Categorical Semantics for Logic-Enriched Type Theories

Robin Adams

Royal Holloway, University of London

TYPES 2011, 8 September 2011

Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 1 / 21

The Curry-Howard Isomorphism

There are two facts that are both sometimes referred to as theCurry-Howard isomorphism. One is trivial, one is not.

Non-trivial Fact

When we do so:

the rules for conjunction are the rules for product type;

the rules for impliciation are the rules for non-dependent functiontype;

the rules for universal quantification are (almost) the rules fordependent function type;

the rules for classical logic are the rules for control operators (usually);

the rules for modal logic are the rules for metavariables;

In this talk, ‘Curry-Howard’ shall mean the second.

The Curry-Howard IsomorphismThere are two facts that are both sometimes referred to as theCurry-Howard isomorphism. One is trivial, one is not.

Trivial Fact

It is possible to write a linear syntax for natural deduction proofs, and thenwrite Γ ` P : φ for ‘P is a proof of φ (that depends on the free variablesand hypotheses Γ)’

Non-trivial Fact

When we do so:

Non-trivial Fact

When we do so:

Non-trivial Fact

When we do so:

My Beliefs on Curry-Howard

I believe:

1 Curry-Howard is surprising,

2 There is something there to be explained. (Why do propositionsbehave like types?)

3 We do not have a good explanation yet. (Propositions are notliterally types.)

4 We are having problems because we tacetly assumepropositions-as-types.

5 We should instead turn Curry-Howard into a mathematical object.

I believe:

IntroductionWe have:

systems of first-orderarithmetic

systems ofsecond-orderarithmetic

set theories

type theories

It is very difficult totranslate between thesystems on the left,and the systems on theright.

Syntax and semanticsare both very different.

Logic-enriched typetheories (LTTs) help.

set theories

type theories

ACA 0PA

set theories

type theories

ACA 0PA

set theories

type theories

set theories

type theories

Introduction

I If propositions reallywere types, itshould be easy.

Introduction

ML(CZF)

Introduction

ML(CZF)

Introduction

LTT ML+EM

Curry−Howard

Introduction

Syntactic translationsare possible.

Curry-Howard becomesjust one of a family.

LTT ML+EM

Curry−Howard

Introduction

Syntactic translationsare possible.

Curry-Howard becomesjust one of a family.

LTT ML+EM

Curry−Howard

Double Negation

Russell−Prawitz

Introduction

Syntactic translationsare not enough.

LTT ML+EM

Curry−Howard

Double Negation

Russell−Prawitz

Introduction

We need a semanticsfor LTTs.

LTT ML+EM

Curry−Howard

Double Negation

Russell−Prawitz

1 Logic-Enriched Type TheoriesSyntax

2 Categorical SemanticsIntroduction to Categorical SemanticsCategorical Semantics for Logic-Enriched Type TheoriesSoundness and Completeness Theorems

3 ApplicationsConservativity of ACA0 over PABounded Quantification

Syntax of an LTT

LTT0 is a system with:Judgement forms:

Γ ` A Type Γ ` M : AΓ ` φ Prop Γ ` P : φ

and associated equality judgements.

arrow types A→ B

product types A× B

natural numbers Na type universe U

classical predicate logic

a propositional universe prop Type

We write Set (A) for A→ prop.

Syntax of an LTT

LTT0 is a system with:

arrow types A→ Bwith objects λx : A.M

product types A× B

Syntax of an LTT

arrow types A→ B

product types A× Bwith objects (M,M)

Syntax of an LTT

arrow types A→ B

product types A× B

natural numbers Nwith objects 0 and S(M)

a type universe U

Syntax of an LTT

arrow types A→ B

product types A× B

natural numbers Na type universe Uwith objects N̂ and M×̂M

Syntax of an LTT

arrow types A→ B

product types A× B

classical predicate logicwith propositions M =A M, ¬φ, φ ∧ ψ, ∀x : A.φ, . . .

Syntax of an LTT

arrow types A→ B

product types A× B

Syntax of an LTT

arrow types A→ B

product types A× B

The Propositional UniverseA type universe is a type whose objects are names of types:

The universe U contains N̂, N̂×̂N̂, . . . .

A type is small iff it has a name in U, large otherwise.

A propositional universe is a type whose objects are names of propositions:

The universe prop contains ¬̂0=̂N̂S(0), ∀̂x : N̂.x=̂N̂x , etc.

In LTT0, prop contains the propositions that do not involvequantification over large types.

A proposition is small iff it has a name in prop, large otherwise.

The strength of an LTT is determined by which types and whichpropositions we can eliminate over.

We can only eliminate N over small types.

We can only use proof by induction with small propositions.

Adding a new type or connective is conservative. Adding it to theuniverses is not.

Adding a new type or connective is conservative. Adding it to theuniverses is not.Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21

Categorical Semantics

We can give semantics to a type theory in a variety of ways:

Map types to sets, ω-sets, PERs, sheaves, domains, . . .

To save repeating work, we:

define the properties a category must have for us to build a semanticsfrom its objects;

give semantics to the theory in an arbitrary category with thoseproperties.

Categorical Semantics for a Dependent Type TheoryTo give semantics to adependent type theory, weneed:

a category B (whoseobjects interpret contextsΓ);

a category E (whoseobjects interprettypes-in-contextΓ ` A Type);

a functor P : E→ B→(mapping Γ ` A to(Γ, x : A)→ Γ).

such that

p = cod ◦P is a fibration

B has a terminal object

EP- B→

such that

P- B→

such that

EP- B→

such that

EP- B→

Categorical Semantics for a LTT

To give semantics to a LTT,we need, in addition:

a category P (whoseobjects interpretpropositions-in-contextΓ ` φ Prop);

a fibration q : P→ B(mapping Γ ` φ Prop toΓ)

for every objectΓ ` A Type in E,a right adjoint π∗ a ∀ anda left adjoint ∃ a π∗

such that P is a locallyCartesian closed category.

P EP- B→

Categorical Semantics for UniversesTo give semantics to a dependent type theory with a universe (U,T ), weneed:

a substructure (intended to represent the small types and smallcontexts)

an object U in E over the empty context (terminal object)a generic object x : U ` T (x) Type in E over domP(` U Type);

D ⊂ - E

⊂ - B?

To give semantics to (prop,V ), we need in addition:

a substructure (intended to represent the small propositions andcontexts consisting solely of small propositions)an object prop in E over the terminal objecta generic object V in E over domPprop.

We require > → 〈x : N〉 → 〈x : N〉 to be a weak fibred natural numberobject in both of these right-hand-sides.

a substructure (intended to represent the small types and smallcontexts)an object U in E over the empty context (terminal object)

a generic object x : U ` T (x) Type in E over domP(` U Type);

D ⊂ - E

⊂ - B?

a substructure (intended to represent the small types and smallcontexts)an object U in E over the empty context (terminal object)a generic object x : U ` T (x) Type in E over domP(` U Type);

D � D×A B

A?� B

Categorical Semantics for Universes

To give semantics to a dependent type theory with a universe (U,T ), weneed:

an object U in E over the empty context (terminal object)

a substructure (intended to represent the small propositions andcontexts consisting solely of small propositions)

an object prop in E over the terminal object

a generic object V in E over domPprop.

an object prop in E over the terminal objecta generic object V in E over domPprop.

Q ⊂ - P

⊂ - B?

a substructure (intended to represent the small propositions andcontexts consisting solely of small propositions)an object prop in E over the terminal object

Q ⊂ - P

⊂ - B?

Q � Q×C B

C?� B

Interpretation

Given an LTTW-category C, define:

for every valid context Γ, an object [[Γ]] of B;

for every type A such that Γ ` A Type, an object [[Γ ` A]] of E suchthat p [[Γ ` A]] = [[Γ]]

for every term M such that Γ ` M : A, an arrow[[Γ ` M]] : [[Γ]]→ domP [[Γ ` A]]

for every proposition φ such that Γ ` φprop, an object [[Γ ` φ]] of Pover [[Γ]]

Soundness Theorem

Theorem

Every judgement is true in any LTTW-category. That is:

1 If Γ ` A = B then [[Γ ` A]] = [[Γ ` B]]

2 If Γ ` M = N : A then [[Γ ` M]] = [[Γ ` N]]

3 If Γ ` φ = ψ then [[Γ ` φ]] = [[Γ ` ψ]]

4 If there is a proof Γ ` P : φ then there is a vertical arrow> → [[Γ ` φ]] in the fibre P/ [[Γ]].

Proof.

Induction on derivations.

Completeness Theorem

Theorem

If a judgement is true in every category C, then it is derivable in T .

Proof.

Define the category Cl (T ), the classifying category of T , thus:

the objects of B are the valid contexts;

the objects of E are the pairs (Γ,A) such that Γ ` A Type,quotiented by equality;

If a judgement is true in Cl (T ), then it is derivable in T .

In fact, Cl (T ) is an initial object in the metacategory ofLTTW-categories. The interpretation given earlier is the unique functorCl (T )→ C. This is the sort of thing that gets category theorists excited.

Theorem

Proof.

In fact, Cl (T ) is an initial object in the metacategory ofLTTW-categories. The interpretation given earlier is the unique functorCl (T )→ C.

This is the sort of thing that gets category theorists excited.

Theorem

Proof.

In fact, Cl (T ) is an initial object in the metacategory ofLTTW-categories. The interpretation given earlier is the unique functorCl (T )→ C. This is the sort of thing that gets category theorists excited.

Conservativity of LTT0 over PAI have previously given syntactic proofs that LTT0 is conservative over PA.We can now give a semantic proof of the same result.

Theorem

LTT0 is conservative over PA.

Proof.

From any model M of PA, we construct a model of LTT0.

The objects of E are the sets built up from |M| by ×, → and P.The objects of B are the sets of all sequences of objects of E. The arrowsare the hor functions.The objects of P over b ∈ B are all subsets of b.

We can similarly prove LTT0 conservative over ACA0.

Corollary

ACA0 is conservative over PA.

Theorem

Proof.

From any model M of PA, we construct a model of LTT0.Define the higher-order recursive (hor) functions to be those built up from0M and SM by composition, primitive recursion, pairing, projection,lambda-abstraction and application.

Corollary

Theorem

Proof.

From any model M of PA, we construct a model of LTT0.Define the higher-order recursive (hor) functions to be those built up from0M and SM by composition, primitive recursion, pairing, projection,lambda-abstraction and application.Define the arithmetic predicates to be those built up from equality byBoolean operations and quantification over |M|.

Corollary

Theorem

Proof.

From any model M of PA, we construct a model of LTT0.The objects of E are the sets built up from |M| by ×, → and P,where A→ B is the set of hor functions from A to B,and PA is the set of arithmetic subsets of A.

The objects of B are the sets of all sequences of objects of E. The arrowsare the hor functions.The objects of P over b ∈ B are all subsets of b.

Corollary

Theorem

Proof.

From any model M of PA, we construct a model of LTT0.The objects of E are the sets built up from |M| by ×, → and P.The objects of B are the sets of all sequences of objects of E. The arrowsare the hor functions.

The objects of P over b ∈ B are all subsets of b.

Corollary

Theorem

Proof.

From any model M of PA, we construct a model of LTT0.The objects of E are the sets built up from |M| by ×, → and P.The objects of B are the sets of all sequences of objects of E. The arrowsare the hor functions.The objects of P over b ∈ B are all subsets of b.

Note that E and P are radically different.

We can similarly prove LTT0

conservative over ACA0.

Corollary

Theorem

Proof.

From any model M of PA, we construct a model of LTT0.The objects of E are the sets built up from |M| by ×, → and P.The objects of B are the sets of all sequences of objects of E. The arrowsare the hor functions.The objects of P over b ∈ B are all subsets of b.

Corollary

Bounded Quantification

Problem: How do we turn prop into the set of Σ0-propositions?

Just close it under bounded quantification? Categorical semantics arehorrible.Answer: Put (a name of) prop in U.We can define bounded quantification by elimination N over prop: