Lecture 3 Relationships Between Sets and Types

Transcription

1 Lecture 3 Relatonshps Between Sets and Types Marktoberdorf Summer School, 2003

2 From ITT (M-L 82) to CTT (Nuprl 2000) 1. subset types { x : A B } 1983 Math 2. quotent types A // E 1983 Math 3. drect computaton ~ 1986 Provng 4. general recurson y 1986 Programmng 5. recursve types µ xt Data types 6. ntersecton A B 1987 Logc 7. partal objects A 1992 Prog languages 8. unon A B 1995 Symmetry 6 9. top Top 1996 Symmetry subtypng 1996 Classes 2

3 CS Math CS abstracton computablty formalsm modularty classes theores Math abstracton constructvty formalsm modularty categores theores 3

4 Types Revew of Set Theoretc Semantcs Structured Sets A = { α} (HOL, PVS, MML 0 ) ω + ω α rank ω ( α ) Cumulatve Herarchy of Sets 4

5 Goal of Howe s Approach Pure Type Theory a, a,, a 1 2 f ( a) f λxb. p < ab, > ( ) λ xb. < ab, > c a U enlarge Type Theory wth Set Terms a, a,, a 1 2 A γ A 1 2 ( ) ( ) ϕ α f a α, α,, α ( ) ( ) λxb. a b < ab, > < α, β > c a c α ( ) ( ) f λxb. f ϕ c α c α 5

6 Issues wth ML-82, Alf, and Nuprl Type Theores 1. The computaton system s untyped; terms are polymorphc. λx. x? A A for any A ncludng vod λx. 1? Z Z λx. 1? N N < 01, >? N < 01, >? N N Z 6

7 Issues wth ML-82, Alf, and Nuprl Type Theores 2. Sequent hypotheses are dependent, and sequents assert functonalty. ( ) ( ) ( ) A ( a) x : A, x : A x G x, x ext g x, x Notce, A s assumed to be a type, and s assumed 1 2 to be a type for all a? A1, and the sequent asserts that G a a a Aa A a and ( 1, 2) s a type for all 1?, 2? 2( 1) (, ) (, ). Moreover, ga1 a2 G a1 a2 g equalty on A, A. 1 2 It would seem that would not make sense. s a functon respectng ( x ) x ( ) ( x ) x : A, : A G x, x ext g x,

8 Issues wth ML-82, Alf, and Nuprl Type Theores 3. There s natural subtypng. { : Z 0 } Z { n: N prme ( n) } N A A// E dscussed below 8

9 Issues wth ML-82, Alf, and Nuprl Type Theores 4. Subtypng n general A B ff a = a' na a = a Here are the propertes: A A' B B ' A A' A' B ' A + B A' + B ' A' B A B ' ' n B 9

10 Issues wth ML-82, Alf, and Nuprl Type Theores 5. Record subtypng s derved. Polymorphc functons and the subtypng relaton allow an elegant and natural defnton of records and dependent records. 10

11 One approach to records s to take labels, L, as ndexes nto components. Records usng labels Gven take Defne { x : A; y : B; z : C} { } L = x, y, z, L Atom Sg : L U by f j = x then A else f j = y then B else C ( ) Defne the record type as x : L Sg x. 11

12 Records as functons We now take { x : A; y : B; z : C} == x : L Sg ( x ). for { } r? x : A; y : B; z : C, let r. == ( ) r s o rx.? A, ry.? B, rz.? C 12

13 Records extenson usng labels Consder { x : A; y : B; z : C; w : D} Is ths a subrecord of To examne ths, let L' =,, { x : A; y : B; z : C } { x y zw, }..? Notce L L '. Defne Notce because ( ) = f = then else ( ) Sg' w D Sg ( ) ( ) x : L' Sg ' x x : L Sg x ( ) = ( ) L L' and Sg' x Sg x for x? L.. 13

14 Record extenson depends on functon polymorphsm ( ) ( ) x : L' Sg' x x : L Sg x ( ) ( ) because any functon r ' n x : L' Sg' x s a functon n x : L Sg x. Gven nputs from L, x and y, r '( x )? Sg' ( x ) ( ) = ( ) ( )? ( ) = ( ) an d Sg ' x Sg x, r ' y Sg ' y Sg y. 14

15 Record subtypng depends on polymorphc functons Let R R 3 4 = { x 1: A1; x 2: A2; x 3: A3} { x : A ; ; x : A }. = Note R R. 4 3 ( x ) If r, then s defned for? R 4 r {,, } x? x x 1 4, { 1,, 3} hence t s defned for x? x x. 15

16 RECORDS Records and varant types { x : A} == { x} A { x : Ay ; : B; z: C} == { x : A} { y : B} { z : C} VARIANT TYPES { xof A} == { x} A ( xof A yof B zof C ) == ( xof A) ( yof B) ( zof C ) 16

17 Equaltes Dffer 6. ML-82, Alf, Nuprl base equalty on sets s structural A B = A' B ' A B = A' B A + B = A' + B ' ff A = A' and B = B ' ff A = A' and B = B ff A = A' and B = B ' ' ' Set equalty s extensonal; γ = γ ' ff x γ ff x γ'. However, we can defne extensonal type equalty A B ff A B & B A. Note, functon equalty s extensonal n type theory: f = gn A B ff f ( x ) = g ( x ) for all x? A. 17

18 Issues wth ML-82, Alf, and Nuprl Type Theores 7. Nuprl uses quotent types. We can defne new equaltes on a type, say Z, to create a new type. Let E denote the equvalence relaton k Let Z == Z// E. k k x = y mod k on Z For example, n Z 2 all elements equal one of 01, ; n Z 6 all elements equal one of ,,,,,. 18

19 Issues wth ML-82, Alf, and Nuprl Type Theores 8. Types are objects n ML-82, Alf, and Nuprl. U1 s a unverse whose objects are (codes) for types. A? U for x? A, B? x : A B? U x : A B? U U F : U U µx. F monotone U N, B, vod? U for all U+ 1? U 19

20 Issues wth Nuprl Type Theores 9. Some Nuprl theorems contradct classcal set theory and classcal logc. (a) Nuprl can solve ths recurson equaton on types: T ( T B) = ( X B) usng µ X. B. (b) Nuprl s doman theory (bar types) ncludes ths result: B ( ( ) ) h: N B.? x : N. h x = tff x 20

21 Nuprl has Unon and Intersecton 10. The types A B, B, A B, B, x : A B x x? A x? A all volate Howe s constructon as outlned. x ({ 0} { 0} ) { 1} { 1} ( ) ncludes 0 0 and 1 1 but no α β can approxmate both. 21

22 Extendng Set Theoretc Semantcs to ML-82, Alf, Nuprl Can we account for these deas n set theory? Can we embed ML-82, Alf, and Nuprl nto the cumulatve herarchy? U j τ 2 τ are naccessble cardnals U 2 U 1 τ 1 22

23 Copng Wth Issues Polymorphsm Functonalty Subtypng Equalty Quotent types Unverses Non-classcal results Unons, ntersectons use ϕ= ϕ' use set equalty and unqueness follows from polymorphsm use extensonal equlaty ntroduce lmted non-determnsm use naccessble cardnals (reflecton) restrct to core Nuprl consult Evan Moran 23

24 Extendng Approxmaton on Functons Defnton For α, β n V, defne the followng preorder, α β, by nducton on the rank of α: 1. γ γ. ( ) ( ) 2. c,, c ',, ' f ' j n. α1 αn α1 αn αj αj 1 3. φ φ' f for all < α, β >? φ, there exst < α', β' >? φ ' such that α ' α and β β ' (note the contravarance n the frst argument). 4. ξ ξ' f for all α'? ξ' there s α? ξ wth α α '. Incorporate nto by concludng α β. 24

25 Capturng Record Polymorphsm n Howe s Model Consder { x : N; y : N; z : N} { x : N; y : N} ( ) = 0, ( ) = 1, ( ) = 2 wth r x r y r z. Let ψ = ϕ ψ 1 2 = ϕ { < x, 0 ><, y, 1> } { < x, 0 ><, y, 1 ><, z, 2 > } Note ψ1 ψ2, snce for each < α, β > n ψ1 there s some α' α, β = β'. Note, ψ2 λxr. ( x ) { x : N; y: N; z: N} ψ1 = ψ2 ψ1 λxr. ( x ) ( )? { N N N} as well. snce, we have. Thus, λxr. x x : ; y: ; z:, and 25

26 Example If ϕ Z B then ϕ? Z B. 2 6 Ths follows from Approx. Theorem (Thm 2.2). If α β and β ϕ,then α ϕ, fndng ϕ? Z B, ϕ? where ϕ = ϕ and ϕ = ϕ Imagne ϕ tests evenness. Z B 26

27 Approxmaton Theorem (Thm 2.2, p.24) If α = β and β e then α e 27

28 Subtypng Fals Wthout Approxmaton Z B Z B 2 6 but ϕ = { < 0, t >, < 1, f > }? Z small to be n Z 6 B; compare to B s too ϕ 2 = { < 0, t ><, 1, f ><, 2, t ><, 3, f ><, 4, t ><, 5, f > } ϕ ϕ and ϕ ϕ

29 Comparng Types and Sets We have seen embeddngs of types nto structured sets. Ths s one relatonshp between types and sets, but not the only nterestng one. Peter Aczel showed n 1976 how to treat sets as types by mappng a verson of constructve set theory, CZF, nto Martn-Löf type theory. Over the years he extended CZF wth nductve defntons and choce prncples so that ZFC = CZF + Choce. Ths holds also for CZF wth unverses (n accessble cardnals), CZF. Jason Hckey formalzed Aczel s constructon n MetaPRL. 29

30 Formally Embeddng Sets nto Types CZF Aczel M-L Types CZF Theory Hckey Nuprl Theory MetaPRL 30

31 Interestng Observaton If we combne Howe s result and Aczel s, we get ths stuaton: CZF Nuprl +1 1 CZF + ZFC + 1 ZFC = CZF + LEM How are CZF and ZFC related? Theorem (Moran): CZF ZFC 31

32 1 Aczel s Embeddng 1 Aczel ntroduces the type Set wth Martn-Löf style rules. If A? U and f ( x)? Set for all x? A, { ( ) } then f x x A Set. For example, f α,, α are sets for? N, and f 1 n { f N } () = α, then () s a set. { f() N } When n = 0, then s the empty set. n n n 32

33 Aczel s Embeddng 2 When looked at as types, sets have the form of wellfounded trees. They look lke Brouwer ordnals

34 Relatng Models of Sets and Types 1 The followng Nuprl recursve type captures Aczel s model of sets as types: ( U ) µ X. I : I x ( U1 ) Let u µ X. I : I x be the untagged Howe set model of ths type. Call t ACZ. 1 34

35 Relatng Models of Sets and Types 2 We can also translate Aczel s type theory defntons of membershp and extensonal equalty. αεα' = u αε A α' φ α = ' A α = u α = A α' φ 35

36 Relatng Models of Sets and Types 3 Let M be the sets below the 1 st naccessble cardnal, the sets n Howe s Z τ. 1 In classcal frst-order logc, L, ACZ and M are models of ZFC. How are they related? 36

37 Theorem : Moran s Theorem ACZ M, that s, there s a surjectve map : ACZ M such that for all αα, ' n ACZ ι (a) αε α' α ε α' A ι ι ( ) ( ) (b) α = α' α = α' A ι ι ( ) ( ) 37

38 Aczel s CZF Axoms Equalty x = yff? z. z? x z? y ( ) Set Inducton Scheme? y.? x? y. P x P y? xp. x ( ( ) ( )) ( ) Parng? z x z y z ( ) Unon? z. y? x. u? y. u? z ( ) Restrcted Separaton ( ( )) ( ) ( ( ))? z. y? z. y? x P y y? x. P y y? z 38

39 Aczel s CZF Axoms (cont.) Strong Collecton? x? a.? y. R x, y? b. R' ab, R' ab, ( ) ( ) where ( ) ( ) ( )? x? a.? y? b. R x, y?? y? b.? x? a. R x, y s Subset Collecton? c?u? x? a? y? b R x y? d? c R ad Infnty?z. Nat z ( ) ( ) (..,. ', ) ( ) 39

40 Summary We know that Classcal Nuprl s consstent and can share theorems wth PVS, HOL, Mzar, provded t avods domans (bar types) and certan recursve types Classcal Nuprl can retan A// E,,,, Top Wth, set theory t becomes a peer theory wth classcal HOL, PVS can consstently add A// E,,,, Top 40