{- |

Module : $EmptyHeader$

Description : <optional short description entry>

Copyright : (c) <Authors or Affiliations>

License : GPLv2 or higher

Maintainer : <email>

Stability : unstable | experimental | provisional | stable | frozen

Portability : portable | non-portable (<reason>)

<optional description>

-}

-- examples for graded logic. To run these examples,

-- try both provable and sat(isfiable) from the (imported) module

-- Generic on the formulas defined here.

-- quickstart: [ghci|hugs] examples.gml.hs

-- then try ``provable phi1''

import Generic

-- gdiam n phi is the GML formula <>_n phi

gdiam :: Int -> L G -> L G; gdiam k x = box (G k) x

-- gbox n phi is the GML formula []_n phi

gbox :: Int -> L G -> L G; gbox k x = neg (gdiam k (neg x))

-- shorthands for propositional atoms

p0 :: L G; p0 = Atom 0

p1 :: L G; p1 = Atom 1

p2 :: L G; p2 = Atom 2

p3 :: L G; p3 = Atom 3

-- gexcl n phi is the GML formula <>!_n phi

-- so that x |= <>!_n phi if x has exactly n successors that satisfy phi

gexcl :: Int -> L G -> L G;

gexcl 0 x = neg $ gdiam 0 x

gexcl k x = (gdiam (k - 1) x) /\ (neg (gdiam k x))

-- example formulas for GML

phi1 :: L G; phi1 = gbox 0 T

phi2 :: L G; phi2 = (gdiam 4 p0) --> (gdiam 3 p1)

phi3 :: L G; phi3 = (gdiam (3 + 5) (p1) --> (gdiam 3 p0) \/ (gdiam 5 p1) )

phi4 :: L G; phi4 = neg ( (gdiam 2 p0) /\ (gbox 1 (neg p1)) /\ (gbox 1 (p1)))

phi5 :: L G; phi5 = ((neg p0) /\ (gexcl 2 T) /\ (gexcl 2 p0)) --> (gbox 0 p0)

phi6 :: L G; phi6 = (g2 12) --> gdiam 2 ((gbox 1 p0) --> gdiam 1 p0)

-- the formulas g3 n k are instances of the G3 axiom schema for GML

g3 :: Int -> Int -> (L G)

g3 n1 n2 = (gexcl 0 (p0/\p1))

-->(((gexcl n1 p0)/\(gexcl n2 p1))

-->(gexcl (n1 + n2) (p0 \/ p1)))

-- the formulas g2 n are instances of the G2 axiom schema for GML

g2 :: Int -> L G;

g2 n = gbox 0 (p0 --> p1) --> (gdiam n p0 --> gdiam n p1)

main = putStr $ show $ provable phi5