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Typed Assembly Languages

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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Typed Assembly Languages

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  1. Typed Assembly Languages COS 441, Fall 2004 Frances Spalding Based on slides from Dave Walker and Greg Morrisett

  2. 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’ • Preservation: each step in evaluation preserves typing If ` e: and e !* e’ then ` e’: COS441 - Typed Assembly Languages

  3. High Level vs. Low Level • If the source language has been proved to be type safe, what do we know about the safety of the code we actually run? • Nothing! • What if the compiler produced a typed assembly language? COS441 - Typed Assembly Languages

  4. Proof Carrying Code • 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 COS441 - Typed Assembly Languages

  5. High-level code for factorial COS441 - Typed Assembly Languages

  6. Assembly code for factorial COS441 - Typed Assembly Languages

  7. TAL0 Syntax • 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 COS441 - Typed Assembly Languages

  8. TAL0 Dynamic Semantics • Machine State  = (H,R,B) • 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:  COS441 - Typed Assembly Languages

  9. TAL0 Dynamic Semantics COS441 - Typed Assembly Languages

  10. TAL0 Static Semantics • ::= 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 COS441 - Typed Assembly Languages

  11. Example Program with Types COS441 - Typed Assembly Languages

  12. Type Checking Basics • We need to keep track of: • 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 COS441 - Typed Assembly Languages

  13. Basic Typing Judgments • ;` n: int • ;` r: (r) • ;` L : (L) COS441 - Typed Assembly Languages

  14. Subtyping on Register Files • 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} COS441 - Typed Assembly Languages

  15. Subtyping on Code Types • Like ordinary function types, code types are contravariant ’ <: ! {} <:’ ! {} COS441 - Typed Assembly Languages

  16. Typing Instructions • The typing judgment for instructions describes the registers on input to the function and the registers on output ` : 12 • is invariant. The types of heap objects cannot change as the program executes COS441 - Typed Assembly Languages

  17. Typing Instructions ;` rs : int ;` v : int ` add rd, rs, v : ![rd := int] ;` r : int ;` v : ! {} ` ble r, v : ! ;` v :  ` mov rd, v : ![rd := ] COS441 - Typed Assembly Languages

  18. Basic Block Typing • All basic blocks end with jump instructions ;` v : ! {} ` jmp v : ! {} • Instruction Sequences ` : 1!2` B : 2! {} `;B : 1! {} COS441 - Typed Assembly Languages

  19. Heap and Register File Typing • Heap Typing Dom(H)=Dom() 8 L 2 Dom(H).` H(L):(L) `H :  • Register File Typing 8r 2 Dom(). ;` R(r) : (r) ` R :  COS441 - Typed Assembly Languages

  20. Machine Typing • Main Typing Judgment `H : ` R : ` B : ! {} `(H,R,B) COS441 - Typed Assembly Languages

  21. Type Safety for TAL0 • Progress: If `1 then 92 such that 12 • Preservation If `1 and 12 then `2 COS441 - Typed Assembly Languages

  22. Limitations of TAL0 • What about situations where we don’t care which specific type is in a register? • Eg. A function that swaps the contents of two registers • We can use polymorphic types to express this. COS441 - Typed Assembly Languages

  23. TAL1 : Polymorphism • ::= … |  8, . { r1:, r2:, r31: {r1:, r2:} ! {}} ! {} • 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[] COS441 - Typed Assembly Languages

  24. Polymorphism Example COS441 - Typed Assembly Languages

  25. Callee-Saved Registers • 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? COS441 - Typed Assembly Languages

  26. TAL1 Semantics • Changes to the static semantics to support polymorphism • Add the judgment ` ok • Modify the other judgments to take this into account (just like homework 8!) • Changes to the dynamic semantics • Add rule to do instantiation (H, R, jmp v[])  (H, R, B[/]) COS441 - Typed Assembly Languages

  27. Full TAL Syntax COS441 - Typed Assembly Languages

  28. Limitations of TAL • Almost every compiler uses a stack • As storage space for local variables and other temporaries when we run out of space in registers • To keep track of control flow • Control Links • Activation Links COS441 - Typed Assembly Languages

  29. STAL: Add a Stack • Machine States: (H, R, S, B) • 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 • push v ´ salloc 1; sst v, 1 • pop v ´ sld r, 1; sfree 1 COS441 - Typed Assembly Languages

  30. Factorial Example COS441 - Typed Assembly Languages

  31. Extensions for STAL • Stack types  ::= nil | :: |  • 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} ! {} COS441 - Typed Assembly Languages

  32. Stack Instruction Typing • Stack Allocation ;` salloc n : [sp := ]![sp := ?::…::?::] • Stack Free ;` sfree : [sp := 1::…::n::]![sp := ] COS441 - Typed Assembly Languages

  33. Stack Instruction Typing • Stack load (sp) = 1 :: … :: n ::  ;` sld r, n : ![r := n] • Stack store ;; ` v : (sp) = 1 :: … :: n ::  ;` sst v, n : ![sp := 1 :: … ::  :: ] COS441 - Typed Assembly Languages

  34. STAL Features • 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 • Arguments on stack or in registers? • Results on stack or in registers? • Return address? Caller pops? Callee pops? • Caller saves? Callee saves? COS441 - Typed Assembly Languages

  35. STAL Syntax COS441 - Typed Assembly Languages

  36. General Limitations of TAL/STAL • More functionality can be added to TAL • There are still some limitations • Must know the order of data allocated on the stack • sp: 1 @  @ 2 @ ’ @ 3 • Must distinguish between heap and stack pointers • Can’t have a function that takes in two generic int pointers and adds their contents COS441 - Typed Assembly Languages

  37. For more information on TAL/STAL • 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. COS441 - Typed Assembly Languages

  38. What Makes Stack Typing So Tricky? • In the heap, we use the abstraction that a location never changes type • 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 COS441 - Typed Assembly Languages

  39. My Research • 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) COS441 - Typed Assembly Languages

  40. Modeling Stack “Evolution” • 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 COS441 - Typed Assembly Languages

  41. k1 TOP k2 k5 k3 k4 k6 Time h t p e ℓ0 D k c a t S ℓ1 ℓ2 ℓ3 … COS441 - Typed Assembly Languages

  42. Tagged Locations • All locations are “tagged” with their version number • (r1) S(k3. ℓ2)) is different from (r1) S(k6. ℓ2)) • The tags act as “guards” on the location • 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 COS441 - Typed Assembly Languages

  43. For more information … Certifying Compilation for a Language with Stack Allocation [Draft]. L. Jia, F. Spalding, D. Walker, N. Glew. COS441 - Typed Assembly Languages

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