Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks
A comparison between ITT and CoC Categorical models of type theories
Category Theory 2015 · Aveiro, PORTUGAL
Nino Guallart
July of 2015
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Index
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Index
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Index
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Index
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Índice
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Introduction
What is a type theory? Formal system of elements (terms), each of one belonging to a certain type. Originally proposed as an alternative to naïve set theory (Russell's paradox), we will see type theories formulated in λ calculus.
Everything is a function Church's λC was intended to study computable functions. It is Turing-complete and can be seen as an abstract model of a functional programming language (Landin, 1965). Whereas λC is untyped, simply typed lambda calculus (STLC) is typed, and therefore can be seen as the basis of typed functional programming languages (Haskell, ML). beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Introduction
What is a type theory? Formal system of elements (terms), each of one belonging to a certain type. Originally proposed as an alternative to naïve set theory (Russell's paradox), we will see type theories formulated in λ calculus.
Everything is a function Church's λC was intended to study computable functions. It is Turing-complete and can be seen as an abstract model of a functional programming language (Landin, 1965). Whereas λC is untyped, simply typed lambda calculus (STLC) is typed, and therefore can be seen as the basis of typed functional programming languages (Haskell, ML). beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Semantics of type theories
In this talk we will focus on the categorial semantics of STLC, before moving on a discussion of the categorical interpretation of more complex type theories.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Índice
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
λ→
e,f := x|c|λxτ .e|ef |
σ,τ := β|σ → τ
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
λσ
τ e,f := x|c|λx .e|ef |(e,f )|π1e|π1e
σ,τ := β|σ → τ|σ × τ
λC syntax: λ→ and →. Type constructors: →, ×,(+). Base types. Void type: 0, with no terms. Unit type: 1, with one term, *. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
λσ
τ e,f := x|c|λx .e|ef |(e,f )|π1e|π1e
σ,τ := β|σ → τ|σ × τ
λC syntax: λ→ and →. Type constructors: →, ×,(+). Base types. Void type: 0, with no terms. Unit type: 1, with one term, *. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
λσ
τ e,f := x|c|λx .e|ef |(e,f )|π1e|π1e
σ,τ := β|σ → τ|σ × τ
λC syntax: λ→ and →. Type constructors: →, ×,(+). Base types. Void type: 0, with no terms. Unit type: 1, with one term, *. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
λσ
τ e,f := x|c|λx .e|ef |(e,f )|π1e|π1e
σ,τ := β|σ → τ|σ × τ
λC syntax: λ→ and →. Type constructors: →, ×,(+). Base types. Void type: 0, with no terms. Unit type: 1, with one term, *. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
There are well-formed (pre)terms which are not well-typed, giving error: (λx : σ.x)(a : τ). From the set of preterms, let's consider the set Λ of well-typed terms, or simply terms.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
Typing rules for → (application and abstraction):
Γ,x : σ ` e : τ ABS Γ ` (λx : σ.e):(σ → τ)
e f Γ ` : σ → τ Γ ` : σ APP Γ ` ef : τ
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
Typing rules for × (introduction and elimination):
t A u B Γ ` : Γ ` : PROD-INT Γ ` (t,u): A × B
Γ ` (t,u): A × B PROD-EL1 π1(t,u): A Γ ` (t,u): A × B PROD-EL2 π2(t,u): B
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Simply typed lambda calculus
Unlike LC, is decidable and non Turing-complete -> its evaluation always halts. Normalising: the reduction of a preterm always terminates. If is a term, it ends giving the normal form of the term. If not, giving out some kind of error or unspeci ed behaviour.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Term evaluation
Based on β−reduction:
(λx : σ.E)F E-APP E[x := F ]
t t0 t t0 → E-APP1 → E-APP2 tu → t0u ut → ut0
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Syntax and semantics
A STLC for natural numbers:
` 0 : N
x N Γ ` : SUCC Γ ` sx : N
Γ ` x : N Γ ` y : N SUM Γ ` x + y : N ... beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Syntax and semantics
A STLC for natural numbers:
` 0 : N
x N Γ ` : SUCC Γ ` sx : N
Γ ` x : N Γ ` y : N SUM Γ ` x + y : N ... beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Syntax and semantics
A STLC for natural numbers:
` 0 : N
x N Γ ` : SUCC Γ ` sx : N
Γ ` x : N Γ ` y : N SUM Γ ` x + y : N ... beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Curry-Howard isomorphism
Predicates-as-types, link between type theory and logical calculus:
Intuitionistic interpretation. If a type A has at least a term a, a : A, a is a proof of A. Lambda abstraction: hypotheses introduction. Term application equates to modus ponens:
e f Γ ` : σ → τ Γ ` : σ APP Γ ` ef : τ
Rules for introduction and elimination of pairs (product) are
akin to those of conjunction of predicates. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Curry-Howard isomorphism
σ,τ ::= β|σ → τ|σ × τ|σ + τ|0|1
p,q ::= b|p → q|p ∧ q|p ∨ q|⊥|> Types can be seen as propositions. Terms can be interpreted intuitionistically (BHK) as proofs (algorithms) of their corresponding types. For example, λxα .x is a proof of α → α, which is a tautology (taulologies equate to combinators). Given a set of basic proofs with their types/propositions, we can build a proof of the propositions derived from them. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Categorical model of STLC
Syntax-semantics duality between STLC and CCC (Lambek, 1970). More precisely, type theories can be interpreted using category theory, and conversely, we can formalise categories in the language of type theories. Generally, we can say that the relation between a type theory and its corresponding category is akin to syntax vs semantics . We can show this studying the relationship between STLC with pairs and Cartesian closed categories (Cfr. Lambek and Scott, 1988).
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Lambek-Lawvere isomorphism
Given Λ, let C be a Cartesian closed category with: Product and exponential. Terminal object 1. Interpretation Λ → C : Types are interpreted as objects, i.e. [[β]] ∈ Ob(C ). A term e : σ is interpreted uniquely as a morphism 1 → [[σ]]. Application of terms: composition of morphisms. Pairs of terms: pair of morphisms.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Lambek-Lawvere isomorphism
Given a CCC C , its internal language is the smallest TT preserving its structure: Types are the objects of C . Terms are the morphisms of C . The composition of morphisms is given by term aplication. A pair of morphisms is described by a pair of terms.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Lambek-Lawvere isomorphism
The functors Syn (or Con) and Lang are dual to each other, so: The interpretation of Λ , Syn(Λ), is a CCC C 0. Conversely, the structure of a given CCC C can be described using a Λ0 of STLC, Λ0 = Lang(C ). Λ0 is the internal language of C .
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Lambek-Lawvere isomorphism
The functors Syn (or Con) and Lang are dual to each other, so: The interpretation of Λ , Syn(Λ), is a CCC C 0. Conversely, the structure of a given CCC C can be described using a Λ0 of STLC, Λ0 = Lang(C ). Λ0 is the internal language of C .
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Lambek-Lawvere isomorphism
The functors Syn (or Con) and Lang are dual to each other, so: The interpretation of Λ , Syn(Λ), is a CCC C 0. Conversely, the structure of a given CCC C can be described using a Λ0 of STLC, Λ0 = Lang(C ). Λ0 is the internal language of C .
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Lambek-Lawvere isomorphism
Type theory Category theory Types Objects Unit type (> or 1) Terminal object Product type A × B Product of objects A × B Function type A → B Exponential object BA Terms Morphisms Pair of terms (f ,g) Pair of morphisms (f ,g) Projections of terms, π1and π2 Projections of morphisms, π1and π2 Abstraction λx A.f : B Arrow f : A → B with a free variable x : A Application fg Composition of arrows fg beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks C-H and L-L isomorphisms
All of them form a kind of triangular relationship between propositional calculus, type theory and category theory. Important in the areas of logic and computation (morphisms as algorithms).
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Índice
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Higher-order type systems
Note the rigid distinction between types and terms in STLC:
λxα .x : α → α
λ(xy)N×N .x + y : N × N → N
Why do we should want to explore these systems?
More expressivity: terms on types, types on terms... Disadvantage: systems can be non-normalising.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Higher-order type systems
Note the rigid distinction between types and terms in STLC:
λxα .x : α → α
λ(xy)N×N .x + y : N × N → N
Why do we should want to explore these systems?
More expressivity: terms on types, types on terms... Disadvantage: systems can be non-normalising.
beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Higher-order type systems
Note the rigid distinction between types and terms in STLC:
λxα .x : α → α
λ(xy)N×N .x + y : N × N → N
Why do we should want to explore these systems?
More expressivity: terms on types, types on terms... Disadvantage: systems can be non-normalising.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Dependent type systems
Some relevant authors: Martin-Löf, P. (1984), Intuitionistic Type Theory, Bibliopolis. Cocquand, T.; Huet, G. (1988), The Calculus of Constructions, Information and Computation, Vol. 76, Issue 2-3. Barendregt, H.P. (1991), Introduction to generalized type systems, Journal of Functional Programming, 1(2):125-154.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Pure type systems
PTS blurrs or erases the distinction between types and terms, thus allowing to predicate over types. The concept of type must be generalised: Sorts: ?, (Prop / Type). Kinds: → ?, → , ... We will focus on a speci c family of normalising type systems.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Barendregt's λ−cube
λω λPω
λ2 λP2
Polymorphism λσ λPσ
Type constructors
λ→ Dependence λP
H.P. Barendregt, Introduction to generalized type systems, Journalbeamer-icsi-logo of Functional Programming, 1(2):125-154, 1991.
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Polymorphism
Universal quanti cation over types.
Identity function Instead of an identity function for each type, there is a single one:
Λαλxα .x : ∀α.α → α
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Dependent types
Types depending of terms. Π types: expands the idea of generalisation and arrow.
Πx:AB(x)
∀x ∈ AB(x)
x : A −→ B(x): If A is a type, B is such that, for each x : A there is a type B(x), i.e., B gives out a family of types indexed over A. A b : B has the form λx .T . If b :Πx:AB(x) and a : A, then ba : B(a). beamer-icsi-logo If B(x) gives out constant types, it equates to A → B.
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Dependent types
All application and abstraction rules of these systems can be summarised as follows if we consider the whole cube system (Cfr. (Roorda, 2000)):
Γ,x : A ` b : B Γ ` (Πx : A.B): t ∈ {?, } ABS Γ ` (λx : A.b):Πx : A.B
Γ ` f : (Πx : A.B): t Γ ` a : A APP Γ ` fa : B[x ::= A]
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Categorical model of dependent types: Locally CCC
A category C whose slice categories C /x are all Cartesian closed. The dependent type Πx:AB(x) can be interpreted as an object in the slice category C /x, since it can be interpreted as a morphism B → A.
C /x B A
In the same way that we did whith STLC, we can consider functors Syn and Lang, so Syn(DTT ) is a lCCC, and for a given lCCC, Lang(lCCC) is a DTT. Also, relationship to intuitionistic quanti ed PL. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Martin-Löf's ITT (impredicative)
Constructive foundations of mathematics (Martin-Löf,1971). Also continuation of the idea of isomorphism between propositions and types within an intuitionistic framework (terms are proofs of propositions). Extension of STLC by adding universes. A universe [idea taken from Grothendieck] contains as objects the previous types. So if σ and τ are elements of universe U, they are types. Universe closed under construction of types:
Γ ` f : A : U Γ ` B : U Γ ` A → B : U
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Martin-Löf's ITT (impredicative)
Showed to be inconsistent when
U : U is considered. (Girard,1971) showed its inconsistency (the expansion of the term does not reach a normal form).
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Girard's paradox
Not at the same time: Universal quanti cation over all propositions. Isomorphism between types and propositions. Alternatives: ITT (predicative): not universal quanti cation over all propositions. Prop and Type are identi ed, a hierarchy of universes. CoC (impredicative): the relationship between propositions and types is weaker than isomorphism: every proposition is identi ed with the type of its roofs, but there are non-propositional types (data types). beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Intuitionistic type theory
The predicative version of this theory (Martin-Löf,1975) has a predicative hierarchy of universes. Dependent and function types. Dependent sum and sum types. Unit and empty types. Strongly normalising. Identi cation of types and (quanti ed) propositions (CH isomorphism). Terms of a certain types are seen as proofs of them. The rules of formation of complex types are akin to the rules of intuitionistic rst-order logic. Equality types and identity types: won't be discussed here. beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Category of ITT: Locally CCC
Extensionally, the aforementioned locally closed Cartesian categories (lCCC) can provide a model for ITT. Intensional versions are more much complicated (won't be considered here).
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Calculus of constructions
Created by Thierry Coquand. Type of propositions P (small types), hierarchy of types of types (large types) and the type T of large types. K ::= ?|i (i ≥ 1)
M N x K x x Mτ MN σ,τ, , ::= | |Π : σ.τ|λ : σ. | τ[x:σ] Proofs have as a type P. Predicates are functions that give out propositions.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Calculus of constructions
Some features: Pure type system. Strongly normalising. Dependent types. Empty and unit type. Non-propositional types: e.g. data types. Distinction between: Impredicative type * (type of propositions). Predicative hierarchy (types of types).
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Calculus of constructions
Propositions can be seen as types, extending thus the CH isomorphism to quanti ed PL, but not all types are to be read as propositions. Rejection of the identi cation between propositions and types (Coquand, 1986):
Typei and Prop are terms, and also for terms M and N, then (MN), ΠM.N and λx.M are terms. T is not a term, is only a type. Predicativity and non-predicativity are not contradictory concepts: simply, the level of proposition may be non-predicative and the level of type must be predicative. (Coquand, 1986) beamer-icsi-logo
Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Model
The model of CoC has to deal with a predicative structure of types and a predicative universe T. Locally CCC for the hierarchy of types. Appropiate structure for the impredicativity: collections of subsets.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Índice
1 Introduction
2 Simply typed lambda calculus
3 Higher-order type systems
4 Concluding remarks
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks Concluding remarks
Concluding remarks Problems with the categorical model of dependent types. Dierences between the predicative hierarchy of ITT and the impredicative universe of CoC. CH isomorphism and its variants: relationship between propositions and types.
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC Introduction Simply typed lambda calculus Higher-order type systems Concluding remarks
That's all! Thank you very much!
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Category Theory 2015 · Aveiro, PORTUGAL A comparison between ITT and CoC