I've been using the following data structure for the representation of propositional logic in Haskell:
data Prop = Pred String | Not Prop | And Prop Prop | Or Prop Prop | Impl Prop Prop | Equiv Prop Prop deriving (Eq, Ord)
Any comments on this structure are welcome.
However, now I want to extend my algoritms to handle FOL - predicate logic. What would be a good way of representing FOL in Haskell?
I've seen versions that are - pretty much - an extension of the above, and versions that are based on more classic context-free grammars. Is there any literature on this, that could be recommended?
A predicate is a function of some value of type a to a Result , i.e. a Bool -like value with Okay as True and Fail as False , which carries additional data in each branch.
Example: P(x,y): “x + 2 = y” is a predicate. It has two variables x and y; Universe of Discourse: x is in {1,2,3}; y is in {4,5,6}. P(1,4) : 1 + 2 = 4 is a proposition (it is F); P(2,4) : 2 + 2 = 4 is a proposition (it is T);
Prolog is a programming language based on predicate logic. A Prolog program attempts to prove a goal, such as brother(Barney,x), from a set of facts and rules.
This is known as higher-order abstract syntax.
First solution: Use Haskell's lambda. A datatype could look like:
data Prop = Not Prop | And Prop Prop | Or Prop Prop | Impl Prop Prop | Equiv Prop Prop | Equals Obj Obj | ForAll (Obj -> Prop) | Exists (Obj -> Prop) deriving (Eq, Ord) data Obj = Num Integer | Add Obj Obj | Mul Obj Obj deriving (Eq, Ord)
You can write a formula as:
ForAll (\x -> Exists (\y -> Equals (Add x y) (Mul x y))))
This is described in detail in in The Monad Reader article. Highly recommended.
Second solution:
Use strings like
data Prop = Not Prop | And Prop Prop | Or Prop Prop | Impl Prop Prop | Equiv Prop Prop | Equals Obj Obj | ForAll String Prop | Exists String Prop deriving (Eq, Ord) data Obj = Num Integer | Var String | Add Obj Obj | Mul Obj Obj deriving (Eq, Ord)
Then you can write a formula like
ForAll "x" (Exists "y" (Equals (Add (Var "x") (Var "y"))) (Mul (Var "x") (Var "y"))))))
The advantage is that you can show the formula easily (it's hard to show a Obj -> Prop
function). The disadvantage is that you have to write changing names on collision (~alpha conversion) and substitution (~beta conversion). In both solutions, you can use GADT instead of two datatypes:
data FOL a where True :: FOL Bool False :: FOL Bool Not :: FOL Bool -> FOL Bool And :: FOL Bool -> FOL Bool -> FOL Bool ... -- first solution Exists :: (FOL Integer -> FOL Bool) -> FOL Bool ForAll :: (FOL Integer -> FOL Bool) -> FOL Bool -- second solution Exists :: String -> FOL Bool -> FOL Bool ForAll :: String -> FOL Bool -> FOL Bool Var :: String -> FOL Integer -- operations in the universe Num :: Integer -> FOL Integer Add :: FOL Integer -> FOL Integer -> FOL Integer ...
Third solution: Use numerals to represent where the variable is bound, where lower means deeper. For example, in ForAll (Exists (Equals (Num 0) (Num 1))) the first variable will bind to Exists, and second to ForAll. This is known as de Bruijn numerals. See I am not a number - I am a free variable.
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