Difference between revisions of "99 questions/46 to 50"

Latest revision as of 02:38, 11 June 2023

This is part of Ninety-Nine Haskell Problems, based on Ninety-Nine Prolog Problems.

Logic and Codes

Problem 46

(**) Truth tables for logical expressions. Solutions

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.

Example:

(table A B (and A (or A B)))
true true true
true fail true
fail true fail
fail fail fail

Example in Haskell:

λ> table (\a b -> (and' a (or' a b)))
True True True
True False True
False True False
False False False

Problem 47

(*) Truth tables for logical expressions (part 2). Solutions

Continue Problem 46 by defining and/2, or/2, etc as being operators. This allows to write the logical expression in the more natural way, as in the example: A and (A or not B). Define operator precedence as usual; i.e. as in Java.

Example:

* (table A B (A and (A or not B)))
true true true
true fail true
fail true fail
fail fail fail

Example in Haskell:

λ> table2 (\a b -> a `and'` (a `or'` not b))
True True True
True False True
False True False
False False False

Problem 48

(*) Truth tables for logical expressions (part 3). Solutions

Generalize Problem 47 in such a way that the logical expression may contain any number of logical variables. Define table/2 in a way that table(List,Expr) prints the truth table for the expression Expr, which contains the logical variables enumerated in List.

Example:

* (table (A,B,C) (A and (B or C) equ A and B or A and C))
true true true true
true true fail true
true fail true true
true fail fail true
fail true true true
fail true fail true
fail fail true true
fail fail fail true

Example in Haskell:

λ> tablen 3 (\[a,b,c] -> a `and'` (b `or'` c) `equ'` a `and'` b `or'` a `and'` c)
-- infixl 3 `equ'`
True  True  True  True
True  True  False True
True  False True  True
True  False False True
False True  True  True
False True  False True
False False True  True
False False False True

-- infixl 7 `equ'`
True  True  True  True
True  True  False True
True  False True  True
True  False False False
False True  True  False
False True  False False
False False True  False
False False False False

Problem 49

(**) Gray codes. Solutions

An n-bit Gray code is a sequence of n-bit strings constructed according to certain rules. For example,

n = 1: C(1) = ['0','1'].
n = 2: C(2) = ['00','01','11','10'].
n = 3: C(3) = ['000','001','011','010',´110´,´111´,´101´,´100´].

Find out the construction rules and write a predicate with the following specification:

% gray(N,C) :- C is the N-bit Gray code

Can you apply the method of "result caching" in order to make the predicate more efficient, when it is to be used repeatedly?

Example in Haskell:

λ> gray 3
["000","001","011","010","110","111","101","100"]

Problem 50

(***) Huffman codes. Solutions

We suppose a set of symbols with their frequencies, given as a list of fr(S,F) terms. Example: [fr(a,45),fr(b,13),fr(c,12),fr(d,16),fr(e,9),fr(f,5)]. Our objective is to construct a list hc(S,C) terms, where C is the Huffman code word for the symbol S. In our example, the result could be Hs = [hc(a,'0'), hc(b,'101'), hc(c,'100'), hc(d,'111'), hc(e,'1101'), hc(f,'1100')] [hc(a,'01'),...etc.]. The task shall be performed by the predicate huffman/2 defined as follows:

% huffman(Fs,Hs) :- Hs is the Huffman code table for the frequency table Fs

Example in Haskell:

λ> huffman [('a',45),('b',13),('c',12),('d',16),('e',9),('f',5)]
[('a',"0"),('b',"101"),('c',"100"),('d',"111"),('e',"1101"),('f',"1100")]

Difference between revisions of "99 questions/46 to 50"

Latest revision as of 02:38, 11 June 2023

Logic and Codes

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

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@@ Line 1: / Line 1: @@
-[[99_Haskell_exercises|Back to 99 Haskell exercises]]
 __NOTOC__
+This is part of [[H-99:_Ninety-Nine_Haskell_Problems|Ninety-Nine Haskell Problems]], based on [https://prof.ti.bfh.ch/hew1/informatik3/prolog/p-99/ Ninety-Nine Prolog Problems].
-These are Haskell translations of [http://www.ic.unicamp.br/~meidanis/courses/mc336/2006s2/funcional/L-99_Ninety-Nine_Lisp_Problems.html Ninety Nine Lisp Problems].
-If you want to work on one of these, put your name in the block so we know someone's working on it. Then, change n in your block to the appropriate problem number, and fill in the <Problem description>,<example in lisp>,<example in Haskell>,<solution in haskell> and <description of implementation> fields.
 == Logic and Codes ==
 == Problem 46 ==
+<div style="border-bottom:1px solid #eee">(**) Truth tables for logical expressions. <span style="float:right"><small>[[99 questions/Solutions/46|Solutions]]</small></span>
+</div>
+&nbsp;<br>
-(**) 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.
+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.
+Example:
 <pre>
-Example:
 (table A B (and A (or A B)))
 true true true
@@ Line 24: / Line 24: @@
 fail true fail
 fail fail fail
+</pre>
 Example in Haskell:
-> table2 (\a b -> (and' a (or' a b))
+<haskell>
+λ> table (\a b -> (and' a (or' a b)))
 True True True
 True False True
 False True False
 False False False
-</pre>
-Solution:
-<haskell>
-not' :: Bool -> Bool
-not' True  = False
-not' False = True
-and',or',nor',nand',xor',impl',equ' :: Bool -> Bool -> Bool
-and' True True = True
-and' _    _    = False
-or' True _    = True
-or' _    True = True
-or' _    _    = False
-nor'  a b = not' $ or'  a b
-nand' a b = not' $ and' a b
-xor' True  False = True
-xor' False True  = True
-xor' _     _     = False
-impl' a b = (not' a) `or'` b
-equ' True  True  = True
-equ' False False = True
-equ' _     _     = False
-table2 :: (Bool -> Bool -> Bool) -> IO ()
-table2 f = putStrLn . unlines $ [show a ++ " " ++ show b ++ " " ++ show (f a b)
-                                | a <- [True, False], b <- [True, False]]
 </haskell>
-The implementations of the logic functions are quite verbose and can be shortened in places (like "equ' = (==)").
-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 :)
 == Problem 47 ==
+<div style="border-bottom:1px solid #eee">(*) Truth tables for logical expressions (part 2). <span style="float:right"><small>[[99 questions/Solutions/47|Solutions]]</small></span>
+</div>
+&nbsp;<br>
+Continue Problem 46 by defining and/2, or/2, etc as being operators. This allows to write the logical expression in the more natural way, as in the example: A and (A or not B). Define operator precedence as usual; i.e. as in Java.
-(*) Truth tables for logical expressions (2).
+Example:
-Continue problem P46 by defining and/2, or/2, etc as being operators. This allows to write the logical expression in the more natural way, as in the example: A and (A or not B). Define operator precedence as usual; i.e. as in Java.
 <pre>
-Example:
 * (table A B (A and (A or not B)))
 true true true
@@ Line 82: / Line 52: @@
 fail true fail
 fail fail fail
+</pre>
 Example in Haskell:
-> table2 (\a b -> a `and'` (a `or'` not b))
+<haskell>
+λ> table2 (\a b -> a `and'` (a `or'` not b))
 True True True
 True False True
 False True False
 False False False
-</pre>
-Solution:
-<haskell>
--- functions as in solution 46
-infixl 4 `or'`
-infixl 6 `and'`
--- "not" has fixity 9 by default
 </haskell>
-Java operator precedence (descending) as far as I could fathom it:
-<pre>
-logical not
-equality
-and
-xor
-or
-</pre>
-Using "not" as a non-operator is a little evil, but then again these problems were designed for languages other than haskell :)
 == Problem 48 ==
+<div style="border-bottom:1px solid #eee">(*) Truth tables for logical expressions (part 3). <span style="float:right"><small>[[99 questions/Solutions/48|Solutions]]</small></span>
+</div>
+&nbsp;<br>
+Generalize Problem 47 in such a way that the logical expression may contain any number of logical variables. Define table/2 in a way that table(List,Expr) prints the truth table for the expression Expr, which contains the logical variables enumerated in List.
-(**) Truth tables for logical expressions (3).
+Example:
-Generalize problem P47 in such a way that the logical expression may contain any number of logical variables. Define table/2 in a way that table(List,Expr) prints the truth table for the expression Expr, which contains the logical variables enumerated in List.
 <pre>
-Example:
 * (table (A,B,C) (A and (B or C) equ A and B or A and C))
 true true true true
@@ Line 127: / Line 84: @@
 fail fail true true
 fail fail fail true
+</pre>
 Example in Haskell:
-> table3 (\a b c -> a `and'` (b `or'` c) `equ'` a `and'` b `or'` a `and'` c)
-True True True True
-True True False True
-True False True True
-True False False True
-False True True True
-False True False True
-False False True True
-False False False True
-</pre>
-Solution:
 <haskell>
+λ> tablen 3 (\[a,b,c] -> a `and'` (b `or'` c) `equ'` a `and'` b `or'` a `and'` c)
--- functions as in solution 46
-infixl 4 `or'`
+-- infixl 3 `equ'`
+True  True  True  True
-infixl 4 `nor'`
+True  True  False True
-infixl 5 `xor'`
+True  False True  True
-infixl 6 `and'`
+True  False False True
-infixl 6 `nand'`
+False True  True  True
-infixl 3 `equ'` -- was 7, changing it to 3 got me the same results as in the original question :(
+False True  False True
+False False True  True
+False False False True
+-- infixl 7 `equ'`
-table3 :: (Bool -> Bool -> Bool -> Bool) -> IO ()
+True  True  True  True
-table3 f = putStrLn . unlines $ [show a ++ " " ++ show b ++ " " ++ show c ++ " " ++ show (f a b c)
+True  True  False True
-                                | a <- [True, False], b <- [True, False], c <- [True, False]]
+True  False True  True
+True  False False False
+False True  True  False
+False True  False False
+False False True  False
+False False False False
 </haskell>
-Using individual table functions for different numbers of variables is even more ugly, but anything else would be a bit of a pain in haskell AFAIK.
 == Problem 49 ==
+<div style="border-bottom:1px solid #eee">(**) Gray codes. <span style="float:right"><small>[[99 questions/Solutions/49|Solutions]]</small></span>
+</div>
-(**) Gray codes.
+&nbsp;<br>
 An n-bit Gray code is a sequence of n-bit strings constructed according to certain rules. For example,
 <pre>
 n = 1: C(1) = ['0','1'].
@@ Line 170: / Line 127: @@
 Find out the construction rules and write a predicate with the following specification:
+<pre>
 % gray(N,C) :- C is the N-bit Gray code
+</pre>
 Can you apply the method of "result caching" in order to make the predicate more efficient, when it is to be used repeatedly?
-<pre>
 Example in Haskell:
-P49> gray 3
-["000","001","011","010","110","111","101","100"]
-</pre>
-Solution:
 <haskell>
+λ> gray 3
-gray :: Int -> [String]
+["000","001","011","010","110","111","101","100"]
-gray 0 = [""]
-gray n = let xs = gray (n-1) in map ('0':) xs ++ map ('1':) (reverse xs)
 </haskell>
-It seems that the Gray code can be recursively defined in the way that for determining the gray code of n we take the Gray code of n-1, prepend a 0 to each word, take the Gray code for n-1 again, reverse it and prepend a 1 to each word. At last we have to append these two lists.
-(The [http://en.wikipedia.org/wiki/Gray_code Wikipedia article] seems to approve this.)
 == Problem 50 ==
+<div style="border-bottom:1px solid #eee">(***) Huffman codes. <span style="float:right"><small>[[99 questions/Solutions/50|Solutions]]</small></span>
+</div>
-(***) Huffman codes.
+&nbsp;<br>
 We suppose a set of symbols with their frequencies, given as a list of fr(S,F) terms. Example: [fr(a,45),fr(b,13),fr(c,12),fr(d,16),fr(e,9),fr(f,5)]. Our objective is to construct a list hc(S,C) terms, where C is the Huffman code word for the symbol S. In our example, the result could be Hs = [hc(a,'0'), hc(b,'101'), hc(c,'100'), hc(d,'111'), hc(e,'1101'), hc(f,'1100')] [hc(a,'01'),...etc.]. The task shall be performed by the predicate huffman/2 defined as follows:
+<pre>
 % huffman(Fs,Hs) :- Hs is the Huffman code table for the frequency table Fs
+</pre>
 Example in Haskell:
-<pre>
+<haskell>
-*Exercises> huffman [('a',45),('b',13),('c',12),('d',16),('e',9),('f',5)]
+λ> huffman [('a',45),('b',13),('c',12),('d',16),('e',9),('f',5)]
 [('a',"0"),('b',"101"),('c',"100"),('d',"111"),('e',"1101"),('f',"1100")]
-</pre>
+</haskell>
-Solution:
-<haskell>
-import Data.List
-data HTree = Leaf Char | Branch HTree HTree
-                deriving Show
-huffman :: (Ord a, Num a) => [(Char,a)] -> [(Char,[Char])]
-huffman freq = sortBy cmpFst $ serialize $
-        htree $ sortBy cmpFst $ [(w, Leaf c) | (c,w) <- freq]
-  where htree [(_, t)] = t
-        htree ((wx,tx):(wy,ty):rest) =
-                htree $ insertBy cmpFst (wx + wy, Branch tx ty) rest
-        cmpFst x y = compare (fst x) (fst y)
-        serialize (Branch l r) =
-                [(c, '0':code) | (c, code) <- serialize l] ++
-                [(c, '1':code) | (c, code) <- serialize r]
-        serialize (Leaf c) = [(c, "")]
-</haskell>
-The argument to <tt>htree</tt> is a list of (weight, tree) pairs, in order of increasing weight.
-The implementation could be made more efficient by using a priority queue instead of an ordered list.
 [[Category:Tutorials]]