sbv-5.7: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Copyright(c) Levent Erkok
Safe HaskellNone




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 = SWord32 Source

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 = SWord8 Source

Mostek was an 8-bit machine.

type Bit = SBool Source

Convenient synonym for symbolic machine bits.

type Registers = Array Register Value Source

Register bank

type Flags = Array Flag Bit Source

Flag bank

type Memory = Model Word32 Word8 Source

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!)




Mergeable Mostek Source

Mergeable instance of Mostek simply pushes the merging into record fields.

type Extract a = Mostek -> a Source

Given a machine state, compute a value out of it

type Program = Mostek -> Mostek Source

Programs are essentially state transformers (on the machine state)

Low-level operations

getReg :: Register -> Extract Value Source

Get the value of a given register

setReg :: Register -> Value -> Program Source

Set the value of a given register

getFlag :: Flag -> Extract Bit Source

Get the value of a flag

setFlag :: Flag -> Bit -> Program Source

Set the value of a flag

peek :: Address -> Extract Value Source

Read memory

poke :: Address -> Value -> Program Source

Write to memory

checkOverflow :: SWord8 -> SWord8 -> SBool -> SBool Source

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 ThmResult Source

Correctness theorem for our checkOverflow implementation.

We have:

>>> checkOverflowCorrect

Instruction set

type Instruction = Program -> Program Source

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

ldx :: Value -> Instruction Source

LDX: Set register X to value v

lda :: Value -> Instruction Source

LDA: Set register A to value v

clc :: Instruction Source

CLC: Clear the carry flag

rorM :: Address -> Instruction Source

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 -> Instruction Source

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

bcc :: Program -> Instruction Source

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

adc :: Address -> Instruction Source

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 :: Instruction Source

DEX: Decrement the value of register X

bne :: Program -> Instruction Source

BNE: Branch if the zero-flag is false

end :: Program Source

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

Legato's algorithm in Haskell/SBV

legato :: Address -> Address -> Address -> Program Source

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 -> Mostek Source

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 -> SBool Source

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 = SFunArray Source

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 ThmResult Source

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, using yices as the SMT solver

C Code generation

legatoInC :: IO () Source

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