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Typed Assembly Languages
COS 441, Fall 2004
Frances Spalding
Based on slides from Dave Walker and Greg Morrisett
Type Safety

Progress: if a state is well-typed then it is not stuck
If  ` e: then either e is a value or
9 e’ such that e !* e’
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Preservation: each step in evaluation preserves
typing
If  ` e: and e !* e’ then  ` e’:
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High Level vs. Low Level
Type Safe Source Code
Very large,
complex compiler
that’s almost
certainly full of all
sorts of insidious
bugs
Assembly Code
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
If the source language has
been proved to be type
safe, what do we know
about the safety of the code
we actually run?
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Nothing!

What if the compiler
produced a typed assembly
language?
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Proof Carrying Code
Source Program

Compilation
&
Certification
Native
Code
Safety
Proof
Code
Producer
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Code
Consumer
Proof Validation

CPU
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The code producer
provides a proof of
safety along with the
executable
The code consumer
checks that proof
before executing the
code
A typing derivation of
the low-level code can
be a safety proof
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High-level code for factorial
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Assembly code for factorial
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TAL0 Syntax
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Registers: r1, r2, …
Labels: L, …
Integers: n
Operands: v ::= r | n | L
Arithmetic Ops: aop ::= add | sub | mul | …
Branch Ops: bop ::= beq | bgt | …
Instructions:  ::= aop rd, rs, v | bop r, v | mov
r, v
Blocks: B ::= jmp v | ; B
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TAL0 Dynamic Semantics
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Machine State  = (H,R,B)
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H maps labels to basic blocks
R maps registers to values (integers and labels)
B is a basic block (corresponding to the current
program counter)
Model evaluation as a transition function
mapping machine states to machine states:

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TAL0 Dynamic Semantics
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TAL0 Static Semantics
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 ::= int |  ! {}
 ::= {r1:1, r2:2, …}
Code labels have type {r1:1, r2:2, …} ! {}
To jump to code with this type, r1 must
contain a value of type , etc
Notice that functions never return – the code
block always ends with a jump to another
label
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Example Program with Types
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Type Checking Basics
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We need to keep track of:
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The types of the registers at each point in the
code
The types of the labels on the code
Heap Types:  will map labels to label types
Register Types:  will map registers to types
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Basic Typing Judgments
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;  ` n: int
;  ` r: (r)
;  ` L : (L)
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Subtyping on Register Files
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Our program will never crash if the register
file contains more values than necessary to
satisfy some typing precondition
Width subtyping:
{r1:1,…,ri-1:i-1,ri:i} <: {r1:1,…,ri-1:i-1}
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Subtyping on Code Types
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Like ordinary function types, code types are
contravariant
’ <: 
 ! {} <: ’ ! {}
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Typing Instructions
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The typing judgment for instructions
describes the registers on input to the
function and the registers on output
 `  : 1  2
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 is invariant. The types of heap objects
cannot change as the program executes
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Typing Instructions
; ` rs : int
; ` v : int
 ` add rd, rs, v :  ! [rd := int]
; ` r : int
; ` v :  ! {}
 ` ble r, v :  ! 
; ` v : 
 ` mov rd, v :  ! [rd := ]
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Basic Block Typing
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All basic blocks end with jump instructions
;  ` v :  ! {}
 ` jmp v :  ! {}
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Instruction Sequences
 `  : 1 ! 2  ` B : 2 ! {}
 ` ;B : 1 ! {}
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Heap and Register File Typing
Heap Typing
Dom(H)=Dom() 8 L 2 Dom(H). ` H(L):(L)
`H:
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Register File Typing
8 r 2 Dom(). ; ` R(r) : (r)
`R:
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Machine Typing
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Main Typing Judgment
` H :   ` R :   ` B :  ! {}
` (H,R,B)
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Type Safety for TAL0
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Progress:
If ` 1 then 9 2 such that 1  2
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Preservation
If ` 1 and 1  2 then ` 2
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Limitations of TAL0
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What about situations where we don’t care
which specific type is in a register?
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Eg. A function that swaps the contents of two
registers
We can use polymorphic types to express
this.
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TAL1 : Polymorphism
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 ::= … | 
8 , . { r1:, r2:, r31: {r1:, r2:} ! {}} ! {}
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Describes a function that swaps the values in r1
and r2 for values of any two types
To jump to polymorphic functions, first
explicitly instantiate type variables: v[]
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Polymorphism Example
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Callee-Saved Registers
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We can use polymorphism to implement
callee-saved registers
We’d like to be able to enforce that a function
doesn’t change certain registers. In other words,
when it returns, certain registers have the same
values as when it started
8 . { r2: int, …, r5:, …., r31: {r1: int, …, r5:, …} ! {}
} ! {}
 Why does this enforce the invariant we want?

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TAL1 Semantics
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Changes to the static semantics to support
polymorphism
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Add the judgment  `  ok
Modify the other judgments to take this into
account (just like homework 8!)
Changes to the dynamic semantics
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Add rule to do instantiation
(H, R, jmp v[])  (H, R, B[/])
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Full TAL Syntax
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Limitations of TAL
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Almost every compiler uses a stack
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As storage space for local variables and other
temporaries when we run out of space in registers
To keep track of control flow
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Control Links
Activation Links
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STAL: Add a Stack
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Machine States: (H, R, S, B)
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Stack S ::= nil | v :: S
A designated register sp points to the top of the
stack
 ::= salloc n | sfree n | sld rd, n | sst rs, n
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push v ´ salloc 1; sst v, 1
pop v ´ sld r, 1; sfree 1
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Factorial Example
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Extensions for STAL
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Stack types  ::= nil | :: | 
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nil represents the empty stack
:: represents a stack v::S where v: and S:
 is a stack variable which we can use to
describe an unknown stack tail
Use polymorphism
8 . {sp: int::, r1: int} ! {}
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Stack Instruction Typing
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Stack Allocation
;  ` salloc n : [sp := ]![sp := ?::…::?::]
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Stack Free
;  ` sfree : [sp := 1::…::n::]![sp := ]
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Stack Instruction Typing
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Stack load
(sp) = 1 :: … :: n :: 
;  ` sld r, n :  ! [r := n]
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Stack store
; ;  ` v :  (sp) = 1 :: … :: n :: 
;  ` sst v, n :  ! [sp := 1 :: … ::  :: ]
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STAL Features
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Polymorphism and polymorphic recursion are
crucial for encoding standard procedure call/return
We didn’t have to bake in the notion of procedure
call/return. Basic jumps were good enough
Can combine features to encode up a variety of
calling conventions
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Arguments on stack or in registers?
Results on stack or in registers?
Return address? Caller pops? Callee pops?
Caller saves? Callee saves?
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STAL Syntax
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General Limitations of TAL/STAL
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More functionality can be added to TAL
There are still some limitations
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Must know the order of data allocated on the
stack
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sp: 1 @  @ 2 @ ’ @ 3
Must distinguish between heap and stack pointers
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Can’t have a function that takes in two generic int
pointers and adds their contents
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For more information on TAL/STAL
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From System F to Typed Assembly
Language. G. Morrisett, D. Walker, K. Crary,
and N. Glew.
Stack-Based Typed Assembly Language. G.
Morrisett, K. Crary, N. Glew, D. Walker.
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What Makes Stack Typing So Tricky?
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In the heap, we use the abstraction that a
location never changes type
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The garbage collector handles the reuse
When modeling instructions like push/pop,
we have to break this abstraction and use
strong updates to change types
Aliasing can make this unsound if we’re not
careful
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My Research
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A different approach to typed assembly
language
Use logic to describe what memory looks like
before and after each instruction
(r1 ) int) (r2 ) S(ℓ)) (ℓ ) bool)
load [r2], r1
(r1 ) int) (r2 ) S(ℓ)) (ℓ ) int)
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Modeling Stack “Evolution”
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Keep track of the different “versions” of each
stack location and how they are related
We can do this using a pair of a tree and a
pointer to a node in the tree
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Stack Depth
Time
ℓ0
k1
TOP
k2
k3
k5
k4
ℓ1
k6
ℓ2
ℓ3
…
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Tagged Locations
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All locations are “tagged” with their version
number
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(r1 ) S(k3. ℓ2)) is different from (r1 ) S(k6. ℓ2))
The tags act as “guards” on the location
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Before accessing a tagged location, you must
prove that it is “live”
In other words, the tag must be on the path
between the root and Top
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For more information…
Certifying Compilation for a Language with
Stack Allocation [Draft]. L. Jia, F. Spalding, D.
Walker, N. Glew.
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