◮ Classical logic CPC is axiomatized the the following axioms x → ( y → x ) ( x → ( y → z )) → (( x → y ) → ( x → z )) ( x ∧ y ) → x A correspondence between logical translations and ( x ∧ y ) → y semantic transformations x → ( y → ( x ∧ y )) x → ( x ∨ y ) Tommaso Moraschini y → ( x ∨ y ) ( x → y ) → (( z → y ) → (( x ∨ z ) → y )) Institute of Computer Science of the Czech Academy of Sciences 0 → x December 1, 2017 ¬¬ x → x and the rule of Modus Ponens x , x → y ⊢ y . ◮ Classical logic is the logic of Boolean reasoning. 1 / 14 2 / 14 ◮ At the beginning of the last century non-classical logics arose ◮ Modal logic K expands the language of classical logic with a for mathematical, philosophical, and linguistic motivations. unary connective ✷ , whose intended to meaning is: ◮ Intuitionistic logic IPC , motivated by constructivism in ✷ ϕ ≡ it is necessary that ϕ. mathematics, is obtained removing ¬¬ x → x from the axiomatization of classical logic: ◮ K is axiomatized by the axioms and rules of classical logic plus x → ( y → x ) the axiom ( x → ( y → z )) → (( x → y ) → ( x → z )) ✷ ( x → y ) → ( ✷ x → ✷ y ) ( x ∧ y ) → x and the Necessitation rule ( x ∧ y ) → y x ⊢ ✷ x . x → ( y → ( x ∧ y )) x → ( x ∨ y ) ◮ Recap: Several logic flourished in the early 20th century, e.g. intuitionistic logic IPC , modal logic K , their axiomatic y → ( x ∨ y ) extensions etc. (and of course classical logic CPC ). ( x → y ) → (( z → y ) → (( x ∨ z ) → y )) ◮ Our understanding of this increasing variety of logics depends 0 → x on the possibility of drawing comparisons between them, and the rule of Modus Ponens typically though logical translations. x , x → y ⊢ y . 3 / 14 4 / 14
Kolmogorov’s translation of CPC into IPC . Gödel’s translation of IPC into S4 . ◮ One of the most important axiomatic extension of the modal ◮ In 1925 Kolmogorov defined a double-negation translation of the formulas ϕ of CPC into formulas ϕ K of IPC as follows: logic K is the system S4 obtained adding the axioms x K := ¬¬ x for variables x ✷ x → x ≡ if ϕ is necessary, then it hlolds 0 K := 0 ✷ x → ✷✷ x ≡ if ϕ is necessary, then it is necessarily so. ( α ∧ β ) K := ¬¬ ( α K ∧ β K ) ◮ In 1933 Gödel defined a translation of IPC into S4 as follows: x G := ¬ ✷ x for variables x ( α ∨ β ) K := ¬¬ ( α K ∨ β K ) 0 G := 0 ( α → β ) K := ¬¬ ( α K → β K ) ( α ∧ β ) G := α G ∧ β G ( ¬ α ) K := ¬ ( α K ) ( α ∨ β ) G := α G ∨ β G where in IPC we define ¬ ϕ := ϕ → 0. ( α → β ) G := ✷ ( α G → β G ) . ◮ Kolmogorov’s translation is logically faithful in the sense that for every set of formulas Γ ∪ { ϕ } , ◮ Gödel’s translation is logically faithful: ⇒ Γ K ⊢ CPC ϕ K . ⇒ Γ G ⊢ S4 ϕ G . Γ ⊢ IPC ϕ ⇐ Γ ⊢ IPC ϕ ⇐ 5 / 14 6 / 14 Semantic dual of Kolmogorov’s translation Semantic dual of Gödel’s translation ◮ The algebraic semantics of CPC are Boolean algebras, i.e. ◮ The algebraic semantics of S4 are interior algebras, i.e. algebras A = � A , ∧ , ¬ , ∨ , ✷ , 0 , 1 � such that � A , ∧ , ∨ , ¬ , 0 , 1 � is algebras A = � A , ∧ , ∨ , ¬ , 0 , 1 � such that � A , ∧ , ∨ , 0 , 1 � is a bounded distributive lattice such that a Boolean algebra and ✷ is an interior operator such that ✷ ( a ∧ b ) = ✷ a ∧ ✷ b and ✷ 1 = 1 , for all a , b ∈ A . a ∨ ¬ a = 1 and a ∧ ¬ a = 0 , for all a ∈ A . ◮ Gödel’s translation of IPC into S4 has a semantic dual, i.e. ◮ The algebraic semantic of IPC are Heyting algebras, i.e. the transformation algebras A = � A , ∧ , ∨ , → , 0 , 1 � such that � A , ∧ , ∨ , 0 , 1 � is a bounded (distributive) lattice and Op : IA → HA a ∧ b ≤ c ⇐ ⇒ a ≤ b → c , for all a , b , c ∈ A . A �→ Op ( A ) := � Op ( A ) , ∧ , ∨ , ⊸ , 0 , 1 � ◮ Kolmogorov’s translation of IPC into CPC has a semantic where Op ( A ) = { a ∈ A : ✷ a = a } and a ⊸ b := ✷ ( a → b ) . dual, i.e. the transformation ◮ Recap: Kolmogorov and Gödel’s logic translations correspond to semantics transformations in the reverse direction: Reg : HA → BA ( · ) K : CPC → IPC and Reg : HA → BA A �→ Reg ( A ) := � Reg ( A ) , ∧ , ⊔ , ¬ , 0 , 1 � ( · ) G : IPC → S4 and Op : IA → HA . where Reg ( A ) = { a ∈ A : ¬¬ a = 1 } and a ⊔ b := ¬¬ ( a ∨ b ) . 7 / 14 8 / 14
Adjoint Functors Matrix powers with infinite exponents ◮ The semantic transformations, dualizing Kolmogorov and ◮ Let X be a class of similar algebras and κ > 0 be a cardinal. ◮ Consider the language L κ Gödel’s translations, are special instances of the following: X whose n -ary operations are the κ -sequences Definition � t i : i < κ � where each t i is a term of X A pair of functors F : X ← → Y : G is an adjunction if there is a pair in variables � x 1 , . . . , � x n . of natural transformation η : 1 X → GF and ǫ : FG → 1 Y such that Definition 1 G ( B ) = G ( ǫ B ) ◦ η G ( B ) and 1 F ( A ) = ǫ F ( A ) ◦ F ( η A ) . Given A ∈ X, let A [ κ ] be the L κ X -algebra with universe A κ s.t. for every A ∈ X and B ∈ Y. � t i : i < κ � A [ κ ] ( � a n ) = � t A a 1 , . . . , � i ( � a 1 /� x 1 , . . . , � a n /� x n ) : i < κ � . ◮ In this case F is left adjoint to G and G right adjoint to F . ◮ Under the identification right adjoints = semantic The κ -th matrix power of X is the class transformations, proving the equivalence X [ κ ] := I { A [ κ ] : A ∈ X } . logical translations ≡ semantic transformations amounts to find a syntactic description of right adjoints. ◮ This construction extends to a functor [ κ ]: X → X [ κ ] . 9 / 14 10 / 14 Compatible Equations Logical description of right adjoints ◮ It turns out that, among quasi-varieties, Definition right adjoints admit a syntactic/logical description. Let X be a class of algebras of language L X and L ⊆ L X . A set ◮ More precisely, we have the following: of equations θ in one variable is compatible with L in X if for every n -ary operation ϕ ∈ L we have that: Theorem θ ( x 1 ) ∪ · · · ∪ θ ( x n ) � X θ ( ϕ ( x 1 , . . . , x n )) . Let X and Y be quasi-varieties. 1. For every non-trivial right adjoint ◮ For every A ∈ X, we let A ( θ, L ) be the algebra of type L G : Y → X with universe A ( θ, L ) = { a ∈ A : A � θ ( a ) } there is a (generalized) quasi-variety K and functors [ κ ]: Y → K and θ L : K → X equipped with the restriction of the operations in L . ◮ We obtain a functor such that G is naturally isomorphic to θ L ◦ [ κ ] . θ L : X → I { A ( θ, L ) : A ∈ X } . 2. Every functor of the form θ L ◦ [ κ ]: Y → X is a right adjoint. 11 / 14 12 / 14
◮ This syntactic description of right adjoints (inspired by work of Finally... McKenzie and others) allows to establish a precise correspondence right adjoints ≡ logical translations where the precise meaning of logical translations come from the syntactic canonical form θ L ◦ [ κ ] of right adjoints. ◮ This new notion of logical translation embraces most known examples, e.g. Kolmogorov and Gödel’s ones. ...thank you for coming! Recap: ◮ One can state a precise correspondence between semantic transformations (understood as right adjoints) and translations between logics (understood as equational consequences). ◮ This yields an algebraic canonical form for right adjoints. ◮ Some computational results follows, e.g. the problem of determining whether two finite algebras are related by an adjunction is decidable. 13 / 14 14 / 14
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