example: Add the section on TBF.

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Sergiu Ivanov 2020-07-12 00:31:24 +02:00
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@ -688,6 +688,84 @@ tab
the - arcs in the graph, as it does not have access to the symbolic
description of the function.
** Threshold Boolean functions (TBF)
=dds= includes some useful definitions for working with threshold
Boolean functions (TBF). A TBF is defined as a vector of weights
and a threshold. For example, the following defines a
TBF implementing the logical AND.
#+BEGIN_SRC racket :results output drawer
(tbf #(1 1) 1)
#+END_SRC
#+RESULTS:
:RESULTS:
(tbf '#(1 1) 1)
:END:
This TBF only returns 1 when both inputs are activated, which
brings their weighted some above 1: 1 \cdot 1 + 1 \cdot 1 = 2 > 1.
#+BEGIN_SRC racket :results output drawer
(apply-tbf (tbf #(1 1) 1) #(1 1))
#+END_SRC
#+RESULTS:
:RESULTS:
1
:END:
Let's actually check out the truth table of this TBF.
#+BEGIN_SRC racket :results table drawer
(tbf-tabulate (tbf #(1 1) 1))
#+END_SRC
#+RESULTS:
:RESULTS:
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
:END:
This truth table corresponds indeed to the logical AND.
=dds= allows reading TBFs from Org tables. In this case, the last
column in each row is treated as the threshold, while the first
values are taken to be the weights. Consider, for example, the
following table:
#+NAME: simple-tbf
| 1 | 1 | 0 |
| 1 | 1 | 1 |
You can read the two TBFs defined in this table in the following
way:
#+BEGIN_SRC racket :results output drawer :var simple-tbf=munch-sexp(simple-tbf)
(read-org-tbfs simple-tbf)
#+END_SRC
#+RESULTS:
:RESULTS:
(list (tbf '#(1 1) 0) (tbf '#(1 1) 1))
:END:
The first TBF implements the logical OR of its inputs, while the
second TBF implements the logical AND. Let's check it by
tabulating both functions:
#+BEGIN_SRC racket :results table drawer :var simple-tbf=munch-sexp(simple-tbf)
(tbf-tabulate* (read-org-tbfs simple-tbf))
#+END_SRC
#+RESULTS:
:RESULTS:
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 |
:END:
The first two columns of this table give the values of the two
inputs. The third column gives the values of the first TBF, and
the fourth column gives the values of the second TBF.
* Reaction systems
:PROPERTIES:
:header-args:racket: :prologue "#lang racket\n(require graph dds/rs dds/utils)"