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### TYPE THEORY *

Zoltán Csörnyei

Eötvös University

Department of General Computer Science

Budapest, Hungary

* Supported by OTKA T037742

Aristotle

Aristotle proposed the now famous Aristotelian syllogistic, form of argument consisting of two premises and a conclusion. His example is:

(i)Every Greek is a person.

(ii) Every person is mortal.

(iii) Every Greek is mortal.

W. of Ockham

Ockham's Razor:

- Frustra fit per plura, quod fieri potest per pauciora.It is vain to do with more what can be done with less.orEssentia non sunt multiplicanda praeter necessitatem.Entities should not be multiplied unnecessarily.

W. of Ockham

- He considered a three valued logic where propositions can take one of three truth values. This became important for mathematics in the 20th Century but it is remarkable that it was first studied by Ockham 600 years earlier.

G. Frege

- one of the founders of modern symbolic logic

He was the first to fully develop the main thesis of logicism, that mathematics is reducible to logic.

( The Russell paradox gave contradiction in Frege's system of axioms. )

D. Hilbert

- Axioms
1. |- A A

2. |- A (B A)

3. |- (A (BC)) ((A B) A C))

4. |- A B |- A

---------------------

|- B

G. Gentzen

- He introduced the notion of
'logical consequence'

B1, B2, …, B n |- A

- Natural deduction
- Sequent calculus
- Derivation tree

L.E.J. Brouwer

- The foundations of intuitionism.
- Judgements about statements are based on the existence of a proof or construction of that statements.
- There are two irrational numbers x and y, such that xy is rational.

Preliminaries II.

- The untyped combinatory logics
Schönfinkel, 1924

- The untyped lambda calculus
Church, 1934

- Turing machines
Turing, 1936

Combinatory logic I.

- expr ::= var
| K | S

| ( expr expr )

- reductions
K x y x

S x y z x z ( y z )

- normal form

Combinatory logic II.

- True K
false K I I S K K

- if E F G E F G
and E F E F false

or E F E true F

not E E false true

Combinatory logic III.

- natural numbers
n ( S B )n ( K I ) B E F G E (F G)

- succ S B C E F G E G F
zero C (CI ( true false))true

- add E F S’ B’ E F S’CEFG C(EG)(FG)
mul E F B’ E F B’CEFG CE(FG)

exp E F F E

Combinatory logic IV.

- Definable functions:
partial recursive numerical function

(Kleene, 1936.)

Lambda calculus I.

- expr ::= var
| λ var . expr

| ( expr expr )

- reductions
(λ x . E1) E2β E1 [ x := E2 ]

λ x . Eλ y . E [ x := y]

λ x . E x E

Lambda calculus II.

- true λ xy.x
false λ xy.y

- if λ xyz.xyz
and λ xy.xy false

or λ xy.x true y

not λ x.x false true

Lambda calculus III.

- natural numbers
n λ fx . f n(x)

- succ λ nfx . f ( nfx )
zero λ x . x ( true false ) true

- add λ xypq . xp ( ypq )
mul λ xyp . x ( yp )

exp λ xy . yx

Lambda calculus IV.

- Definable functions:
partial recursive numerical function

(undefinedunsolvable)

solvable: F, (λ x.E) F = I

(Kleene, 1936., Wadsworth, 1971.)

Turing machine

- Turing Theorem (1936.)
combinatory logics

lambda calculus

Turing machine

Formal Type System

- Formal Type System: ( S, J, R )
S syntax, J judgements, R rules

- Type environment Γ
- Rule
Γ |- I1 … Γ |- In

---------------------------

Γ |- I

Type System F1

- type ::= basic_type
| type type

- expr ::= var
| λvar : type . expr

| ( expr expr )

„type checking”

(Pascal, C++, …)

Rules ( F1 )

Γ , x:A , Γ’ |- wf

- ------------------------- [Val x]
Γ , x:A , Γ’ |- x:A

Γ , x:A |- E : B

- ----------------------------- [Val Fun]
Γ |- λ:A . E : A B

Γ |- E : A B Γ |- F : A

- ------------------------------------- [Val Appl]
Γ |- E F : B

Theorems ( F1 )

- Type preservation
If E : A and E ->> F then F : A .

- Type unumbigouos
If Γ|- E : A andΓ|- E : B then A B .

- Finite reductions
thereisno infinite reduction sequence.

The power ( F1 )

- (Schwichtenberg, 1976.)
The lambda definable functions are

exactly the extended polinomials.

- constant functions, projections, signum
function, addition, multiplication.

(Type System F1 / Curry)

- type ::= type_var
| type type

- expr ::= var
| λvar . expr

| ( expr expr )

„type inference”

(ML, Haskell, …)

Type System F2

- Id-Nat λx : Nat . x
Id-Bool λx : Bool . x

Id-Int λx : Int. x

- Id-λx : . x
Id . λx : . x polimorphic function

- Id : . type ofpolimorphic function

Type System F2 II.

- type ::= type_var
| type type

| type_var. type

- expr ::= var
| var : type . expr

| ( expr expr )

| type_var. expr

| ( expr ) [ type ]

Type System F2 III.

- Id [ Nat ] (. x : . x)[ Nat ] β
x : Nat . x : NatNat

- Type of Id [ Nat ] ?
.

- type of type-expression kind

Type System F3

- * type of types of expressions
- kind ::= *
| (* kind )

- type ::= type_var
| type type

| type_var: kind . type

| type_var: kind . type

| ( type type )

Type System F4,F5,...

- F4 kind ::= *
| (* kind )

| ( kind *)

- F5 kind ::= *
| (* kind )

| ( kind *)

| ((* * ) kind )

Type System F

- F = Fi
- kind ::= *
| ( kind kind )

Extension of Type System, F I.

- F (F-omega-sub)
subtyping

- Top
- type Top
- type_var type . expr

Extension of Type System, F II.

- Subtyping
Γ |- E : A Γ |- A B

- -------------------------------- [Subsumption]
Γ |- E : B

Γ |- A BΓ |- C D

- ------------------------------------- [Sub Arrow]
Γ |- B C A D

Extension of Type System, F III.

- Existential type
- .A
pack, unpack (open)

- Recursive type
- .A
fold, unfold

The power of F

- class of functions definiable in F
is much larger than the primitive recursive

functions.

- Ackermann’s function is definiable.

Isomorphism (Curry, Feys 1956.)

- Isomorphism between
combinators Hilbert’s axioms

1. x.x |- A A

2. xy.x |- A (B A)

3. xyz.xz(yz) |- (A(BC))((AB)(AC))

4. function |- A B |- A

application --------------------

|- B

The Curry-Howard isomorphism(Howard, 1959.)

- isomorphism between
lambda calculus intuitive prop.logic

term variable assumption

term proof

type formula

type constructor connective

reduction normalization

… …

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