While working on reverse-engineering the Microchip ATF15xx CPLD family, I found myself deriving minimal logic functions from a truth table. This useful because while it is easy to sample all possible states of a black box combinatorial function using e.g. boundary scan, these truth tables are unwieldy and don’t provide much insight into the hardware. While a minimal function with the same truth table would not necessarily be the function implemented by the hardware (which may have hidden variables, or simply use a larger equivalent function that is more convenient to implement), deriving one still provides great insight. In this note I explore this process.
My chosen approach (thanks to John Regehr for the suggestion) I got for an earlier project is to implement an interpreter for a simple logic expression abstract syntax tree in Racket and then use Rosette to translate assertions about the results of interpreting an arbitrary logic expression, as well as a cost function, into a query for an SMT solver.
Although I could use an off-the-shelf logic minimizer here (like Espresso), most logic minimizers solve a different problem: quickly translating large designs to simple netlists. However, I would like to have a complex output netlist: the ATF15xx CPLDs have a hardware XOR gate that I would like the minimizer to infer on its own. On the other hand, I don’t really care about the runtime of the minimizer as long as it’s on the order of minutes to hours. Rosette’s flexibility is a perfect match for this task.
The following code demonstrates the approach and its ability to derive a XOR gate from3 the input expression. It can be easily modified for a particular application by extending (or reducing, e.g. for translation to an and-inverter graph) the logic language, or altering the cost function.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 #lang rosette/safe (require rosette/lib/angelic rosette/lib/match) (define (^^ x y) (|| (&& x (! y)) (&& (! x) y))) (struct lnot (a) #:transparent) (struct land (a b) #:transparent) (struct lor (a b) #:transparent) (struct lxor (a b) #:transparent) (struct lvar (v) #:transparent) (struct llit (v) #:transparent) (define (ldump e) (match e [(lnot a) `(! ,(ldump a))] [(land a b) `(&& ,(ldump a) ,(ldump b))] [(lor a b) `(\|\| ,(ldump a) ,(ldump b))] [(lxor a b) `(^^ ,(ldump a) ,(ldump b))] [(lvar v) v] [(llit v) v])) (define (leval e) (match e [(lnot a) (! (leval a))] [(land a b) (&& (leval a) (leval b))] [(lor a b) (|| (leval a) (leval b))] [(lxor a b) (^^ (leval a) (leval b))] [(lvar v) v] [(llit v) v])) (define (lcost e) (match e [(lnot a) (+ 1 (lcost a))] [(land a b) (+ 2 (lcost a) (lcost b))] [(lor a b) (+ 2 (lcost a) (lcost b))] [(lxor a b) (+ 2 (lcost a) (lcost b))] [(lvar v) 0] [(llit v) 1])) (define (??lexpr terminals #:depth depth) (apply choose* (if (<= depth 0) terminals (let [(a (??lexpr terminals #:depth (- depth 1))) (b (??lexpr terminals #:depth (- depth 1)))] (append terminals (list (lnot a) (land a b) (lor a b) (lxor a b))))))) (define (lmincost #:forall inputs #:tactic template #:equiv behavior) (define model (optimize #:minimize (list (lcost template)) #:guarantee (assert (forall inputs (equal? (leval template) behavior))))) (if (unsat? model) model (evaluate template model))) (define-symbolic a b c boolean?) (define f (lmincost #:forall (list a b c) #:tactic (??lexpr (list (lvar a) (lvar b) (lvar c) (llit #f)) #:depth 3) #:equiv (&& (|| a (! (&& b c))) (! (&& a (|| (! b) (! c))))))) (displayln (ldump f)) ; (! (^^ (&& c b) a))