sbv-7.0: 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 Flag Source #

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




Bounded Flag Source # 
Enum Flag Source # 


succ :: Flag -> Flag #

pred :: Flag -> Flag #

toEnum :: Int -> Flag #

fromEnum :: Flag -> Int #

enumFrom :: Flag -> [Flag] #

enumFromThen :: Flag -> Flag -> [Flag] #

enumFromTo :: Flag -> Flag -> [Flag] #

enumFromThenTo :: Flag -> Flag -> Flag -> [Flag] #

Eq Flag Source # 


(==) :: Flag -> Flag -> Bool #

(/=) :: Flag -> Flag -> Bool #

Ord Flag Source # 


compare :: Flag -> Flag -> Ordering #

(<) :: Flag -> Flag -> Bool #

(<=) :: Flag -> Flag -> Bool #

(>) :: Flag -> Flag -> Bool #

(>=) :: Flag -> Flag -> Bool #

max :: Flag -> Flag -> Flag #

min :: Flag -> Flag -> Flag #

Ix Flag Source # 


range :: (Flag, Flag) -> [Flag] #

index :: (Flag, Flag) -> Flag -> Int #

unsafeIndex :: (Flag, Flag) -> Flag -> Int

inRange :: (Flag, Flag) -> Flag -> Bool #

rangeSize :: (Flag, Flag) -> Int #

unsafeRangeSize :: (Flag, Flag) -> Int

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!)

Mostek is equipped with an automatically derived Mergeable instance because each field is Mergeable.




Generic Mostek Source # 

Associated Types

type Rep Mostek :: * -> * #


from :: Mostek -> Rep Mostek x #

to :: Rep Mostek x -> Mostek #

Mergeable Mostek Source # 


symbolicMerge :: Bool -> SBool -> Mostek -> Mostek -> Mostek Source #

select :: (SymWord b, Num b) => [Mostek] -> Mostek -> SBV b -> Mostek Source #

type Rep Mostek Source # 
type Rep Mostek = D1 (MetaData "Mostek" "Data.SBV.Examples.BitPrecise.Legato" "sbv-7.0-Dr9nfrJTFD01AI4IYir1JV" False) (C1 (MetaCons "Mostek" PrefixI True) ((:*:) (S1 (MetaSel (Just Symbol "memory") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Memory)) ((:*:) (S1 (MetaSel (Just Symbol "registers") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Registers)) (S1 (MetaSel (Just Symbol "flags") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Flags)))))

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, 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 2011 MacBook, this proof takes about 1 minute 45 seconds with the SFunArray memory model using boolector as the solver.

C Code generation

legatoInC :: IO () Source #

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