Game Semantics for Dependent Types

Transcription

1 Bath, 20 ctober, 2015

2 verview Game theoretic model of dependent type theory (DTT): refines model in domains and (total) continuous functions; call-by-name evaluation; faithful model of (total) DTT with Σ-, Π-, intensional Id-types and finite inductive type families; fully complete if Id-types limited;

3 verview Game theoretic model of dependent type theory (DTT): refines model in domains and (total) continuous functions; call-by-name evaluation; faithful model of (total) DTT with Σ-, Π-, intensional Id-types and finite inductive type families; fully complete if Id-types limited; Id-types more intensional than domain model: function extensionality fails; intensional in orthogonal way to HoTT (time vs space): UI holds.

4 Game Semantics? Interpolates between operational and denotational semantics: very intensional with structural clarity of categorical model; Unified framework for intensional, computational semantics: CF (H, N, AJM);

5 Game Semantics? Interpolates between operational and denotational semantics: very intensional with structural clarity of categorical model; Unified framework for intensional, computational semantics: CF (H, N, AJM); references, non-local control, dynamically generated local names, probability, non-determinism, concurrency...; various evaluation strategies; sums, recursive types, polymorphism; propositional logic, impredicative 2 nd order quantification, external 1 st order quantification;

6 Game Semantics? Interpolates between operational and denotational semantics: very intensional with structural clarity of categorical model; Unified framework for intensional, computational semantics: CF (H, N, AJM); references, non-local control, dynamically generated local names, probability, non-determinism, concurrency...; various evaluation strategies; sums, recursive types, polymorphism; propositional logic, impredicative 2 nd order quantification, external 1 st order quantification; internal 1 st order quantification / dependent types surprisingly absent (and surprisingly hard!);

7 Game Semantics? Interpolates between operational and denotational semantics: very intensional with structural clarity of categorical model; Unified framework for intensional, computational semantics: CF (H, N, AJM); references, non-local control, dynamically generated local names, probability, non-determinism, concurrency...; various evaluation strategies; sums, recursive types, polymorphism; propositional logic, impredicative 2 nd order quantification, external 1 st order quantification; internal 1 st order quantification / dependent types surprisingly absent (and surprisingly hard!); Tight correspondence with syntax (full abstraction, full faithful completeness): often unique semantics in this respect.

8 Computation / Logic data type / proposition computational process / argument program / proof environment / refutation variable declarations / axiom links substitution / cut termination / correctness pure sequential functional behaviour (no state, parallelism, control ops) /.. Games 2-player game (duality!) play: alternating seq. of moves layer () strategy pponent () strategy copycat strategies interaction winning history-freeness/innocence + determinism + well-bracketing of strategies

9 Computation / Logic data type / proposition computational process / argument program / proof environment / refutation variable declarations / axiom links substitution / cut termination / correctness pure sequential functional behaviour (no state, parallelism, control ops) /.. Games 2-player game (duality!) play: alternating seq. of moves layer () strategy pponent () strategy copycat strategies interaction winning history-freeness/innocence + determinism + well-bracketing of strategies

10 An example: layer: pponent: x : B, y : (B B) y(x) : B [tt/x, (λ z:b z)/y] B (B B ) B tt tt ff ff

11 An example: layer: pponent: x : B, y : (B B) y(x) : B [(λ z:b ff)/y] B (B B ) B ff ff

12 Games and winning strategies form a smcc Game: I: the game with one play of length 0; A B: playing A and B simultaneously through interleaving, where only pponent can switch games; A B: swap, (A) and B simultaneously, layer switches.

13 Games and winning strategies form a smcc Game: I: the game with one play of length 0; A B: playing A and B simultaneously through interleaving, where only pponent can switch games; A B: swap, (A) and B simultaneously, layer switches. Also model simple type theory (STT): have a ccc Game! : A B :=!A B;!A: playing ω equivalent copies of A simultaneously, where only pponent can switch games; roduct A&B: pponent chooses to play A or B (unit: I).

14 Games and winning strategies form a smcc Game: I: the game with one play of length 0; A B: playing A and B simultaneously through interleaving, where only pponent can switch games; A B: swap, (A) and B simultaneously, layer switches. Also model simple type theory (STT): have a ccc Game! : A B :=!A B;!A: playing ω equivalent copies of A simultaneously, where only pponent can switch games; roduct A&B: pponent chooses to play A or B (unit: I). Ground types (finite inductive types): for a set X, game X with one pponent move, followed by any of the layer moves x X.

15 An example: copycats (identity morphisms) layer: pponent: x : A x : A A A a a a. a.

16 An example: interaction (composition) layer: pponent: y : N 2y : N x : N x + 1 : N parallel composition... N N x x + 1 N N y 2y

17 An example: interaction (composition) layer: pponent: x : N x + 1 : N 2(x + 1) : N N N N x x + 1 2(x + 1)... plus hiding similarly in general

18 Use the term simple type theory (STT) to refer to a simple λ-calculus with binary products, function types and finite inductive types {a i 1 i n}, or a total finitary CF with binary products, with βη-rules and CF commutative conversions for case-constructs. Straightforward consequence of AJM: Theorem The interpretation of STT in Game! is fully and faithfully complete.

19 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). rder in context matters! No clean separation syntax types and terms.

20 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). rder 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. rogramming logic as a proof assistant. Logical Frameworks.

21 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)

22 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(ctober).

23 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 translation of DTT into STT. Let DTT inherit the equational theory of STT to make translation faithful.

24 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 translation of DTT into STT. Let DTT inherit the equational theory of STT to make translation faithful. 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 B T type x : A B(a) type B(x) T type + translation on terms.

25 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.

26 Dependent Games and Strategies Game B with dependency on A consists of: game (B) (without dependency); continuous function str(a) B Sub( (B)) from strategies on A to subgames ( -closed subsets of plays) of (B). Alternatively: continuous function str(a) B Game.

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

28 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.

29 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. pponent can determine fibre B(τ) of B: explicitly, revealing winning strategy τ on A, by playing in!a; implicitly, by playing in (B); layer has to stay within B(τ) for all τ consistent with pponent s behaviour.

30 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. pponent can determine fibre B(τ) of B: explicitly, revealing winning strategy τ on A, by playing in!a; implicitly, by playing in (B); layer has to stay within B(τ) for all τ consistent with pponent s behaviour. That is, as long as there is such a τ; otherwise, anything goes. Indeed, pponent is totally free and might not respect the type dependency, as ( ) should be faithful (to match ( ) T ).

31 Non-example of dependently typed algorithm: scheduling finance meetings layer/academic: pponent/education Finance Business Manager Manager: x : months 31 : days(x) Π( months, days ) 31 layer chooses fibre: e.g. February doesn t have a 31 st day. Mr Manager shouldn t allow that...

32 Example of dependently typed algorithm: Academic/layer: pponent/education Finance Business Manager Manager: x : months case days,months (x, {31, 1,..., 1, 31}) : days(x) [January/x] Π( months, days ) January 31

33 Example (fibre-wise identities): layer: pponent: x : A, y : B(x) y : B(x) choose your favourite Π( [[A]], [[B]] [[B]]) b b b. b

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

35 Theorem Have indexed ccc Game op! ob(dgame! (A)) := {continuous str(a) DGame CCCat of dependent games: B Sub( (B))} hom-sets DGame! (A)(B, C) := wstr(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.

36 Theorem Have indexed ccc Game op! ob(dgame! (A)) := {continuous str(a) DGame CCCat of dependent games: B Sub( (B))} hom-sets DGame! (A)(B, C) := wstr(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! bj: lists [X k ] k, where str( (X 1 )& & (X k 1 )) X k Sub( (X k )). Mor: lists of winning strategies on Π-games of more variables.

37 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 Id B (s, t) type: σ (σ; [[s]]) (σ; [[t]]) ([[B]]).

38 Non-Example: layer: x : B, y : B p : Id B (x, y) pponent : [tt/x, ff/y] Π( B, Π( B, Id B ) tt ff tt...as tt does not lie in intersection tt ff.

39 Example: layer: x : B refl x : Id B (x, x) pponent: Π( B, Id B {diag B }) x x...as x lies in x x.

40 lace 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 roofs (UI). Discrete ground types (0-types), but function hierarchy generates (open) propositional identities: observational equivalences.

41 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!.

42 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.

43 Future Work Ultimate goal: intensional, computational analysis of HoTT. game semantics of higher inductive types / quotient types; examining function extensionality and univalence; universes and a more intensional notion of type family; infinite inductive type families and their definability results; examining completeness properties of the model for the complete type hierarchy, including Id-types; constructing models of DTT with effects.

44 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. wstr(a)/obs.equiv.

45 Bonus Example (Higher order dependent functions): approving holiday plans layer/hd Student: pponent/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 ) International Talk Like a irate Day ff ff (ITLD happens every year, so don t need to ask about the year. Moreover, layer is also in charge of supplying the date, so we don t have to determine the date before specifying the holiday.)

46 Bonus Example (Higher order dependent functions): approving holiday plans layer/hd Student: pponent/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 irate Day ff tt

47 Bonus Example (Higher order dependent functions): approving holiday plans layer/hd Student: pponent/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 ) Holi tt tt (Note that Holi happens every year with variable Gregorian date and that layer gets to choose the day so can even specify the holiday before the day.)