Higher-Order Categorical Substructural Logics Yoshihiro Maruyama - - PowerPoint PPT Presentation

Full Lambek Hyperdoctrine Full Lambek Tripos

Higher-Order Categorical Substructural Logics

Yoshihiro Maruyama

Oxford University and Kyoto University

CT 2015, Aveiro, 18 June

Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 1 / 8

Full Lambek Hyperdoctrine Full Lambek Tripos

Overview

No concept of substructural topos; no topos-style semantics for general substructural logics (fuzzy, linear, relevant, etc.) so far. In this talk I present the concept of substructural tripos, giving tripos-style semantics for a wide variety of substructural logics.

Full Lambek Calculus FL is a standard system allowing us to represent various logics as axiomatic extensions incl. CL and IL. Full Lambek hyperdoctrines and triposes are introduced to give complete semantics for first-order and higher-order FL.

The framework thus developed allows us to compare different categorical logics, in particular to give functorial accounts of logical translations (Gödel, Girard, GTM, Baaz, etc.).

Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 2 / 8

Full Lambek Hyperdoctrine Full Lambek Tripos

Outline

1

Full Lambek Hyperdoctrine

2

Full Lambek Tripos

Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 3 / 8

Full Lambek Hyperdoctrine Full Lambek Tripos

Outline

1

Full Lambek Hyperdoctrine

2

Full Lambek Tripos

Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 4 / 8

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Typed Full Lambek Calculus

TFLq has the following logical connectives: ⌦, ^, _, \, /, 1, 0, >, ?, 8, 9. Note: there are two kinds of implication connectives \ and /. In TFLq, every variable x comes with its type σ. That is, TFLq has basic types, which are denoted by letters like σ, τ, and x : σ is a formal expression meaning that a variable x is of type σ. A (type) context is a finite list of type declarations on variables: x1 : σ1, ..., xn : σn. A context is often denoted Γ.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Typed Full Lambek Calculus (cont.)

Accordingly, TFLq has typed predicate symbols (aka. predicates in context) and typed function symbols (aka. function symbols in context): R(x1, ..., xn) [x1 : σ1, ..., xn : σn] is a formal expression meaning that R is a predicate with n variables x1, ..., xn of types σ1, ..., σn respectively; likewise, f : τ [x1 : σ1, ..., xn : σn] is a formal expression meaning that f is a function symbol with n variables x1, ..., xn of types σ1, ..., σn and with its values in τ. Then, formulae-in-context ϕ [Γ] and terms-in-context t : τ [Γ] are defined in the usual, inductive way.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Typed Full Lambek Calculus (cont.)

TFLq thus has both a type structure and a logic structure, dealing with sequents-in-contexts: Φ ` ϕ [Γ] (or Γ | Φ ` ϕ). There are two kinds of conjunction in TFLq: multiplicative or monoidal ⌦ and additive or cartesian ^: Φ, ϕ, ψ, Ψ ` χ [Γ] Φ, ϕ ⌦ ψ, Ψ ` χ [Γ] (⌦L) Φ ` ϕ [Γ] Ψ ` ψ [Γ] Φ, Ψ ` ϕ ⌦ ψ [Γ] (⌦R) Φ, ϕ, Ψ ` χ [Γ] Φ, ϕ ^ ψ, Ψ ` χ [Γ] (^L1) Φ, ϕ, Ψ ` χ [Γ] Φ, ψ ^ ϕ, Ψ ` χ [Γ] (^L2) Φ ` ϕ [Γ] Φ ` ψ [Γ] Φ ` ϕ ^ ψ [Γ] (^R) The underlying type system is basically the same as that of: A. Pitts, Categorical Logic, Handbook of Logic in Comput. Sci.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Typed Full Lambek Calculus (cont.)

Due to non-commutativity, there are two kinds of implication in TFLq, \ and /, which are a right adjoint of ϕ ⌦ (-) and a right adjoint of (-) ⌦ ψ respectively. Φ ` ϕ [Γ] Ψ1, ψ, Ψ2 ` χ [Γ] Ψ1, Φ, ϕ\ψ, Ψ2 ` χ [Γ] (\L) ϕ, Φ ` ψ [Γ] Φ ` ϕ\ψ [Γ] (\R) Φ ` ϕ [Γ] Ψ1, ψ, Ψ2 ` χ [Γ] Ψ1, ψ/ϕ, Φ, Ψ2 ` χ [Γ] (/L) Φ, ϕ ` ψ [Γ] Φ ` ψ/ϕ [Γ] (/R) We omit the rules for the other propositional connectives.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Typed Full Lambek Calculus (cont.)

Finally, we have the following rules for quantifiers 8 and 9, in which type contexts explicitly change; notice that type contexts do not change in the rest of the rules presented above. Φ1, ϕ, Φ2 ` ψ [x : σ, Γ] Φ1, 8xϕ, Φ2 ` ψ [x : σ, Γ] (8L) Φ ` ϕ [x : σ, Γ] Φ ` 8xϕ [Γ] (8R) Φ1, ϕ, Φ2 ` ψ [x : σ, Γ] Φ1, 9xϕ, Φ2 ` ψ [Γ] (9L) Φ ` ϕ [x : σ, Γ] Φ ` 9xϕ [x : σ, Γ] (9R) As usual, there are eigenvariable conditions on the rules above. The other two rules do not have eigenvariable conditions, and this is why contexts do not change in them.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Typed Full Lambek Calculus (cont.)

The following sequents-in-context are deducible in TFLq: ϕ ⌦ (9xψ) ` 9x(ϕ ⌦ ψ) [Γ] and 9x(ϕ ⌦ ψ) ` ϕ ⌦ (9xψ) [Γ]. (9xψ) ⌦ ϕ ` 9x(ψ ⌦ ϕ) [Γ] and 9x(ψ ⌦ ϕ) ` (9xψ) ⌦ ϕ [Γ]. where it is supposed that ϕ does not contain x as a free variable, and Γ contains type declarations on those free variables that appear in ϕ and 9xψ.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Full Lambek Algebra

(A, ⌦, ^, _, \, /, 1, 0, >, ?) is called an FL algebra iff (A, ⌦, 1) is a monoid; 0 is an element of A; (A, ^, _, >, ?) is a bounded lattice; for any a 2 A, a\(-) : A ! A is a right adjoint of a ⌦ (-): a ⌦ b  c iff b  a\c; for any b 2 A, (-)/b : A ! A is a right adjoint of (-) ⌦ b: a ⌦ b  c iff a  c/b. FL denotes the category of FL algebras and homomorphisms. This is propositional. How to extend all this for predicate logic?

• Alg. Log.: A propositional logic is a single algebra A.
• Cat. Log.: A predicate logic is a fibred algebra (AC)C∈C;

the base cat. C accounts for the underlying type theory.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Full Lambek Hyperdoctrine

An FL hyperdoctrine is an FL-valued presheaf P : Cop → FL such that C is a cat. with finite products, and for any projection π : X × Y → Y, P(π) : P(Y) → P(X × Y) has a right adjoint ∀π : P(X × Y) → P(Y) with the corresponding Beck-Chevalley condition P(X × Y) P(Y) P(X × Z) P(Z)

?

P(X×f)

• ∀π

?

P(f)

• ∀π0

(This means (∀xϕ)[t/y] = ∀x(ϕ[t/y]) in the syntactic hypdoc.)

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Full Lambek Hyperdoctrine (def. cont’d)

Furthermore, for any projection π : X × Y → Y in C, P(π) : P(Y) → P(X × Y) has a left adjoint ∃π : P(X × Y) → P(Y) with the corresponding Beck-Chevalley condition: P(X × Y) P(Y) P(X × Z) P(Z)

?

P(X×f)

• ∃π

?

P(f)

• ∃π0

This comes from Lawvere’s idea of quantifiers as adjoints. Equality, comprehension, and the like can be treated as well.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Full Lambek Hyperdoctrine (def. cont’d)

Furthermore, the Frobenius Reciprocity conditions hold: for any π : X × Y → Y in C, any a ∈ P(Y), and any b ∈ P(X × Y), a ⊗ (∃πb) = ∃π(P(π)(a) ⊗ b) (∃πb) ⊗ a = ∃π(b ⊗ P(π)(a)). For an axiomatic extension FLX of FL, an FLX hyperdoctrine is defined by restricting the value cat. FL into FLX. An FL (resp. FLX) hypdoc. is also called a fibred FL (resp. FLX) algebra. In full generality we conceive of universally algebraised hyperdoctrines P : Cop → Alg(T) (or P : Cop → V).

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Interpretations in Hyperdoctrines

An interpretation J-K of TFLq in P : Cop ! FL consists of: assignment of an object JσK in C to each basic type σ; assignment of an arrow Jf : τ [Γ]K : Jσ1K ⇥ ... ⇥ JσnK ! JσK in C to each typed function symbol f : τ [Γ] in TFLq; assignment of an element JR [Γ]K in P(JΓK), which is an FL algebra, to each typed predicate symbol R [Γ] in TFLq. Then, terms and formulae are interpreted inductively: e.g., Jx : σ [x : σ, Γ]K := π : JσK ⇥ JΓK ! JσK. Jf(t1, ..., tn) : τ [Γ]K := JfK hJt1 : σ1 [Γ]K, ..., Jtn : σn [Γ]Ki. J8xϕ [Γ]K := 8π(Jϕ [x : σ, Γ]K). J9xϕ [Γ]K := 9π(Jϕ [x : σ, Γ]K).

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Generic Completeness

ϕ1, ..., ϕn ` ψ [Γ] is satisfied in an interpretation J-K in an FLX hyperdoctrine P iff the following holds in P(JΓK): Jϕ1 [Γ]K ⌦ ... ⌦ Jϕn [Γ]K  Jψ [Γ]K. We then have the generic completeness thm.: A sequent-in-context ϕ ` ψ [Γ] is deducbile in TFLq

X iff it is

valid in any interpretation in any FLX hyperdoctrine.

This is proven via the combination of the Lindenbaum algebra and syntactic category constructions.

This yields various categorical completeness thms., e.g., for classical, int., fuzzy, relevant, linear logics. Ref.: M., Full Lambek

Hyperdoctrine: Categ. Semantics for First-Order Substr. Logics, LNCS, 2013.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Tarskian Completeness

The Tarski semantics precisely amount to interpretations in the powerset hyperdoctrine HomSet(-, 2). Φ ` ψ [Γ] is deducible in TFLq iff it is satisfied in any interpretation in HomSet(-, Ω) for any complete Ω 2 FL. Φ ` ψ [Γ] is deducible in TFLq

X iff it is satisfied in any safe

interpretation in HomSet(-, Ω) for any Ω 2 FLX. In the presence of prelinearity (ϕ ! ψ) _ (ψ ! ϕ), it is even sufficient to consider totally ordered Ω’s only. In the presence of all the structural rules and LEM, it is sufficient to consider 2 only (the Tarski completeness). Such FL hypdocs. give models of substructural set theories, and universes of quantale-valued sets via the HJP const.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Full Lambek Hyperdoctrine Full Lambek Tripos

Outline

1

Full Lambek Hyperdoctrine

2

Full Lambek Tripos

Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 6 / 8

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Higher-Order Full Lambek Calculus

HoFLq has the following type constructors: ⇥, !, Prop. Recall that TFLq has no type constructor.

Prop is a constant corresponding to truth value objects Ω. The logical meaning of truth value objects SubE(-) ' HomE(-, Ω) is that props. (preds.) in SubE(σ) can be seen as terms from σ to Ω. Logic is thereby reflected into the type structure.

The term constructors of Prop are the logical constants, and the equality axiom is given by extensionality: ϕ ` ψ [Γ] ψ ` ϕ [Γ] ϕ = ψ : Prop [Γ] ext. The term constructors and eqs. for ⇥, ! are added as well.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Higher-Order Full Lambek Calculus

Summing up, the underlying type theory of HoFLq is λ×,→ expanded with the basic type Prop. In HoFLq, we can define the power type Pσ of each type σ by Pσ := σ → Ω. We then have the membership predicate t ∈ s : Ω as application (evaluation). Also have the comprehension term {x : σ | ϕ(x)} : Pσ as abstraction (transpose). E.g., x ∈ {x : σ | ϕ(x)} = ϕ(x). In such a way, we can do set theory in HoFLq. The HOL of toposes is an axiomatic extension of HoFLq, apart from minor syntactic differences (e.g., choice of primitives).

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Higher-Order Completeness Theorem

Definition (Full Lambek Tripos) An FL tripos is an FL hyperdoctrine P : Cop ! FL such that C is a CCC, and there is Ω 2 C with nat. equiv. P(-) ' HomC(-, Ω). “Tripos" originally meant “topos representing indexed partially

• rdered set" (HJP). Toposes categorically embed into triposes.

Not all triposes arise from toposes. Triposes are more general. Theorem (Higher-Order Completeness) A sequent-in-context Φ ` ψ [Γ] is deducbile in HoFLq

X iff it is

valid in any interpretation in any FLX tripos. The topos-HOL completeness is just a special case of the thm., which subsumes higher-order categorical completeness thms. for classical, int., linear, relevant, and fuzzy logics in particular.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Duality as Semantics

Duality induced by a Janus (aka. schizophrenic) object Ω: HomD(-, Ω) a HomC(-, Ω) : Cop ! D.

M., Categorical Duality Theory: Domains, Convexity, and the Distribution Monad, Proc. CSL ’13. M., Natural Duality, Modality, and Coalgebra, J. Pure Appl. Algebra, 2012.

D := Alg as in many cases. HomC(-, Ω) : Cop ! Alg give Alg-relativized hyperdoctrines under certain conditions. The right adjoint functor HomTop(-, 2) : Topop ! Frm is a geometric hyperdoctrine.

• Adj. functor HomSet(-, Ω) : Setop ! Alg is an Alg-hyprdoc.

In the Heyting case, these give rise to sheaf topoi via the tripos-topos construction by Hyland-Johnstone-Pitts. In the quantum case, these yield quantum-valued sets by Takeuti and Ozawa. (substr. case, quantale-valued sets)

Duality hypdoc. always have truth val. obj. as dualising obj. Ω.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Hyperdoctrinal Gödel Translation

We can think of ¬¬ as a functor Fix¬¬ from HA, the cat. of heyting algs., to BA, the cat. of boolean algs.: Fix¬¬(A) = {a 2 A | ¬¬a = a}. Let P : Cop ! HA be an intuitionistic hyperdoctrine. Then, the composed functor Fix¬¬ P : Cop ! BA forms a classical hyperdoctrine. This is a first-order and hyperdoctrinal analogue of the construction of boolean toposes from given toposes via double negation Gro. topologies on them (it is the HOL case only).

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Hyperdoctrinal Girard Translation

We can treat Girard’s translation along a similar line. We denote by FL!

c the cat. of com. FL algs. with exp. !; they give

the algebraic counterpart of intuitionistic linear logic ILL with !. We regard exp. ! as a functor Fix! from FL!

c to HA:

Fix!(A) = {a 2 A | !a = a} Fix!(A) is the set of those elements of A that admit the structural rules, thus forming a heyting algebra. Let P : Cop ! FL!

c be an ILL hyperdoctrine. Then, the

composed functor Fix! P : Cop ! HA forms an intuitionistic hyperdoctrine. Slightly more general than Girard’s translation thm, in the sense the latter corresp. to the case of syntactic hypdoc. in the former.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Lawvere-Tierney Topology

A Lawvere-Tierney topology j : Ω → Ω in a topos E may be regarded as a natural transformation j : SubE(-) → SubE(-) such that jC is a left-exact monad on SubC(C) for every C ∈ C. Definition A Lawvere-Tierney topology (or operator) on a T-hyperdoctrine P : C → Alg(T) is a natural transformation j : P → P s.t. jC : P(C) → P(C) is a left-exact monad on P(C) for ∀C ∈ C. Co-topology (or co-operator) is defined in a similar way.

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic

Universal Translation Theorem

Fix a Lawvere-Tierney toplogy j on P : Cop ! Alg(T). Suppose the image of jC : P(C) ! P(C) is an algebra of a monad S. Then, we define an Alg(S)-valued presheaf Pj : Cop ! Alg(S) as follows: Pj(C) = Fix(jC) = {jC(ϕ) | ϕ 2 P(C)}. Theorem Assume: for π : C ⇥ D ! D, ϕ 2 Pj(C ⇥ D), and ψ 2 Pj(D), jD(8π(jC×D(ϕ))) ` 8π(jC×D(ϕ)). If P : Cop ! Alg(T) has quantifiers 8, 9, then Pj : Cop ! Alg(S) has quantifiers 8, 9 as well (i.e., j pres. quantifier structures).

Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics

Full Lambek Hyperdoctrine Full Lambek Tripos

Conclusions

Summing up: Tripos semantics for higher-order intuitionistic logic can be extended so as to accommodate a variety of substructural logics.

Generic/set-theoretical completeness theorems. Can be applied to classical, intuitionistic, linear, relevant, and fuzzy logics.

And this allows us to systematise different logical translations between different logics, including Gödel’s and Girard’s translation. References:

• Y. Maruyama, Full Lambek Hyperdoctrine, Springer LNCS, 2013
• Y. Maruyama, Duality Theory and Categorical Universal Logic,

EPTCS, 2014

• Y. Maruyama, Full Lambek Tripos, preprint, 2015.

Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 7 / 7