(Effectul) Game Semantics for Dependent Types

Transcription

1 Eindhoven, 3 April 2016

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3 Overview Game theoretic models of dependent type theory (DTT): get faithful model of (total) DTT with Σ-, Π-, intensional Id-types and finite inductive type families; fully complete if Id-types limited;

4 Overview Game theoretic models of dependent type theory (DTT): get faithful model of (total) DTT with Σ-, Π-, intensional Id-types and finite inductive type families; fully complete if Id-types limited; very intensional model - refines domain model; Id-types more intensional: function extensionality fails; intensional orthogonally to HoTT (time vs space): UIP holds;

5 Overview Game theoretic models of dependent type theory (DTT): get faithful model of (total) DTT with Σ-, Π-, intensional Id-types and finite inductive type families; fully complete if Id-types limited; very intensional model - refines domain model; Id-types more intensional: function extensionality fails; intensional orthogonally to HoTT (time vs space): UIP holds; call-by-name evaluation; can (dis)allow effects by varying condition on strategies: combining dependent types and effects!

6 The Semantic (Hyper)Cube Consider subfamily str of all strategies to exclude effects: No non-termination: winning No state: history-freeness / innocence + visibility No control operators: question-answer discipline No erratic choice: determinacy for Player.

7 The Semantic (Hyper)Cube Consider subfamily str of all strategies to exclude effects: No non-termination: winning No state: history-freeness / innocence + visibility No control operators: question-answer discipline No erratic choice: determinacy for Player. Primarily, here, for str we demand all these conditions (for completeness results), but can vary to model effects.

8 Games and Simple Type Theory (STT) Write (Game, I, &, ) for ccc of games and strategies.

9 Games and Simple Type Theory (STT) Write (Game, I, &, ) for ccc of games and strategies. Ground types (finite inductive types): for a set X, game X with one Opponent move, followed by any of the Player moves x X.

10 Games and Simple Type Theory (STT) Write (Game, I, &, ) for ccc of games and strategies. Ground types (finite inductive types): for a set X, game X with one Opponent move, followed by any of the Player moves x X. Call simple type theory (STT): λ-calculus with binary products, function types and finite inductive types {a i 1 i n}, or a total finitary PCF with binary products. Straightforward consequence of AJM2000: Theorem Interpretation in Game fully and faithfully complete: [[ ]] Syntax STT Game full and faithful.

11 Dependent type theory (DTT)? What is it? Curry-Howard for predicate logic: types with free (term) variables, constructions Σ, Π, Id (cf.,, =) on types. Judgements like x : A B(x) type and x : A b(x) : B(x). Order in context matters! No clean separation syntax types and terms.

12 Dependent type theory (DTT)? What is it? Curry-Howard for predicate logic: types with free (term) variables, constructions Σ, Π, Id (cf.,, =) on types. Judgements like x : A B(x) type and x : A b(x) : B(x). Order in context matters! No clean separation syntax types and terms. Why care? Move towards richer type systems: e.g. GADTs in Haskell. Types allowed to refer to data: e.g. n : N List(n) type. Specification by typing: certification by type checking. Programming logic as a proof assistant. Logical Frameworks.

13 Dependent types and their terms: n : N List(n) type n : N, x : List(n) reverse(x) : List(n) Π- and Σ-types (generalising and ): λ n λ x reverse(x) : Π n:n List(n) List(n) mult : Π k,l,m:n matrix(k, l) matrix(l, m) matrix(k, m) 9, [No, wise, fish, would, go, anywhere, without, a, porpoise] : Σ n:n List(n)

14 Dependent types and their terms: n : N List(n) type n : N, x : List(n) reverse(x) : List(n) Π- and Σ-types (generalising and ): λ n λ x reverse(x) : Π n:n List(n) List(n) mult : Π k,l,m:n matrix(k, l) matrix(l, m) matrix(k, m) 9, [No, wise, fish, would, go, anywhere, without, a, porpoise] : Σ n:n List(n) Id-type: refl : Id N (0, (λ x x)(0)) x, y : B case(x,case(y,refl,refl),case(y,refl,refl)) : Id B (lsor(x,y),rsor(x,y)) Finite inductive type families: x : months days(x) type 31 : days(october).

15 A Faithful Translation to Simple Type Theory (STT) Idea: DTT talks about same algorithms as STT but can assign them a more precise type/specification. Formally: have faithful translation ( ) T of DTT into STT.

16 A Faithful Translation to Simple Type Theory (STT) Idea: DTT talks about same algorithms as STT but can assign them a more precise type/specification. Formally: have faithful translation ( ) T of DTT into STT. x : A (a i i {b i,j j}) type {b i,j i, j} type x : A Σ y:b C type B T C T type x : A Π y:b C type B T C T type x : A, y : B, y : B Id B (y, y ) type {refl} type x : A B(a) type B(x) T type + translation on terms.

17 The idea of our interpretation [[ ]] will be to construct a category CtxtGame of dependent context games with a functor ( ) to Game, such that Syntax DTT ( ) T Syntax STT [[ ]] CtxtGame ( ) [[ ]] Game. Games model of DTT will therefore automatically be faithful.

18 Dependent Games and Strategies Game B with dependency on A consists of: game (B) (without dependency); function str(a) B Sub( (B)) from strategies on A to subgames ( -closed subsets of plays) of (B). Can define I, & and on games with dependency and make into ccc with homset DGame(A)(B, C) := str(π A (B C)).

19 We define a game Π A B A (B) of dependent functions. Idea: the choice of a fibre B(a) for the output of a dependent function f : Π A B is entirely the responsibility of the context that provides the argument a.

20 We define a game Π A B A (B) of dependent functions. Idea: the choice of a fibre B(a) for the output of a dependent function f : Π A B is entirely the responsibility of the context that provides the argument a. Opponent can determine fibre B(τ) of B: explicitly, revealing strategy τ on A, by playing in A; implicitly, by playing in (B); Player has to stay within B(τ) for all τ consistent with Opponent s behaviour.

21 We define a game Π A B A (B) of dependent functions. Idea: the choice of a fibre B(a) for the output of a dependent function f : Π A B is entirely the responsibility of the context that provides the argument a. Opponent can determine fibre B(τ) of B: explicitly, revealing strategy τ on A, by playing in A; implicitly, by playing in (B); Player has to stay within B(τ) for all τ consistent with Opponent s behaviour. That is, as long as there is such a τ; otherwise, anything goes. Indeed, Opponent is totally free and might not respect the type dependency, as ( ) should be faithful (to match ( ) T ).

22 Non-example of dependently typed algorithm: scheduling finance meetings Player/Academic: Opponent/Education Finance Business Manager Manager: x : months 31 : days(x) Π( months, days ) O 31 P Player chooses fibre: e.g. February doesn t have a 31 st day. Mr Manager shouldn t allow that...

23 Example of dependently typed algorithm: Academic/Player: Opponent/Education Finance Business Manager Manager: x : months case days,months (x, {31, 1,..., 1, 31}) : days(x) [January/x] Π( months, days ) O P January O 31 P

24 Example (fibre-wise identities): Player: Opponent: x : A, y : B(x) y : B(x) choose your favourite Π( [[A]], [[B]] [[B]]) b b b. b O P O P

25 Theorem Have indexed ccc Game op DGame CCCat of dependent games: ob(dgame(a)) := {str(a) B Sub( (B))} hom-sets DGame(A)(B, C) := str(π A (B C)) fibrewise identities as in example; fibrewise comp. & change of base: usual AJM-composition.

26 Theorem Have indexed ccc Game op DGame CCCat of dependent games: ob(dgame(a)) := {str(a) B Sub( (B))} hom-sets DGame(A)(B, C) := str(π A (B C)) fibrewise identities as in example; fibrewise comp. & change of base: usual AJM-composition. As we don t have (additive) Σ-types, not a model of DTT! Well! I ve often seen a cat without a grin, thought Alice; but a grin without a cat! It s the most curious thing I ever saw in all my life.

27 Theorem Have indexed ccc Game op DGame CCCat of dependent games: ob(dgame(a)) := {str(a) B Sub( (B))} hom-sets DGame(A)(B, C) := str(π A (B C)) fibrewise identities as in example; fibrewise comp. & change of base: usual AJM-composition. As we don t have (additive) Σ-types, not a model of DTT! Well! I ve often seen a cat without a grin, thought Alice; but a grin without a cat! It s the most curious thing I ever saw in all my life. Theorem Formally add them: get model CtxtGame of DTT with ΣΠ-types! Obj: lists [X k ] k, where str( (X 1 )& & (X k 1 )) X k Sub( (X k )). Mor: lists of strategies on Π-games of more variables.

28 We can also interpret finite inductive type families x : A (a i i {b i,j j} type: [[a i ]] {b i,j j} {b i,j i, j} else {b i,j i, j} ; identity types x : A, x : A Id A (x, x ) type: (σ, σ ) {refl σ = σ } {refl}.

29 Non-Example: Player: x : B, y : B p : Id B (x, y) Opponent : [tt/x, ff/y] Π( B, Π( B, Id B ) tt ff refl O P O P O P...as tt ff.

30 Example: Player: x : B refl x : Id B (x, x) Opponent:...as x = x. Π( B, Id B {diag B }) refl O P

31 Example, given x : A f : B(x, x, refl x ): Player: Opponent: x : A, x : A, y : Id A (x, x ) J(f ) : B Π( A, Π( A, Π( Id A, B))) b refl f (b) a f (a).. O P O P O P...as refl move by Opponent has excluded all pairs (x, x ) with x x.

32 Place in intensionality spectrum Id-types: Domains HoTT Games Failure of Equality Reflection Streicher Intensionality Criteria (I1) and (I2) Streicher Intensionality Criterion (I3) Failure of Function Extensionality (FunExt) Failure of Uniqueness of Identity Proofs (UIP). Discrete ground types (0-types), but function hierarchy generates (open) propositional identities: applicative equivalences.

33 Completeness Results Summarising, we have Theorem (Soundness and Faithfulness) We have a faithful model of DTT with Σ-, Π-, intensional Id-types and finite inductive type families: faithful functor Syntax DTT [[ ]] CtxtGame.

34 Completeness Results Summarising, we have Theorem (Soundness and Faithfulness) We have a faithful model of DTT with Σ-, Π-, intensional Id-types and finite inductive type families: faithful functor Syntax DTT [[ ]] CtxtGame. Actually, the model has strong completeness properties. Theorem (Full Completeness) This interpretation is full when restricted to the types of the form A or Π A Id B (f, g) with A and B built without Id-types: Restricted Types SyntaxDTT [[ ]] CtxtGame full.

35 Alternative Id-types and Function Extensionality Interpretation intensional Id-types not unique! Try alternative, mimicking the domain semantics, validating function extensionality: Π x:a Id B (f (x), g(x)) = Id Πx:A B(f, g). Demand that dep. games respect applicative equivalence. Define Id A : (σ, σ ) σ AE σ AE (A), where σ AE is the set of plays of (A) that is compatible with a strategy that is applicatively equivalent to σ.

36 Alternative Id-types and Function Extensionality Interpretation intensional Id-types not unique! Try alternative, mimicking the domain semantics, validating function extensionality: Π x:a Id B (f (x), g(x)) = Id Πx:A B(f, g). Demand that dep. games respect applicative equivalence. Define Id A : (σ, σ ) σ AE σ AE (A), where σ AE is the set of plays of (A) that is compatible with a strategy that is applicatively equivalent to σ. Interpret x : A refl x : Id A (x, x) as the copycat. For x : A f : B(x, x, refl x ), interpret x : A, x : A, y : Id A (x, x ) J(f ) : B(x, x, y) by playing [[f ]] between [[Id A ]] and [[B]]. Incompatible with partial or non-deterministic strategies!

37 Future Work game semantics of higher inductive types / quotient types; combining function extensionality with partiality and non-determinism; modelling universes and univalence; infinite inductive type families and their definability results; examining full completeness (w.r.t. DTT + UIP) of the model for the type hierarchy including unrestricted Id-types; what do these models teach us about DTT with effects? construct CBV and CBPV models; compare with realizability models.

38 In particular, for the first item: HtpyGame... Gpd Set Game Game A collapsing time-like identity, a.k.a. extensional collapse Gpd Set collapsing space-like identity, a.k.a. 0-truncation X X 0. str(a)/obs.equiv.

39 Bonus Example (Higher order dependent functions): approving holiday plans Player/PhD Student: Opponent/Supervisor : w : years, x : Π(y : days(w), Π(z : holidays(w, y), B)) going on holidays? : B [2015/w, acceptable holidays/x] Π(years, Π( Π(days, Π(holidays, B ))), B ) O P O International Talk Like a Pirate Day P ff O ff P (ITLPD happens every year, so don t need to ask about the year. Moreover, Player is also in charge of supplying the date, so we don t have to determine the date before specifying the holiday.)

40 Bonus Example (Higher order dependent functions): approving holiday plans Player/PhD Student: Opponent/Supervisor : w : years, x : Π(y : days(w), Π(z : holidays(w, y), B)) going on holidays? : B [2015/w, acceptable holidays/x] disobedient student... Π(years, Π( Π(days, Π(holidays, B ))), B ) International Talk Like a Pirate Day ff tt O P O P O P

41 Bonus Example (Higher order dependent functions): approving holiday plans Player/PhD Student: Opponent/Supervisor : w : years, x : Π(y : days(w), Π(z : holidays(w, y), B)) going on holidays? : B [2015/w, acceptable holidays/x] Π(years, Π( Π(days, Π(holidays, B ))), B ) O P O Holi P O P 2015 O 65 P tt O tt P (Note that Holi happens every year with variable Gregorian date and that Player gets to choose the day so can even specify the holiday before the day.)