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Now, write a predicate table/3 which prints the truth table of a given logical expression in two variables.
Now, write a predicate table/3 which prints the truth table of a given logical expression in two variables.
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The first step in this problem is to define the Boolean predicates:
<haskell>
<haskell>
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-- NOT negates a single Boolean argument
not' :: Bool -> Bool
not' :: Bool -> Bool
not' True = False
not' True = False
not' False = True
not' False = True
 +
-- Type signature for remaining logic functions
and',or',nor',nand',xor',impl',equ' :: Bool -> Bool -> Bool
and',or',nor',nand',xor',impl',equ' :: Bool -> Bool -> Bool
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 +
-- AND is True if both a and b are True
and' True True = True
and' True True = True
and' _ _ = False
and' _ _ = False
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-- OR is True if a or b or both are True
or' False False = False
or' False False = False
or' _ _ = True
or' _ _ = True
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-- NOR is the negation of 'or'
nor' a b = not' $ or' a b
nor' a b = not' $ or' a b
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 +
-- NAND is the negation of 'and'
nand' a b = not' $ and' a b
nand' a b = not' $ and' a b
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-- XOR is True if either a or b is True, but not if both are True
xor' True False = True
xor' True False = True
xor' False True = True
xor' False True = True
xor' _ _ = False
xor' _ _ = False
 +
-- IMPL is True if a implies b, equivalent to (not a) or (b)
impl' a b = (not' a) `or'` b
impl' a b = (not' a) `or'` b
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-- EQU is True if a and b are equal
equ' True True = True
equ' True True = True
equ' False False = True
equ' False False = True
equ' _ _ = False
equ' _ _ = False
 +
</haskell>
-
table2 :: (Bool -> Bool -> Bool) -> IO ()
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The above implementations build each logic function from scratch; they could be shortened using Haskell's builtin equivalents:
-
table2 f = mapM_ putStrLn [show a ++ " " ++ show b ++ " " ++ show (f a b)
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-
| a <- [True, False], b <- [True, False]]
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<haskell>
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and' a b = a && b
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or' a b = a || b
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nand' a b = not (and' a b)
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nor' a b = not (or' a b)
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xor' a b = not (equ' a b)
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impl' a b = or' (not a) b
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equ' a b = a == b
</haskell>
</haskell>
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The implementations of the logic functions are quite verbose and can be shortened in places (like "equ' = (==)").
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Some could be reduced even further using [[Pointfree]] style:
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 +
<haskell>
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and' = (&&)
 +
or' = (||)
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equ' = (==)
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</haskell>
 +
 
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The only remaining task is to generate the truth table; most of the complexity here comes from the string conversion and IO. The approach used here accepts a Boolean function <tt>(Bool -> Bool -> Bool)</tt>, then calls that function with all four combinations of two Boolean values, and converts the resulting values into a list of space-separated strings. Finally, the strings are printed out by mapping <hask>putStrLn</hask> across the list of strings:
 +
 
 +
<haskell>
 +
table :: (Bool -> Bool -> Bool) -> IO ()
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table f = mapM_ putStrLn [show a ++ " " ++ show b ++ " " ++ show (f a b)
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| a <- [True, False], b <- [True, False]]
 +
</haskell>
The table function in Lisp supposedly uses Lisp's symbol handling to substitute variables on the fly in the expression. I chose passing a binary function instead because parsing an expression would be more verbose in haskell than it is in Lisp. Template Haskell could also be used :)
The table function in Lisp supposedly uses Lisp's symbol handling to substitute variables on the fly in the expression. I chose passing a binary function instead because parsing an expression would be more verbose in haskell than it is in Lisp. Template Haskell could also be used :)
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 +
The table function can be generalized to work for any given binary function and domain.
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<haskell>
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table :: (Bool -> Bool -> Bool) -> String
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table f = printBinary f [True, False]
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printBinary :: (Show a, Show b) => (a -> a -> b) -> [a] -> String
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printBinary f domain = concatMap (++ "\n") [printBinaryInstance f x y | x <- domain, y <- domain]
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printBinaryInstance :: (Show a, Show b) => (a -> a -> b) -> a -> a -> String
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printBinaryInstance f x y = show x ++ " " ++ show y ++ " " ++ show (f x y)
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</haskell>

Current revision

(**) Define predicates and/2, or/2, nand/2, nor/2, xor/2, impl/2 and equ/2 (for logical equivalence) which succeed or fail according to the result of their respective operations; e.g. and(A,B) will succeed, if and only if both A and B succeed.

A logical expression in two variables can then be written as in the following example: and(or(A,B),nand(A,B)).

Now, write a predicate table/3 which prints the truth table of a given logical expression in two variables.

The first step in this problem is to define the Boolean predicates: 

-- NOT negates a single Boolean argument
not' :: Bool -> Bool
not' True  = False
not' False = True
 
-- Type signature for remaining logic functions
and',or',nor',nand',xor',impl',equ' :: Bool -> Bool -> Bool
 
-- AND is True if both a and b are True
and' True True = True
and' _    _    = False
 
-- OR is True if a or b or both are True
or' False False = False
or' _     _     = True
 
-- NOR is the negation of 'or'
nor'  a b = not' $ or'  a b
 
-- NAND is the negation of 'and'
nand' a b = not' $ and' a b
 
-- XOR is True if either a or b is True, but not if both are True
xor' True  False = True
xor' False True  = True
xor' _     _     = False
 
-- IMPL is True if a implies b, equivalent to (not a) or (b)
impl' a b = (not' a) `or'` b
 
-- EQU is True if a and b are equal
equ' True  True  = True
equ' False False = True
equ' _     _     = False

The above implementations build each logic function from scratch; they could be shortened using Haskell's builtin equivalents: 

and'  a b = a && b
or'   a b = a || b
nand' a b = not (and' a b)
nor'  a b = not (or' a b)
xor'  a b = not (equ' a b)
impl' a b = or' (not a) b
equ'  a b = a == b

Some could be reduced even further using Pointfree style: 

and' = (&&)
or'  = (||)
equ' = (==)
The only remaining task is to generate the truth table; most of the complexity here comes from the string conversion and IO. The approach used here accepts a Boolean function (Bool -> Bool -> Bool), then calls that function with all four combinations of two Boolean values, and converts the resulting values into a list of space-separated strings. Finally, the strings are printed out by mapping
putStrLn
across the list of strings: 
table :: (Bool -> Bool -> Bool) -> IO ()
table f = mapM_ putStrLn [show a ++ " " ++ show b ++ " " ++ show (f a b)
                                | a <- [True, False], b <- [True, False]]

The table function in Lisp supposedly uses Lisp's symbol handling to substitute variables on the fly in the expression. I chose passing a binary function instead because parsing an expression would be more verbose in haskell than it is in Lisp. Template Haskell could also be used :)

The table function can be generalized to work for any given binary function and domain. 

table :: (Bool -> Bool -> Bool) -> String
table f = printBinary f [True, False]
 
printBinary :: (Show a, Show b) => (a -> a -> b) -> [a] -> String
printBinary f domain = concatMap (++ "\n") [printBinaryInstance f x y | x <- domain, y <- domain]
 
printBinaryInstance :: (Show a, Show b) => (a -> a -> b) -> a -> a -> String
printBinaryInstance f x y = show x ++ " " ++ show y ++ " " ++ show (f x y)
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