This module defines three altenative representations for certain propositional normal forms, namely

 data XPDNF a          -- a representation for Prime Disjunctive Normal Forms or PDNF's on a given atom type a
 data XPCNF a          -- a representation for Prime Disjunctive Normal Forms or PDNF's on a given atom type a
 data MixForm a        -- a type made of pairwise minimal DNF's and CNF's on a given atom type a

For each of these types there is a converter from and a converter to propositional formulas

    fromXPDNF :: Ord a => XPDNF a -> PropForm a             toXPDNF :: Ord a => PropForm a -> XPDNF a
    fromXPCNF :: Ord a => XPCNF a -> PropForm a             toXPCNF :: Ord a => PropForm a -> XPCNF a
  fromMixForm :: Ord a => MixForm a -> PropForm a         toMixForm :: Ord a => PropForm a -> MixForm a

Each of these three types is turned into a propositional algebra PropAlg, i.e. for every ordered type a of atoms we have three instances

 PropAlg a (XPDNF a)
 PropAlg a (XPCNF a)
 PropAlg a (MixForm a)

Different to the two default propositional algebras on propositional formulas and truth tables, these three algebras comprise fast function implementations and thus provide practical versions for propositional algebras, where propositions of arbitrary size are processed in reasonable time. In more detail the involved complexities are given in the table below (see ......). It also explains, which of the three algebras should be chosen in an actual application.

Actually, this module is essentially a re-implementation of already explained concepts from PropLogicCore and DefaultPropLogic and for the user it shouldn't be necessary to further explain how the algorithms work. The remainder of this document is an attempt to do just that. However, if you at least want an idea of what is going on here, it may suffice to read the first section with the introductory example below.

Recall, that we already defined Disjunctive Normal Forms and Prime Disjunctive Normal Forms in DefaultPropLogic as special versions of propositional formulas, along with a canonizer pdnf to obtain these normal forms

 type DNF a = PropForm a
 type PDNF a = DNF a
 pdnf :: PropForm a -> PDNF a

For a simple example formula p, given by

 > p = DJ [EJ [A "x", A "y"], N (A "z")]  ::  PropForm String

more conveniently displayed by

 > display p
 [[x <-> y] + -z]

the PDNF of p is then generated by

 > pdnf p
 DJ [CJ [EJ [A "x",F],EJ [A "y",F]],CJ [EJ [A "x",T],EJ [A "y",T]],CJ [EJ [A "z",F]]]
 > display (pdnf p)
 [[[x <-> false] * [y <-> false]] + [[x <-> true] * [y <-> true]] + [* [z <-> false]]]

or more conveniently displayed in its evaluated form

 > display (eval (pdnf p))
 [[-x * -y] + [x * y] + -z]

.............!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!..............................

(Actually, each converter pair is also part of each of the given algebras. For example, in the XPDNF instance holds: fromXPDNF = toPropForm and toXPDNF = fromPropForm.)

iTr [i_1,...,i_n] iform replaces each index j in iform by i_j. For example,

 > let iform =  iForm [[-1,3,-4,5],[-2,-3,4,6]] :: IForm
 > iform
 COSTACK [COSTACK [-1,3,-4,5],COSTACK [-2,-3,4,6]]
 > iTr [7,8,9,10,11,12,13] iform
 COSTACK [COSTACK [-7,9,-10,11],COSTACK [-8,-9,10,12]]
 > iTr [2,4] iform
 -- error, because the index list [2,4] must at least be of length 6 to cover the indices 1,..,6 of iform.

idx [i_1,...,i_n] i_k returns k, i.e. the index of the index in the given index list. Note, that the first member of the list starts with index 1, not 0. For example,

 > idx [2,4,6,8] 6
 3

nth [i_1,...,i_n] k returns i_k, i.e. the k's element in the list [i_1,...,i_n], counting from 1. For example,

 > nth [2,4,6,8] 3
 6

                                         DefaultPropLogic.                               FastPropLogic
                                         --------------------------------------         -----------------------------------
                                         PropForm            TruthTable                 XPDNF       XPCNF          MixForm
 --------------------------------------------------------------------------------------------------------------------------
 at
 false
 true
 neg
 conj, disj, subj, equij
 valid
 satisfiable
 subvalent
 equivalent
 covalent, disvalent,
 properSubvalent, properDisvalent
 atoms
 redAtoms, irrAtoms
 nullatomic
 subatomic, equiatomic
 .........
 .........