VirtualSubstitution.hs

{-# OPTIONS_GHC -Wall #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  ToySolver.Arith.VirtualSubstitution
-- Copyright   :  (c) Masahiro Sakai 2014
-- License     :  BSD-style
-- 
-- Maintainer  :  masahiro.sakai@gmail.com
-- Stability   :  provisional
-- Portability :  portable
--
-- Naive implementation of virtual substitution
--
-- Reference:
-- 
-- * V. Weispfenning. The complexity of linear problems in fields.
--   Journal of Symbolic Computation, 5(1-2): 3-27, Feb.-Apr. 1988.
-- 
-- * Hirokazu Anai, Shinji Hara. Parametric Robust Control by Quantifier Elimination.
--   J.JSSAC, Vol. 10, No. 1, pp. 41-51, 2003.
--
-----------------------------------------------------------------------------
module ToySolver.Arith.VirtualSubstitution
  ( QFFormula
  , evalQFFormula

  -- * Projection
  , project
  , projectN
  , projectCases
  , projectCasesN

  -- * Constraint solving
  , solve
  , solveQFFormula
  ) where

import Control.Exception
import Control.Monad
import qualified Data.Foldable as Foldable
import qualified Data.IntMap as IM
import qualified Data.IntSet as IS
import Data.Set (Set)
import qualified Data.Set as Set
import Data.Maybe
import Data.VectorSpace hiding (project)

import ToySolver.Data.ArithRel
import ToySolver.Data.Boolean
import ToySolver.Data.BoolExpr (BoolExpr (..))
import qualified ToySolver.Data.BoolExpr as BoolExpr
import qualified ToySolver.Data.LA as LA
import ToySolver.Data.Var

-- | Quantifier-free formula of LRA
type QFFormula = BoolExpr (LA.Atom Rational)

-- | @'evalQFFormula' M φ@ returns whether @M ⊧_LRA φ@ or not.
evalQFFormula :: Model Rational -> QFFormula -> Bool
evalQFFormula m = BoolExpr.fold f
  where
    f (ArithRel lhs op rhs) = evalOp op (LA.evalExpr m lhs) (LA.evalExpr m rhs)

{-| @'project' x φ@ returns @(ψ, lift)@ such that:

* @⊢_LRA ∀y1, …, yn. (∃x. φ) ↔ ψ@ where @{y1, …, yn} = FV(φ) \\ {x}@, and

* if @M ⊧_LRA ψ@ then @lift M ⊧ φ@.
-}
project :: Var -> QFFormula -> (QFFormula, Model Rational -> Model Rational)
project x formula = (formula', mt)
  where
    xs = projectCases x formula
    formula' = orB' [phi | (phi,_) <- xs]
    mt m = head $ do
      (phi, mt') <- xs
      guard $ evalQFFormula m phi
      return $ mt' m
    orB' = orB . concatMap f
      where
        f (Or xs) = concatMap f xs
        f x = [x]

{-| @'projectN' {x1,…,xm} φ@ returns @(ψ, lift)@ such that:

* @⊢_LRA ∀y1, …, yn. (∃x1, …, xm. φ) ↔ ψ@ where @{y1, …, yn} = FV(φ) \\ {x1,…,xm}@, and

* if @M ⊧_LRA ψ@ then @lift M ⊧_LRA φ@.
-}
projectN :: VarSet -> QFFormula -> (QFFormula, Model Rational -> Model Rational)
projectN vs2 = f (IS.toList vs2)
  where
    f :: [Var] -> QFFormula -> (QFFormula, Model Rational -> Model Rational)
    f [] formula     = (formula, id)
    f (v:vs) formula = (formula3, mt1 . mt2)
      where
        (formula2, mt1) = project v formula
        (formula3, mt2) = f vs formula2

{-| @'projectCases' x φ@ returns @[(ψ_1, lift_1), …, (ψ_m, lift_m)]@ such that:

* @⊢_LRA ∀y1, …, yn. (∃x. φ) ↔ (ψ_1 ∨ … ∨ φ_m)@ where @{y1, …, yn} = FV(φ) \\ {x}@, and

* if @M ⊧_LRA ψ_i@ then @lift_i M ⊧_LRA φ@.
-}
projectCases :: Var -> QFFormula -> [(QFFormula, Model Rational -> Model Rational)]
projectCases v phi = [(psi, \m -> IM.insert v (LA.evalExpr m t) m) | (psi, t) <- projectCases' v phi]

{-| @'projectCases' x φ@ returns @[(ψ_1, lift_1), …, (ψ_m, lift_m)]@ such that:

* @⊢_LRA ∀y1, …, yn. (∃x. φ) ↔ (ψ_1 ∨ … ∨ φ_m)@ where @{y1, …, yn} = FV(φ) \\ {x}@, and

* if @M ⊧_LRA ψ_i@ then @lift_i M ⊧_LRA φ@.

Note that

> (∃x. φ(x)) ⇔ ∨_{t∈S} φ(t)
>   where
>     Ψ = {a_i x - b_i ρ_i 0 | i ∈ I, ρ_i ∈ {=, ≠, ≤, <}} the set of atomic subformulas in φ(x)
>     S = { b_i / a_i, b_i / a_i + 1, b_i / a_i - 1 | i∈I } ∪ {1/2 (b_i / a_i + b_j / a_j) | i,j∈I, i≠j}
-}
projectCases' :: Var -> QFFormula -> [(QFFormula, LA.Expr Rational)]
projectCases' v phi
  | phi' == false = []
  | Set.null xs = [(phi', LA.constant 0)]
  | otherwise   = [(phi'', t) | t <- Set.toList s, let phi'' = applySubst1 v t phi', phi'' /= false]
  where
    phi' = simplify phi
    xs = collect v phi'
    s = Set.unions
        [ xs
        , Set.fromList [e ^+^ LA.constant 1 | e <- Set.toList xs]
        , Set.fromList [e ^-^ LA.constant 1 | e <- Set.toList xs]
        , Set.fromList [(e1 ^+^ e2) ^/ 2 | (e1,e2) <- pairs (Set.toList xs)]
        ]

{-| @'projectCasesN' {x1,…,xm} φ@ returns @[(ψ_1, lift_1), …, (ψ_n, lift_n)]@ such that:

* @⊢_LRA ∀y1, …, yp. (∃x. φ) ↔ (ψ_1 ∨ … ∨ φ_n)@ where @{y1, …, yp} = FV(φ) \\ {x1,…,xm}@, and

* if @M ⊧_LRA ψ_i@ then @lift_i M ⊧_LRA φ@.
-}
projectCasesN :: VarSet -> QFFormula -> [(QFFormula, Model Rational -> Model Rational)]
projectCasesN vs = f (IS.toList vs) 
  where
    f [] phi = return (phi, id)
    f (v:vs) phi = do
      (phi2, mt1) <- projectCases v phi
      (phi3, mt2) <- f vs phi2
      return (phi3, mt1 . mt2)

simplify :: QFFormula -> QFFormula
simplify = BoolExpr.simplify . BoolExpr.fold simplifyLit

simplifyLit :: LA.Atom Rational -> QFFormula
simplifyLit (ArithRel lhs op rhs) =
  case LA.asConst e of
    Just c -> if evalOp op c 0 then true else false
    Nothing -> Atom (ArithRel e op (LA.constant 0))
  where
    e = lhs ^-^ rhs

collect :: Var -> QFFormula -> Set (LA.Expr Rational)
collect v = Foldable.foldMap f
  where
    f (ArithRel lhs _ rhs) = assert (rhs == LA.constant 0) $
      case LA.extractMaybe v lhs of
        Nothing -> Set.empty
        Just (a,b) -> Set.singleton (negateV (b ^/ a))

applySubst1 :: Var -> LA.Expr Rational -> QFFormula -> QFFormula
applySubst1 v t = BoolExpr.fold f
  where
    f rel = Atom (LA.applySubst1Atom v t rel)

pairs :: [a] -> [(a,a)]
pairs [] = []
pairs (x:xs) = [(x,x2) | x2 <- xs] ++ pairs xs

-- | @'solveQFFormula' {x1,…,xn} φ@ returns @Just M@ such that @M ⊧_LRA φ@ when
-- such @M@ exists, returns @Nothing@ otherwise.
--
-- @FV(φ)@ must be a subset of @{x1,…,xn}@.
-- 
solveQFFormula :: VarSet -> QFFormula -> Maybe (Model Rational)
solveQFFormula vs formula = listToMaybe $ do
  (formula2, mt) <- projectCasesN vs formula
  let m = IM.empty
  guard $ evalQFFormula m formula2
  return $ mt m

-- | @'solve' {x1,…,xn} φ@ returns @Just M@ such that @M ⊧_LRA φ@ when
-- such @M@ exists, returns @Nothing@ otherwise.
--
-- @FV(φ)@ must be a subset of @{x1,…,xn}@.
-- 
solve :: VarSet -> [LA.Atom Rational] -> Maybe (Model Rational)
solve vs cs = solveQFFormula vs (andB [Atom c | c <- cs])