PropLogic-0.9.0.1: A system for propositional logic with default and fast instances of propositional algebras.

FastPropLogic

Contents

Description

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.

Synopsis

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

type ILine = Costack ILitSource

type IForm = Costack ILineSource

type XLit a = (Olist a, ILit)Source

type XLine a = (Olist a, ILine)Source

type XForm a = (Olist a, IForm)Source

data XPDNF a Source

Constructors

XPDNF (XForm a) 

Instances

Ord a => PropAlg a (XPDNF a) 
Eq a => Eq (XPDNF a) 
Read a => Read (XPDNF a) 
Show a => Show (XPDNF a) 
Display a => Display (XPDNF a) 

data XPCNF a Source

Constructors

XPCNF (XForm a) 

Instances

Ord a => PropAlg a (XPCNF a) 
Eq a => Eq (XPCNF a) 
Read a => Read (XPCNF a) 
Show a => Show (XPCNF a) 
Display a => Display (XPCNF a) 

data MixForm a Source

Constructors

M2DNF (XForm a) 
M2CNF (XForm a) 
PDNF (XForm a) 
PCNF (XForm a) 

Instances

Ord a => PropAlg a (MixForm a) 
Eq a => Eq (MixForm a) 
Read a => Read (MixForm a) 
Show a => Show (MixForm a) 
Display a => Display (MixForm a) 

Conversions

IdxPropForm -- indexed propositional formulas

tr :: (s -> t) -> PropForm s -> PropForm tSource

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 :: Ord a => Olist a -> a -> IAtomSource

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 :: Ord a => Olist a -> IAtom -> aSource

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

itr :: Ord a => Olist a -> Olist a -> Maybe (Olist IAtom)Source

iUni :: Ord a => [Olist a] -> (Olist a, [Maybe (Olist IAtom)])Source

unifyXForms :: Ord a => [XForm a] -> (Olist a, [IForm])Source

Purely syntactical conversions to propositional formulas

xLIT :: Ord a => XLit a -> PropForm aSource

xNLC :: Ord a => XLine a -> PropForm aSource

xNLD :: Ord a => XLine a -> PropForm aSource

xCNF :: Ord a => XForm a -> PropForm aSource

xDNF :: Ord a => XForm a -> PropForm aSource

Conversions to and from propositional formulas

The IForm algebra

Basic operations

isILine :: Costack Int -> BoolSource

isIForm :: Costack (Costack Int) -> BoolSource

The propositional algebra operations

Generation of pairwise minimal, minimal and prime forms

Generation of prime and pairwise minimal forms of two lines

Implementation of the M- and the P-Procedure

allPairs :: [a] -> [(a, a)]Source

The XForm operations

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