Wolfgang Jeltsch. Seminar talk at the Institute of Cybernetics Tallinn, Estonia

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1 in in Brandenburgische Technische Universität Cottbus Cottbus, Germany Seminar talk at the Institute of Cybernetics Tallinn, Estonia February 10, 2011

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4 in trueness of a proposition depends on time times are natural numbers propositional logic extended with four new constructs: ϕ ϕ will hold at the next time ϕ ϕ will always hold ϕ ϕ will eventually hold ϕ ψ ϕ will hold for some time, and then ψ will hold for now only and : restricted LTL continuous time also possible

5 Embedding into predicate logic temporal formula ϕ can be translated into predicate logic formula ϕ ϕ may contain a single free variable t that denotes the time atomic propositions p correspond to predicates ˆp that take a time argument translation for propositional logic fragment: p = ˆp(t) = = ϕ ψ = ϕ ψ ϕ ψ = ϕ ψ ϕ ψ = ϕ ψ in translation for and : ϕ = t [t, ). ϕ [t /t] ϕ = t [t, ). ϕ [t /t]

6 as a type system type inhabitation depends on time simple type system extended with two new type constructors and temporal type α can be translated into dependent type α α may contain a single-free variable t that denotes the time translation for and : α = Πt [t, ). α [t /t] α = Σt [t, ). α [t /t] concepts from (FRP): behaviors events restricted LTL corresponds to a strongly typed form of FRP t denotes start times of behaviors and events in

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8 Basics categorical models should be CCCCs: LTL extends propositional logic FRP extends simply-typed λ-calculus components of a categorical model: objects propositions/types morphisms time-independent proofs/functions: and are (endo)functors: f : α β f : Πt. α β in f : α β f : α β start time consistency is ensured: f : α β f : α β : (Πt. α β ) (Πt. α β ) : (Πt. α β ) (Πt. α β )

9 Operations on behaviors is a comonad: head : α α tails : α α is a strong cartesian functor: units : 1 1 zip : α β (α β) is not an applicative functor: lifting of pure values would have to be possible: const : α α would break start time consistency: however, this is possible: const : Πt. α Πt [t, ). α [t /t] f : 1 α f units : 1 α in

10 Operations on events is a monad: now : α α join : α α is not a strong monad: time shifting of values would have to be possible: shift : α β (α β) would break start time consistency: shift : Πt. α β Σt [t, ). α [t /t] β [t /t] however, is -strong: in age : α β ( α β) sampling can be derived: sample : α β (α β) sample = (head id) age

11 From S4 to restricted LTL until now, we have categorical models for CS4/IS4 no big surprise: classically, restricted LTL is a specialization of S4 intuitionistically, it is too classical S4 and restricted LTL differ in their restrictions on the accessibility relation: S4 reflexive order restr. LTL total reflexive order add a further operation that ensures totality of time: in race : α β (α β + α β + α β) possible outcomes of time comparison represented by the different alternatives: = α β < α β > α β

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13 -LTL and its corresponding FRP dialect translation of -formulas into predicate logic formulas: ϕ ψ = t (t, ). ( t [t, t ). ϕ [t /t]) ψ [t /t] as a type constructor of FRP: α β = Σt (t, ). (Πt [t, t ). α [t /t]) β [t /t] components of a value of type α β: a finite behavior with values of type α a terminating event with a value of type β introduction of weak variant of that does not guarantee termination notation: strong variant ( as defined above) weak variant and now derivable: in α = α 0 β = β + 1 β

14 Applications of -types -types are useful as such: temperatures from some sensor that may be detached from the computer: R 1 dialog window: etc. UI α -types are useful in combination with (co)induction: audio signal that may switch between stereo and mono: νσ. (R R) R σ in positions of a pen that might be taken off from the drawing area: etc. νσ. (R R) 1 σ

15 The -functor categorical model C is a CCCC derive a category U from C: Obj U = Obj C Obj C {, } hom((α 1, β 1, w 1 ), (α 2, β 2, w 2 )) hom(α 1, α 2 ) hom( β 1, β 2 ) if w 1 w 2 = otherwise in is a functor from U to C notation: α w β = (α, β, w) applying to morphisms allows for several things: mapping of values of the behavior part mapping of value of the terminating event weakening

16 Comonadic and monadic structure _ w β is a comonad: head : α w β α tails : α w β (α w β) w β β = 0 and w = leads to comonadic structure of α w _ is an ideal monad: optjoin : α w ( β + α w β) α w β in monad can be derived: now : β ( β + α w β) join : ( β + α w β) + α w ( β + α w β) β + α w β α = 1 and w = leads to monadic structure of

17 Monoidal structure in make U a symmetric monoidal category: (α 1, β 1, w 1 ) (α 2, β 2, w 2 ) = (α 1 α 2, ρ, w 1 w 2 ) I = (1, 0, ) where ρ = β 1 β 2 + β 1 α 2 w2 β 2 + α 1 w1 β 1 β 2 is a strong symmetric monoidal functor from U to C: merge : α 1 w1 β 1 α 2 w2 β 2 α 1 α 2 w1 w 2 ρ never : 1 0

18 Specializations is a strong symmetric monoidal functor from U to C: merge : α 1 w1 β 1 α 2 w2 β 2 α 1 α 2 w1 w 2 ρ never : 1 0 where ρ = β 1 β 2 + β 1 α 2 w2 β 2 + α 1 w1 β 1 β 2 strong cartesian functor structure of : β 1 = β 2 = 0 w 1 = w 2 = in from merge to age: β 1 = 0 α 2 = 1 w 1 = w 2 = from merge to race: α 1 = α 2 = 1 w 1 = w 2 =

19 The inverse of merge the type of the terminating event: in ρ = β 1 β 2 + β 1 α 2 w2 β 2 + α 1 w1 β 1 β 2 drop information from the terminating event: restrict i : ρ β i + α i wi β i restrict i = [ι 1 π i, ι i π i, ι 1 i π i ] recover the original -values: recover i : α 1 α 2 w1 w 2 ρ α i wi β i recover i = optjoin (π i restrict i ) combine the recovered values: merge 1 : α 1 α 2 w1 w 2 ρ α 1 β 1 α 2 β 2 merge 1 = recover 1, recover 2

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21 in LTL use N as the set of times translation of -formulas into predicate logic formulas: ϕ = ϕ [t + 1/t] as a type constructor of FRP: α = α [t + 1/t] in value of type α is a value of type α occurring at the next time semantically, is just a strong cartesian functor: f : α β f : α β unit : 1 1 pair : α β (α β)

22 Deriving the other constructs in,, and derivable via induction and coinduction: α = νσ. α σ β = µσ. β + σ α β = µσ. α ( β + σ) α β = νσ. α ( β + σ) interesting exercise: derive all operations of -FRP from the -operations proof that the derived operations fulfill the necessary laws

23 Advanced dataflow programming in -FRP is a kind of dataflow language: streams over α: α partial streams over α: (1 + α) νσ. 1 (α σ) more powerful than traditional dataflow languages: productive partial streams over α: (1 + α) νσ. 1 (α σ) streams with values of different type

24 Shifting fby operator appends a stream to an initial value: fby : α α α needs to shift values to the future cannot be done implicitely, since it would break start time consistency can be made possible by introducing tensorial strength: in shift : α β (α β) simpler operator is sufficient: later : α α is now an applicative functor

25 in in Brandenburgische Technische Universität Cottbus Cottbus, Germany Seminar talk at the Institute of Cybernetics Tallinn, Estonia February 10, 2011