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This is going to be a brief post of a small program that I wrote in Scheme (a dialect of Lisp). It converts a propositional formula into conjunctive normal form I hope that this may be useful to someone, it was for me just to get it in my head how the algorithm works, and to check my Logic assignments against ;). The code should be fairly portable across most Scheme implementations, if not drop me a line. So without further ado, here is the code:- (define (cnf x)
(cnfp (nnf (impl-free x))))
(define (cnfp x)
(cond ((symbol? x) x)
((equal? (fst x) 'and) `(and ,(cnfp (scnd x)) ,(cnfp (thrd x))))
((equal? (fst x) 'or) (distr (cnfp (scnd x)) (cnfp (thrd x))))))
(define (distr x y)
(cond ((equal? (fst x) 'and) `(and ,(distr (scnd x) y) ,(distr (thrd x) y)))
((equal? (fst y) 'and) `(and ,(distr x (scnd y)) ,(distr x (thrd y))))
(#t `(or ,x ,y))))
(define (impl-free x)
(cond ((symbol? x) x)
((equal? (fst x) 'impl) `(or (not ,(impl-free (scnd x))),(impl-free (thrd x))))
((equal? (fst x) 'not) `(not ,(impl-free (scnd x))))
(#t `(,(fst x) ,(impl-free (scnd x)) ,(impl-free (thrd x))))))
(define (nnf x)
(cond ((symbol? x) x)
((and (equal? (fst x) 'not) (equal? (fst (scnd x)) 'not)) (nnf (scnd (scnd x))))
((equal? (fst x) 'and) `(and ,(nnf (scnd x)) ,(nnf (thrd x))))
((equal? (fst x) 'or) `(or ,(nnf (scnd x)) ,(nnf (thrd x))))
((and (equal? (fst x) 'not) (equal? (fst (scnd x)) 'and)) (nnf `(or (not ,(scnd (scnd x))) (not ,(thrd (scnd x))))))
((and (equal? (fst x) 'not) (equal? (fst (scnd x)) 'or)) (nnf `(and (not ,(scnd (scnd x))) (not ,(thrd (scnd x))))))
((equal? (fst x) 'not) `(not ,(nnf (scnd x))))))
(define (fst x)
(if (symbol? x) x
(if (list? (car x))
(car (car x))
(car x))))
(define (scnd x)
(if (list? (car x))
(cadr (car x))
(cadr x)))
(define (thrd x)
(if (list? (car x))
(car (cddr (car x)))
(car (cddr x))))
Example usage > (cnf '(not (impl p (not (and q (impl (not p) q)))))) (and p (and q (or p q))) > (cnf '(or a (and b c))) (and (or a b) (or a c)) Notes on code: The program recognises implication as the symbol 'impl'. It is split into a bunch of functions, 'impl-free' takes a formula and removes all implications (a -> b), replacing them with "not a" or "b" 'nnf' removes extraneous 'not's and performs deMorgans law. If you notice a bug, or mistake, or have any suggestions, please tell me. Comments
Extra note by Riftor Sun 17:50 13th November In some implementations of scheme you may not have access to the predicate 'symbol?', but you may have 'atom?', replace symbol? with atom? should be fine.
reply by someone...? Wed 03:02 22nd March Do you know a way to do the distribution with multiple arguments for AND , OR ??
reply by someone...? Wed 03:13 22nd March multiple variable is referring to "more than 2"
reply by someone...? Thu 18:00 23rd March Do you mean something of the form (and a b c d)?
You probably want a function like:-
And use that when dealing with 'and' in the distr function. I will write this properly a bit later when I have time (I think I may have also spotted a bug with the original code).
reply by someone...? Sun 21:15 21st May Yes, there is a bug in the cnfp function.
In your Scheme implementation, what is the result of: (cnf '(and x (not y))) ? I'm using mzscheme, which returns: (and x # The cond expression of cnfp returns no value when '(not y) is encountered. So, here's the modified cnfp, with a clause to handle '(not expr). (define (cnfp x) (cond ((symbol? x) x) ((equal? (fst x) 'and) `(and ,(cnfp (scnd x)) ,(cnfp (thrd x)))) ((equal? (fst x) 'not) x) ((equal? (fst x) 'or) (distr (cnfp (scnd x)) (cnfp (thrd x)))))) -- Christopher Brannon
OCaml version by someone...? Wed 13:18 23rd August Here's an OCaml version.
I've omitted the impl-free step because implications have already been removed from the statements I'm working with. I've also fixed the bug in th e original cnfp. type unknown = [ `X of int ] type expr = [ | `True | `False | unknown | `Not of expr | `Or of expr * expr | `And of expr * expr ] let cnf_expr expr = let rec cnfp = function | (`True | `False | #unknown) as x -> x | (`Not _) as x -> x | `And (x, y) -> `And (cnfp x, cnfp y) | `Or (x, y) -> distr (cnfp x, cnfp y) and distr = function | (`And (x2, x3), y) -> `And (distr (x2, y), distr (x3, y)) | (x, `And (y2, y3)) -> `And (distr (x, y2), distr (x, y3)) | (x, y) -> `Or (x, y) and nnf = function | (`True | `False | #unknown) as x -> x | `Not (`Not x) -> x | `And (x2, x3) -> `And (nnf x2, nnf x3) | `Or (x2, x3) -> `Or (nnf x2, nnf x3) | `Not (`And (x2, x3)) -> nnf (`Or (`Not x2, `Not x3)) | `Not (`Or (x2, x3)) -> nnf (`And (`Not x2, `Not x3)) | `Not x -> `Not (nnf x) in cnfp (nnf expr) (My limited testing seems to show it's correct - but E&OE). Rich (http://merjis.com/)
super by Riftor Thu 16:51 24th August That's a really nice implementation, well done!
It's nice to see other people out there doing the same things as me!
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