Undecidability of { , 1 , } -equations in subvarieties of - PowerPoint PPT Presentation

Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. Gavin St.John Under the advisement of Nikolaos Galatos University of Denver Department of Mathematics Topology, Algebra, and Categories in Logic 2017 Institute of Computer Science, Czech Academy of Sciences 27 June 2017 Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 1 / 28

Residuated Latices Definition A (commutative) residuated latice is a structure R = ( R, · , ∨ , ∧ , \ , /, 1) , such that ◮ ( R, ∨ , ∧ ) is a latice ◮ ( R, · , 1) is a (commutative) monoid ◮ For all x, y, z ∈ R x · y ≤ z ⇐ ⇒ y ≤ x \ z ⇐ ⇒ x ≤ z/y, where ≤ is the latice order. We denote the variety of (commutative) residuated latices by ( CRL ) RL . If (r) is a rule (axiom), then ( C ) RL r := ( C ) RL + (r) . Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 2 / 28

Known results for Qasi-Equational Theory n ) represents the knoted rule x n ≤ x m (k m Undecidable Q.Eq. Theory Decidable Q.Eq. Theory Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 3 / 28

Known results for Qasi-Equational Theory n ) represents the knoted rule x n ≤ x m (k m Undecidable Q.Eq. Theory Decidable Q.Eq. Theory RL CRL Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 3 / 28

Known results for Qasi-Equational Theory n ) represents the knoted rule x n ≤ x m (k m Undecidable Q.Eq. Theory Decidable Q.Eq. Theory RL CRL RL + (k m n ) , 1 ≤ n < m Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 3 / 28

Known results for Qasi-Equational Theory n ) represents the knoted rule x n ≤ x m (k m Undecidable Q.Eq. Theory Decidable Q.Eq. Theory RL CRL RL + (k m n ) , 1 ≤ n < m CRL + (k m n ) Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 3 / 28

Known results for Qasi-Equational Theory n ) represents the knoted rule x n ≤ x m (k m Undecidable Q.Eq. Theory Decidable Q.Eq. Theory RL CRL RL + (k m n ) , 1 ≤ n < m CRL + (k m n ) CRL + (?) Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 3 / 28

◮ Van Alten (2005) showed CRL in the presence of any knoted rule has the FEP. Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 4 / 28

◮ Van Alten (2005) showed CRL in the presence of any knoted rule has the FEP. ◦ Consequently, extensions of CRL in the signatures {≤ , · , 1 } have been fully characterized. Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 4 / 28

◮ Van Alten (2005) showed CRL in the presence of any knoted rule has the FEP. ◦ Consequently, extensions of CRL in the signatures {≤ , · , 1 } have been fully characterized. ◮ We inspect (in)equations in the signature {· , 1 , ∨} . Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 4 / 28

◮ Van Alten (2005) showed CRL in the presence of any knoted rule has the FEP. ◦ Consequently, extensions of CRL in the signatures {≤ , · , 1 } have been fully characterized. ◮ We inspect (in)equations in the signature {· , 1 , ∨} . ◦ Proof theoretically, such axioms correspond to inference rules, e.g., X, Y, Y, Z ⊢ C X, Z ⊢ C x ≤ x 2 ∨ 1 ⇐ ⇒ X, Y, Z ⊢ C Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 4 / 28

◮ Van Alten (2005) showed CRL in the presence of any knoted rule has the FEP. ◦ Consequently, extensions of CRL in the signatures {≤ , · , 1 } have been fully characterized. ◮ We inspect (in)equations in the signature {· , 1 , ∨} . ◦ Proof theoretically, such axioms correspond to inference rules, e.g., X, Y, Y, Z ⊢ C X, Z ⊢ C x ≤ x 2 ∨ 1 ⇐ ⇒ X, Y, Z ⊢ C ◦ The work of Chvalovský & Horčík (2016) implies the undecidability for many such extensions in RL . Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 4 / 28

◮ Van Alten (2005) showed CRL in the presence of any knoted rule has the FEP. ◦ Consequently, extensions of CRL in the signatures {≤ , · , 1 } have been fully characterized. ◮ We inspect (in)equations in the signature {· , 1 , ∨} . ◦ Proof theoretically, such axioms correspond to inference rules, e.g., X, Y, Y, Z ⊢ C X, Z ⊢ C x ≤ x 2 ∨ 1 ⇐ ⇒ X, Y, Z ⊢ C ◦ The work of Chvalovský & Horčík (2016) implies the undecidability for many such extensions in RL . ◦ So we restrict our investigation to the commutative case. Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 4 / 28

Linearization Any equation s = t in the signature {· , 1 , ∨} is equivalent to some conjunction of linear inequations we call “ d -rules ” of the form: � m x d j (1) · · · x d j ( n ) (d) x 1 · · · x n ≤ , n 1 j =1 where d := { d 1 , ..., d m } ⊂ N n . Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 5 / 28

Linearization Any equation s = t in the signature {· , 1 , ∨} is equivalent to some conjunction of linear inequations we call “ d -rules ” of the form: � m x d j (1) · · · x d j ( n ) (d) x 1 · · · x n ≤ , n 1 j =1 where d := { d 1 , ..., d m } ⊂ N n . Such conjoins can be determined by the properties of CRL : ◮ x ≤ y ⇐ ⇒ x ∨ y = y ◮ x ∨ y ≤ z ⇐ ⇒ x ≤ z and y ≤ z ◮ linearization Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 5 / 28

Linearization Any equation s = t in the signature {· , 1 , ∨} is equivalent to some conjunction of linear inequations we call “ d -rules ” of the form: � m x d j (1) · · · x d j ( n ) (d) x 1 · · · x n ≤ , n 1 j =1 where d := { d 1 , ..., d m } ⊂ N n . Such conjoins can be determined by the properties of CRL : ◮ x ≤ y ⇐ ⇒ x ∨ y = y ◮ x ∨ y ≤ z ⇐ ⇒ x ≤ z and y ≤ z ◮ linearization E.g., the rule ( ∀ u )( ∀ v ) u 2 v ≤ u 3 ∨ uv is equivalent to, via the substitutions u = x ∨ y and v = z, ( ∀ x )( ∀ y )( ∀ z ) xyz ≤ x 3 ∨ x 2 y ∨ xy 2 ∨ y 3 ∨ xz ∨ yz Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 5 / 28

Conditions on d ⊂ N n ◮ If (d) implies a knoted rule, then CRL + (d) is decidable. Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 6 / 28

Conditions on d ⊂ N n ◮ If (d) implies a knoted rule, then CRL + (d) is decidable. E.g., if (d) is xy ≤ xy 2 ∨ x 2 y , then = x 2 ≤ x 3 . CRL + (d) | Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 6 / 28

Conditions on d ⊂ N n ◮ If (d) implies a knoted rule, then CRL + (d) is decidable. E.g., if (d) is xy ≤ xy 2 ∨ x 2 y , then = x 2 ≤ x 3 . CRL + (d) | ◮ If CRL + (d) is to be undecidable, d ⊂ N n must refute certain conditions with respect to the set of vectors representing the exponents of the variables. Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 6 / 28

Conditions on d ⊂ N n ◮ If (d) implies a knoted rule, then CRL + (d) is decidable. E.g., if (d) is xy ≤ xy 2 ∨ x 2 y , then = x 2 ≤ x 3 . CRL + (d) | ◮ If CRL + (d) is to be undecidable, d ⊂ N n must refute certain conditions with respect to the set of vectors representing the exponents of the variables. ◮ We view d = { d j } m j =1 as a set of linear subspaces of R n . Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 6 / 28

Conditions on d ⊂ N n ◮ If (d) implies a knoted rule, then CRL + (d) is decidable. E.g., if (d) is xy ≤ xy 2 ∨ x 2 y , then = x 2 ≤ x 3 . CRL + (d) | ◮ If CRL + (d) is to be undecidable, d ⊂ N n must refute certain conditions with respect to the set of vectors representing the exponents of the variables. ◮ We view d = { d j } m j =1 as a set of linear subspaces of R n . ( ⋆ ) Given any nonempty A ⊆ { 1 , ..., n } , and any nontrivial valuation of variables x 1 , ..., x n in N , there exists j � = j ′ ≤ m such that the supports of d j and d j ′ intersect A , and � � n n d j ( i ) x i � = d j ′ ( i ) x i i =1 i =1 Gavin St.John Undecidability of {· , 1 , ∨} -equations in subvarieties of commutative residuated latices. 6 / 28