diff --git a/Language/Graphs.agda b/Language/Graphs.agda index 33f6c1e..73be572 100644 --- a/Language/Graphs.agda +++ b/Language/Graphs.agda @@ -110,8 +110,7 @@ cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es (e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁))) } -record Relaxable (T : Graph → Set) : Set where - field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂ +open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction) instance IndexRelaxable : Relaxable Graph.Index @@ -127,17 +126,7 @@ instance ) } - ProdRelaxable : ∀ {P : Graph → Set} {Q : Graph → Set} → - {{ PRelaxable : Relaxable P }} → {{ QRelaxable : Relaxable Q }} → - Relaxable (P ⊗ Q) - ProdRelaxable {{pr}} {{qr}} = record - { relax = (λ { g₁⊆g₂ (p , q) → - ( Relaxable.relax pr g₁⊆g₂ p - , Relaxable.relax qr g₁⊆g₂ q) } - ) - } - -open Relaxable {{...}} public +open Relaxable {{...}} relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) → g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ] @@ -145,81 +134,6 @@ relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNode rewrite cast-is-id refl (Graph.nodes g₂) with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _) --- Tools for graph construction. The most important is a 'monotonic function': --- one that takes a graph, and produces another graph, such that the --- new graph includes all the information from the old one. - -MonotonicGraphFunction : (Graph → Set) → Set -MonotonicGraphFunction T = (g₁ : Graph) → Σ Graph (λ g₂ → T g₂ × g₁ ⊆ g₂) - --- Now, define some operations on monotonic functions; these are useful --- to save the work of threading intermediate graphs in and out of operations. - -infixr 2 _⟨⊗⟩_ -_⟨⊗⟩_ : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }} → - MonotonicGraphFunction T₁ → MonotonicGraphFunction T₂ → - MonotonicGraphFunction (T₁ ⊗ T₂) -_⟨⊗⟩_ {{r}} f₁ f₂ g - with (g' , (t₁ , g⊆g')) ← f₁ g - with (g'' , (t₂ , g'⊆g'')) ← f₂ g' = - (g'' , ((Relaxable.relax r g'⊆g'' t₁ , t₂) , ⊆-trans g⊆g' g'⊆g'')) - -infixl 2 _update_ -_update_ : ∀ {T : Graph → Set} {{ TRelaxable : Relaxable T }} → - MonotonicGraphFunction T → (∀ (g : Graph) → T g → Σ Graph (λ g' → g ⊆ g')) → - MonotonicGraphFunction T -_update_ {{r}} f mod g - with (g' , (t , g⊆g')) ← f g - with (g'' , g'⊆g'') ← mod g' t = - (g'' , ((Relaxable.relax r g'⊆g'' t , ⊆-trans g⊆g' g'⊆g''))) - -infixl 2 _map_ -_map_ : ∀ {T₁ T₂ : Graph → Set} → - MonotonicGraphFunction T₁ → (∀ (g : Graph) → T₁ g → T₂ g) → - MonotonicGraphFunction T₂ -_map_ f fn g = let (g' , (t₁ , g⊆g')) = f g in (g' , (fn g' t₁ , g⊆g')) - - --- To reason about monotonic functions and what we do, we need a way --- to describe values they produce. A 'graph-value predicate' is --- just a predicate for some (dependent) value. - -GraphValuePredicate : (Graph → Set) → Set₁ -GraphValuePredicate T = ∀ (g : Graph) → T g → Set - -Both : {T₁ T₂ : Graph → Set} → GraphValuePredicate T₁ → GraphValuePredicate T₂ → - GraphValuePredicate (T₁ ⊗ T₂) -Both P Q = (λ { g (t₁ , t₂) → (P g t₁ × Q g t₂) }) - --- Since monotnic functions keep adding on to a function, proofs of --- graph-value predicates go stale fast (they describe old values of --- the graph). To keep propagating them through, we need them to still --- on 'bigger graphs'. We call such predicates monotonic as well, since --- they respect the ordering of graphs. - -MonotonicPredicate : ∀ {T : Graph → Set} {{ TRelaxable : Relaxable T }} → - GraphValuePredicate T → Set -MonotonicPredicate {T} P = ∀ (g₁ g₂ : Graph) (t₁ : T g₁) (g₁⊆g₂ : g₁ ⊆ g₂) → - P g₁ t₁ → P g₂ (relax g₁⊆g₂ t₁) - - --- A 'map' has a certain property if its ouputs satisfy that property --- for all inputs. - -always : ∀ {T : Graph → Set} → GraphValuePredicate T → MonotonicGraphFunction T → Set -always P m = ∀ g₁ → let (g₂ , t , _) = m g₁ in P g₂ t - -⟨⊗⟩-reason : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }} - {P : GraphValuePredicate T₁} {Q : GraphValuePredicate T₂} - {P-Mono : MonotonicPredicate P} - {m₁ : MonotonicGraphFunction T₁} {m₂ : MonotonicGraphFunction T₂} → - always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂) -⟨⊗⟩-reason {P-Mono = P-Mono} {m₁ = m₁} {m₂ = m₂} aP aQ g - with p ← aP g - with (g' , (t₁ , g⊆g')) ← m₁ g - with q ← aQ g' - with (g'' , (t₂ , g'⊆g'')) ← m₂ g' = (P-Mono _ _ _ g'⊆g'' p , q) - pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index pushBasicBlock bss g = ( record diff --git a/MonotonicState.agda b/MonotonicState.agda new file mode 100644 index 0000000..4d4c3c8 --- /dev/null +++ b/MonotonicState.agda @@ -0,0 +1,116 @@ +open import Agda.Primitive using (lsuc) + +module MonotonicState {s} {S : Set s} + (_≼_ : S → S → Set s) + (≼-trans : ∀ {s₁ s₂ s₃ : S} → s₁ ≼ s₂ → s₂ ≼ s₃ → s₁ ≼ s₃) where + +open import Data.Product using (Σ; _×_; _,_) + +open import Utils using (_⊗_; _,_) + +-- Sometimes, we need a state monad whose values depend on the state. However, +-- one trouble with such monads is that as the state evolves, old values +-- in scope are over the 'old' state, and don't get updated accordingly. +-- Apparently, a related version of this problem is called 'demonic bind'. +-- +-- One solution to the problem is to also witness some kind of relationtion +-- between the input and output states. Using this relationship makes it possible +-- to 'bring old values up to speed'. +-- +-- Motivated primarily by constructing a Control Flow Graph, the 'relationship' +-- I've chosen is a 'less-than' relation. Thus, 'MonotonicState' is just +-- a (dependent) state "monad" that also witnesses that the state keeps growing. + +MonotonicState : (S → Set s) → Set s +MonotonicState T = (s₁ : S) → Σ S (λ s₂ → T s₂ × s₁ ≼ s₂) + +-- It's not a given that the (arbitrary) _≼_ relationship can be used for +-- updating old values. The Relaxable typeclass represents type constructor +-- that support the operation. + +record Relaxable (T : S → Set s) : Set (lsuc s) where + field relax : ∀ {s₁ s₂ : S} → s₁ ≼ s₂ → T s₁ → T s₂ + +instance + ProdRelaxable : ∀ {P : S → Set s} {Q : S → Set s} → + {{ PRelaxable : Relaxable P }} → {{ QRelaxable : Relaxable Q }} → + Relaxable (P ⊗ Q) + ProdRelaxable {{pr}} {{qr}} = record + { relax = (λ { g₁≼g₂ (p , q) → + ( Relaxable.relax pr g₁≼g₂ p + , Relaxable.relax qr g₁≼g₂ q) } + ) + } + +-- In general, the "MonotonicState monad" is not even a monad; it's not +-- even applicative. The trouble is that functions in general cannot be +-- 'relaxed', and to apply an 'old' function to a 'new' value, you'd thus +-- need to un-relax the value (which also isn't possible in general). +-- +-- However, we _can_ combine pairs from two functions into a tuple, which +-- would equivalent to the applicative operation if functions were relaxable. +-- +-- TODO: Now that I think about it, the swapped version of the applicative +-- operation is possible, since it doesn't require lifting functions. + +infixr 4 _⟨⊗⟩_ +_⟨⊗⟩_ : ∀ {T₁ T₂ : S → Set s} {{ _ : Relaxable T₁ }} → + MonotonicState T₁ → MonotonicState T₂ → MonotonicState (T₁ ⊗ T₂) +_⟨⊗⟩_ {{r}} f₁ f₂ s + with (s' , (t₁ , s≼s')) ← f₁ s + with (s'' , (t₂ , s'≼s'')) ← f₂ s' = + (s'' , ((Relaxable.relax r s'≼s'' t₁ , t₂) , ≼-trans s≼s' s'≼s'')) + +infixl 4 _update_ +_update_ : ∀ {T : S → Set s} {{ _ : Relaxable T }} → + MonotonicState T → (∀ (s : S) → T s → Σ S (λ s' → s ≼ s')) → + MonotonicState T +_update_ {{r}} f mod s + with (s' , (t , s≼s')) ← f s + with (s'' , s'≼s'') ← mod s' t = + (s'' , ((Relaxable.relax r s'≼s'' t , ≼-trans s≼s' s'≼s''))) + +infixl 4 _map_ +_map_ : ∀ {T₁ T₂ : S → Set s} → + MonotonicState T₁ → (∀ (s : S) → T₁ s → T₂ s) → MonotonicState T₂ +_map_ f fn s = let (s' , (t₁ , s≼s')) = f s in (s' , (fn s' t₁ , s≼s')) + + +-- To reason about MonotonicState instances, we need predicates over their +-- values. But such values are dependent, so our predicates need to accept +-- the state as argument, too. + +DependentPredicate : (S → Set s) → Set (lsuc s) +DependentPredicate T = ∀ (s₁ : S) → T s₁ → Set s + +Both : {T₁ T₂ : S → Set s} → DependentPredicate T₁ → DependentPredicate T₂ → + DependentPredicate (T₁ ⊗ T₂) +Both P Q = (λ { s (t₁ , t₂) → (P s t₁ × Q s t₂) }) + +-- Since monotnic functions keep adding on to the state, proofs of +-- predicates over their outputs go stale fast (they describe old values of +-- the state). To keep them relevant, we need them to still hold on 'bigger +-- states'. We call such predicates monotonic as well, since they respect the +-- ordering relation. + +MonotonicPredicate : ∀ {T : S → Set s} {{ _ : Relaxable T }} → + DependentPredicate T → Set s +MonotonicPredicate {T} {{r}} P = ∀ (s₁ s₂ : S) (t₁ : T s₁) (s₁≼s₂ : s₁ ≼ s₂) → + P s₁ t₁ → P s₂ (Relaxable.relax r s₁≼s₂ t₁) + +-- A MonotonicState "monad" m has a certain property if its ouputs satisfy that +-- property for all inputs. + +always : ∀ {T : S → Set s} → DependentPredicate T → MonotonicState T → Set s +always P m = ∀ s₁ → let (s₂ , t , _) = m s₁ in P s₂ t + +⟨⊗⟩-reason : ∀ {T₁ T₂ : S → Set s} {{ _ : Relaxable T₁ }} + {P : DependentPredicate T₁} {Q : DependentPredicate T₂} + {P-Mono : MonotonicPredicate P} + {m₁ : MonotonicState T₁} {m₂ : MonotonicState T₂} → + always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂) +⟨⊗⟩-reason {P-Mono = P-Mono} {m₁ = m₁} {m₂ = m₂} aP aQ s + with p ← aP s + with (s' , (t₁ , s≼s')) ← m₁ s + with q ← aQ s' + with (s'' , (t₂ , s'≼s'')) ← m₂ s' = (P-Mono _ _ _ s'≼s'' p , q)