add constant propagation example

dataflow.jl

@@ -82,7 +82,7 @@ show(io::IO, ::BotElement) = print(io, "⊥")
 # LatticeElement.
 
 
-# Part 3: dataflow analysis
+# Part 3: abstract value
 
 # Note: the paper (https://api.semanticscholar.org/CorpusID:28519618) uses U+1D56E MATHEMATICAL BOLD FRAKTUR CAPITAL C for this
 const AbstractValue = Dict{Symbol,LatticeElement}
@@ -97,11 +97,44 @@ const AbstractValue = Dict{Symbol,LatticeElement}
 <=(X::AbstractValue, Y::AbstractValue) = X⊓Y == X
 <(X::AbstractValue, Y::AbstractValue)  = X<=Y && X!=Y
 
+
+#########################################################
+# example problem 1. - find uses of undefined variables #
+#########################################################
+
+# flat lattice of variable definedness
+
+struct IsDefined <: LatticeElement
+    is::Bool
+end
+show(io::IO, isdef::IsDefined) = print(io, isdef.is ? "defined" : "undefined")
+
+const undef = IsDefined(false)
+const def   = IsDefined(true)
+
+# abstract semantics
+
+abstract_eval(x::Sym, s::AbstractValue) = get(s, x.name, ⊥)
+
+abstract_eval(x::Num, s::AbstractValue) = def
+
+function abstract_eval(x::Call, s::AbstractValue)
+    if any(a->(abstract_eval(a,s) == ⊥), x.args)
+        return ⊥
+    end
+    return def
+end
+
+# data flow analysis
+
+# Note:
+# - in this problem, we make sure that states will always move to higher position in lattice, so we use ⊔ (join) operator for state update
+# - and the condition we use to check whether or not the statement makes a change is `!(new <= prev)`
 function max_fixed_point(P::Program, a₁::AbstractValue, eval) where {Program<:AbstractVector{Stmt}}
     n = length(P)
     bot = AbstractValue( v => ⊥ for v in keys(a₁) )
     s = [ a₁; [ bot for i = 2:n ] ]
-    W = BitSet(1)
+    W = BitSet(1:n)
 
     while !isempty(W)
         pc = first(W)
@@ -137,38 +170,112 @@ function max_fixed_point(P::Program, a₁::AbstractValue, eval) where {Program<:
     s
 end
 
+prog1 = [Assign(:x, 0),                       # 1
+         GotoIf(5, Call(:randbool, Exp[])),   # 2
+         Assign(:y, 1),                       # 3
+         Goto(5),                             # 4
+         Assign(:z, Call(:pair, Exp[:x,:y])), # 5
+         Ret()]
 
-# example problem - find uses of undefined variables
+# variables initially undefined
+l = AbstractValue(:x => undef, :y => undef, :z => undef)
 
-# flat lattice of variable definedness
+max_fixed_point(prog1, l, abstract_eval)
 
-struct IsDefined <: LatticeElement
-    is::Bool
+
+#########################################################################
+# example problem 2. - constant folding propagation (the paper example) #
+#########################################################################
+
+# lattice
+
+# Note: intuitively, each lattice element can be interpreted in the following way:
+# - `Int` means "constant" value
+# - `⊤` means "not constant due to missing information"
+# - `⊥` means "not constant due to conflict"
+
+struct Const <: LatticeElement
+    val::Int
 end
-show(io::IO, isdef::IsDefined) = print(io, isdef.is ? "defined" : "undefined")
 
-const undef = IsDefined(false)
-const def   = IsDefined(true)
+# abstract semantics
 
-abstract_eval(x::Sym, s::AbstractValue) = get(s, x.name, ⊥)
+abstract_eval(x::Num, s::AbstractValue) = Const(x.val)
 
-abstract_eval(x::Num, s::AbstractValue) = def
+abstract_eval(x::Sym, s::AbstractValue) = get(s, x.name, ⊤)
 
 function abstract_eval(x::Call, s::AbstractValue)
-    if any(a->(abstract_eval(a,s) == ⊥), x.args)
-        return ⊥
+    f = getfield(@__MODULE__, x.head.name)
+
+    to_val(x::Num)   = x.val::Int
+    to_val(x::Const) = x.val::Int
+    argvals = Int[]
+    for arg in x.args
+        arg = abstract_eval(arg, s)
+        arg === ⊤ && return ⊤
+        arg === ⊥ && return ⊥
+        push!(argvals, to_val(arg))
     end
-    return def
+
+    return Const(f(argvals...))
 end
 
-prog1 = [Assign(:x, 0),                       # 1
-         GotoIf(5, Call(:randbool, Exp[])),   # 2
-         Assign(:y, 1),                       # 3
-         Goto(5),                             # 4
-         Assign(:z, Call(:pair, Exp[:x,:y])), # 5
-         Ret()]
+# Note: in this problem, we make sure that states will always move to _lower_ position in lattice, so
+# - initialize states with `⊤`
+# - we use `⊓` (meet) operator to update states,
+# - and the condition we use to check whether or not the statement makes a change is `!(new >= prev)`
+function max_fixed_point(P::Program, a₁::AbstractValue, eval) where {Program<:AbstractVector{Stmt}}
+    n = length(P)
+    init = AbstractValue( v => ⊤ for v in keys(a₁) )
+    s = [ a₁; [ init for i = 2:n ] ]
+    W = BitSet(1:n)
 
-# variables initially undefined
-l = AbstractValue(:x => undef, :y => undef, :z => undef)
+    while !isempty(W)
+        pc = first(W)
+        while pc != n+1
+            delete!(W, pc)
+            I = P[pc]
+            new = s[pc]
+            if isa(I, Assign)
+                # for an assignment, outgoing value is different from incoming
+                new = copy(new)
+                new[I.lhs.name] = eval(I.rhs, new)
+            end
+            if isa(I, Goto)
+                pc´ = I.label
+            else
+                pc´ = pc+1
+                if isa(I, GotoIf)
+                    l = I.label
+                    if !(new >= s[l])
+                        push!(W, l)
+                        s[l] = s[l] ⊓ new
+                    end
+                end
+            end
+            if pc´<=n && !(new >= s[pc´])
+                s[pc´] = s[pc´] ⊓ new
+                pc = pc´
+            else
+                pc = n+1
+            end
+        end
+    end
+    s
+end
 
-max_fixed_point(prog1, l, abstract_eval)
+prog1 = [Assign(:x, 1),                    # 1
+         Assign(:y, 2),                    # 2
+         Assign(:z, 3),                    # 3
+         Goto(9),                          # 4
+         Assign(:r, Call(:(+), [:y, :z])), # 5
+         GotoIf(8, Call(:(≤), [:x, :z])),  # 6
+         Assign(:r, Call(:(+), [:z, :y])), # 7
+         Assign(:x, Call(:(+), [:x, 1])),  # 8
+         GotoIf(5, Call(:(<), [:x, 10])),  # 9
+         ]
+
+# initially values are not constant (due to missing information)
+a₁ = AbstractValue(:x => ⊤, :y => ⊤, :z => ⊤, :r => ⊤)
+
+max_fixed_point(prog1, a₁, abstract_eval) # The solution contains the `:r => Const(5)`, which is not found in the program