Idris2/libs/papers/Search/CTL.idr
Thomas E. Hansen 8c76118f2f [ papers ] Finish Search.CTL
* Switch to `Inf` to actually use codata/corecursion.
* Add `%hint`s to mark the interface implementations as such, despite
  use of a record for `DepthInv` (this is necessary for other stuff).
* Pass in `Oh` to `reaches10.evidence` in order for things to work.

With huge thanks to gallais for helping me put the final things in
place!

Co-authored-by: Guillaume Allais <guillaume.allais@ens-lyon.org>
2022-10-05 14:30:08 +02:00

418 lines
14 KiB
Idris

||| The content of this module is based on the paper
||| Applications of Applicative Proof Search
||| by Liam O'Connor
||| https://doi.org/10.1145/2976022.2976030
module Search.CTL
import Data.So
import Data.Nat
import Data.List.Quantifiers
import Decidable.Equality
import public Search.Negation
import public Search.HDecidable
import public Search.Properties
%default total
------------------------------------------------------------------------
-- Type and some basic functions
||| Labeled transition diagram
public export
record Diagram (labels : Type) (state : Type) where
constructor TD
||| Transition function
transFn : (labels, state) -> List (labels, state)
||| Initial state
iState : labels
%name Diagram td,td1,td2
||| Parallel composition of transition diagrams
public export
pComp : {Lbls1, Lbls2 : _} -> {Sts : _}
-> (td1 : Diagram Lbls1 Sts)
-> (td2 : Diagram Lbls2 Sts)
-> Diagram (Lbls1, Lbls2) Sts
pComp (TD transFn1 iState1) (TD transFn2 iState2) =
TD compTransFn (iState1, iState2)
where
compTransFn : ((Lbls1, Lbls2), Sts) -> List ((Lbls1, Lbls2), Sts)
compTransFn = (\ ((l1, l2), st) =>
map (\ (l1', st') => ((l1', l2), st')) (transFn1 (l1, st)) ++
map (\ (l2', st') => ((l1, l2'), st')) (transFn2 (l2, st)))
||| A process which always increases the shared number.
public export
HiHorse : Diagram () Nat
HiHorse = TD transFn ()
where
transFn : ((), Nat) -> List ((), Nat)
transFn (l, st) = [(l, (S st))]
||| A process which always decreases the shared number.
public export
LoRoad : Diagram () Nat
LoRoad = TD transFn ()
where
transFn : ((), Nat) -> List ((), Nat)
transFn (l, st) = [(l, pred st)]
-- different formulation of LTE, see also:
-- https://agda.github.io/agda-stdlib/Data.Nat.Base.html#4636
-- thanks @gallais!
public export
data LTE' : (n : Nat) -> (m : Nat) -> Type where
LTERefl : LTE' m m
LTEStep : LTE' n m -> LTE' n (S m)
||| Convert LTE' to LTE
lteAltToLTE : {m : _} -> LTE' n m -> LTE n m
lteAltToLTE {m=0} LTERefl = LTEZero
lteAltToLTE {m=(S k)} LTERefl = LTESucc (lteAltToLTE LTERefl)
lteAltToLTE {m=(S m)} (LTEStep s) = lteSuccRight (lteAltToLTE s)
parameters (Lbls, Sts : Type)
||| A computation tree (corecursive rose tree?)
public export
data CT : Type where
At : (Lbls, Sts) -> Inf (List CT) -> CT
||| Given a transition diagram and a starting value for the shared state,
||| construct the computation tree of the given transition diagram.
public export
model : Diagram Lbls Sts -> (st : Sts) -> CT
model (TD transFn iState) st = follow (iState, st)
where
follow : (Lbls, Sts) -> CT
followAll : List (Lbls, Sts) -> List CT
follow st = At st (Delay (followAll (transFn st)))
followAll [] = []
followAll (st :: sts) = follow st :: followAll sts
||| A formula has a bound (for Bounded Model Checking; BMC) and a computation
||| tree to check against.
public export
Formula : Type
Formula = (depth : Nat) -> (tree : CT) -> Type
||| A tree models a formula if there exists a depth d0 for which the property
||| holds for all depths d >= d0.
-- Called "satisfies" in the paper
public export
data Models : (m : CT) -> (f : Formula) -> Type where
ItModels : (d0 : Nat) -> ({d : Nat} -> (d0 `LTE'` d) -> f d m) -> Models m f
------------------------------------------------------------------------
-- Depth invariance
||| Depth-invariance (DI) is when a formula cannot be falsified by increasing
||| the search depth.
public export
record DepthInv (f : Formula) where
constructor DI
prf : {n : Nat} -> {m : CT} -> f n m -> f (S n) m
||| A DI-formula holding for a specific depth means the CT models the formula
||| in general (we could increase the search depth and still be fine).
public export
diModels : {n : Nat} -> {m : CT} -> {f : Formula} -> {auto d : DepthInv f}
-> (p : f n m) -> Models m f
diModels {n} {f} {m} @{(DI diPrf)} p = ItModels n (\ q => diLTE p q)
where
diLTE : {n, n' : _} -> f n m -> (ltePrf' : n `LTE'` n') -> f n' m
diLTE p LTERefl = p
diLTE p (LTEStep x) = diPrf (diLTE p x)
||| A trivially true (TT) formula.
public export
data TrueF : Formula where
TT : {n : _} -> {m : _} -> TrueF n m
||| A tt formula is depth-invariant.
public export
%hint
TrueDI : DepthInv TrueF
TrueDI = DI (const TT)
------------------------------------------------------------------------
-- Guards
namespace Guards
||| The formula `Guarded g` is true when the current state satisfies the
||| guard `g`.
public export
data Guarded : (g : (st : Sts) -> (l : Lbls) -> Type) -> Formula where
Here : {st : _} -> {l : _}
-> {ms : Inf (List CT)} -> {depth : Nat}
-> {g : _}
-> (guardOK : g st l)
-> Guarded g depth (At (l, st) ms)
||| Guarded expressions are depth-invariant as the guard does not care about
||| depth.
public export
%hint
diGuarded : {p : _} -> DepthInv (Guarded p)
diGuarded {p} = DI prf
where
prf : {n : _} -> {m : _} -> Guarded p n m -> Guarded p (S n) m
prf (Here x) = Here x -- can be interactively generated!
------------------------------------------------------------------------
-- Conjunction / And
||| Conjunction of two `Formula`s
public export
record AND' (f, g : Formula) (depth : Nat) (tree : CT) where
constructor MkAND' --: {n : _} -> {m : _} -> f n m -> g n m -> (AND' f g) n m
fst : f depth tree
snd : g depth tree
||| Conjunction is depth-invariant
public export
%hint
diAND' : {f, g : Formula}
-> {auto p : DepthInv f}
-> {auto q : DepthInv g}
-> DepthInv (AND' f g)
diAND' @{(DI diP)} @{(DI diQ)} = DI (\ a' => MkAND' (diP a'.fst) (diQ a'.snd))
------------------------------------------------------------------------
-- Always Until
namespace AU
||| A proof that for all paths in the tree, f holds until g does.
public export
data AlwaysUntil : (f, g : Formula) -> Formula where
||| We've found a place where g holds, so we're done.
Here : {t : _} -> {n : _} -> g n t -> AlwaysUntil f g (S n) t
||| If f still holds and we can recursively show that g holds for all
||| possible subpaths in the CT, then all branches have f hold until g
||| does.
There : {st : _} -> {infCTs : Inf _} -> {n : _}
-> f n (At st infCTs)
-> All ((AlwaysUntil f g) n) (Force infCTs)
-> AlwaysUntil f g (S n) (At st infCTs)
||| Provided `f` and `g` are depth-invariant, AlwaysUntil is
||| depth-invariant.
public export
%hint
diAU : {f,g : _} -> {auto p : DepthInv f} -> {auto q : DepthInv g}
-> DepthInv (AlwaysUntil f g)
diAU @{(DI diP)} @{(DI diQ)} = DI prf
where
prf : {d : _} -> {t : _}
-> AlwaysUntil f g d t
-> AlwaysUntil f g (S d) t
prf (Here au) = Here (diQ au)
prf (There au aus) = There (diP au) (mapAllAU prf aus)
where
-- `All.mapProperty` erases the list and so won't work here
mapAllAU : {d : _} -> {lt : _}
-> (prf : AlwaysUntil f g d t -> AlwaysUntil f g (S d) t)
-> All (AlwaysUntil f g d) lt
-> All (AlwaysUntil f g (S d)) lt
mapAllAU prf [] = []
mapAllAU prf (au :: aus) = (prf au) :: mapAllAU prf aus
------------------------------------------------------------------------
-- Exists Until
namespace EU
||| A proof that somewhere in the tree, there is a path for which f holds
||| until g does.
public export
data ExistsUntil : (f, g : Formula) -> Formula where
||| If g holds here, we've found a branch where we can stop.
Here : {t : _} -> {n : _} -> g n t -> ExistsUntil f g (S n) t
||| If f holds here and any of the further branches have a g, then there
||| is a branch where f holds until g does.
There : {st : _} -> {infCTs : Inf _} -> {n : _}
-> f n (At st infCTs)
-> Any (ExistsUntil f g n) (Force infCTs)
-> ExistsUntil f g (S n) (At st infCTs)
||| Provided `f` and `g` are depth-invariant, ExistsUntil is
||| depth-invariant.
public export
%hint
diEU : {f, g : _} -> {auto p : DepthInv f} -> {auto q : DepthInv g}
-> DepthInv (ExistsUntil f g)
diEU @{(DI diP)} @{(DI diQ)} = DI prf
where
prf : {d : _} -> {t : _}
-> ExistsUntil f g d t
-> ExistsUntil f g (S d) t
prf (Here eu) = Here (diQ eu)
prf (There eu eus) = There (diP eu) (mapAnyEU prf eus)
where
-- `Any.mapProperty` erases the list and so won't work here
mapAnyEU : {d : _} -> {lt : _}
-> (prf : ExistsUntil f g d t -> ExistsUntil f g (S d) t)
-> Any (ExistsUntil f g d) lt
-> Any (ExistsUntil f g (S d)) lt
mapAnyEU prf (Here x) = Here (prf x)
mapAnyEU prf (There x) = There (mapAnyEU prf x)
------------------------------------------------------------------------
-- Finally, Completed, and the finite forms of Global
||| "Always finally" means that for all paths, the formula f will eventually
||| hold.
|||
||| This is equivalent to saying `A [TT U f]` (where TT is trivially true).
public export
AlwaysFinally : Formula -> Formula
AlwaysFinally f = AlwaysUntil TrueF f
||| "Exists finally" means that for some pathe, the formula f will eventually
||| hold.
|||
||| This is equivalent to saying `E [TT U f]` (where TT is trivially true).
public export
ExistsFinally : Formula -> Formula
ExistsFinally f = ExistsUntil TrueF f
||| A completed formula is a formula for which no more successor states exist.
public export
data Completed : Formula where
IsCompleted : {st : _} -> {n : _} -> {infCTs : Inf _}
-> (Force infCTs) === []
-> Completed n (At st infCTs)
||| A completed formula is depth-invariant (there is nothing more to do).
public export
%hint
diCompleted : DepthInv Completed
diCompleted = DI prf
where
prf : {d : _} -> {t : _} -> Completed d t -> Completed (S d) t
prf (IsCompleted p) = IsCompleted p
||| We can only handle always global checks on finite paths.
public export
AlwaysGlobal : (f : Formula) -> Formula
AlwaysGlobal f = AlwaysUntil f (f `AND'` Completed)
||| We can only handle exists global checks on finite paths.
public export
ExistsGlobal : (f : Formula) -> Formula
ExistsGlobal f = ExistsUntil f (f `AND'` Completed)
------------------------------------------------------------------------
-- Proof search (finally!)
||| Model-checking is a half-decider for the formula `f`
public export
MC : (f : Formula) -> Type
MC f = (t : CT) -> (d : Nat) -> HDec (f d t)
||| Proof-search combinator for guards.
public export
now : {g : (st : Sts) -> (l : Lbls) -> Type}
-> {hdec : _}
-> {auto p : AnHDec hdec}
-> ((st : Sts) -> (l : Lbls) -> hdec (g st l))
-> MC (Guarded g)
now f (At (l, st) ms) d = [| Guards.Here (toHDec (f st l)) |]
||| Check if the current state has any successors.
public export
isCompleted : MC Completed
isCompleted (At st ms) _ = IsCompleted <$> isEmpty (Force ms)
where
||| Half-decider for whether a list is empty
isEmpty : {x : _} -> (n : List x) -> HDec (n === [])
isEmpty [] = yes Refl
isEmpty (_ :: _) = no
||| Conjunction of model-checking procedures.
public export
mcAND' : {f, g : Formula} -> MC f -> MC g -> MC (f `AND'` g)
mcAND' mcF mcG t d = [| MkAND' (mcF t d) (mcG t d) |]
||| Proof-search for `AlwaysUntil`.
|||
||| Evaluates the entire `Inf (List CT)` of the state-space, since we need
||| `f U g` to hold across every path.
public export
auSearch : {f, g : Formula} -> MC f -> MC g -> MC (AlwaysUntil f g)
auSearch _ _ _ Z = no
auSearch p1 p2 t@(At st ms) (S n) = [| AU.Here (p2 t n) |]
<|> [| AU.There (p1 t n) rest |]
where
-- in AlwaysUntil searches, we have to check the entire `Inf (List CT)`
rest : HDec (All (AlwaysUntil f g n) (Force ms))
rest = HDecidable.List.all (toList ms) (\ m => auSearch p1 p2 m n)
||| Proof-search for `ExistsUntil`.
|||
||| `Inf` over the state-space, since `E [f U g]` holds as soon as `f U g` is
||| found.
public export
euSearch : {f, g : Formula} -> MC f -> MC g -> MC (ExistsUntil f g)
euSearch _ _ _ Z = no
euSearch p1 p2 t@(At st ms) (S n) = [| EU.Here (p2 t n) |]
<|> [| EU.There (p1 t n) rest |]
where
rest : HDec (Any (ExistsUntil f g n) ms)
rest = HDecidable.List.any (Force ms) (\ m => euSearch p1 p2 m n)
||| Proof-search for Exists Finally
public export
efSearch : {f : _} -> MC f -> MC (ExistsFinally f)
efSearch g = euSearch (\ _, _ => yes TT) g
||| Proof-search for Always Finally
public export
afSearch : {f : _} -> MC f -> MC (AlwaysFinally f)
afSearch g = auSearch (\ _, _ => yes TT) g
||| Proof-search for Exists Global
public export
egSearch : {f : _} -> MC f -> MC (ExistsGlobal f)
egSearch g = euSearch g (g `mcAND'` isCompleted)
||| Proof-search for Always Global
public export
agSearch : {f : _} -> MC f -> MC (AlwaysGlobal f)
agSearch g = auSearch g (g `mcAND'` isCompleted)
------------------------------------------------------------------------
-- Proof search example
||| This CT is a model of composing the `HiHorse` and `LoRoad` programs.
public export
Tree : CT ((), ()) Nat
Tree = model ((), ()) Nat (HiHorse `pComp` LoRoad) 0
||| Prove that there exists a path where `HiHorse || LoRoad`'s state reaches 10.
public export
reaches10 : ? -- HDec (ExistsFinally [...])
reaches10 =
efSearch ((), ()) Nat
(now ((), ()) Nat
(\ st, _ => fromDec $ decEq st 10)) Tree 20
export
r10Proof : Models ((), ()) Nat
Tree
(ExistsFinally ((), ()) Nat
(Guarded ((), ()) Nat (\ st, _ => st === 10)))
r10Proof = diModels ((), ()) Nat (reaches10.evidence Oh)