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Weak Bases - an encoding of Post's lattice in Logtalk.
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Release 1.0.
Copyright (c) 2014  Victor Lagerkvist. All Rights Reserved.
Weak Bases is free software. You can redistribute it and/or
modify it under the terms of the simplified BSD license.

CONTENTS

 1. License
 2. About
 3. Weak Bases web site
 4. Installation and running
 5. Examples
 6. Authors

1. LICENSE

Copyright 2014 Victor Lagerkvist. All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:

   1. Redistributions of source code must retain the above copyright
      notice, this list of conditions and the following disclaimer.

   2. Redistributions in binary form must reproduce the above copyright
      notice, this list of conditions and the following disclaimer in the
      documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS ``AS IS'' AND ANY EXPRESS
OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
EVENT SHALL THE COPYRIGHT HOLDERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT,
INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE,
EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

The views and conclusions contained in the software and documentation
are those of the authors and should not be interpreted as representing
official policies, either expressed or implied, of the copyright holders.

2. ABOUT
Weak Bases is an encoding of Post's lattice in Prolog, which main
function is to calculate so called weak bases of a Boolean co-clone,
but it can also be used for queries about the inclusion structure of
the lattice, such as finding the smallest co-clone of a given
relation.

3. WEAK BASES WEB SITE

Visit the Weak Bases GitHub page at:
https://github.com/Joelbyte/weak-bases.

4. INSTALLATION AND RUNNING

Weak Bases requires Logtalk 2.40.0 or a later version.

To use the latest version of Weak Bases, fetch the latest source
code, either as an archive or from the git repository, extract it to
a directory of your choice, and:

* Start Logtalk from that directory.
* Type {loader}. (Including '.').

5. PROGRAM STRUCTURE
The bulk of the program is spread across the two files relations.lgt
and operators.lgt. As an Argus-eyed reader might guess from the file
names the former contains predicates related (pun not intended) to
relations while the latter contains predicates related to
functions. More specifically relations.lgt consists of four important
predicates:

* relation/2. 
* pol/2.
* weak_base/2.
* smallest_co_clone/2.

relation(IC, R) is true if R is a base of the co-clone IC, where R is
represented as a list of Boolean tuples. pol(IC, F) is true if F is a
base of pol(IC), where F is a list of functions. weak_base(IC, R) is
true if R is a weak base of IC which does not contain any redundant
columns. smallest_co_clone(R1, R2) is true if R2 is a base of the
smallest co-clone containing R1. There are also a few predicates for
printing relations of which the most important are write_latex/1, which
prints its single argument as a pmatrix environment in LaTeX.

The file operators.lgt consists of the
following two important predicates:

* operator/3.
* close_relation/3.

operator(F, Args, Res) is true if the function F applied to the
arguments of Args is Res. close_relation(R, Fs, R1) is true if R1 is
the smallest superset of R closed under the functions Fs.

6. EXAMPLES

Follow the previous instructions to get everything up and
running. Many of the predicates can be used both for testing and
finding solutions. For example, to find a weak base of the co-clone
BR, one issues the following commands.

[1]  ?- relations::weak_base(br/_, R).

Which gives the solution R =
[[0,0,0,0,1,1,1,1],[0,0,1,1,0,0,1,1],[0,1,0,1,0,1,0,1]].

If one has a relation and want to find its generating co-clone the
following command can be issued.

[1]  ?- R1 = [[0,0,1],[0,1,0],[1,0,0],[1,1,0],[1,0,1],[0,1,1]], relations::smallest_co_clone(R1, R2).

Which gives the answer R2 = in2/3, i.e. the set of all Boolean
relations closed under complement.

7. AUTHORS

Weak Bases was written by Victor Lagerkvist during the spring of 2013
and 2014.