In fact, the algorithm is expressed over the flat sequence of opcodes as occurring in the binary format, and performs only a single pass over it.
+Consequently, it can be integrated directly into a decoder.
+The algorithm is expressed in typed pseudo code whose semantics is intended to be self-explanatory.
+
+Data Structures
+Types are representable as an enumeration.
+type val_type = I32 | I64 | F32 | F64 | V128 | Funcref | Externref
+
+func is_num(t : val_type | Unknown) : bool =
+ return t = I32 || t = I64 || t = F32 || t = F64 || t = Unknown
+
+func is_vec(t : val_type | Unknown) : bool =
+ return t = V128 || t = Unknown
+
+func is_ref(t : val_type | Unknown) : bool =
+ return t = Funcref || t = Externref || t = Unknown
+
+
The algorithm uses two separate stacks: the value stack and the control stack.
+The former tracks the types of operand values on the stack,
+the latter surrounding structured control instructions and their associated blocks.
+type val_stack = stack(val_type | Unknown)
+
+type ctrl_stack = stack(ctrl_frame)
+type ctrl_frame = {
+ opcode : opcode
+ start_types : list(val_type)
+ end_types : list(val_type)
+ height : nat
+ unreachable : bool
+}
+
+
For each value, the value stack records its value type, or Unknown when the type is not known.
+For each entered block, the control stack records a control frame with the originating opcode, the types on the top of the operand stack at the start and end of the block (used to check its result as well as branches), the height of the operand stack at the start of the block (used to check that operands do not underflow the current block), and a flag recording whether the remainder of the block is unreachable (used to handle stack-polymorphic typing after branches).
+For the purpose of presenting the algorithm, the operand and control stacks are simply maintained as global variables:
+var vals : val_stack
+var ctrls : ctrl_stack
+
+
However, these variables are not manipulated directly by the main checking function, but through a set of auxiliary functions:
+func push_val(type : val_type | Unknown) =
+ vals.push(type)
+
+func pop_val() : val_type | Unknown =
+ if (vals.size() = ctrls[0].height && ctrls[0].unreachable) return Unknown
+ error_if(vals.size() = ctrls[0].height)
+ return vals.pop()
+
+func pop_val(expect : val_type | Unknown) : val_type | Unknown =
+ let actual = pop_val()
+ error_if(actual =/= expect && actual =/= Unknown && expect =/= Unknown)
+ return actual
+
+func push_vals(types : list(val_type)) = foreach (t in types) push_val(t)
+func pop_vals(types : list(val_type)) : list(val_type) =
+ var popped := []
+ foreach (t in reverse(types)) popped.prepend(pop_val(t))
+ return popped
+
+
Pushing an operand value simply pushes the respective type to the value stack.
+Popping an operand value checks that the value stack does not underflow the current block and then removes one type.
+But first, a special case is handled where the block contains no known values, but has been marked as unreachable.
+That can occur after an unconditional branch, when the stack is typed polymorphically.
+In that case, an unknown type is returned.
+A second function for popping an operand value takes an expected type, which the actual operand type is checked against.
+The types may differ in case one of them is Unknown.
+The function returns the actual type popped from the stack.
+Finally, there are accumulative functions for pushing or popping multiple operand types.
+
+
Note
+
The notation stack[i] is meant to index the stack from the top,
+so that, e.g., ctrls[0] accesses the element pushed last.
+
The control stack is likewise manipulated through auxiliary functions:
+func push_ctrl(opcode : opcode, in : list(val_type), out : list(val_type)) =
+ let frame = ctrl_frame(opcode, in, out, vals.size(), false)
+ ctrls.push(frame)
+ push_vals(in)
+
+func pop_ctrl() : ctrl_frame =
+ error_if(ctrls.is_empty())
+ let frame = ctrls[0]
+ pop_vals(frame.end_types)
+ error_if(vals.size() =/= frame.height)
+ ctrls.pop()
+ return frame
+
+func label_types(frame : ctrl_frame) : list(val_types) =
+ return (if frame.opcode == loop then frame.start_types else frame.end_types)
+
+func unreachable() =
+ vals.resize(ctrls[0].height)
+ ctrls[0].unreachable := true
+
+
Pushing a control frame takes the types of the label and result values.
+It allocates a new frame record recording them along with the current height of the operand stack and marks the block as reachable.
+Popping a frame first checks that the control stack is not empty.
+It then verifies that the operand stack contains the right types of values expected at the end of the exited block and pops them off the operand stack.
+Afterwards, it checks that the stack has shrunk back to its initial height.
+The type of the label associated with a control frame is either that of the stack at the start or the end of the frame, determined by the opcode that it originates from.
+Finally, the current frame can be marked as unreachable.
+In that case, all existing operand types are purged from the value stack, in order to allow for the stack-polymorphism logic in pop_val to take effect.
+
+
Note
+
Even with the unreachable flag set, consecutive operands are still pushed to and popped from the operand stack.
+That is necessary to detect invalid examples like \((\href{../syntax/instructions.html#syntax-instr-control}{\mathsf{unreachable}}~(\href{../syntax/types.html#syntax-valtype}{\mathsf{i32}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}})~\href{../syntax/types.html#syntax-valtype}{\mathsf{i64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}})\).
+However, a polymorphic stack cannot underflow, but instead generates Unknown types as needed.
+