Copyright | (c) Levent Erkok |
---|---|

License | BSD3 |

Maintainer | erkokl@gmail.com |

Stability | experimental |

Safe Haskell | None |

Language | Haskell2010 |

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: http://www.cs.utexas.edu/~moore/acl2/workshop-2004/contrib/legato/Weakest-Preconditions-Report.pdf

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.

- type Address = SWord32
- data Register
- data Flag
- type Value = SWord8
- type Bit = SBool
- type Registers = Array Register Value
- type Flags = Array Flag Bit
- type Memory = Model Word32 Word8
- data Mostek = Mostek {}
- type Extract a = Mostek -> a
- type Program = Mostek -> Mostek
- getReg :: Register -> Extract Value
- setReg :: Register -> Value -> Program
- getFlag :: Flag -> Extract Bit
- setFlag :: Flag -> Bit -> Program
- peek :: Address -> Extract Value
- poke :: Address -> Value -> Program
- checkOverflow :: SWord8 -> SWord8 -> SBool -> SBool
- checkOverflowCorrect :: IO ThmResult
- type Instruction = Program -> Program
- ldx :: Value -> Instruction
- lda :: Value -> Instruction
- clc :: Instruction
- rorM :: Address -> Instruction
- rorR :: Register -> Instruction
- bcc :: Program -> Instruction
- adc :: Address -> Instruction
- dex :: Instruction
- bne :: Program -> Instruction
- end :: Program
- legato :: Address -> Address -> Address -> Program
- runLegato :: (Address, Value) -> (Address, Value) -> Address -> Mostek -> (Value, Value)
- type InitVals = (Value, Value, Bit, Bit)
- initMachine :: Memory -> InitVals -> Mostek
- legatoIsCorrect :: Memory -> (Address, Value) -> (Address, Value) -> Address -> InitVals -> SBool
- type Model = SFunArray
- correctnessTheorem :: IO ThmResult
- legatoInC :: IO ()

# Mostek architecture

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

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.)

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!)

`Mostek`

is equipped with an automatically derived `Mergeable`

instance
because each field is `Mergeable`

.

type Program = Mostek -> Mostek Source #

Programs are essentially state transformers (on the machine state)

# Low-level operations

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`

.

# 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

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, 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.

# Verification

correctnessTheorem :: IO ThmResult Source #

The correctness theorem.
On a 2011 MacBook, this proof takes about 1 minute 45 seconds with the `SFunArray`

memory model
using boolector as the solver.