Categorical Models for Two Intuitionistic Modal Logics Categorical Models Wolfgang Jeltsch for Two Intuitionistic Modal Logics Introduction Intuitionistic S4 categories Temporal Wolfgang Jeltsch categories References TT¨ U K¨ uberneetika Instituut Teooriaseminar May 10, 2012
Categorical Models for Two Intuitionistic Modal Logics Introduction 1 Wolfgang Jeltsch Introduction Intuitionistic S4 categories 2 Intuitionistic S4 categories Temporal categories Temporal categories 3 References References 4
Categorical Models for Two Intuitionistic Modal Logics Introduction 1 Wolfgang Jeltsch Introduction Intuitionistic S4 categories 2 Intuitionistic S4 categories Temporal Temporal categories 3 categories References References 4
Modal logics Categorical Models for Two used to deal with things like possibility, belief, and time Intuitionistic Modal Logics in this talk only time Wolfgang Jeltsch two new operators ✷ and ✸ : Introduction ✷ ϕ now and at every future time, ϕ holds Intuitionistic ✸ ϕ now or at some future time, ϕ holds S4 categories Temporal later also future-only variants: categories ✷ ′ ϕ at every future time, ϕ holds References ✸ ′ ϕ at some future time, ϕ holds ✷ and ✸ dual and interdefinable in classical modal logics: ✷ ϕ := ¬ ✸ ¬ ϕ ✸ ϕ := ¬ ✷ ¬ ϕ
Kripke semantics Categorical Models used for classical modal logics for Two Intuitionistic Modal Logics Kripke frame: Wolfgang set W of worlds Jeltsch accessibility relation R ⊆ W × W Introduction Kripke model assigns truth values to formulas for each world Intuitionistic S4 categories semantics of modal operators: Temporal ✷ ϕ true at w if ϕ is true at every w ′ with categories ( w , w ′ ) ∈ R References ✸ ϕ true at w if ϕ is true at some w ′ with ( w , w ′ ) ∈ R Kripke frames in the temporal case: worlds are times accessibility relation is reflexive order of times
Concrete modal logics Categorical Models for Two Intuitionistic some classical logics: Modal Logics Wolfgang K axioms that have to hold in every modal logic Jeltsch S4 additional axioms that ensure that Introduction the accessibility relation is reflexive Intuitionistic and transitive S4 categories Temporal some intuitionistic logics and their categorical models: categories IK BCCCs with additional structure References for modeling ✷ and ✸ CS4/IS4 additional structure that corresponds to reflexivity and transitivity of accessibility relations in the classical case
This talk Categorical Models for Two Intuitionistic Modal Logics Wolfgang Jeltsch categorical models of intuitionistic S4 based on categorical Introduction models of CS4 and IS4 Intuitionistic categorical models for an intuitionistic temporal logic: S4 categories Temporal additional structure for modeling future-only operators categories additional structure that corresponds to totality References of accessibility orders in the classical case
Categorical Models for Two Intuitionistic Modal Logics Introduction 1 Wolfgang Jeltsch Introduction Intuitionistic S4 categories 2 Intuitionistic S4 categories Temporal Temporal categories 3 categories References References 4
Basic structure Categorical remember: Models for Two objects model propositions Intuitionistic Modal Logics if objects A and B model propositions ϕ and ψ , morphisms f : A → B model proofs of ϕ ⊢ ψ Wolfgang Jeltsch BCCCs as models of intuitionistic propositional logic: Introduction Intuitionistic 1 ˆ = ⊤ × ˆ = ∧ 0 ˆ = ⊥ + ˆ = ∨ → ˆ = ⇒ S4 categories Temporal BCCCs with additional structure as models of modal logics categories functors ✷ and ✸ for modeling logical operators ✷ and ✸ References morphism maps correspond to the following logical rules: ϕ ⊢ ψ ϕ ⊢ ψ ✷ ϕ ⊢ ✷ ψ ✸ ϕ ⊢ ✸ ψ ϕ ⊢ ψ shall mean that at all times, ϕ implies ψ
Monoidal functors Categorical ✷ is a strong monoidal functor on the cartesian structure Models for Two (cartesian functor): Intuitionistic Modal Logics ✷ A × ✷ B ∼ = ✷ ( A × B ) Wolfgang Jeltsch 1 ∼ = ✷ 1 Introduction duality of ✷ and ✸ would mean that ✸ is a strong Intuitionistic S4 categories monoidal functor on the cocartesian structure: Temporal ✸ ( A + B ) ∼ categories = ✸ A + ✸ B References ✸ 0 ∼ = 0 do not require this: left-to-right transformations would transport information about the future into the present would make it impossible to use temporal logic as a language for programs that run in real time (FRP)
Comonads and monads Categorical ✷ is a comonad: Models for Two ε A : ✷ A → A δ A : ✷ A → ✷✷ A Intuitionistic Modal Logics classical analog is that accessibility relations are orders: Wolfgang Jeltsch type of ε corresponds to reflexivity axiom type of δ corresponds to transitivity axiom Introduction ✸ is a monad: Intuitionistic S4 categories η A : A → ✸ A µ A : ✸✸ A → ✸ A Temporal categories classical analog is also that accessibility relations are orders: References type of η corresponds to reflexivity axiom type of µ corresponds to transitivity axiom classically, only one reflexivity and one transitivity axiom necessary (because ✷ and ✸ are interdefinable) need both the comonad and the monad structure in the intuitionistic case
Relative tensorial strength Categorical Models for Two Intuitionistic Modal Logics ✸ is ✷ -strong: Wolfgang Jeltsch natural transformation s with Introduction s A , B : ✷ A × ✸ B → ✸ ( ✷ A × B ) Intuitionistic S4 categories exists Temporal categories s is compatible with cartesian functor, comonad, References and monad structure proposition corresponding to s holds automatically in classical logic (because ✷ and ✸ are interdefinable)
Categorical Models for Two Intuitionistic Modal Logics Introduction 1 Wolfgang Jeltsch Introduction Intuitionistic S4 categories 2 Intuitionistic S4 categories Temporal Temporal categories 3 categories References References 4
Future only logic with future-only operators ✷ ′ and ✸ ′ : Categorical Models for Two Intuitionistic ✷ ϕ = ϕ ∧ ✷ ′ ϕ Modal Logics ✸ ϕ = ϕ ∨ ✸ ′ ϕ Wolfgang Jeltsch functors ✷ ′ and ✸ ′ with the following properties: Introduction ✷ A = A × ✷ ′ A Intuitionistic S4 categories ✸ A = A + ✸ ′ A Temporal categories ✷ ′ is an ideal comonad, and ✸ ′ is an ideal monad: References natural transformations δ ′ and µ ′ with δ ′ : ✷ ′ A → ✷ ′ ✷ A µ ′ : ✸ ′ ✸ A → ✸ ′ A exist comonad and monad structure derived from δ ′ and µ ′
Linear time Categorical Models for Two Intuitionistic Modal Logics Wolfgang classically, accessibility order must be total Jeltsch introduction of a natural transformation r with Introduction Intuitionistic S4 categories r A , B : ✸ A × ✸ B → ✸ ( A ⊙ B ) , Temporal categories where References A ⊙ B := A × B + A × ✸ ′ B + ✸ ′ A × B
A nicer solution Categorical an operator � �· , ·� � with Models for Two f : C → ✸ A g : C → ✸ B Intuitionistic Modal Logics � � f , g � � : C → ✸ ( A ⊙ B ) Wolfgang Jeltsch looks a bit like the �· , ·� -operator of a product require A ⊙ B to be a product in the Kleisli category of ✸ Introduction � �· , ·� � is now the �· , ·� -operator of that product Intuitionistic S4 categories projections: Temporal categories ̟ 1 : A × B + A × ✸ ′ B + ✸ ′ A × B → A + ✸ ′ A References ̟ 2 : A × B + A × ✸ ′ B + ✸ ′ A × B → B + ✸ ′ B product axioms (in the Kleisli category) ensure that proofs of ✸ A and ✸ B can be recovered from proof of ✸ ( A ⊙ B ): µ ( ✸ ̟ 1 ) � � f , g � � = f µ ( ✸ ̟ 2 ) � � f , g � � = g as a result, r is an isomorphism
Categorical Models for Two Intuitionistic Modal Logics Introduction 1 Wolfgang Jeltsch Introduction Intuitionistic S4 categories 2 Intuitionistic S4 categories Temporal Temporal categories 3 categories References References 4
References Categorical Models for Two Intuitionistic Satoshi Kobayashi Modal Logics Monad as Modality Wolfgang Jeltsch Theoretical Computer Science 175 (1997), pp. 29–74 Introduction Gavin Bierman and Valeria de Paiva Intuitionistic On an intuitionistic modal logic S4 categories Studia Logica 65 (2000), pp. 383–416 Temporal categories References Wolfgang Jeltsch Towards a Common Categorical Semantics for Linear-Time Temporal Logic and Functional Reactive Programming Electronic Notes in Theoretical Computer Science ( to appear )
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