A Categorical Foundation of Functional Reactive Programming with Mutable State
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1 page.1 A Categorical Foundation of Functional Reactive Programming with Mutable State Wolfgang Jeltsch TTÜ Küberneetika Instituut Teooriapäevad Sakal 27 October 2013 Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 1 / 16
2 page.2 1 The linear adjunction 2 The temporal adjunction 3 Integration of linearity and temporality 4 Outlook Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 2 / 16
3 page.3 The linear adjunction 1 The linear adjunction 2 The temporal adjunction 3 Integration of linearity and temporality 4 Outlook Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 3 / 16
4 page.4 The linear adjunction Linear functional programming values have to be used exactly once: a value can represent the current state of an object changes to the state (destructive updates) expressible as pure functions some functions that describe destructive updates: newtmpfile : World File World writenewline : File File disposetmpfile : File World World Curry Howard correspondent of intuitionistic linear logic modeled by symmetric monoidal closed categories (SMCCs) Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 4 / 16
5 page.5 The linear adjunction Models of linear functional programming SMCC (L,, I, ): L for modeling types and linear functions for modeling linear pairs: I for modeling the linear unit: for modeling linear functions: (A B) C A (B C) A B B A I A A A I A Hom(A B, C) Hom(A, B C) Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 5 / 16
6 page.6 The linear adjunction Models of non-linear functional programming cartesian closed category (CCC) C: C for modeling types and (non-linear) functions for modeling (non-linear) pairs: Hom(Z, X) Hom(Z, Y) Hom(Z, X Y) 1 for modeling the (non-linear) unit: for modeling (non-linear) functions: 1 Hom(Z, 1) Hom(X Y, Z) Hom(X, Y Z) (C,, 1, ) is an SMCC with specific structure: Z Z Z Z 1 Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 6 / 16
7 page.7 The linear adjunction Interaction of non-linear and linear functional programming (C,, 1, ) and (L,, I, ) connected by a lax symmetric monoidal adjunction (LSMA) F G: F for using ordinary values in linear functional programming G for modeling actions that create objects Hom(FX, A) Hom(X, GA) FX FY F(X Y) GA GB G(A B) I F1 implies isomorphisms: see the work of Benton (1994) FX FY F(X Y) I F1 1 GI Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 7 / 16
8 page.8 The temporal adjunction 1 The linear adjunction 2 The temporal adjunction 3 Integration of linearity and temporality 4 Outlook Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 8 / 16
9 page.9 The temporal adjunction Functional reactive programming (FRP) and its models FRP captures temporal aspects of programs: contains constructs that describe temporal behavior type-inhabitance is time-dependent in general, times can be any totally ordered set here restriction to discrete time Curry Howard correspondent of intuitionistic temporal logic modeled by CCCs T with a cartesian endofunctor : T for modeling types and time-universal functions, 1, for modeling pairs, the unit, and functions for modeling values at the next time: A B (A B) 1 1 Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 9 / 16
10 page.10 The temporal adjunction Interaction of functional programming and FRP C and T connected by an adjunction F G: F for modeling types with time-independent type inhabitance G for modeling time-universal values Hom(FX, A) Hom(X, GA) require that adjunction interacts sensibly with next functors: F( X) (FX) (GA) G( A) for the CCC C (which models functional programming), we take the identity functor as a specific next functor: FX (FX) GA G( A) Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 10 / 16
11 page.11 Integration of linearity and temporality 1 The linear adjunction 2 The temporal adjunction 3 Integration of linearity and temporality 4 Outlook Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 11 / 16
12 page.12 Integration of linearity and temporality Unifying the linear and the temporal adjunction what we have so far: (C,, 1, ) and (L,, I, ) connected by an LSMA (C, Id) and (T, ) connected by an adjunction that respects cartesian endofunctors generalize cartesian endofunctors to symmetric monoidal endofunctors (SMEs): work also in a linear setting, where we do not have products in general for L, we use the identity functor, like we did for C now we require both adjunctions to be LSMAs and to respect SMEs makes both adjoint functors of the C T adjunction cartesian: FX FY F(X Y) GA GB G(A B) 1 F1 1 G1 Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 12 / 16
13 page.13 Integration of linearity and temporality Adding models of linear FRP consider the following category: objects SMCCs with an SME morphisms LSMAs that respect SMEs Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 13 / 16
14 page.14 Integration of linearity and temporality Adding models of linear FRP consider the following category: objects SMCCs with an SME morphisms LSMAs that respect SMEs the unified adjunctions from the last slide form a span in this category C F G L F G T Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 13 / 16
15 page.15 Integration of linearity and temporality Adding models of linear FRP consider the following category: objects SMCCs with an SME morphisms LSMAs that respect SMEs the unified adjunctions from the last slide form a span in this category is there a category that models a linear variant of FRP? F C G F G L T? Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 13 / 16
16 page.16 Integration of linearity and temporality Adding models of linear FRP consider the following category: objects SMCCs with an SME morphisms LSMAs that respect SMEs the unified adjunctions from the last slide form a span in this category is there a category that models a linear variant of FRP? define it to be the pushout of the span C L F G G F T F G G F S Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 13 / 16
17 page.17 Integration of linearity and temporality Adding models of linear FRP consider the following category: objects SMCCs with an SME morphisms LSMAs that respect SMEs the unified adjunctions from the last slide form a span in this category is there a category that models a linear variant of FRP? define it to be the pushout of the span interaction of T S adjunction with makes sense: C L F G G F T F G G F S F( X) (FX) (GA) G( A) Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 13 / 16
18 page.18 Outlook 1 The linear adjunction 2 The temporal adjunction 3 Integration of linearity and temporality 4 Outlook Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 14 / 16
19 page.19 Outlook Outlook extend this work to continuous time: categorical models of FRP that also cover continuous time exist are based on process functor interaction of adjunctions with unclear in general in the discrete case, can be defined in terms of idea: make the discrete case continuous by using hyperreal numbers (see Beauxis and Mimram 2011) turn the semantics into an API of an FRP library Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 15 / 16
20 page.20 References R. Beauxis and S. Mimram. A Non-Standard Semantics for Kahn Networks in Continuous Time. In Proceedings of CSL 11, pages 35 50, Schloss Dagstuhl. P. N. Benton. A mixed linear and non-linear logic: Proofs, terms and models. Technical report UCAM-CL-TR-352, University of Cambridge, Oct Wolfgang Jeltsch (TTÜ Küberneetika Instituut) FRP with Mutable State Teooriapäevad Sakal 16 / 16
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