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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;
etc.
In this talk, ‘Curry-Howard’ shall mean the second.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 2 / 21
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:
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;
etc.
In this talk, ‘Curry-Howard’ shall mean the second.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 2 / 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;
etc.
In this talk, ‘Curry-Howard’ shall mean the second.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 2 / 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;
etc.
In this talk, ‘Curry-Howard’ shall mean the second.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 2 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 3 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 3 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 3 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 3 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 3 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 3 / 21
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.
PA
I 0Σ
IΣ2
I 1Σ
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
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.
I 0Σ
IΣ2
I 1Σ
ATR 0
WKL0
RCA 0
ACA 0PA
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
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.
I 0Σ
IΣ2
I 1Σ
ATR 0
WKL0
RCA 0
ACA 0PA
CZF
IZF
ZF
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
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.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
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.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
It is very difficult totranslate between thesystems on the left,and the systems on theright.
I If propositions reallywere types, itshould be easy.
Syntax and semanticsare both very different.
Logic-enriched typetheories (LTTs) help.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
LTT
LTTW
0
ML(CZF)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
I 0Σ
IΣ2
I 1Σ
WKL0
RCA 0
ACA 0
ATR 0
PA
CZF
IZF
ZF
1
ML 2
ML
ML
MLW
V1ML
LTT
LTTW
0
ML(CZF)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
LTT ML+EM
Curry−Howard
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
Syntactic translationsare possible.
Curry-Howard becomesjust one of a family.
LTT ML+EM
Curry−Howard
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
Syntactic translationsare possible.
Curry-Howard becomesjust one of a family.
LTT ML+EM
Curry−Howard
Double Negation
Russell−Prawitz
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
Syntactic translationsare not enough.
LTT ML+EM
Curry−Howard
Double Negation
Russell−Prawitz
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
Introduction
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.
We need a semanticsfor LTTs.
LTT ML+EM
Curry−Howard
Double Negation
Russell−Prawitz
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 4 / 21
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
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 5 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
arrow types A→ Bwith objects λx : A.M
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
arrow types A→ B
product types A× Bwith objects (M,M)
natural numbers Na type universe U
classical predicate logic
a propositional universe prop Type
We write Set (A) for A→ prop.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
arrow types A→ B
product types A× B
natural numbers Nwith objects 0 and S(M)
a type universe U
classical predicate logic
a propositional universe prop Type
We write Set (A) for A→ prop.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
arrow types A→ B
product types A× B
natural numbers Na type universe Uwith objects N̂ and M×̂M
classical predicate logic
a propositional universe prop Type
We write Set (A) for A→ prop.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
arrow types A→ B
product types A× B
natural numbers Na type universe U
classical predicate logicwith propositions M =A M, ¬φ, φ ∧ ψ, ∀x : A.φ, . . .
a propositional universe prop Type
We write Set (A) for A→ prop.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
Syntax of an LTT
LTT0 is a system with:
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 6 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 7 / 21
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.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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 8 / 21
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→
B
p
?
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 9 / 21
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
. . .
E
P- B→
B
p
?
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 9 / 21
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→
B
p
?�
cod
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 9 / 21
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→
B
p
?�
cod
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 9 / 21
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→
B
p
?�
cod
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 10 / 21
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→
B
p
?�
codq
-
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 10 / 21
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→
B
p
?�
codq
-
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 10 / 21
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→
B
p
?�
codq
-
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 10 / 21
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
A?
⊂ - 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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
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
A?
⊂ - 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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
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 � D×A B
A?� 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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
Categorical Semantics for Universes
To 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);
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 object
a 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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
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);
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.
Q ⊂ - P
C?
⊂ - B?
We require > → 〈x : N〉 → 〈x : N〉 to be a weak fibred natural numberobject in both of these right-hand-sides.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
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);
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 object
a generic object V in E over domPprop.
Q ⊂ - P
C?
⊂ - B?
We require > → 〈x : N〉 → 〈x : N〉 to be a weak fibred natural numberobject in both of these right-hand-sides.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
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);
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.
Q � Q×C B
C?� B
?
We require > → 〈x : N〉 → 〈x : N〉 to be a weak fibred natural numberobject in both of these right-hand-sides.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
Categorical Semantics for Universes
To 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);
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 object
a 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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
Categorical Semantics for Universes
To 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);
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 object
a 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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 11 / 21
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]]
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 12 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 13 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 14 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 14 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 14 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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.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.
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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.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|.
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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,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.
We can similarly prove LTT0 conservative over ACA0.
Corollary
ACA0 is conservative over PA.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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.
Note that E and P are radically different.
We can similarly prove LTT0
conservative over ACA0.
Corollary
ACA0 is conservative over PA.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 15 / 21
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:
∀x < 0.φ(x) = >∀x < S(n).φ(x) = ∀x < n.φ(x) ∧ φ(n)
Conversely, any formula in prop in the new LTT corresponds to aΣ0-formula in I Σ0(exp).(Show that the functions in N→ N are all defined by a Σ0-formula inI Σ0(exp). Use the fact that the Σ0-definable functions are closed underprimitive recursion.)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 16 / 21
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:
∀x < 0.φ(x) = >∀x < S(n).φ(x) = ∀x < n.φ(x) ∧ φ(n)
Conversely, any formula in prop in the new LTT corresponds to aΣ0-formula in I Σ0(exp).(Show that the functions in N→ N are all defined by a Σ0-formula inI Σ0(exp). Use the fact that the Σ0-definable functions are closed underprimitive recursion.)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 16 / 21
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:
∀x < 0.φ(x) = >∀x < S(n).φ(x) = ∀x < n.φ(x) ∧ φ(n)
Conversely, any formula in prop in the new LTT corresponds to aΣ0-formula in I Σ0(exp).(Show that the functions in N→ N are all defined by a Σ0-formula inI Σ0(exp). Use the fact that the Σ0-definable functions are closed underprimitive recursion.)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 16 / 21
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:
∀x < 0.φ(x) = >∀x < S(n).φ(x) = ∀x < n.φ(x) ∧ φ(n)
Conversely, any formula in prop in the new LTT corresponds to aΣ0-formula in I Σ0(exp).(Show that the functions in N→ N are all defined by a Σ0-formula inI Σ0(exp). Use the fact that the Σ0-definable functions are closed underprimitive recursion.)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 16 / 21
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:
∀x < 0.φ(x) = >∀x < S(n).φ(x) = ∀x < n.φ(x) ∧ φ(n)
Conversely, any formula in prop in the new LTT corresponds to aΣ0-formula in I Σ0(exp).
(Show that the functions in N→ N are all defined by a Σ0-formula inI Σ0(exp). Use the fact that the Σ0-definable functions are closed underprimitive recursion.)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 16 / 21
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:
∀x < 0.φ(x) = >∀x < S(n).φ(x) = ∀x < n.φ(x) ∧ φ(n)
Conversely, any formula in prop in the new LTT corresponds to aΣ0-formula in I Σ0(exp).(Show that the functions in N→ N are all defined by a Σ0-formula inI Σ0(exp). Use the fact that the Σ0-definable functions are closed underprimitive recursion.)
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 16 / 21
Conclusion
Logic-enriched type theories are the right setting for investigating manyfoundational questions in type theory, and in orthodox logic.
At themoment, they are a solution looking for a problem.
I don’t want to carry on finding new proofs of old results.
Questions I plan to investigate:
What is the proof-theoretic ordinal of this LTT?
What is the set of functions definable in this LTT?
Some logical features work nicely in LTTs (Σ0-induction,Σ1-induction)
Some do not (Σ2-induction)
What’s the difference between these?
Please bring me some more.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 17 / 21
Conclusion
Logic-enriched type theories are the right setting for investigating manyfoundational questions in type theory, and in orthodox logic. At themoment, they are a solution looking for a problem.
I don’t want to carry on finding new proofs of old results.
Questions I plan to investigate:
What is the proof-theoretic ordinal of this LTT?
What is the set of functions definable in this LTT?
Some logical features work nicely in LTTs (Σ0-induction,Σ1-induction)
Some do not (Σ2-induction)
What’s the difference between these?
Please bring me some more.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 17 / 21
Conclusion
Logic-enriched type theories are the right setting for investigating manyfoundational questions in type theory, and in orthodox logic. At themoment, they are a solution looking for a problem.
I don’t want to carry on finding new proofs of old results.
Questions I plan to investigate:
What is the proof-theoretic ordinal of this LTT?
What is the set of functions definable in this LTT?
Some logical features work nicely in LTTs (Σ0-induction,Σ1-induction)
Some do not (Σ2-induction)
What’s the difference between these?
Please bring me some more.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 17 / 21
Conclusion
Logic-enriched type theories are the right setting for investigating manyfoundational questions in type theory, and in orthodox logic. At themoment, they are a solution looking for a problem.
I don’t want to carry on finding new proofs of old results.
Questions I plan to investigate:
What is the proof-theoretic ordinal of this LTT?
What is the set of functions definable in this LTT?
Some logical features work nicely in LTTs (Σ0-induction,Σ1-induction)
Some do not (Σ2-induction)
What’s the difference between these?
Please bring me some more.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 17 / 21
Syntax of an LTTLTT0 is a system with:Judgement forms:
Γ ` A type Γ ` M : AΓ ` φ Prop Γ ` P : φ
and associated equality judgements.
Type A ::=
A→ A | A× A | N | U | T (M) | Set (A)
Term M ::= x
| λx : A.M | MM | (M,M) | π1(M) | π2(M) |0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |{x : A | P}
Proposition φ ::=
M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::=
· · · | M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A
| A× A | N | U | T (M) | Set (A)
Term M ::= x | λx : A.M | MM
| (M,M) | π1(M) | π2(M) |0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |{x : A | P}
Proposition φ ::=
M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::=
· · · | M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A | A× A
| N | U | T (M) | Set (A)
Term M ::= x | λx : A.M | MM | (M,M) | π1(M) | π2(M)
|0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |{x : A | P}
Proposition φ ::=
M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::=
· · · | M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A | A× A | N
| U | T (M) | Set (A)
Term M ::= x | λx : A.M | MM | (M,M) | π1(M) | π2(M) |0 | S(M) | EN(M,M,M,M)
| N̂ | M×̂M |{x : A | P}
Proposition φ ::=
M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::=
· · · | M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×
classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A | A× A | N | U | T (M)
| Set (A)
Term M ::= x | λx : A.M | MM | (M,M) | π1(M) | π2(M) |0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |
{x : A | P}
Proposition φ ::=
M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::=
· · · | M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A | A× A | N | U | T (M)
| Set (A)
Term M ::= x | λx : A.M | MM | (M,M) | π1(M) | π2(M) |0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |
{x : A | P}
Proposition φ ::= M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ
| prop | V (P)
Proof P ::= · · ·
| M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A | A× A | N | U | T (M)
| Set (A)
Term M ::= x | λx : A.M | MM | (M,M) | π1(M) | π2(M) |0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |
{x : A | P}
Proposition φ ::= M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::= · · · | M=̂MM | ¬̂φ | φ∧̂φ | · · ·
| M ∈ M
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
Syntax of an LTTLTT0 is a system with:
arrow types
product types
natural numbers
a type universe closed under N and ×classical predicate logic
a propositional universe
typed sets
Type A ::= A→ A | A× A | N | U | T (M) | Set (A)Term M ::= x | λx : A.M | MM | (M,M) | π1(M) | π2(M) |
0 | S(M) | EN(M,M,M,M) | N̂ | M×̂M |{x : A | P}
Proposition φ ::= M =A M | ¬φ | φ ∧ φ | φ ∨ φ | φ→ φ |∀x : A.φ | ∃x : A.φ | prop | V (P)
Proof P ::= · · · | M=̂MM | ¬̂φ | φ∧̂φ | · · · | M ∈ MRobin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 18 / 21
We can give a semantic proof of this result:A function f : Nn → N is definable in PA iff there is a formulaφ[x1, . . . , xn, y ] such that:
1 for all a1, . . . , an, PA ` φ[a1, . . . , an, f (a1, . . . , an)];
2 PA ` ∀x1 · · · ∀xn∃!yφ[x1, . . . , xn, y ]
Theorem
The functions definable in PA are exactly the ε0-recursive functions.
Proof.
Construct a model of LTT0 in which the arrows are the ε0-recursivefunctions. Then apply conservativity.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 19 / 21
History of LTTs
2002 Aczel and Gambino [?] define translations betweenConstructive ZF (CZF) and the type theory ML1V.
2006 Gambino and Aczel [?] introduce the logic-enriched typetheory ML(CZF) as a half-way stage.
2007 Adams and Luo [?] show how an LTT can represent Weyl’sschool of predicativism.
2010 Adams and Luo [?] show their system is not conservativeover PA.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 20 / 21
History of LTTs
2002 Aczel and Gambino [?] define translations betweenConstructive ZF (CZF) and the type theory ML1V.
2006 Gambino and Aczel [?] introduce the logic-enriched typetheory ML(CZF) as a half-way stage.
2007 Adams and Luo [?] show how an LTT can represent Weyl’sschool of predicativism.
2010 Adams and Luo [?] show their system is not conservativeover PA.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 20 / 21
History of LTTs
2002 Aczel and Gambino [?] define translations betweenConstructive ZF (CZF) and the type theory ML1V.
2006 Gambino and Aczel [?] introduce the logic-enriched typetheory ML(CZF) as a half-way stage.
2007 Adams and Luo [?] show how an LTT can represent Weyl’sschool of predicativism.
2010 Adams and Luo [?] show their system is not conservativeover PA.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 20 / 21
History of LTTs
2002 Aczel and Gambino [?] define translations betweenConstructive ZF (CZF) and the type theory ML1V.
2006 Gambino and Aczel [?] introduce the logic-enriched typetheory ML(CZF) as a half-way stage.
2007 Adams and Luo [?] show how an LTT can represent Weyl’sschool of predicativism.
2010 Adams and Luo [?] show their system is not conservativeover PA.
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 20 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21
The Moral of the Story
From this work, I take the message:
LTTs can do some things better than either orthodox logics or typetheories.
LTTs are very useful as an intermediary between orthodox logics andtype theories.
LTTs turn Curry-Howard into a mathematical object — a translationfrom an LTT to a type theory;
. . . which becomes just one of a family of translations.
But I need semantics to guide future research.
I also need to think of better names.
I do not claim:
LTTs are ‘better’ than predicate logics or type theories.
Curry-Howard is ‘bad’
Robin Adams (RHUL) Categorical Semantics for LTTs TYPES 2011 21 / 21