Safe Haskell | Safe-Infered |
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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.

- type IAtom = Int
- type ILit = Int
- type ILine = Costack ILit
- type IForm = Costack ILine
- type XLit a = (Olist a, ILit)
- type XLine a = (Olist a, ILine)
- type XForm a = (Olist a, IForm)
- data XPDNF a = XPDNF (XForm a)
- data XPCNF a = XPCNF (XForm a)
- data MixForm a
- type IdxPropForm a = (Olist a, PropForm IAtom)
- tr :: (s -> t) -> PropForm s -> PropForm t
- iTr :: Olist IAtom -> IForm -> IForm
- idx :: Ord a => Olist a -> a -> IAtom
- nth :: Ord a => Olist a -> IAtom -> a
- itr :: Ord a => Olist a -> Olist a -> Maybe (Olist IAtom)
- iUni :: Ord a => [Olist a] -> (Olist a, [Maybe (Olist IAtom)])
- unifyIdxPropForms :: Ord a => [IdxPropForm a] -> (Olist a, [PropForm IAtom])
- unifyXForms :: Ord a => [XForm a] -> (Olist a, [IForm])
- fromIdxPropForm :: Ord a => IdxPropForm a -> PropForm a
- toIdxPropForm :: Ord a => PropForm a -> IdxPropForm a
- newAtomsXForm :: Ord a => XForm a -> Olist a -> XForm a
- iLIT :: ILit -> PropForm IAtom
- iNLC :: ILine -> PropForm IAtom
- iNLD :: ILine -> PropForm IAtom
- iCNF :: IForm -> PropForm IAtom
- iDNF :: IForm -> PropForm IAtom
- xLIT :: Ord a => XLit a -> PropForm a
- xNLC :: Ord a => XLine a -> PropForm a
- xNLD :: Ord a => XLine a -> PropForm a
- xCNF :: Ord a => XForm a -> PropForm a
- xDNF :: Ord a => XForm a -> PropForm a
- toXPDNF :: Ord a => PropForm a -> XPDNF a
- toXPCNF :: Ord a => PropForm a -> XPCNF a
- toM2DNF :: Ord a => PropForm a -> MixForm a
- toM2CNF :: Ord a => PropForm a -> MixForm a
- fromXPDNF :: Ord a => XPDNF a -> PropForm a
- fromXPCNF :: Ord a => XPCNF a -> PropForm a
- fromMixForm :: Ord a => MixForm a -> PropForm a
- isIAtom :: Int -> Bool
- isILit :: Int -> Bool
- isILine :: Costack Int -> Bool
- isIForm :: Costack (Costack Int) -> Bool
- iLine :: [Int] -> ILine
- iForm :: [[Int]] -> IForm
- iAtom :: ILit -> IAtom
- iBool :: ILit -> Bool
- negLit :: ILit -> ILit
- lineIndices :: ILine -> Olist IAtom
- formIndices :: IForm -> Olist IAtom
- lineLength :: ILine -> Int
- formLength :: IForm -> Int
- volume :: IForm -> Int
- isOrderedForm :: IForm -> Bool
- orderForm :: IForm -> IForm
- atomForm :: IAtom -> IForm
- botForm :: IForm
- topForm :: IForm
- formJoinForm :: IForm -> IForm -> IForm
- formListJoin :: [IForm] -> IForm
- lineMeetLine :: ILine -> ILine -> IForm
- lineMeetForm :: ILine -> IForm -> IForm
- formMeetForm :: IForm -> IForm -> IForm
- formListMeet :: [IForm] -> IForm
- dualLine :: ILine -> IForm
- dualForm :: IForm -> IForm
- invertLine :: ILine -> ILine
- invertForm :: IForm -> IForm
- negLine :: ILine -> IForm
- negForm :: IForm -> IForm
- formCojoinLine :: IForm -> ILine -> IForm
- formCojoinForm :: IForm -> IForm -> IForm
- formAntijoinLine :: IForm -> ILine -> IForm
- formAntijoinForm :: IForm -> IForm -> IForm
- elimLine :: ILine -> Olist IAtom -> ILine
- elimForm :: IForm -> Olist IAtom -> IForm
- lineCovLine :: ILine -> ILine -> Bool
- lineCovForm :: ILine -> IForm -> Bool
- formCovForm :: IForm -> IForm -> Bool
- pairPartition :: ILine -> ILine -> (ILine, ILine, ILine, ILine)
- data CaseSymbol
- caseSymbol :: ILine -> ILine -> CaseSymbol
- pairPrim' :: ILine -> ILine -> IForm
- pairMin' :: ILine -> ILine -> IForm
- xprim' :: ILine -> ILine -> (CaseSymbol, IForm)
- xmin' :: ILine -> ILine -> (CaseSymbol, IForm)
- xprim :: ILine -> ILine -> (CaseSymbol, IForm)
- xmin :: ILine -> ILine -> (CaseSymbol, IForm)
- pairPrim :: ILine -> ILine -> IForm
- pairMin :: ILine -> ILine -> IForm
- isMinimalPair :: ILine -> ILine -> Bool
- allPairs :: [a] -> [(a, a)]
- isPairwiseMinimal :: IForm -> Bool
- cPrime :: ILine -> ILine -> Maybe ILine
- cPrimes :: IForm -> IForm
- mrec :: (IForm, IForm, IForm) -> IForm
- m2form :: IForm -> IForm
- iformJoinM2form :: IForm -> IForm -> IForm
- primForm :: IForm -> IForm
- iformJoinPrimForm :: IForm -> IForm -> IForm
- xformAtoms :: Ord a => XForm a -> Olist a
- xformRedAtoms :: Ord a => XForm a -> Olist a
- xformIrrAtoms :: Ord a => XForm a -> Olist a
- mixToPDNF :: Ord a => MixForm a -> MixForm a
- mixToPCNF :: Ord a => MixForm a -> MixForm a

# Introductory example

## Generating a Prime Disjunctive Normal Form, the default and the fast way

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`

.)

## XPDNF as a propositional algebra

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

## The canonization steps

# Syntax

# Conversions

## IdxPropForm -- indexed propositional formulas

type IdxPropForm a = (Olist a, PropForm IAtom)Source

`tr f form`

replaces each atom form occurrence `(A x)`

in the formula `form`

by the new atom `(A (f x))`

. Everything else remains.

`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

unifyIdxPropForms :: Ord a => [IdxPropForm a] -> (Olist a, [PropForm IAtom])Source

fromIdxPropForm :: Ord a => IdxPropForm a -> PropForm aSource

toIdxPropForm :: Ord a => PropForm a -> IdxPropForm aSource

## Purely syntactical conversions to propositional formulas

## Conversions to and from propositional formulas

fromMixForm :: Ord a => MixForm a -> PropForm aSource

# The IForm algebra

## Basic operations

lineIndices :: ILine -> Olist IAtomSource

formIndices :: IForm -> Olist IAtomSource

lineLength :: ILine -> IntSource

formLength :: IForm -> IntSource

isOrderedForm :: IForm -> BoolSource

## The propositional algebra operations

formJoinForm :: IForm -> IForm -> IFormSource

formListJoin :: [IForm] -> IFormSource

lineMeetLine :: ILine -> ILine -> IFormSource

lineMeetForm :: ILine -> IForm -> IFormSource

formMeetForm :: IForm -> IForm -> IFormSource

formListMeet :: [IForm] -> IFormSource

invertLine :: ILine -> ILineSource

invertForm :: IForm -> IFormSource

formCojoinLine :: IForm -> ILine -> IFormSource

formCojoinForm :: IForm -> IForm -> IFormSource

formAntijoinLine :: IForm -> ILine -> IFormSource

formAntijoinForm :: IForm -> IForm -> IFormSource

lineCovLine :: ILine -> ILine -> BoolSource

lineCovForm :: ILine -> IForm -> BoolSource

formCovForm :: IForm -> IForm -> BoolSource

## Generation of pairwise minimal, minimal and prime forms

### Generation of prime and pairwise minimal forms of two lines

data CaseSymbol Source

caseSymbol :: ILine -> ILine -> CaseSymbolSource

### Implementation of the M- and the P-Procedure

isMinimalPair :: ILine -> ILine -> BoolSource

isPairwiseMinimal :: IForm -> BoolSource

iformJoinM2form :: IForm -> IForm -> IFormSource

iformJoinPrimForm :: IForm -> IForm -> IFormSource

# The XForm operations

xformAtoms :: Ord a => XForm a -> Olist aSource

xformRedAtoms :: Ord a => XForm a -> Olist aSource

xformIrrAtoms :: Ord a => XForm a -> Olist aSource

# The propositional algebras

## XPDNF and XPCNF

## MixForm

# Complexities and choice of a algebra

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 ......... .........