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I have the boolean expression (a && !b). I want to add variables c, d into the expression and then have Mathematica transform the expression into conjunctive normal form. I would like it to go something like this:

a&&!b
  = (a && !b && c) || (a && !b && !c)
  = (a && !b && c && d) || (a && !b && c && !d) || (a && !b && !c && d) || 
    (a && !b && !c && !d)

Although a human can do this easily, how can can I get Mathematica to do it? What are the commands?

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    $\begingroup$ Take a look at BooleanConvert and forms it has available. But I don't know how to use it for your purpose :) $\endgroup$
    – Kuba ♦
    Aug 29, 2013 at 8:40

2 Answers 2

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I don't now that I really understand your question, but from a pure expression manipulation perspective this might be useful:

Or @@ And @@@ Tuples[{{a && ! b}, {c, ! c}, {d, ! d}}]

(a && ! b && c && d) || (a && ! b && c && ! d) || (a && ! b && ! c && d) || (a && ! b && ! c && ! d)

The same operation a bit more concisely:

Or @@ Tuples[{a && ! b} && {c, ! c} && {d, ! d}]
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  • $\begingroup$ I still don't believe that this is not automatic :P Are you really going to put && {sym, ! sym} each time? But +1 for Tuples I don't know why I decided to use Outer.:) $\endgroup$
    – Kuba ♦
    Aug 29, 2013 at 10:57
  • $\begingroup$ @Kuba Frankly I don't understand the question so I wasn't sure what operation was to be done. I voted for your answer by the way. I think my code is open ended in that you could have any number of variations in each sublist. $\endgroup$
    – Mr.Wizard
    Aug 29, 2013 at 10:59
  • $\begingroup$ I have a CNF and BooleanMinimize can simplify it to a&&!b. Now I want to perform the revert process. $\endgroup$
    – Gqqnbig
    Aug 29, 2013 at 11:19
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Since I don't know how to incorporate BooleanConvert here is a walkaround for this case:

conv = Or @@ Flatten[Outer[
                           And,
                           {#1},
                           Sequence @@ Transpose[{Not /@ #2, #2}]]
                    , 2] &

.

conv @@ {a && ! b, {c}}

(a && ! b && ! c) || (a && ! b && c)

conv @@ {a && ! b, {c, d}}

(a && ! b && ! c && ! d) || (a && ! b && ! c && d) || (a && ! b && c && ! d) || 
(a && ! b && c && d)
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