frame semantics A frame is a pair F = (W ; R) where W 6= ; 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 j= ' between a model M with a point w and a formula ' is defined inductively as follows: Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas 1 M; w j= p iff w 2 V (p). 2 M; w 6j= ?. 3 M; w j= ' ⊃ iff M; w 6j= ' or M; w j= . 4 M; w j= o('1;:::;'n) iff there are points u1;:::; un 2 W such that Rwu1 ::: un and M; ui j= 'i for 1 ≤ i ≤ n. 5 M; w j= (o=i)('1;:::;'n) iff for all u1;:::; un 2 W , if Rui u1 ::: w ::: un and M; uj j= 'j for all 1 ≤ j ≤ n and j 6= i, then M; ui j= 'i . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Unary case: 1 M; w j= ♦A iff there exists u 2 W with R(w; u) and M; u j= A. # 2 M; w j= A iff for every u 2 W , if R(u; w), then M; u j= A. Binary case: 1 J; u j= A=B iff for all v; w 2 W with S(w; u; v), if J; v j= B, then J; w j= A 2 J; u j= AnB iff for all v; w 2 W with S(v; w; u), if J; w j= A, then J; v j= B. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas From RBL to MRBNL # The translation (:) : LRBL(Prop) !LMRBNL(Prop) is defined as below: z oi (α1; : : : αn) = (::: (α1 ·i α2) :::) ·i αn) :::) (oi =j)(α1; : : : ; αn) = (::: (α1 ·i α2) :::) ·i αj−1)ni (::: (αj =i αn) : : : =i αj+1) z ((Γ1;:::; Γn)oi ) = (::: (Γ1 ◦i Γ2) :::) ◦i Γn) :::) Theorem For any LRBL-sequent Γ ) α, `RBL Γ ) α if and only if y y `MRBNL ((Γ)) ⊃ α . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas From MRBNL to MKt # The translation (:) : LMBFNL(Prop) !LMKt (Prop) is defined as below: p# = p; ># = >; ?# = ?; (:α)# = :α#; (α ^ β)# = α# ^ β#; # # # # # # (α _ β) = α _ β ; (α ·i β) = ♦i1(♦i1α ^ ♦i2β ); # # # # # # # # # # (αni β) = i2(♦i1α ⊃ i1β ); (α=i β) = i1(♦i2β ⊃ i1α ): Theorem For any LMBFNL-sequent Γ ) α, `MBFNL Γ ) α if and only if # # `MKt (f (Γ)) ⊃ α . Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Figure: Translation # Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas From MKt to Kt Let P ⊆ Prop and fx; q1;:::; qng 6⊆ P be a distinguished propositional variable. Define a translation ∗ (:) : L t (P) !LK:t(P [ fx; q1;:::; qng) recursively as follows: K12 p∗ = p; ?∗ = ?; (A ⊃ B)∗ = A∗ ⊃ B∗: ∗ ∗ (♦i A) = :x ^ ♦(qi ^ A ); # ∗ # ∗ (i A) = :x ⊃ (qi ⊃ A ); Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Theorem For any LMKt -sequent Γ ) α, `MKt Γ ) α if and only if ∗ ∗ `Kt (f (Γ)) ⊃ α . Figure: Translation ∗ Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas (Id) A ) A; and inference rules Γ[A ◦ B] ) C Γ ) A ∆ ) B (·L) ; (·R) ; Γ[A · B] ) C Γ ◦ ∆ ) A · B ∆ ) A; Γ[A] ) B (Cut) Γ[∆] ) B Γ[A ] ) B Γ ) A Γ ) B (^L) i ; (^R) ; Γ[A1 ^ A2] ) B Γ ) A ^ B Γ[A ] ) B Γ[A ] ) B Γ ) A (_L) 1 2 ; (_R) i : Γ[A1 _ A2] ) B Γ ) A1 _ A2 (· L), (· R), (^R) and (_L) are invertible. Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas 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-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas By σ(e) we denote a formula structure z1 ◦ (z2 ··· (zn−1 ◦ zn) ··· ) such that 8 <xj if e(xj ) = 1 zj = :xj if e(xj ) = 0 σ(A) = σ(D1) _ ::: _ σ(Dm) and σ(Di ) = y1 · (y2 ··· (yn−1 · yn) ··· ) such that 8 >xj if xj 2 Di <> yj = xj if :xj 2 Di > > :xj _ xj o:w: Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Lemma e(A) = 1 iff `LG σ(e) ) σ(A) Zhe Lin and Minghui MaInstitute of Logic and Cognition, Sun Yat-SenOn University the complexity of the equational theory of generalized residuated boolean algerbas Let us consider a quantified Boolean formula φ in DNF form i.e.