Copyright | (c) Gershom Bazerman, Jeff Polakow 2011 |
---|---|
License | BSD 3 Clause |
Maintainer | gershomb@gmail.com |
Stability | experimental |
Safe Haskell | None |
Language | Haskell98 |
Machinery for representing and simplifying simple propositional formulas. Type families are used to maintain a simple normal form, taking advantage of the duality between "And" and "Or". Additional tools are provided to pull out common atoms in subformulas and otherwise iterate until a simplified fixpoint. Full and general simplification is NP-hard, but the tools here can take typical machine-generated formulas and perform most simplifications that could be spotted and done by hand by a reasonable programmer.
While there are many functions below, only qAtom
, andq
(s), orq
(s), and qNot
need be used directly to build expressions. simplifyQueryRep
performs a basic simplification, simplifyIons
works on expressions with negation to handle their reduction, and fixSimplifyQueryRep
takes a function built out of any combination of basic simplifiers (you can write your own ones taking into account any special properties of your atoms) and runs it repeatedly until it ceases to reduce the size of your target expression.
The general notion is either that you build up an expression with these combinators directly, simplify it further, and then transform it to a target semantics, or that an expression in some AST may be converted into a normal form expression using such combinators, and then simplified and transformed back to the original AST.
Here are some simple interactions:
Prelude Data.BoolSimplifier> (qAtom "A") `orq` (qAtom "B") QOp | fromList [QAtom Pos "A",QAtom Pos "B"] fromList []
Prelude Data.BoolSimplifier> ppQueryRep $ (qAtom "A") `orq` (qAtom "B") "(A | B)"
Prelude Data.BoolSimplifier> ppQueryRep $ ((qAtom "A") `orq` (qAtom "B") `andq` (qAtom "A")) "(A)"
Prelude Data.BoolSimplifier> ppQueryRep $ ((qAtom "A") `orq` (qAtom "B") `andq` (qAtom "A" `orq` qAtom "C")) "((A | B) & (A | C))"
Prelude Data.BoolSimplifier> ppQueryRep $ simplifyQueryRep $ ((qAtom "A") `orq` (qAtom "B") `andq` (qAtom "A" `orq` qAtom "C")) "((A | (B & C)))"
- data QOrTyp
- data QAndTyp
- data QAtomTyp
- type family QFlipTyp t :: *
- data QueryRep qtyp a where
- extractAs :: QueryRep qtyp a -> Set (QueryRep QAtomTyp a)
- extractCs :: QueryRep qtyp a -> Set (QueryRep (QFlipTyp qtyp) a)
- qOr :: Ord a => Set (QueryRep QAtomTyp a) -> Set (QueryRep QAndTyp a) -> QueryRep QOrTyp a
- qAnd :: Ord a => Set (QueryRep QAtomTyp a) -> Set (QueryRep QOrTyp a) -> QueryRep QAndTyp a
- class PPQueryRep a where
- ppQueryRep :: a -> String
- qop :: (Ord a, Show qtyp, Show (QFlipTyp qtyp), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => Set (QueryRep QAtomTyp a) -> Set (QueryRep (QFlipTyp qtyp) a) -> QueryRep qtyp a
- extractAtomCs :: (Ord a, Show qtyp, Show (QFlipTyp qtyp), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => Set (QueryRep qtyp a) -> (Set (QueryRep qtyp a), Set (QueryRep QAtomTyp a))
- class HasClause fife qtyp where
- andqs :: Ord a => CombineQ a qtyp QAndTyp => [QueryRep qtyp a] -> QueryRep QAndTyp a
- orqs :: Ord a => CombineQ a qtyp QOrTyp => [QueryRep qtyp a] -> QueryRep QOrTyp a
- class CombineQ a qtyp1 qtyp2 where
- simplifyQueryRep :: (Ord a, Show (QFlipTyp qtyp), Show (QFlipTyp (QFlipTyp qtyp)), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => QueryRep qtyp a -> QueryRep qtyp a
- getCommonClauseAs :: Ord a => Set (QueryRep fife a) -> Maybe (QueryRep QAtomTyp a, Set (QueryRep fife a), Set (QueryRep fife a))
- getCommonClauseCs :: Ord a => Set (QueryRep fife a) -> Maybe (QueryRep (QFlipTyp fife) a, Set (QueryRep fife a), Set (QueryRep fife a))
- fixSimplifyQueryRep :: (QueryRep qtyp a -> QueryRep qtyp a) -> QueryRep qtyp a -> QueryRep qtyp a
- data Ion a
- qAtom :: Ord a => a -> QueryRep QAtomTyp (Ion a)
- isEmptyQR :: QueryRep qtyp a -> Bool
- isConstQR :: QueryRep qtyp a -> Bool
- class PPConstQR qtyp where
- class QNot qtyp where
- simplifyIons :: (Ord a, Show (QFlipTyp qtyp), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => QueryRep qtyp (Ion a) -> QueryRep qtyp (Ion a)
- maximumByNote :: String -> (a -> a -> Ordering) -> [a] -> a
Documentation
We'll start with three types of formulas: disjunctions, conjunctions, and atoms
Show QOrTyp | |
QNot QOrTyp | |
PPConstQR QOrTyp | |
Ord a => CombineQ a QAtomTyp QOrTyp | |
Ord a => CombineQ a QOrTyp QAtomTyp | |
Ord a => CombineQ a QOrTyp QOrTyp | |
Ord a => CombineQ a QOrTyp QAndTyp | |
Ord a => CombineQ a QAndTyp QOrTyp | |
PPQueryRep (QueryRep QOrTyp (Ion String)) | |
type QNeg QOrTyp = QAndTyp | |
type QFlipTyp QOrTyp = QAndTyp |
Show QAndTyp | |
QNot QAndTyp | |
PPConstQR QAndTyp | |
Ord a => CombineQ a QAtomTyp QAndTyp | |
Ord a => CombineQ a QOrTyp QAndTyp | |
Ord a => CombineQ a QAndTyp QAtomTyp | |
Ord a => CombineQ a QAndTyp QOrTyp | |
Ord a => CombineQ a QAndTyp QAndTyp | |
PPQueryRep (QueryRep QAndTyp (Ion String)) | |
type QNeg QAndTyp = QOrTyp | |
type QFlipTyp QAndTyp = QOrTyp |
data QueryRep qtyp a where Source
A formula is either an atom (of some type, e.g. String
).
A non-atomic formula (which is either a disjunction or a conjunction) is
n-ary and consists of a Set
of atoms and a set of non-atomic subformulas of
dual connective, i.e. the non-atomic subformulas of a disjunction must all
be conjunctions. The type system enforces this since there is no QFlipTyp
instance for QAtomTyp
.
qOr :: Ord a => Set (QueryRep QAtomTyp a) -> Set (QueryRep QAndTyp a) -> QueryRep QOrTyp a Source
convenience constructors, not particularly smart
class PPQueryRep a where Source
pretty printer class
ppQueryRep :: a -> String Source
PPQueryRep (QueryRep qtyp String) | |
PPQueryRep (QueryRep QAtomTyp (Ion String)) | |
PPQueryRep (QueryRep QAndTyp (Ion String)) | |
PPQueryRep (QueryRep QOrTyp (Ion String)) |
qop :: (Ord a, Show qtyp, Show (QFlipTyp qtyp), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => Set (QueryRep QAtomTyp a) -> Set (QueryRep (QFlipTyp qtyp) a) -> QueryRep qtyp a Source
smart constructor for QOp
does following optimization: a /\ (a \/ b) <-> a, or dually: a \/ (a /\ b) <-> a
extractAtomCs :: (Ord a, Show qtyp, Show (QFlipTyp qtyp), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => Set (QueryRep qtyp a) -> (Set (QueryRep qtyp a), Set (QueryRep QAtomTyp a)) Source
class HasClause fife qtyp where Source
QueryReps can be queried for clauses within them, and clauses within them can be extracted.
andqs :: Ord a => CombineQ a qtyp QAndTyp => [QueryRep qtyp a] -> QueryRep QAndTyp a Source
convenience functions
class CombineQ a qtyp1 qtyp2 where Source
smart constructors for QueryRep
andq :: QueryRep qtyp1 a -> QueryRep qtyp2 a -> QueryRep QAndTyp a Source
orq :: QueryRep qtyp1 a -> QueryRep qtyp2 a -> QueryRep QOrTyp a Source
Ord a => CombineQ a QAtomTyp QAtomTyp | |
Ord a => CombineQ a QAtomTyp QOrTyp | |
Ord a => CombineQ a QAtomTyp QAndTyp | |
Ord a => CombineQ a QOrTyp QAtomTyp | |
Ord a => CombineQ a QOrTyp QOrTyp | |
Ord a => CombineQ a QOrTyp QAndTyp | |
Ord a => CombineQ a QAndTyp QAtomTyp | |
Ord a => CombineQ a QAndTyp QOrTyp | |
Ord a => CombineQ a QAndTyp QAndTyp |
simplifyQueryRep :: (Ord a, Show (QFlipTyp qtyp), Show (QFlipTyp (QFlipTyp qtyp)), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => QueryRep qtyp a -> QueryRep qtyp a Source
(a /\ b) \/ (a /\ c) \/ d <-> (a /\ (b \/ c)) \/ d (and also the dual)
getCommonClauseAs :: Ord a => Set (QueryRep fife a) -> Maybe (QueryRep QAtomTyp a, Set (QueryRep fife a), Set (QueryRep fife a)) Source
Given a set of QueryReps, extracts a common clause if possible, returning the clause, the terms from which the clause has been extracted, and the rest.
getCommonClauseCs :: Ord a => Set (QueryRep fife a) -> Maybe (QueryRep (QFlipTyp fife) a, Set (QueryRep fife a), Set (QueryRep fife a)) Source
fixSimplifyQueryRep :: (QueryRep qtyp a -> QueryRep qtyp a) -> QueryRep qtyp a -> QueryRep qtyp a Source
Takes any given simplifier and repeatedly applies it until it ceases to reduce the size of the query reprepresentation.
We can wrap any underying atom dype in an Ion to give it a "polarity" and add handling of "not" to our simplification tools.
simplifyIons :: (Ord a, Show (QFlipTyp qtyp), QFlipTyp (QFlipTyp qtyp) ~ qtyp) => QueryRep qtyp (Ion a) -> QueryRep qtyp (Ion a) Source
a /\ (b \/ ~b) /\ (c \/ d) <-> a /\ (c \/ d) a /\ ~a /\ (b \/ c) <-> False (a \/ ~a) /\ (b \/ ~b) <-> True (*)
and duals
N.B. 0-ary \/ is False and 0-ary /\ is True
maximumByNote :: String -> (a -> a -> Ordering) -> [a] -> a Source