diff --git a/ChangeLog.md b/ChangeLog.md new file mode 100644 --- /dev/null +++ b/ChangeLog.md @@ -0,0 +1,5 @@ +# Revision history for heyting-algebra + +## 0.0.1.0 -- YYYY-mm-dd + +* First version. Released on an unsuspecting world. diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,373 @@ +Mozilla Public License Version 2.0 +================================== + +1. Definitions +-------------- + +1.1. "Contributor" + means each individual or legal entity that creates, contributes to + the creation of, or owns Covered Software. + +1.2. "Contributor Version" + means the combination of the Contributions of others (if any) used + by a Contributor and that particular Contributor's Contribution. + +1.3. "Contribution" + means Covered Software of a particular Contributor. + +1.4. "Covered Software" + means Source Code Form to which the initial Contributor has attached + the notice in Exhibit A, the Executable Form of such Source Code + Form, and Modifications of such Source Code Form, in each case + including portions thereof. + +1.5. "Incompatible With Secondary Licenses" + means + + (a) that the initial Contributor has attached the notice described + in Exhibit B to the Covered Software; or + + (b) that the Covered Software was made available under the terms of + version 1.1 or earlier of the License, but not also under the + terms of a Secondary License. + +1.6. "Executable Form" + means any form of the work other than Source Code Form. + +1.7. "Larger Work" + means a work that combines Covered Software with other material, in + a separate file or files, that is not Covered Software. + +1.8. "License" + means this document. + +1.9. "Licensable" + means having the right to grant, to the maximum extent possible, + whether at the time of the initial grant or subsequently, any and + all of the rights conveyed by this License. + +1.10. "Modifications" + means any of the following: + + (a) any file in Source Code Form that results from an addition to, + deletion from, or modification of the contents of Covered + Software; or + + (b) any new file in Source Code Form that contains any Covered + Software. + +1.11. "Patent Claims" of a Contributor + means any patent claim(s), including without limitation, method, + process, and apparatus claims, in any patent Licensable by such + Contributor that would be infringed, but for the grant of the + License, by the making, using, selling, offering for sale, having + made, import, or transfer of either its Contributions or its + Contributor Version. + +1.12. "Secondary License" + means either the GNU General Public License, Version 2.0, the GNU + Lesser General Public License, Version 2.1, the GNU Affero General + Public License, Version 3.0, or any later versions of those + licenses. + +1.13. "Source Code Form" + means the form of the work preferred for making modifications. + +1.14. "You" (or "Your") + means an individual or a legal entity exercising rights under this + License. For legal entities, "You" includes any entity that + controls, is controlled by, or is under common control with You. For + purposes of this definition, "control" means (a) the power, direct + or indirect, to cause the direction or management of such entity, + whether by contract or otherwise, or (b) ownership of more than + fifty percent (50%) of the outstanding shares or beneficial + ownership of such entity. + +2. License Grants and Conditions +-------------------------------- + +2.1. Grants + +Each Contributor hereby grants You a world-wide, royalty-free, +non-exclusive license: + +(a) under intellectual property rights (other than patent or trademark) + Licensable by such Contributor to use, reproduce, make available, + modify, display, perform, distribute, and otherwise exploit its + Contributions, either on an unmodified basis, with Modifications, or + as part of a Larger Work; and + +(b) under Patent Claims of such Contributor to make, use, sell, offer + for sale, have made, import, and otherwise transfer either its + Contributions or its Contributor Version. + +2.2. Effective Date + +The licenses granted in Section 2.1 with respect to any Contribution +become effective for each Contribution on the date the Contributor first +distributes such Contribution. + +2.3. Limitations on Grant Scope + +The licenses granted in this Section 2 are the only rights granted under +this License. No additional rights or licenses will be implied from the +distribution or licensing of Covered Software under this License. +Notwithstanding Section 2.1(b) above, no patent license is granted by a +Contributor: + +(a) for any code that a Contributor has removed from Covered Software; + or + +(b) for infringements caused by: (i) Your and any other third party's + modifications of Covered Software, or (ii) the combination of its + Contributions with other software (except as part of its Contributor + Version); or + +(c) under Patent Claims infringed by Covered Software in the absence of + its Contributions. + +This License does not grant any rights in the trademarks, service marks, +or logos of any Contributor (except as may be necessary to comply with +the notice requirements in Section 3.4). + +2.4. Subsequent Licenses + +No Contributor makes additional grants as a result of Your choice to +distribute the Covered Software under a subsequent version of this +License (see Section 10.2) or under the terms of a Secondary License (if +permitted under the terms of Section 3.3). + +2.5. Representation + +Each Contributor represents that the Contributor believes its +Contributions are its original creation(s) or it has sufficient rights +to grant the rights to its Contributions conveyed by this License. + +2.6. Fair Use + +This License is not intended to limit any rights You have under +applicable copyright doctrines of fair use, fair dealing, or other +equivalents. + +2.7. Conditions + +Sections 3.1, 3.2, 3.3, and 3.4 are conditions of the licenses granted +in Section 2.1. + +3. Responsibilities +------------------- + +3.1. Distribution of Source Form + +All distribution of Covered Software in Source Code Form, including any +Modifications that You create or to which You contribute, must be under +the terms of this License. You must inform recipients that the Source +Code Form of the Covered Software is governed by the terms of this +License, and how they can obtain a copy of this License. You may not +attempt to alter or restrict the recipients' rights in the Source Code +Form. + +3.2. Distribution of Executable Form + +If You distribute Covered Software in Executable Form then: + +(a) such Covered Software must also be made available in Source Code + Form, as described in Section 3.1, and You must inform recipients of + the Executable Form how they can obtain a copy of such Source Code + Form by reasonable means in a timely manner, at a charge no more + than the cost of distribution to the recipient; and + +(b) You may distribute such Executable Form under the terms of this + License, or sublicense it under different terms, provided that the + license for the Executable Form does not attempt to limit or alter + the recipients' rights in the Source Code Form under this License. + +3.3. Distribution of a Larger Work + +You may create and distribute a Larger Work under terms of Your choice, +provided that You also comply with the requirements of this License for +the Covered Software. If the Larger Work is a combination of Covered +Software with a work governed by one or more Secondary Licenses, and the +Covered Software is not Incompatible With Secondary Licenses, this +License permits You to additionally distribute such Covered Software +under the terms of such Secondary License(s), so that the recipient of +the Larger Work may, at their option, further distribute the Covered +Software under the terms of either this License or such Secondary +License(s). + +3.4. Notices + +You may not remove or alter the substance of any license notices +(including copyright notices, patent notices, disclaimers of warranty, +or limitations of liability) contained within the Source Code Form of +the Covered Software, except that You may alter any license notices to +the extent required to remedy known factual inaccuracies. + +3.5. Application of Additional Terms + +You may choose to offer, and to charge a fee for, warranty, support, +indemnity or liability obligations to one or more recipients of Covered +Software. However, You may do so only on Your own behalf, and not on +behalf of any Contributor. You must make it absolutely clear that any +such warranty, support, indemnity, or liability obligation is offered by +You alone, and You hereby agree to indemnify every Contributor for any +liability incurred by such Contributor as a result of warranty, support, +indemnity or liability terms You offer. You may include additional +disclaimers of warranty and limitations of liability specific to any +jurisdiction. + +4. Inability to Comply Due to Statute or Regulation +--------------------------------------------------- + +If it is impossible for You to comply with any of the terms of this +License with respect to some or all of the Covered Software due to +statute, judicial order, or regulation then You must: (a) comply with +the terms of this License to the maximum extent possible; and (b) +describe the limitations and the code they affect. Such description must +be placed in a text file included with all distributions of the Covered +Software under this License. Except to the extent prohibited by statute +or regulation, such description must be sufficiently detailed for a +recipient of ordinary skill to be able to understand it. + +5. Termination +-------------- + +5.1. The rights granted under this License will terminate automatically +if You fail to comply with any of its terms. However, if You become +compliant, then the rights granted under this License from a particular +Contributor are reinstated (a) provisionally, unless and until such +Contributor explicitly and finally terminates Your grants, and (b) on an +ongoing basis, if such Contributor fails to notify You of the +non-compliance by some reasonable means prior to 60 days after You have +come back into compliance. Moreover, Your grants from a particular +Contributor are reinstated on an ongoing basis if such Contributor +notifies You of the non-compliance by some reasonable means, this is the +first time You have received notice of non-compliance with this License +from such Contributor, and You become compliant prior to 30 days after +Your receipt of the notice. + +5.2. If You initiate litigation against any entity by asserting a patent +infringement claim (excluding declaratory judgment actions, +counter-claims, and cross-claims) alleging that a Contributor Version +directly or indirectly infringes any patent, then the rights granted to +You by any and all Contributors for the Covered Software under Section +2.1 of this License shall terminate. + +5.3. In the event of termination under Sections 5.1 or 5.2 above, all +end user license agreements (excluding distributors and resellers) which +have been validly granted by You or Your distributors under this License +prior to termination shall survive termination. + +************************************************************************ +* * +* 6. Disclaimer of Warranty * +* ------------------------- * +* * +* Covered Software is provided under this License on an "as is" * +* basis, without warranty of any kind, either expressed, implied, or * +* statutory, including, without limitation, warranties that the * +* Covered Software is free of defects, merchantable, fit for a * +* particular purpose or non-infringing. The entire risk as to the * +* quality and performance of the Covered Software is with You. * +* Should any Covered Software prove defective in any respect, You * +* (not any Contributor) assume the cost of any necessary servicing, * +* repair, or correction. This disclaimer of warranty constitutes an * +* essential part of this License. No use of any Covered Software is * +* authorized under this License except under this disclaimer. * +* * +************************************************************************ + +************************************************************************ +* * +* 7. Limitation of Liability * +* -------------------------- * +* * +* Under no circumstances and under no legal theory, whether tort * +* (including negligence), contract, or otherwise, shall any * +* Contributor, or anyone who distributes Covered Software as * +* permitted above, be liable to You for any direct, indirect, * +* special, incidental, or consequential damages of any character * +* including, without limitation, damages for lost profits, loss of * +* goodwill, work stoppage, computer failure or malfunction, or any * +* and all other commercial damages or losses, even if such party * +* shall have been informed of the possibility of such damages. This * +* limitation of liability shall not apply to liability for death or * +* personal injury resulting from such party's negligence to the * +* extent applicable law prohibits such limitation. Some * +* jurisdictions do not allow the exclusion or limitation of * +* incidental or consequential damages, so this exclusion and * +* limitation may not apply to You. * +* * +************************************************************************ + +8. Litigation +------------- + +Any litigation relating to this License may be brought only in the +courts of a jurisdiction where the defendant maintains its principal +place of business and such litigation shall be governed by laws of that +jurisdiction, without reference to its conflict-of-law provisions. +Nothing in this Section shall prevent a party's ability to bring +cross-claims or counter-claims. + +9. Miscellaneous +---------------- + +This License represents the complete agreement concerning the subject +matter hereof. If any provision of this License is held to be +unenforceable, such provision shall be reformed only to the extent +necessary to make it enforceable. Any law or regulation which provides +that the language of a contract shall be construed against the drafter +shall not be used to construe this License against a Contributor. + +10. Versions of the License +--------------------------- + +10.1. New Versions + +Mozilla Foundation is the license steward. Except as provided in Section +10.3, no one other than the license steward has the right to modify or +publish new versions of this License. Each version will be given a +distinguishing version number. + +10.2. Effect of New Versions + +You may distribute the Covered Software under the terms of the version +of the License under which You originally received the Covered Software, +or under the terms of any subsequent version published by the license +steward. + +10.3. Modified Versions + +If you create software not governed by this License, and you want to +create a new license for such software, you may create and use a +modified version of this License if you rename the license and remove +any references to the name of the license steward (except to note that +such modified license differs from this License). + +10.4. Distributing Source Code Form that is Incompatible With Secondary +Licenses + +If You choose to distribute Source Code Form that is Incompatible With +Secondary Licenses under the terms of this version of the License, the +notice described in Exhibit B of this License must be attached. + +Exhibit A - Source Code Form License Notice +------------------------------------------- + + This Source Code Form is subject to the terms of the Mozilla Public + License, v. 2.0. If a copy of the MPL was not distributed with this + file, You can obtain one at http://mozilla.org/MPL/2.0/. + +If it is not possible or desirable to put the notice in a particular +file, then You may include the notice in a location (such as a LICENSE +file in a relevant directory) where a recipient would be likely to look +for such a notice. + +You may add additional accurate notices of copyright ownership. + +Exhibit B - "Incompatible With Secondary Licenses" Notice +--------------------------------------------------------- + + This Source Code Form is "Incompatible With Secondary Licenses", as + defined by the Mozilla Public License, v. 2.0. diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/heyting-algebras.cabal b/heyting-algebras.cabal new file mode 100644 --- /dev/null +++ b/heyting-algebras.cabal @@ -0,0 +1,72 @@ +name: heyting-algebras +version: 0.0.1.0 +synopsis: Heyting and Boolean algebras +description: + This package provides Heyting and Boolean operations together + with various constructions of Heyting algebras. +license: MPL-2.0 +license-file: LICENSE +author: Marcin Szamotulski +maintainer: profunctor@pm.me +copyright: (c) 2018 Marcin Szamotulski +category: Math +build-type: Simple +extra-source-files: ChangeLog.md +cabal-version: >=1.10 +tested-with: GHC==8.2.2, GHC==8.4.3 + +flag export-properties + description: + Export quickcheck properties from library; this adds QuickCheck + as a dependency. + manual: True + default: False + +library + exposed-modules: Algebra.Heyting + Algebra.Heyting.Free + Algebra.Heyting.Layered + Algebra.Boolean + Algebra.Boolean.Free + -- other-modules: + -- other-extensions: + build-depends: base >= 4.9 && < 4.13 + , containers >= 0.4.2 && < 0.7 + , free-algebras >= 0.0.4 && < 0.0.6 + , hashable >= 1.2.6.1 && < 1.3 + , lattices >= 1.0 && < 1.11 + , tagged >= 0.8.5 && < 0.9 + , unordered-containers + >= 0.2.6.0 && < 0.3 + , universe-base >= 1.0 && < 1.1 + hs-source-dirs: src + default-language: Haskell2010 + default-extensions: FlexibleInstances + RankNTypes + ghc-options: -Wall + if flag(export-properties) + build-depends: + QuickCheck >= 2.10 && < 2.13 + cpp-options: + -DEXPORT_PROPERTIES + +test-suite heyting-algebras-test + type: exitcode-stdio-1.0 + hs-source-dirs: test + main-is: Main.hs + build-depends: base + , containers + , lattices + , universe-base + , QuickCheck + , heyting-algebras + , tasty + , tasty-quickcheck + default-language: Haskell2010 + default-extensions: FlexibleInstances + , TypeApplications + +source-repository head + type: git + location: https://github.com/coot/heyting-algebras + diff --git a/src/Algebra/Boolean.hs b/src/Algebra/Boolean.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Boolean.hs @@ -0,0 +1,153 @@ +{-# LANGUAGE CPP #-} +{-# LANGUAGE DeriveDataTypeable #-} +{-# LANGUAGE DeriveGeneric #-} +{-# LANGUAGE GeneralizedNewtypeDeriving #-} +module Algebra.Boolean + ( BooleanAlgebra + , (==>) + , not + , iff + , iff' + -- * Adjunction between Boolean and Heyting algebras + , Boolean + , runBoolean + , boolean + + -- * Properties + -- $properties + , prop_not + , prop_BooleanAlgebra + ) where + +import Prelude hiding (not) + +import Control.Applicative (Const (..)) +import Data.Data (Data, Typeable) +import Data.Functor.Identity (Identity (..)) +import Data.Proxy (Proxy (..)) +import Data.Semigroup (All (..), Any (..), Endo (..)) +import Data.Tagged (Tagged (..)) +import Data.Universe.Class (Finite) +import qualified Data.Set as S +import GHC.Generics (Generic) +#ifdef EXPORT_PROPERTIES +import Test.QuickCheck hiding ((==>)) +#endif + +import Algebra.Lattice ( Lattice + , BoundedLattice + , JoinSemiLattice (..) + , BoundedJoinSemiLattice + , MeetSemiLattice (..) + , BoundedMeetSemiLattice + , bottom + , top + ) + +import Algebra.Heyting ( HeytingAlgebra (..) + , iff + , iff' + , not + , toBoolean + , prop_HeytingAlgebra + ) + +-- | +-- Boolean algebra is a Heyting algebra which negation satisfies the law of +-- excluded middle, i.e. either of the following: +-- +-- prop> not . not == not +-- +-- or +-- +-- prop> x ∨ not x == top +-- +-- Another characterisation of Boolean algebras is as +-- [complemented](https://en.wikipedia.org/wiki/Complemented_lattice) +-- [distributive lattices](https://ncatlab.org/nlab/show/distributive+lattice) +-- where the complement satisfies the following three properties: +-- +-- prop> (not a) ∧ a == bottom and (not a) ∨ a == top -- excluded middle law +-- prop> not (not a) == a -- involution law +-- prop> a ≤ b ⇒ not b ≤ not a -- order-reversing +class HeytingAlgebra a => BooleanAlgebra a + +-- | +-- @'Boolean'@ is the left adjoint functor from the category of Heyting algebras +-- to the category of Boolean algebras; its right adjoint is the inclusion. +newtype Boolean a = Boolean + { runBoolean :: a -- ^ extract value from @'Boolean'@ + } + deriving + ( JoinSemiLattice, BoundedJoinSemiLattice, MeetSemiLattice + , BoundedMeetSemiLattice, Lattice, BoundedLattice, HeytingAlgebra + , Eq, Ord, Read, Show, Bounded, Typeable, Data, Generic + ) + +instance HeytingAlgebra a => BooleanAlgebra (Boolean a) + +-- TODO: move to tests +instance (Arbitrary a, HeytingAlgebra a) => Arbitrary (Boolean a) where + arbitrary = boolean <$> arbitrary + shrink (Boolean a) = [ boolean a' | a' <- shrink a ] + +-- | +-- Smart constructro of the @'Boolean'@ type. +boolean :: HeytingAlgebra a => a -> Boolean a +boolean = Boolean . toBoolean + +-- +-- Instances +-- + +instance BooleanAlgebra Bool + +instance BooleanAlgebra All + +instance BooleanAlgebra Any + +instance BooleanAlgebra () + +instance BooleanAlgebra (Proxy a) + +instance BooleanAlgebra a => BooleanAlgebra (Tagged t a) + +instance BooleanAlgebra b => BooleanAlgebra (a -> b) + +instance BooleanAlgebra a => BooleanAlgebra (Identity a) + +instance BooleanAlgebra a => BooleanAlgebra (Const a b) + +instance BooleanAlgebra a => BooleanAlgebra (Endo a) + +instance (BooleanAlgebra a, BooleanAlgebra b) => BooleanAlgebra (a, b) + +-- +-- containers +-- + +instance (Ord a, Finite a) => BooleanAlgebra (S.Set a) + +-- +-- $properties +-- +-- /Properties are exported only if @export-properties@ cabal flag is defined./ +#ifdef EXPORT_PROPERTIES + +-- | +-- Test that @'not'@ satisfies Boolean algebra axioms. +prop_not :: (HeytingAlgebra a, Eq a, Show a) => a -> Property +prop_not a = + counterexample "not (not a) /= a" (not (not a) === a) + .&&. counterexample "not a ∧ a /= bottom" (not a /\ a === bottom) + .&&. counterexample "not a ∨ a /= top" (not a \/ a === top) + +-- | +-- Test that @a@ is satisfy both @'Algebra.Heyting.prop_HeytingAlgebra'@ and +-- @'prop_not'@. +prop_BooleanAlgebra :: (BooleanAlgebra a, Eq a, Show a) + => a -> a -> a -> Property +prop_BooleanAlgebra a b c = + prop_HeytingAlgebra a b c + .&&. prop_not a +#endif diff --git a/src/Algebra/Boolean/Free.hs b/src/Algebra/Boolean/Free.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Boolean/Free.hs @@ -0,0 +1,72 @@ +{-# LANGUAGE TypeFamilies #-} +module Algebra.Boolean.Free + ( FreeBoolean (..) + ) where + +import Control.Monad (ap) +import Algebra.Lattice + ( BoundedJoinSemiLattice (..) + , JoinSemiLattice (..) + , BoundedMeetSemiLattice (..) + , MeetSemiLattice (..) + , BoundedLattice + , Lattice + ) +import Data.Algebra.Free + ( AlgebraType0 + , AlgebraType + , FreeAlgebra (..) + , proof + , fmapFree + , bindFree + ) + +import Algebra.Boolean (BooleanAlgebra) +import Algebra.Heyting (HeytingAlgebra (..)) + +-- | +-- Free Boolean algebra. @'FreeAlgebra'@ instance provides all the usual +-- combinators for a free algebra. +newtype FreeBoolean a = FreeBoolean + { runFreeBoolean :: forall h . BooleanAlgebra h => (a -> h) -> h } + +instance JoinSemiLattice (FreeBoolean a) where + FreeBoolean f \/ FreeBoolean g = FreeBoolean (\inj -> f inj \/ g inj) + +instance BoundedJoinSemiLattice (FreeBoolean a) where + bottom = FreeBoolean (\_ -> bottom) + +instance MeetSemiLattice (FreeBoolean a) where + FreeBoolean f /\ FreeBoolean g = FreeBoolean (\inj -> f inj /\ g inj) + +instance BoundedMeetSemiLattice (FreeBoolean a) where + top = FreeBoolean (\_ -> top) + +instance Lattice (FreeBoolean a) + +instance BoundedLattice (FreeBoolean a) + +instance HeytingAlgebra (FreeBoolean a) where + FreeBoolean f ==> FreeBoolean g = FreeBoolean (\inj -> f inj ==> g inj) + +instance BooleanAlgebra (FreeBoolean a) + +type instance AlgebraType0 FreeBoolean a = () +type instance AlgebraType FreeBoolean a = BooleanAlgebra a +instance FreeAlgebra FreeBoolean where + returnFree a = FreeBoolean (\inj -> inj a) + foldMapFree f (FreeBoolean inj) = inj f + + codom = proof + forget = proof + +instance Functor FreeBoolean where + fmap = fmapFree + +instance Applicative FreeBoolean where + pure = returnFree + (<*>) = ap + +instance Monad FreeBoolean where + return = returnFree + (>>=) = bindFree diff --git a/src/Algebra/Heyting.hs b/src/Algebra/Heyting.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Heyting.hs @@ -0,0 +1,257 @@ +{-# LANGUAGE CPP #-} +module Algebra.Heyting + ( HeytingAlgebra (..) + , iff + , iff' + , toBoolean + + -- * Properties + -- + -- $properties + , prop_BoundedMeetSemiLattice + , prop_BoundedJoinSemiLattice + , prop_HeytingAlgebra + , prop_implies + ) + where + +import Prelude hiding (not) +import qualified Prelude + +import Control.Applicative (Const (..)) +import Data.Functor.Identity (Identity (..)) +import Data.Hashable (Hashable) +import Data.Proxy (Proxy (..)) +import Data.Semigroup (All (..), Any (..), Endo (..)) +import Data.Tagged (Tagged (..)) +import Data.Universe.Class (Finite, universe) +import qualified Data.Map as M +#if __GLASGOW_HASKELL__ >= 822 +import qualified Data.Map.Merge.Lazy as Merge +#endif +import qualified Data.Set as S +import qualified Data.HashMap.Lazy as HM +import qualified Data.HashSet as HS + +import Algebra.Lattice ( BoundedJoinSemiLattice (..) + , BoundedMeetSemiLattice (..) + , BoundedLattice + , Meet (..) + , Join (..) + , (/\) + , (\/) + ) +import Algebra.Lattice.Dropped (Dropped (..)) +import Algebra.Lattice.Lifted (Lifted (..)) +import Algebra.Lattice.Levitated (Levitated) +import qualified Algebra.Lattice.Levitated as L +import Algebra.PartialOrd (leq) +#ifdef EXPORT_PROPERTIES +import Test.QuickCheck hiding ((==>)) +import qualified Test.QuickCheck as QC +#endif + +-- | +-- Heyting algebra is a bounded semi-lattice with implication which should +-- satisfy the following law: +-- +-- prop> x ∧ a ≤ b ⇔ x ≤ (a ⇒ b) +-- +-- This means @a ⇒ b@ is an +-- [exponential object](https://ncatlab.org/nlab/show/exponential%2Bobject), +-- which makes any Heyting algebra +-- a [cartesian +-- closed category](https://ncatlab.org/nlab/show/cartesian%2Bclosed%2Bcategory). +-- +-- Some useful properties of Heyting algebras: +-- +-- prop> (a ⇒ b) ∧ a ≤ b +-- prop> b ≤ a ⇒ a ∧ b +-- prop> a ≤ b iff a ⇒ b = ⊤ +-- prop> b ≤ b' then a ⇒ b ≤ a ⇒ b' +-- prop> a'≤ a then a' ⇒ b ≤ a ⇒ b +class BoundedLattice a => HeytingAlgebra a where + -- | + -- Default implementation: @a ==> b = not a \/ b@, it requires @not@ to + -- satisfy Boolean axioms, which will make it into a Boolean algebra. + (==>) :: a -> a -> a + (==>) a b = not a \/ b + + -- | + -- Default implementation: @not a = a '==>' 'bottom'@ + -- + -- It is useful for optimisation reasons. + not :: a -> a + not a = a ==> bottom + + {-# MINIMAL (==>) | not #-} + +-- | +-- Less than fixity of both @'\/'@ and @'/\'@. +infixr 4 ==> + +iff :: HeytingAlgebra a => a -> a -> a +iff a b = (a ==> b) /\ (b ==> a) + +iff' :: (Eq a, HeytingAlgebra a) => a -> a -> Bool +iff' a b = Meet top `leq` Meet (iff a b) + +-- | +-- Every Heyting algebra contains a Boolean algebra. @'toBoolean'@ maps onto it; +-- moreover it is a monad (Heyting algebra is a category as every poset is) which +-- preserves finite infima. +toBoolean :: HeytingAlgebra a => a -> a +toBoolean = not . not + +instance HeytingAlgebra Bool where + not = Prelude.not + +instance HeytingAlgebra All where + All a ==> All b = All (a ==> b) + not (All a) = All (not a) + +instance HeytingAlgebra Any where + Any a ==> Any b = Any (a ==> b) + not (Any a) = Any (not a) + +instance HeytingAlgebra () where + _ ==> _ = () + +instance HeytingAlgebra (Proxy a) where + _ ==> _ = Proxy + +instance HeytingAlgebra a => HeytingAlgebra (Tagged t a) where + Tagged a ==> Tagged b = Tagged (a ==> b) + +instance HeytingAlgebra b => HeytingAlgebra (a -> b) where + f ==> g = \a -> f a ==> g a + +instance HeytingAlgebra a => HeytingAlgebra (Identity a) where + (Identity a) ==> (Identity b) = Identity (a ==> b) + +instance HeytingAlgebra a => HeytingAlgebra (Const a b) where + (Const a) ==> (Const b) = Const (a ==> b) + +instance HeytingAlgebra a => HeytingAlgebra (Endo a) where + (Endo f) ==> (Endo g) = Endo (f ==> g) + +instance (HeytingAlgebra a, HeytingAlgebra b) => HeytingAlgebra (a, b) where + (a0, b0) ==> (a1, b1) = (a0 ==> a1, b0 ==> b1) + +-- +-- Dropped, Lifted, Levitated +-- + +instance (Eq a, HeytingAlgebra a) => HeytingAlgebra (Dropped a) where + (Drop a) ==> (Drop b) | Meet a `leq` Meet b = Top + | otherwise = Drop (a ==> b) + Top ==> a = a + _ ==> Top = Top + +instance HeytingAlgebra a => HeytingAlgebra (Lifted a) where + (Lift a) ==> (Lift b) = Lift (a ==> b) + Bottom ==> _ = Lift top + _ ==> Bottom = Bottom + +instance (Eq a, HeytingAlgebra a) => HeytingAlgebra (Levitated a) where + (L.Levitate a) ==> (L.Levitate b) | Meet a `leq` Meet b = L.Top + | otherwise = L.Levitate (a ==> b) + L.Top ==> a = a + _ ==> L.Top = L.Top + L.Bottom ==> _ = L.Top + _ ==> L.Bottom = L.Bottom + +-- +-- containers +-- + +-- | +-- Power set: the cannoical example of a Boolean algebra +instance (Ord a, Finite a) => HeytingAlgebra (S.Set a) where + not a = S.fromList universe `S.difference` a + +instance (Eq a, Finite a, Hashable a) => HeytingAlgebra (HS.HashSet a) where + not a = HS.fromList universe `HS.difference` a + +-- | +-- It is not a boolean algebra (even if values are). +instance (Ord k, Finite k, HeytingAlgebra v) => HeytingAlgebra (M.Map k v) where + -- _xx__ + -- __yy_ + -- T_iTT where i = x ==> y; T = top; _ missing (or removed key) +#if __GLASOW_HASKELL__ >= 822 + a ==> b = + Merge.merge + Merge.dropMissing -- drop if an element is missing in @b@ + (Merge.mapMissing (\_ _ -> top)) -- put @top@ if an element is missing in @a@ + (Merge.zipWithMatched (\_ -> (==>))) -- merge matching elements with @==>@ + a b + + \/ M.fromList [(k, top) | k <- universe, not (M.member k a), not (M.member k b) ] + -- for elements which are not in a, nor in b add + -- a @top@ key +#else + a ==> b = + M.intersectionWith (==>) a b + `M.union` M.map (const top) (M.difference b a) + `M.union` M.fromList [(k, top) | k <- universe, not (M.member k a), not (M.member k b)] +#endif + +instance (Eq k, Finite k, Hashable k, HeytingAlgebra v) => HeytingAlgebra (HM.HashMap k v) where + a ==> b = HM.intersectionWith (==>) a b + `HM.union` HM.map (const top) (HM.difference b a) + `HM.union` HM.fromList [(k, top) | k <- universe, not (HM.member k a), not (HM.member k b)] + +-- +-- $properties +-- +-- /Properties are exported only if @export-properties@ cabal flag is defined./ +#ifdef EXPORT_PROPERTIES + +-- | +-- Verfifies bounded meet semilattice laws. +prop_BoundedMeetSemiLattice :: (BoundedMeetSemiLattice a, Eq a, Show a) + => a -> a -> a -> Property +prop_BoundedMeetSemiLattice a b c = + counterexample "meet associativity" ((a /\ (b /\ c)) === ((a /\ b) /\ c)) + .&&. counterexample "meet commutativity" ((a /\ b) === (b /\ a)) + .&&. counterexample "meet idempotent" ((a /\ a) === a) + .&&. counterexample "meet identity" ((top /\ a) === a) + .&&. counterexample "meet order" (Meet (a /\ b) `leq` Meet a) + +-- | +-- Verfifies bounded join semilattice laws. +prop_BoundedJoinSemiLattice :: (BoundedJoinSemiLattice a, Eq a, Show a) => a -> a -> a -> Property +prop_BoundedJoinSemiLattice a b c = + counterexample "join associativity" ((a \/ (b \/ c)) === ((a \/ b) \/ c)) + .&&. counterexample "join commutativity" ((a \/ b) === (b \/ a)) + .&&. counterexample "join idempotent" ((a \/ a) === a) + .&&. counterexample "join identity" ((bottom \/ a) === a) + .&&. counterexample "join order" (Join a `leq` Join (a \/ b)) + +-- | +-- Verifies the Heyting algebra law for @==>@: +-- for all @a@: @_ /\ a@ is left adjoint to @a ==>@ +prop_implies :: (HeytingAlgebra a, Eq a, Show a) + => a -> a -> a -> Property +prop_implies x a b = + counterexample ("Failed: x ≤ (a ⇒ b) then x ∧ a ≤ b\n\ta ⇒ b = " ++ show (a ==> b)) + (Meet x `leq` Meet (a ==> b) QC.==> (Meet (x /\ a) `leq` Meet b)) + .&&. + counterexample ("Failed: x ∧ a ≤ b then x ≤ (a ⇒ b)\n\ta ⇒ b = " ++ show (a ==> b)) + (Meet (x /\ a) `leq` Meet b QC.==> (Meet x `leq` Meet (a ==> b))) + + +-- | +-- Usefull for testing valid instances of @'HeytingAlgebra'@ type class. It +-- validates: +-- +-- * bounded lattice laws +-- * @'prop_implies'@ +prop_HeytingAlgebra :: (HeytingAlgebra a, Eq a, Show a) + => a -> a -> a -> Property +prop_HeytingAlgebra a b c = + prop_BoundedJoinSemiLattice a b c + .&&. prop_BoundedMeetSemiLattice a b c + .&&. prop_implies a b c +#endif diff --git a/src/Algebra/Heyting/Free.hs b/src/Algebra/Heyting/Free.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Heyting/Free.hs @@ -0,0 +1,76 @@ +{-# LANGUAGE TypeFamilies #-} +module Algebra.Heyting.Free + ( FreeHeyting (..) + ) where + +import Prelude hiding (not) + +import Control.Monad (ap) +import Algebra.Lattice + ( BoundedJoinSemiLattice (..) + , JoinSemiLattice (..) + , BoundedMeetSemiLattice (..) + , MeetSemiLattice (..) + , BoundedLattice + , Lattice + ) +import Data.Algebra.Free + ( AlgebraType0 + , AlgebraType + , FreeAlgebra (..) + , proof + , fmapFree + , bindFree + ) + +import Algebra.Heyting (HeytingAlgebra (..)) + +-- | +-- Free Heyting algebra. @'FreeAlgebra'@ instance provides all the usual +-- combinators for a free algebra. +-- +-- The +-- [graph](https://en.wikipedia.org/wiki/Heyting_algebra#/media/File:Rieger-Nishimura.svg) +-- of free Heyting algebra with one generator, i.e. @'FreeHeyting' ()@. +newtype FreeHeyting a = FreeHeyting + { runFreeHeyting :: forall h . HeytingAlgebra h => (a -> h) -> h } + +instance JoinSemiLattice (FreeHeyting a) where + FreeHeyting f \/ FreeHeyting g = FreeHeyting (\inj -> f inj \/ g inj) + +instance BoundedJoinSemiLattice (FreeHeyting a) where + bottom = FreeHeyting (\_ -> bottom) + +instance MeetSemiLattice (FreeHeyting a) where + FreeHeyting f /\ FreeHeyting g = FreeHeyting (\inj -> f inj /\ g inj) + +instance BoundedMeetSemiLattice (FreeHeyting a) where + top = FreeHeyting (\_ -> top) + +instance Lattice (FreeHeyting a) + +instance BoundedLattice (FreeHeyting a) + +instance HeytingAlgebra (FreeHeyting a) where + FreeHeyting f ==> FreeHeyting g = FreeHeyting (\inj -> f inj ==> g inj) + not (FreeHeyting f) = FreeHeyting (\inj -> not (f inj)) + +type instance AlgebraType0 FreeHeyting a = () +type instance AlgebraType FreeHeyting a = HeytingAlgebra a +instance FreeAlgebra FreeHeyting where + returnFree a = FreeHeyting (\inj -> inj a) + foldMapFree f (FreeHeyting inj) = inj f + + codom = proof + forget = proof + +instance Functor FreeHeyting where + fmap = fmapFree + +instance Applicative FreeHeyting where + pure = returnFree + (<*>) = ap + +instance Monad FreeHeyting where + return = returnFree + (>>=) = bindFree diff --git a/src/Algebra/Heyting/Layered.hs b/src/Algebra/Heyting/Layered.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Heyting/Layered.hs @@ -0,0 +1,68 @@ +module Algebra.Heyting.Layered + ( Layered (..) + , layer + ) where + +import Prelude + +import Algebra.Lattice + ( BoundedJoinSemiLattice (..) + , JoinSemiLattice (..) + , BoundedMeetSemiLattice (..) + , MeetSemiLattice (..) + , BoundedLattice + , Lattice + ) + +import Algebra.Heyting (HeytingAlgebra (..)) + +-- | +-- Layer one Heyting algebra on top of the other. Note: this is not +-- a categorial sum. +data Layered a b + = Lower a + | Upper b + deriving (Show, Eq, Ord) + +instance (JoinSemiLattice a, JoinSemiLattice b) => JoinSemiLattice (Layered a b) where + Lower _ \/ u@Upper{} = u + u@Upper{} \/ Lower _ = u + Lower l \/ Lower l' = Lower $ l \/ l' + Upper u \/ Upper u' = Upper $ u \/ u' + +instance (BoundedJoinSemiLattice a, JoinSemiLattice b) => BoundedJoinSemiLattice (Layered a b) where + bottom = Lower bottom + +instance (MeetSemiLattice a, MeetSemiLattice b) => MeetSemiLattice (Layered a b) where + l@Lower{} /\ Upper _ = l + Upper _ /\ l@Lower{} = l + Lower l /\ Lower l' = Lower $ l /\ l' + Upper u /\ Upper u' = Upper $ u /\ u' + +instance (MeetSemiLattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (Layered a b) where + top = Upper top + +instance ( Lattice a + , Lattice b + ) => Lattice (Layered a b) + +instance ( Lattice a + , Lattice b + , BoundedJoinSemiLattice a + , BoundedMeetSemiLattice b + ) => BoundedLattice (Layered a b) + +instance ( HeytingAlgebra a + , HeytingAlgebra b + , Eq a + ) => HeytingAlgebra (Layered a b) where + Lower _ ==> Upper _ = Upper top + Upper _ ==> l@Lower{} = l + Upper u ==> Upper u' = Upper $ u ==> u' + Lower l ==> Lower l' = case l ==> l' of + ll' | ll' == top -> Upper top + | otherwise -> Lower ll' + +layer :: (a -> c) -> (b -> c) -> Layered a b -> c +layer f _ (Lower a) = f a +layer _ g (Upper b) = g b diff --git a/test/Main.hs b/test/Main.hs new file mode 100644 --- /dev/null +++ b/test/Main.hs @@ -0,0 +1,197 @@ +{-# LANGUAGE GeneralizedNewtypeDeriving #-} +{-# LANGUAGE ScopedTypeVariables #-} +module Main where + +import Algebra.Lattice ( JoinSemiLattice + , BoundedJoinSemiLattice + , MeetSemiLattice + , BoundedMeetSemiLattice + , Lattice + , BoundedLattice + ) +import Algebra.Lattice.Dropped (Dropped (..)) +import Algebra.Lattice.Lifted (Lifted (..)) +import Algebra.Lattice.Levitated (Levitated) +import qualified Algebra.Lattice.Levitated as L +import Data.Universe.Class (Universe (..), Finite) +import qualified Data.Set as S +import qualified Data.Map as M + +import Test.QuickCheck + +import Algebra.Boolean +import Algebra.Heyting +import Algebra.Heyting.Layered + +import Test.Tasty +import Test.Tasty.QuickCheck + +-- | Arbitrary wrapper +newtype Arb a = Arb a + deriving ( JoinSemiLattice + , BoundedJoinSemiLattice + , MeetSemiLattice + , BoundedMeetSemiLattice + , Lattice + , BoundedLattice + , BooleanAlgebra + , HeytingAlgebra + , Eq + , Ord + ) + +instance Show a => Show (Arb a) where + show (Arb a) = show a + +instance (Finite k, Arbitrary k, Arbitrary v, Ord k) => Arbitrary (Arb (M.Map k v)) where + arbitrary = frequency + [ (1, Arb . M.fromList <$> arbitrary) + , (4, Arb . M.fromList . zip universe <$> vectorOf (length (universe @k)) arbitrary) + , (6, return $ Arb M.empty) + ] + +instance Arbitrary a => Arbitrary (Arb (Lifted a)) where + arbitrary = Arb . maybe Bottom Lift <$> arbitrary + shrink (Arb Bottom) = [] + shrink (Arb (Lift a)) = + Arb Bottom : [ Arb (Lift a') | a' <- shrink a ] + +instance Arbitrary a => Arbitrary (Arb (Dropped a)) where + arbitrary = Arb . maybe Top Drop <$> arbitrary + shrink (Arb Top) = [] + shrink (Arb (Drop a)) = + Arb Top : [ Arb (Drop a') | a' <- shrink a ] + +instance Arbitrary a => Arbitrary (Arb (Levitated a)) where + arbitrary = frequency + [ (1, return $ Arb L.Top) + , (1, return $ Arb L.Bottom) + , (2, Arb . L.Levitate <$> arbitrary) + ] + + shrink (Arb L.Bottom) = [] + shrink (Arb (L.Levitate a)) + = Arb L.Bottom + : Arb L.Top + : [ Arb (L.Levitate a') | a' <- shrink a ] + shrink (Arb L.Top) = [] + +data S5 = S1 | S2 | S3 | S4 | S5 + deriving (Ord, Eq, Show) + +instance Universe S5 where + universe = [S1, S2, S3, S4, S5] + +instance Finite S5 + +instance Arbitrary S5 where + arbitrary = elements universe + +instance (Arbitrary a, Ord a) => Arbitrary (Arb (S.Set a)) where + arbitrary = Arb . S.fromList <$> arbitrary + shrink (Arb as) = [ Arb (S.fromList as') | as' <- shrink (S.toList as) ] + +instance (Arbitrary a, Arbitrary b) => Arbitrary (Arb (Layered a b)) where + arbitrary = oneof + [ Arb . Lower <$> arbitrary + , Arb . Upper <$> arbitrary + ] + shrink (Arb (Lower a)) = [ Arb (Lower a') | a' <- shrink a ] + shrink (Arb (Upper b)) = [ Arb (Upper b') | b' <- shrink b ] + +-- Another arbitrary newtype wrapper; using tagged type let us avoid +-- overlapping instances. +newtype Composed f g a = Composed (f (g a)) + deriving ( JoinSemiLattice + , BoundedJoinSemiLattice + , MeetSemiLattice + , BoundedMeetSemiLattice + , Lattice + , BoundedLattice + , BooleanAlgebra + , HeytingAlgebra + , Eq + , Ord + ) + +instance Show (f (g a)) => Show (Composed f g a) where + show (Composed fga) = show fga + +instance Arbitrary a => Arbitrary (Composed Dropped Lifted a) where + arbitrary = frequency + [ (1, return $ Composed Top) + , (1, return $ Composed (Drop Bottom)) + , (2, Composed . Drop . Lift <$> arbitrary) + ] + + shrink (Composed Top) = [] + shrink (Composed (Drop Bottom)) = [Composed Top] + shrink (Composed (Drop (Lift a))) = + [ Composed Top, Composed (Drop Bottom) ] + ++ [ Composed (Drop (Lift a')) | a' <- shrink a ] + +instance Arbitrary a => Arbitrary (Composed Lifted Dropped a) where + arbitrary = frequency + [ (1, return $ Composed Bottom) + , (1, return $ Composed (Lift Top)) + , (2, Composed . Lift . Drop <$> arbitrary) + ] + +instance Arbitrary a => Arbitrary (Composed Lifted Lifted a) where + arbitrary = frequency + [ (1, return $ Composed Bottom) + , (1, return $ Composed (Lift Bottom)) + , (2, Composed . Lift . Lift <$> arbitrary) + ] + + shrink (Composed Bottom) = [] + shrink (Composed (Lift Bottom)) = [Composed Bottom] + shrink (Composed (Lift (Lift a))) = + [ Composed Bottom, Composed (Lift Bottom) ] + ++ [ Composed (Lift (Lift a')) | a' <- shrink a ] + +instance Arbitrary a => Arbitrary (Composed Dropped Dropped a) where + arbitrary = frequency + [ (1, return $ Composed Top) + , (1, return $ Composed (Drop Top)) + , (2, Composed . Drop . Drop <$> arbitrary) + ] + + shrink (Composed Top) = [] + shrink (Composed (Drop Top)) = [Composed Top] + shrink (Composed (Drop (Drop a))) = + [ Composed Top, Composed (Drop Top) ] + ++ [ Composed (Drop (Drop a')) | a' <- shrink a ] + + +main :: IO () +main = defaultMain tests + +tests :: TestTree +tests = + testGroup "heyting-algebras tests" + [ testGroup "BooleanAlgebra tests" + [ testProperty "Bool" $ prop_BooleanAlgebra @Bool + , testProperty "(Bool, Bool)" $ prop_BooleanAlgebra @(Bool, Bool) + , testProperty "Boolean (Lifted Bool)" $ prop_BooleanAlgebra @(Boolean (Arb (Lifted Bool))) + , testProperty "(Set S5)" $ prop_BooleanAlgebra @(Arb (S.Set S5)) + ] + , testGroup "Not BooleanAlgebra tests" + [ testProperty "Not BooleanAlgebra (Lifted Bool)" $ expectFailure $ prop_not @(Arb (Lifted Bool)) + , testProperty "Not BooleanAlgebra (Dropped Bool)" $ expectFailure $ prop_not @(Arb (Dropped Bool)) + , testProperty "Not BooleanAlgebra (Levitated Bool)" $ expectFailure $ prop_not @(Arb (Levitated Bool)) + ] + , testGroup "HeytingAlgebra tests" + [ testProperty "Lifted Bool" $ prop_HeytingAlgebra @(Arb (Lifted Bool)) + , testProperty "Dropped Bool" $ prop_HeytingAlgebra @(Arb (Dropped Bool)) + , testProperty "Layered Bool Bool" $ prop_HeytingAlgebra @(Arb (Layered Bool Bool)) + , testProperty "Levitated Bool" $ prop_HeytingAlgebra @(Arb (Levitated Bool)) + , testProperty "Sum (Lifted Bool) (Dropped Bool)" + $ prop_HeytingAlgebra @(Arb (Layered (Arb (Lifted Bool)) (Arb (Dropped Bool)))) + , testProperty "Map S5 Bool" $ prop_HeytingAlgebra @(Arb (M.Map S5 Bool)) + , testProperty "Dropped (Lifted Bool)" $ prop_HeytingAlgebra @(Composed Dropped Lifted Bool) + , testProperty "Lifted (Dropped Bool)" $ prop_HeytingAlgebra @(Composed Lifted Dropped Bool) + , testProperty "Lifted (Lifted Bool)" $ prop_HeytingAlgebra @(Composed Lifted Lifted Bool) + , testProperty "Dropped (Dropped Bool)" $ prop_HeytingAlgebra @(Composed Dropped Dropped Bool) + ] + ]