Subset sum
=======================

Description
-----------

This example tries to solve the well-known NP-complete problem of subset sum, by means of the so called simple Kernel P systems (skP systems).

Model
-----

The corresponding P-Lingua file is the following::

	@model<simple_kernel_psystems>

	def main()
	{
	  call subset_sum();
	}

	def subset_sum()
	{
	  call init_cells();  
	  call init_multisets();  
	  call init_rules();
	}

	def init_cells()
	{
	  @mu = [[]'1 []'2]'0;
	}

	def init_multisets()
	{
	  @ms(1) = S;
	  @ms(2) = A{1};
	  @ms(2) += v{i}*w{i} : 1<=i<=n;

	  /* Environment alphabet: */
	}

	def init_rules()
	{
	  /* C1 */
	  /* r1,1 */ @guard {>=+T} ? [S]'1 --> [yes]'0;
	  /* r1,2 */ @guard {>=+F} && {>=-T} ? [S]'1 --> [no]'0;

	  /* C2 */
	  /* r2,i */ [A{i}]'2 |--> [B{i},A{i+1}]'2 [A{i+1}]'2 : 1<=i<n;
	  /* r2,n */ [A{n}]'2 |--> [B{n},X]'2 [X]'2;
	  /* r2,n+i */ @guard {=+B{i}}&&{=+X} ? [v{i}]'2 --> [v]'2 : 1<=i<=n;
	  /* r2,2n+1 */ [X]'2 --> [Y]'2;
	  /* r2,2n+2 */ @guard {=-v*k} ? [Y]'2 --> [F]'1;
	  /* r2,2n+3 */ @guard {=+v*k} ? [Y]'2 --> [T]'1;
	}

Files
-----

.. container:: boldlink

    * `Custom application <http://www.gcn.us.es/redmine/dmsf/files/2957/download>`_, defining the needed input and output tables for subset sum problem.
    * `Model file <http://www.gcn.us.es/redmine/dmsf/files/2959/download>`_, P-Lingua file with the parametrized file specifying the family of P systems for solving the subset sum problem. The custom interface enable the user to introduce the data corresponding to each different scenario.
    * `Scenario file <http://www.gcn.us.es/redmine/dmsf/files/2958/download>`_, for a specific instance of the problem (that is, a specific graph to colour), for a specific instance of the problem.


References
----------

* `A Linear Solution of Subset Sum Problem by Using Membrane Creation <http://www.springerlink.com/content/hhk559wmhg0qw87l/>`_.
* `Solving the independent set problem by using tissue-like p systems with cell division <http://www.springerlink.com/content/4k32v721u7725mx7/>`_ (`detailed publication info <http://www.gcn.us.es/?q=node/491>`_).