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Extract common parts of forward analyses into Forward.agda 

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-03-22 17:50:29 -07:00
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Analysis/Forward.agda Normal file
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open import Language
open import Lattice
module Analysis.Forward
{L : Set} {h}
{_≈ˡ_ : L → L → Set} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
(isFiniteHeightLatticeˡ : IsFiniteHeightLattice L h _≈ˡ_ _⊔ˡ_ _⊓ˡ_)
(≈ˡ-dec : IsDecidable _≈ˡ_) where
open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Data.Nat using (suc)
open import Data.Product using (_×_; proj₁; _,_)
open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Unit using (⊤)
open import Function using (_∘_)
open import Utils using (Pairwise)
import Lattice.FiniteValueMap
open IsFiniteHeightLattice isFiniteHeightLatticeˡ
using ()
renaming
( isLattice to isLatticeˡ
; fixedHeight to fixedHeightˡ
; _≼_ to _≼ˡ_
)
module WithProg (prog : Program) where
open Program prog
-- The variable -> sign map is a finite value-map with keys strings. Use a bundle to avoid explicitly specifying operators.
module VariableValuesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeˡ vars
open VariableValuesFiniteMap
using ()
renaming
( FiniteMap to VariableValues
; isLattice to isLatticeᵛ
; _≈_ to _≈ᵛ_
; _⊔_ to _⊔ᵛ_
; _≼_ to _≼ᵛ_
; ≈₂-dec⇒≈-dec to ≈ˡ-dec⇒≈ᵛ-dec
; _∈_ to _∈ᵛ_
; _∈k_ to _∈kᵛ_
; _updating_via_ to _updatingᵛ_via_
; locate to locateᵛ
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
; ∈k-dec to ∈k-decᵛ
; all-equal-keys to all-equal-keysᵛ
)
public
open IsLattice isLatticeᵛ
using ()
renaming
( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
; ⊔-idemp to ⊔ᵛ-idemp
)
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ
using ()
renaming
( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
)
≈ᵛ-dec = ≈ˡ-dec⇒≈ᵛ-dec ≈ˡ-dec
joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
⊥ᵛ = proj₁ (proj₁ (proj₁ fixedHeightᵛ))
-- Finally, the map we care about is (state -> (variables -> sign)). Bring that in.
module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
open StateVariablesFiniteMap
using (_[_]; m₁≼m₂⇒m₁[ks]≼m₂[ks])
renaming
( FiniteMap to StateVariables
; isLattice to isLatticeᵐ
; _∈k_ to _∈kᵐ_
; locate to locateᵐ
; _≼_ to _≼ᵐ_
; ≈₂-dec⇒≈-dec to ≈ᵛ-dec⇒≈ᵐ-dec
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
)
public
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟_ isLatticeᵛ states-Unique ≈ᵛ-dec _ fixedHeightᵛ
using ()
renaming
( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
)
≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
-- build up the 'join' function, which follows from Exercise 4.26's
--
-- L₁ → (A → L₂)
--
-- Construction, with L₁ = (A → L₂), and f = id
joinForKey : State → StateVariables → VariableValues
joinForKey k states = foldr _⊔ᵛ_ ⊥ᵛ (states [ incoming k ])
-- The per-key join is made up of map key accesses (which are monotonic)
-- and folds using the join operation (also monotonic)
joinForKey-Mono : ∀ (k : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (joinForKey k)
joinForKey-Mono k {fm₁} {fm₂} fm₁≼fm₂ =
foldr-Mono joinSemilatticeᵛ joinSemilatticeᵛ (fm₁ [ incoming k ]) (fm₂ [ incoming k ]) _⊔ᵛ_ ⊥ᵛ ⊥ᵛ
(m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ (incoming k) fm₁≼fm₂)
(⊔ᵛ-idemp ⊥ᵛ) ⊔ᵛ-Monotonicʳ ⊔ᵛ-Monotonicˡ
-- The name f' comes from the formulation of Exercise 4.26.
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) joinForKey joinForKey-Mono states
renaming
( f' to joinAll
; f'-Monotonic to joinAll-Mono
)
-- With 'join' in hand, we need to perform abstract evaluation.
module WithEvaluator (eval : Expr → VariableValues → L)
(eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ˡ_ (eval e)) where
-- For a particular evaluation function, we need to perform an evaluation
-- for an assignment, and update the corresponding key. Use Exercise 4.26's
-- generalized update to set the single key's value.
private module _ (k : String) (e : Expr) where
open VariableValuesFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
renaming
( f' to updateVariablesFromExpression
; f'-Monotonic to updateVariablesFromExpression-Mono
)
public
states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
-- The per-state update function makes use of the single-key setter,
-- updateVariablesFromExpression, for the case where the statement
-- is an assignment.
--
-- This per-state function adjusts the variables in that state,
-- also monotonically; we derive the for-each-state update from
-- the Exercise 4.26 again.
updateVariablesForState : State → StateVariables → VariableValues
updateVariablesForState s sv
with code s
... | k ← e =
let
(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
in
updateVariablesFromExpression k e vs
updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
with code s
... | k ← e =
let
(vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
(vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
in
updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
renaming
( f' to updateAll
; f'-Monotonic to updateAll-Mono
)
-- Finally, the whole sign analysis consists of getting the 'join'
-- of all incoming states, then applying the per-state evaluation
-- function. This is just a composition, and is trivially monotonic.
analyze : StateVariables → StateVariables
analyze = updateAll ∘ joinAll
analyze-Mono : Monotonic _≼ᵐ_ _≼ᵐ_ analyze
analyze-Mono {sv₁} {sv₂} sv₁≼sv₂ = updateAll-Mono {joinAll sv₁} {joinAll sv₂} (joinAll-Mono {sv₁} {sv₂} sv₁≼sv₂)
-- The fixed point of the 'analyze' function is our final goal.
open import Fixedpoint ≈ᵐ-dec isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
using ()
renaming (aᶠ to result)
public

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Analysis/Sign.agda
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@ -1,21 +1,16 @@
module Analysis.Sign where
open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Data.Nat using (suc)
open import Data.Product using (_×_; proj₁; _,_)
open import Data.Product using (proj₁; _,_)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Unit using (⊤)
open import Function using (_∘_)
open import Relation.Nullary using (yes; no)
open import Language
open import Lattice
open import Utils using (Pairwise)
open import Showable using (Showable; show)
import Lattice.FiniteValueMap
import Analysis.Forward
data Sign : Set where
+ : Sign
@ -114,101 +109,10 @@ postulate minus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ
module WithProg (prog : Program) where
open Program prog
-- The variable -> sign map is a finite value-map with keys strings. Use a bundle to avoid explicitly specifying operators.
module VariableSignsFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
open VariableSignsFiniteMap
using ()
renaming
( FiniteMap to VariableSigns
; isLattice to isLatticeᵛ
; _≈_ to _≈ᵛ_
; _⊔_ to _⊔ᵛ_
; _≼_ to _≼ᵛ_
; ≈₂-dec⇒≈-dec to ≈ᵍ-dec⇒≈ᵛ-dec
; _∈_ to _∈ᵛ_
; _∈k_ to _∈kᵛ_
; _updating_via_ to _updatingᵛ_via_
; locate to locateᵛ
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
; all-equal-keys to all-equal-keysᵛ
; ∈k-dec to ∈k-decᵛ
)
open IsLattice isLatticeᵛ
using ()
renaming
( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
; ⊔-idemp to ⊔ᵛ-idemp
)
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeᵍ vars-Unique ≈ᵍ-dec _ fixedHeightᵍ
using ()
renaming
( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
)
module ForwardWithProg = Analysis.Forward.WithProg (record { isLattice = isLatticeᵍ; fixedHeight = fixedHeightᵍ }) ≈ᵍ-dec prog
open ForwardWithProg
≈ᵛ-dec = ≈ᵍ-dec⇒≈ᵛ-dec ≈ᵍ-dec
joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
⊥ᵛ = proj₁ (proj₁ (proj₁ fixedHeightᵛ))
-- Finally, the map we care about is (state -> (variables -> sign)). Bring that in.
module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
open StateVariablesFiniteMap
using (_[_]; m₁≼m₂⇒m₁[ks]≼m₂[ks])
renaming
( FiniteMap to StateVariables
; isLattice to isLatticeᵐ
; _∈k_ to _∈kᵐ_
; locate to locateᵐ
; _≼_ to _≼ᵐ_
; ≈₂-dec⇒≈-dec to ≈ᵛ-dec⇒≈ᵐ-dec
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
)
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟_ isLatticeᵛ states-Unique ≈ᵛ-dec _ fixedHeightᵛ
using ()
renaming
( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
)
≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
-- build up the 'join' function, which follows from Exercise 4.26's
--
-- L₁ → (A → L₂)
--
-- Construction, with L₁ = (A → L₂), and f = id
joinForKey : State → StateVariables → VariableSigns
joinForKey k states = foldr _⊔ᵛ_ ⊥ᵛ (states [ incoming k ])
-- The per-key join is made up of map key accesses (which are monotonic)
-- and folds using the join operation (also monotonic)
joinForKey-Mono : ∀ (k : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (joinForKey k)
joinForKey-Mono k {fm₁} {fm₂} fm₁≼fm₂ =
foldr-Mono joinSemilatticeᵛ joinSemilatticeᵛ (fm₁ [ incoming k ]) (fm₂ [ incoming k ]) _⊔ᵛ_ ⊥ᵛ ⊥ᵛ
(m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ (incoming k) fm₁≼fm₂)
(⊔ᵛ-idemp ⊥ᵛ) ⊔ᵛ-Monotonicʳ ⊔ᵛ-Monotonicˡ
-- The name f' comes from the formulation of Exercise 4.26.
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) joinForKey joinForKey-Mono states
renaming
( f' to joinAll
; f'-Monotonic to joinAll-Mono
)
-- With 'join' in hand, we need to perform abstract evaluation.
vars-in-Map : ∀ (k : String) (vs : VariableSigns) →
k ∈ˡ vars → k ∈kᵛ vs
vars-in-Map k vs@(m , kvs≡vars) k∈vars rewrite kvs≡vars = k∈vars
states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
eval : ∀ (e : Expr) → VariableSigns → SignLattice
eval : ∀ (e : Expr) → VariableValues → SignLattice
eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
eval (` k) vs
@ -257,50 +161,7 @@ module WithProg (prog : Program) where
eval-Mono (# 0) _ = ≈ᵍ-refl
eval-Mono (# (suc n')) _ = ≈ᵍ-refl
private module _ (k : String) (e : Expr) where
open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
renaming
( f' to updateVariablesFromExpression
; f'-Monotonic to updateVariablesFromExpression-Mono
)
public
open ForwardWithProg.WithEvaluator eval eval-Mono using (result)
updateVariablesForState : State → StateVariables → VariableSigns
updateVariablesForState s sv
with code s
... | k ← e =
let
(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
in
updateVariablesFromExpression k e vs
updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
with code s
... | k ← e =
let
(vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
(vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
in
updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
renaming
( f' to updateAll
; f'-Monotonic to updateAll-Mono
)
analyze : StateVariables → StateVariables
analyze = updateAll ∘ joinAll
analyze-Mono : Monotonic _≼ᵐ_ _≼ᵐ_ analyze
analyze-Mono {sv₁} {sv₂} sv₁≼sv₂ = updateAll-Mono {joinAll sv₁} {joinAll sv₂} (joinAll-Mono {sv₁} {sv₂} sv₁≼sv₂)
open import Fixedpoint ≈ᵐ-dec isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
using ()
renaming (aᶠ to signs)
-- Debugging code: print the resulting map.
output = show signs
-- For debugging purposes, print out the result.
output = show result