agda-spa/Analysis/Forward.agda

199 lines
9.1 KiB
Agda
Raw Normal View History

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 -> abstract value (e.g. 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 -> value)). 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.
updateVariablesFromStmt : BasicStmt VariableValues VariableValues
updateVariablesFromStmt (k e) vs = updateVariablesFromExpression k e vs
updateVariablesFromStmt noop vs = vs
updateVariablesFromStmt-Monoʳ : (bs : BasicStmt) Monotonic _≼ᵛ_ _≼ᵛ_ (updateVariablesFromStmt bs)
updateVariablesFromStmt-Monoʳ (k e) {vs₁} {vs₂} vs₁≼vs₂ = updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
updateVariablesFromStmt-Monoʳ noop vs₁≼vs₂ = vs₁≼vs₂
updateVariablesForState : State StateVariables VariableValues
updateVariablesForState s sv =
let
bss = code s
(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
in
foldr updateVariablesFromStmt vs bss
updateVariablesForState-Monoʳ : (s : State) Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂ =
let
bss = code s
(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
foldr-Mono' (IsLattice.joinSemilattice isLatticeᵛ) bss
updateVariablesFromStmt updateVariablesFromStmt-Monoʳ
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 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