Applications.lidr 10.6 KB
 Nicola Botta committed Mar 02, 2021 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 % -*-Latex-*- %if False > module Applications > import Syntax.PreorderReasoning > import Data.Vect > import About > import Functors > import Monads > %default total > %auto_implicits off > %access public export > %hide Prelude.List.last > DetSys : Type -> Type > DetSys X = X -> X > flowDet : {X : Type} -> DetSys X -> Nat -> DetSys X > flowDet f Z = id > flowDet f ( S n) = flowDet f n . f > lastLemma : {A : Type} -> {n : Nat} -> > (x : A) -> (xs : Vect (S n) A) -> last (x :: xs) = last xs > lastLemma x (y :: ys) = Refl %endif \section{Applications in dynamical systems and control theory} \label{section:applications} In this section we discuss applications of the principle of preservation of extensional equality to dynamical systems and control theory. % We have seen in section \ref{section:example} that time discrete deterministic dynamical systems on a set |X| are functions of type |X -> X| < DetSys : Type -> Type < DetSys X = X -> X and that generalizing this notion to systems with uncertainties leads to monadic systems > MonSys : (Type -> Type) -> Type -> Type > MonSys M X = X -> M X where |M| is an \emph{uncertainty} monad: |List|, |Maybe|, |Dist| \citep{10.1017/S0956796805005721}, |SimpleProb| \citep{2017_Botta_Jansson_Ionescu}, etc. % For monadic systems, one can derive a number of general results. % One is that every deterministic system can be embedded in a monadic systems: > embed : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => DetSys X -> MonSys M X > embed f = pure . f A more interesting result is that the flow of a monadic system is a monoid morphism from |(Nat, (+), 0)| to |(MonSys M X, (>=>), pure)|. % As discussed in section~\ref{section:example}, |flowMonL <=> flowMonR| and here we write just |flow|. %if False > flow : {M : Type -> Type} -> {X : Type} -> Fat.VeriMonadFat M => > MonSys M X -> Nat -> MonSys M X > flow f Z = pure > flow f ( S n) = f >=> flow f n %endif The two parts of the monoid morphism proof are > flowLemma1 : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => > (f : MonSys M X) -> flow f Z <=> pure > > flowLemma2 : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => {m, n : Nat} -> > (f : MonSys M X) -> flow f (m + n) <=> (flow f m >=> flow f n) Proving |flowLemma1| is immediate (because |flow f Z = pure|): > flowLemma1 f = reflEE % We prove |flowLemma2| by induction on |m| using the properties from section \ref{section:monads}: |pure| is a left and right identity of Kleisli composition and Kleisli composition is associative. % The base case is straightforward > flowLemma2 {m = Z} {n} f x = > ( flow f (Z + n) x ) ={ Refl }= > ( flow f n x ) ={ sym (pureLeftIdKleisli (flow f n) x) }= > ( (pure >=> flow f n) x ) ={ Refl }= > ( (flow f Z >=> flow f n) x ) QED but the induction step again relies on Kleisli composition preserving extensional equality. > flowLemma2 f {m = S l} {n} x = > ( flow f (S l + n) x ) ={ Refl }= > ( (f >=> flow f (l + n)) x ) ={ kleisliPresEE f f (flow f (l + n)) (flow f l >=> flow f n) > reflEE (flowLemma2 f) x }= > ( (f >=> (flow f l >=> flow f n)) x ) ={ sym (kleisliAssoc f (flow f l) (flow f n) x) }= > ( ((f >=> flow f l) >=> flow f n) x ) ={ Refl }= > ( (flow f (S l) >=> flow f n) x ) QED As seen in section \ref{section:monads}, this follows directly from the monad ADT and from the preservation of extensional equality for functors. \paragraph*{A representation theorem.} % Another important result for monadic systems is a representation theorem: any monadic system |f : MonSys M X| can be represented by a deterministic system on |M X|. With > repr : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => MonSys M X -> DetSys (M X) > repr f = id >=> f % in other symbols: repr f x = x >>= f % and for an arbitrary monadic system |f|, |repr f| is equivalent to |f| in the sense that % %format (flowDet f n) = f "^{" n "}" > reprLemma : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => > (f : MonSys M X) -> (n : Nat) -> repr (flow f n) <=> flowDet (repr f) n where |flowDet| is the flow of a deterministic system < flowDet : {X : Type} -> DetSys X -> Nat -> DetSys X < flowDet f Z = id < flowDet f ( S n) = flowDet f n . f As for |flowLemma2|, proving the representation lemma is straightforward but crucially relies on associativity of Kleisli composition and thus, as seen in section \ref{section:monads}, on preservation of extensional equality: % > reprLemma f Z mx = % > ( (repr (flow f Z)) mx ) ={ Refl }= % > ( (id >=> flow f Z) mx ) ={ Refl }= % > ( (id >=> pure) mx ) ={ Fat.pureRightIdKleisli id mx }= % > ( id mx ) ={ Refl }= % > ( flowDet (repr f) Z mx ) QED > reprLemma f Z mx = Fat.pureRightIdKleisli id mx > reprLemma f (S m) mx = > ( repr (flow f (S m)) mx ) ={ Refl }= > ( (id >=> flow f (S m)) mx ) ={ Refl }= > ( (id >=> (f >=> flow f m)) mx ) ={ sym (Fat.kleisliAssoc id f (flow f m) mx) }= > ( ((id >=> f) >=> flow f m) mx ) ={ kleisliLeapfrog (id >=> f) (flow f m) mx }= > ( (id >=> flow f m) ((id >=> f) mx) ) ={ Refl }= > ( repr (flow f m) ((id >=> f) mx) ) ={ reprLemma f m ((id >=> f) mx) }= > ( flowDet (repr f) m ((id >=> f) mx) ) ={ Refl }= > ( flowDet (repr f) m (repr f mx) ) ={ Refl }= > ( flowDet (repr f) (S m) mx ) QED Notice also the application of |kleisliLeapfrog| to deduce |(id >=> flow f m) ((id >=> f) mx)| from |((id >=> f) >=> flow f m) mx|. % If we had formulated the theory in terms of bind instead of Kleisli composition, the two expressions would be intensionally equal. \paragraph*{Flows and trajectories.} % The last application of preservation of extensional equality in the context of dynamical systems theory is a result about flows and trajectories. % For a monadic system |f|, the trajectories of length |n+1| starting at state |x : X| are > trj : {M : Type -> Type} -> {X : Type} -> Fat.VeriMonadFat M => > MonSys M X -> (n : Nat) -> X -> M (Vect (S n) X) > trj f Z x = map (x ::) (pure Nil) > trj f ( S n) x = map (x ::) ((f >=> trj f n) x) In words, the trajectory obtained by making zero steps starting at |x| is an |M|-structure containing just |[x]|. % To compute the trajectories for |S n| steps, we first bind the outcome of a single step |f x : M X| into |trj f n|. % This results in an |M|-structure of vectors of length |S n|. % Finally, we prepend these possible trajectories with the initial state |x|. Since |trj f n x| is an |M|-structure of vectors of states, we can compute the last state of each trajectory. % It turns out that this is point-wise equal to |flow f n|: > flowTrjLemma : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => > (f : MonSys M X) -> (n : Nat) -> > flow f n <=> map {A = Vect (S n) X} last . trj f n To prove this result, we first derive the auxiliary lemma > mapLastLemma : {F : Type -> Type} -> {X : Type} -> {n : Nat} -> VeriFunctor F => > (x : X) -> (mvx : F (Vect (S n) X)) -> > (map last . map (x ::)) mvx = map last mvx > mapLastLemma {X} {n} x mvx = > ( map {A = Vect (S (S n)) X} last (map (x ::) mvx) ) > ={ sym (mapPresComp {A = Vect (S n) X} last (x ::) mvx) }= > ( map (last . (x ::)) mvx ) > ={ mapPresEE (last . (x ::)) last (lastLemma x) mvx }= > ( map last mvx ) QED where |lastLemma x : last . (x ::) <=> last|. In the implementation of |mapLastLemma| we have applied both preservation of composition and preservation of extensional equality. % With |mapLastLemma| in place, |flowTrjLemma| is readily implemented by induction on the number of steps %**PJ: Only added to avoid bad page-break %\newpage > flowTrjLemma {X} f Z x = > ( flow f Z x ) ={ Refl }= (pure x) ={ Refl }= > ( pure (last (x :: Nil)) ) ={ sym (Fat.pureNatTrans last (x :: Nil)) }= > ( map last (pure (x :: Nil)) ) ={ cong {f = map last} > (sym (Fat.pureNatTrans {A = Vect Z X} (x ::) Nil)) }= > ( map last (map (x ::) (pure Nil)) ) ={ Refl }= > ( map last (trj f Z x) ) QED > > flowTrjLemma f (S m) x = > ( flow f (S m) x ) ={ Refl }= > ( (f >=> flow f m) x ) ={ kleisliPresEE f f (flow f m) (map (last {len = m}) . trj f m) > reflEE (flowTrjLemma f m) x }= > > ( (f >=> map (last {len = m}) . trj f m) x ) ={ sym (mapKleisliLemma f (trj f m) last x) }= > ( map last ((f >=> trj f m) x) ) ={ sym (mapLastLemma x ((f >=> trj f m) x)) }= > ( map last (map (x ::) ((f >=> trj f m) x)) ) ={ Refl }= > ( map last (trj f (S m) x) ) QED Again, preservation of extensional equality proves essential for the induction step. \paragraph*{Dynamic programming (DP).} The relationship between the flow and the trajectory of a monadic dynamical system also plays a crucial role in the \emph{semantic verification} of dynamic programming. % DP \citep{bellman1957} is a method for solving sequential decision problems. % These problems are at the core of many applications in economics, logistics and computer science and are, in principle, well understood~\citep{bellman1957, de_moor1995, %de_moor1999, gnesi1981dynamic, 2017_Botta_Jansson_Ionescu}. Proving that dynamic programming is semantically correct boils down to showing that the value function |val| that is at the core of the backwards induction algorithm of DP is extensionally equal to a specification |val'|. The |val| function of DP takes |n| policies or decision rules and is computed by iterating |n| times a monadic dynamical system similar to the function argument of |flow| but with an additional \emph{control} argument. % At each iteration, a \emph{reward} function is mapped on the states %TiRi: can this be left out?: obtained by iterating the system and the result is reduced with a \emph{measure} function. % In this computation, the measure function is applied a number of times that is exponential in |n|. By contrast, |val'| is computed by applying the measure function only once, but to a structure of a size exponential in |n| that is obtained by adding up the rewards along all the trajectories. The equivalence between |val| and |val'| is established by structural induction. % As in the |flowTrjLemma| discussed above, |map| preserving extensional equality turns out to be pivotal in applying the induction hypothesis, see \citep{brede2020} for details.