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3. Logic and Codes


Solutions can be found here.
3.01 (**) Truth tables for logical expressions.
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. Note that A and B can be Prolog goals (not only the constants true and fail).

A logical expression in two variables can then be written in prefix notation, 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


3.02 (*) Truth tables for logical expressions (2).
Continue problem 3.01 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


3.03 (**) Truth tables for logical expressions (3).
Generalize problem 3.02 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


3.04 (**) Gray code.
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?

3.05 (***) Huffman code.
First of all, study a good book on discrete mathematics or algorithms for a detailed description of Huffman codes, or consult Wikipedia

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

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