ToySolver.Data.LA

Some definition for Theory of Linear Arithmetics.

Expression of linear arithmetics

data Expr r

Linear combination of variables and constants. Non-negative keys are used for variables's coefficients. key unitVar is used for constants.

Instances

Eq r => Eq (Expr r)

Methods

(==) :: Expr r -> Expr r -> Bool

(/=) :: Expr r -> Expr r -> Bool

Ord r => Ord (Expr r)

Methods

compare :: Expr r -> Expr r -> Ordering

(<) :: Expr r -> Expr r -> Bool

(<=) :: Expr r -> Expr r -> Bool

(>) :: Expr r -> Expr r -> Bool

(>=) :: Expr r -> Expr r -> Bool

max :: Expr r -> Expr r -> Expr r

min :: Expr r -> Expr r -> Expr r

(Num r, Eq r, Read r) => Read (Expr r)

Methods

readsPrec :: Int -> ReadS (Expr r)

readList :: ReadS [Expr r]

readPrec :: ReadPrec (Expr r)

readListPrec :: ReadPrec [Expr r]

Show r => Show (Expr r)

Methods

showsPrec :: Int -> Expr r -> ShowS

show :: Expr r -> String

showList :: [Expr r] -> ShowS

NFData r => NFData (Expr r)

Methods

rnf :: Expr r -> ()

(Num r, Eq r) => VectorSpace (Expr r)

Associated Types

type Scalar (Expr r) :: *

Methods

(*^) :: Scalar (Expr r) -> Expr r -> Expr r

(Num r, Eq r) => AdditiveGroup (Expr r)

Methods

zeroV :: Expr r

(^+^) :: Expr r -> Expr r -> Expr r

negateV :: Expr r -> Expr r

(^-^) :: Expr r -> Expr r -> Expr r

Variables (Expr r)

Methods

vars :: Expr r -> VarSet

IsOrdRel (Expr Integer) QFFormula

Methods

(.<.) :: Expr Integer -> Expr Integer -> QFFormula

(.<=.) :: Expr Integer -> Expr Integer -> QFFormula

(.>.) :: Expr Integer -> Expr Integer -> QFFormula

(.>=.) :: Expr Integer -> Expr Integer -> QFFormula

ordRel :: RelOp -> Expr Integer -> Expr Integer -> QFFormula

IsEqRel (Expr Integer) QFFormula

Methods

(.==.) :: Expr Integer -> Expr Integer -> QFFormula

(./=.) :: Expr Integer -> Expr Integer -> QFFormula

type Scalar (Expr r)

type Scalar (Expr r) = r

type Var = Int

Variables are represented as non-negative integers

Conversion

var :: Num r => Var -> Expr r

variable

constant :: (Num r, Eq r) => r -> Expr r

constant

terms :: Expr r -> [(r, Var)]

terms contained in the expression.

fromTerms :: (Num r, Eq r) => [(r, Var)] -> Expr r

Create a Expr from a list of terms.

coeffMap :: Expr r -> IntMap r

a mapping from variables to coefficients

fromCoeffMap :: (Num r, Eq r) => IntMap r -> Expr r

Create a Expr from a mapping from variables to coefficients.

unitVar :: Var

Special variable that should always be evaluated to 1.

asConst :: Num r => Expr r -> Maybe r

Query

coeff :: Num r => Var -> Expr r -> r

lookup a coefficient of the variable. coeff v e == fst (extract v e)

lookupCoeff :: Num r => Var -> Expr r -> Maybe r

lookup a coefficient of the variable. It returns Nothing if the expression does not contain v. lookupCoeff v e == fmap fst (extractMaybe v e)

extract :: Num r => Var -> Expr r -> (r, Expr r)

extract v e returns (c, e') such that e == c *^ v ^+^ e'

extractMaybe :: Num r => Var -> Expr r -> Maybe (r, Expr r)

extractMaybe v e returns Just (c, e') such that e == c *^ v ^+^ e' if e contains v, and returns Nothing otherwise.

Operations

mapCoeff :: (Num b, Eq b) => (a -> b) -> Expr a -> Expr b

map coefficients.

mapCoeffWithVar :: (Num b, Eq b) => (a -> Var -> b) -> Expr a -> Expr b

map coefficients.

evalExpr :: Num r => Model r -> Expr r -> r

evaluate the expression under the model.

evalLinear :: VectorSpace a => Model a -> a -> Expr (Scalar a) -> a

evaluate the expression under the model.

lift1 :: VectorSpace x => x -> (Var -> x) -> Expr (Scalar x) -> x

applySubst :: (Num r, Eq r) => VarMap (Expr r) -> Expr r -> Expr r

applySubst1 :: (Num r, Eq r) => Var -> Expr r -> Expr r -> Expr r

applySubst1 x e e1 == e1[e/x]

showExpr :: (Num r, Eq r, Show r) => Expr r -> String

Atomic formula of linear arithmetics

type Atom r = OrdRel (Expr r)

Atomic Formula of Linear Arithmetics

showAtom :: (Num r, Eq r, Show r) => Atom r -> String

evalAtom :: (Num r, Ord r) => Model r -> Atom r -> Bool

evaluate the formula under the model.

applySubstAtom :: (Num r, Eq r) => VarMap (Expr r) -> Atom r -> Atom r

applySubst1Atom :: (Num r, Eq r) => Var -> Expr r -> Atom r -> Atom r

applySubst1 x e phi == phi[e/x]

solveFor :: (Real r, Fractional r) => Atom r -> Var -> Maybe (RelOp, Expr r)

Solve linear (in)equation for the given variable.

solveFor a v returns Just (op, e) such that Atom v op e is equivalent to a.

module ToySolver.Data.OrdRel

misc

type BoundsEnv r = VarMap (Interval r)

computeInterval :: (Real r, Fractional r) => BoundsEnv r -> Expr r -> Interval r

compute bounds for a Expr with respect to BoundsEnv.