sbv-0.9.22: Symbolic bit vectors: Bit-precise verification and automatic C-code generation.





An encoding and correctness proof of Legato's multiplier in Haskell. Bill Legato came up with an interesting way to multiply two 8-bit numbers on Mostek, as described here:

Here's Legato's algorithm, as coded in Mostek assembly:

    step1 :       LDX #8         ; load X immediate with the integer 8 
    step2 :       LDA #0         ; load A immediate with the integer 0 
    step3 : LOOP  ROR F1         ; rotate F1 right circular through C 
    step4 :       BCC ZCOEF      ; branch to ZCOEF if C = 0 
    step5 :       CLC            ; set C to 0 
    step6 :       ADC F2         ; set A to A+F2+C and C to the carry 
    step7 : ZCOEF ROR A          ; rotate A right circular through C 
    step8 :       ROR LOW        ; rotate LOW right circular through C 
    step9 :       DEX            ; set X to X-1 
    step10:       BNE LOOP       ; branch to LOOP if Z = 0 

This program came to be known as the Legato's challenge in the community, where the challenge was to prove that it indeed does perform multiplication. This file formalizes the Mostek architecture in Haskell and proves that Legato's algorithm is indeed correct.


Mostek architecture

type Address = SWord32Source

The memory is addressed by 32-bit words.

data Register Source

We model only two registers of Mostek that is used in the above algorithm, can add more.



data Flag Source

The carry flag (FlagC) and the zero flag (FlagZ)



type Value = SWord8Source

Mostek was an 8-bit machine.

type Bit = SBoolSource

Convenient synonym for symbolic machine bits.

type Registers = Array Register ValueSource

Register bank

type Flags = Array Flag BitSource

Flag bank

type Memory = Model Word32 Word8Source

The memory maps 32-bit words to 8-bit words. (The Model data-type is defined later, depending on the verification model used.)

data Mostek Source

Abstraction of the machine: The CPU consists of memory, registers, and flags. Unlike traditional hardware, we assume the program is stored in some other memory area that we need not model. (No self modifying programs!)




type Extract a = Mostek -> aSource

Given a machine state, compute a value out of it

type Program = Mostek -> MostekSource

Programs are essentially state transformers (on the machine state)

Low-level operations

getReg :: Register -> Extract ValueSource

Get the value of a given register

setReg :: Register -> Value -> ProgramSource

Set the value of a given register

getFlag :: Flag -> Extract BitSource

Get the value of a flag

setFlag :: Flag -> Bit -> ProgramSource

Set the value of a flag

peek :: Address -> Extract ValueSource

Read memory

poke :: Address -> Value -> ProgramSource

Write to memory

checkOverflow :: SWord8 -> SWord8 -> SBool -> SBoolSource

Checking overflow. In Legato's multipler the ADC instruction needs to see if the expression x + y + c overflowed, as checked by this function. Note that we verify the correctness of this check separately below in checkOverflowCorrect.

checkOverflowCorrect :: IO ThmResultSource

Correctness theorem for our checkOverflow implementation.

We have:

>>> checkOverflowCorrect

Instruction set

type Instruction = Program -> ProgramSource

An instruction is modeled as a Program transformer. We model mostek programs in direct continuation passing style.

ldx :: Value -> InstructionSource

LDX: Set register X to value v

lda :: Value -> InstructionSource

LDA: Set register A to value v

clc :: InstructionSource

CLC: Clear the carry flag

rorM :: Address -> InstructionSource

ROR, memory version: Rotate the value at memory location a to the right by 1 bit, using the carry flag as a transfer position. That is, the final bit of the memory location becomes the new carry and the carry moves over to the first bit. This very instruction is one of the reasons why Legato's multiplier is quite hard to understand and is typically presented as a verification challenge.

rorR :: Register -> InstructionSource

ROR, register version: Same as rorM, except through register r.

bcc :: Program -> InstructionSource

BCC: branch to label l if the carry flag is false

adc :: Address -> InstructionSource

ADC: Increment the value of register A by the value of memory contents at address a, using the carry-bit as the carry-in for the addition.

dex :: InstructionSource

DEX: Decrement the value of register X

bne :: Program -> InstructionSource

BNE: Branch if the zero-flag is false

end :: ProgramSource

The end combinator stops our program, providing the final continuation that does nothing.

Legato's algorithm in Haskell/SBV

legato :: Address -> Address -> Address -> ProgramSource

Parameterized by the addresses of locations of the factors (F1 and F2), the following program multiplies them, storing the low-byte of the result in the memory location lowAddr, and the high-byte in register A. The implementation is a direct transliteration of Legato's algorithm given at the top, using our notation.

Verification interface

runLegato :: (Address, Value) -> (Address, Value) -> Address -> Mostek -> (Value, Value)Source

Given address/value pairs for F1 and F2, and the location of where the low-byte of the result should go, runLegato takes an arbitrary machine state m and returns the high and low bytes of the multiplication.

type InitVals = (Value, Value, Value, Bit, Bit)Source

Helper synonym for capturing relevant bits of Mostek

initMachine :: Memory -> InitVals -> MostekSource

Create an instance of the Mostek machine, initialized by the memory and the relevant values of the registers and the flags

legatoIsCorrect :: Memory -> (Address, Value) -> (Address, Value) -> Address -> InitVals -> SBoolSource

The correctness theorem. For all possible memory configurations, the factors (x and y below), the location of the low-byte result and the initial-values of registers and the flags, this function will return True only if running Legato's algorithm does indeed compute the product of x and y correctly.


type Model = SFunArraySource

Choose the appropriate array model to be used for modeling the memory. (See Memory.) The SFunArray is the function based model. SArray is the SMT-Lib array's based model.

correctnessTheorem :: IO ThmResultSource

The correctness theorem. On a decent MacBook Pro, this proof takes about 3 minutes with the SFunArray memory model and about 30 minutes with the SArray model.

C Code generation

legatoInC :: IO ()Source

Generate a C program that implements Legato's algorithm automatically.