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Or, more generally, you can chain an arbitrary series of adders to form an N-bit adder. First you define an abbreviation for the AndL function to represent the composition of multiple circuit blocks: let Blocks l = AndL l And here is the definition of an N-bit adder with a halfAdder at one end: let nBitCarryRippleAdder (n:int) (x:bitvec) (y:bitvec) (sum:bitvec) (carry:bitvec) = Blocks [ for i in 0 .. n-1 -> if i = 0 then halfAdder x.[i] y.[i] sum.[i] carry.[i] else fullAdder x.[i] y.[i] carry.[i-1] sum.[i] carry.[i] ] Using a similar approach, you get the following satisfying specification of a symmetric N-bit adder that accepts a carry as input and also gives a carry as output: let rippleAdder (n:int) (x:bitvec) (y:bitvec) (sum:bitvec) (carry:bitvec) Blocks [ for i in 0 .. n-1 -> fullAdder x.[i] y.[i] carry.[i] sum.[i] carry.[i+1] ] =
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Let s now take a look at the propositional formula for a halfAdder with variable inputs and outputs: > halfAdder (var "x") (var "y") (var "sum") (var "carry");; > val it : Prop = And(Not And (Not And (Var "sum", Not Not And (Not And (Var "x",Var "y"),Not And (Not Var "x",Not Var "y"))), Not And (Not Var "sum", Not Not Not And (Not And (Var "x",Var "y"),Not And (Not Var "x",Not Var "y")))), Not And (Not And (Var "carry",And (Var "x",Var "y")), Not And (Not Var "carry",Not And (Var "x",Var "y")))) Clearly, you don t want to be doing too much of that! You will see better ways of inspecting circuits and the symbolic values of bit vectors in the section Representing Propositional Formulae Efficiently Using BDDs.
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In passing, we note that the twoBitAdder uses an internal wire. You could model this using an existential formula: let twoBitAdderWithHiding (x1,x2) (y1,y2) (sum1,sum2) carry = let carryInnerVar = fresh "carry" let carryInner = var(carryInnerVar) Exists(carryInnerVar, halfAdder x1 y1 sum1 carryInner &&& fullAdder x2 y2 carryInner sum2 carry) However, this brings up issues beyond the scope of this chapter, and instead we take an approach to modeling where there are no boundaries to the circuits and where all internal wires are exposed.
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Checking Simple Properties of Circuits
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Now that you have modeled the initial hardware circuits, you can check simple properties of these circuits. For example, you can check that if you give a fullAdder all low (that is, false) inputs, then the output wires may be low as well and, conversely, that you have a contradiction if one of the output wires is high: > tautology (fullAdder Lo Lo Lo Lo Lo);; val it : bool = true > satisfiable (fullAdder Lo Lo Lo Hi Lo);; val it : bool = false It is of course much better to check these results symbolically by giving symbolic inputs. For example, you can check that if the same value is given to the two inputs of a halfAdder, then the sum output is low and the carry output is the same as the input: > tautology (halfAdder (var "x") (var "x") Lo (var "x"));; val it : bool = true Likewise, you can check that a 2-bit adder is commutative, in other words, that it doesn t matter if you swap the x and y inputs. > tautology (nBitCarryRippleAdder 2 (vec 2 "x") (vec 2 "y") (vec 2 "sum") (vec 3 "carry") === nBitCarryRippleAdder 2 (vec 2 "y") (vec 2 "x") (vec 2 "sum") (vec 3 "carry"));; val it : bool = true However, if you repeat the same for sizes of 5 or bigger, things start to slow down a little, and the naive implementation of propositional logic tautology checking based on truth tables begins to break down. Hence, you have to turn to more efficient techniques for processing propositional formulae.
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