在Haskell中将命题逻辑扩展为模态逻辑 [英] Extending propositional logic to modal logic in Haskell
问题描述
我已经在Haskell中编写了一些代码,用于对命题逻辑进行建模
I have written some code in Haskell for modeling propositional logic
data Formula = Prop {propName :: String}
| Neg Formula
| Conj Formula Formula
| Disj Formula Formula
| Impl Formula Formula
| BiImpl Formula Formula
deriving (Eq,Ord)
但是,由于数据类型是封闭的,因此没有自然的方法可以将此扩展到Modal Logic.因此,我认为我应该改用类.这样,以后我可以轻松地在不同的模块中添加新的语言功能.问题是我不知道该怎么写.我想要以下类似的东西
However, there is no natural way to extend this to Modal Logic, since the data type is closed. Therefore, I thought I should use classes instead. That way, I can easily add new language features in different modules later on. The problem is that I don't exactly know how to write it. I would like something like the following
type PropValue = (String,Bool) -- for example ("p",True) states that proposition p is true
type Valuation = [PropValue]
class Formula a where
evaluate :: a -> Valuation -> Bool
data Proposition = Prop String
instance Formula Proposition where
evaluate (Prop s) val = (s,True) `elem` val
data Conjunction = Conj Formula Formula -- illegal syntax
instance Formula Conjunction where
evaluate (Conj φ ψ) v = evaluate φ v && evaluate ψ v
错误当然是在连接词的定义中.但是,我不清楚如何重写它才能使其正常工作.
The mistake is of course in the definition of Conjunction. However, it is unclear to me how I could rewrite it so that it works.
推荐答案
这应该有效:
data Conjunction f = Conj f f
instance Formula f => Formula (Conjunction f) where
evaluate (Conj φ ψ) v = evaluate φ v && evaluate ψ v
但是,我不确定类型类是否是您要实现的目标的正确工具.
However, I am not sure type classes are the right tool for what you are trying to achieve.
也许您可以旋转使用显式类型级别的仿函数并对其进行重复操作:
Maybe you could give a whirl to using explicit type level functors and recurring over them:
-- functor for plain formulae
data FormulaF f = Prop {propName :: String}
| Neg f
| Conj f f
| Disj f f
| Impl f f
| BiImpl f f
-- plain formula
newtype Formula = F {unF :: FormulaF Formula}
-- functor adding a modality
data ModalF f = Plain f
| MyModality f
-- modal formula
newtype Modal = M {unM :: ModalF Modal}
是的,这并不是十分方便,因为诸如 F,M,Plain
之类的构造函数有时会妨碍您的工作.但是,与类型类不同,您可以在此处使用模式匹配.
Yes, this is not terribly convenient since constructors such as F,M,Plain
get sometimes in the way. But, unlike type classes, you can use pattern matching here.
另一种选择是使用GADT:
As another option, use a GADT:
data Plain
data Mod
data Formula t where
Prop {propName :: String} :: Formula t
Neg :: Formula t -> Formula t
Conj :: Formula t -> Formula t -> Formula t
Disj :: Formula t -> Formula t -> Formula t
Impl :: Formula t -> Formula t -> Formula t
BiImpl :: Formula t -> Formula t -> Formula t
MyModality :: Formula Mod -> Formula Mod
type PlainFormula = Formula Plain
type ModalFormula = Formula Mod
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