# -*- coding: utf-8 -*-
#+TITLE: Playing with explicit Kripke structures
#+DESCRIPTION: Kripke structures that are fully stored in memory, as explicit graphs.
#+INCLUDE: setup.org
#+HTML_LINK_UP: tut.html

Kripke structures, can be defined as ω-automata in which labels are on
states, and where all runs are accepting (i.e., the acceptance
condition is =t=).  They are typically used by model checkers to
represent the state space of the model to verify.

On [[file:tut51.org][another page]] we have described how to implement a Kripke structure
that can be explored on the fly, by implementing a function that take
the current state, and produce its successor states.  While
implementing an on-the-fly Kripke structure is good for large example,
it requires some implementation effort we are not always willing to
put for a toy example.

This document shows how to create a Kripke structure that is stored as
an explicit graph.  The class for those is =spot::kripke_graph= and
works similarly to the class =spot::twa_graph= (used for
automata).  The main difference between those two classes is that
Kripke structures labels the states instead of the transitions.  Using
=spot::kripke_graph= instead of =spot::twa_graph= saves a bit of
memory.

Our Kripke structure represent a model whose states consist of pairs
of modulo-3 integers $(x,y)$. At any state the possible actions
will be to increment any one of the two integers (nondeterministically).
That increment is obviously done modulo 3.  For instance state $(1,2)$
has two possible successors:
  - $(2,2)$ if =x= was incremented, or
  - $(1,0)$ if =y= was incremented.
Initially both variables will be 0.  The complete state space has
9 states, that we will store explicitly as a graph.

In addition, we would like to label each state by atomic propositions
=odd_x= and =odd_y= that are true only when the corresponding
variables are odd.  Using such variables, we could try to verify
whether if =odd_x= infinitely often holds, then =odd_y= infinitely
often holds as well.


* C++ implementation
** Building the state space

Here is how we could create the Kripke structure:

#+NAME: create-kripke
#+BEGIN_SRC C++
  #include <spot/kripke/kripkegraph.hh>

  spot::kripke_graph_ptr create_kripke(spot::bdd_dict_ptr dict,
                                       bool named_states = false)
  {
    spot::kripke_graph_ptr k = spot::make_kripke_graph(dict);

    bdd odd_x = bdd_ithvar(k->register_ap("odd_x"));
    bdd odd_y = bdd_ithvar(k->register_ap("odd_y"));

    unsigned x0y0 = k->new_state((!odd_x) & !odd_y);
    unsigned x0y1 = k->new_state((!odd_x) & odd_y);
    unsigned x0y2 = k->new_state((!odd_x) & !odd_y);
    unsigned x1y0 = k->new_state(odd_x & !odd_y);
    unsigned x1y1 = k->new_state(odd_x & odd_y);
    unsigned x1y2 = k->new_state(odd_x & !odd_y);
    unsigned x2y0 = k->new_state((!odd_x) & !odd_y);
    unsigned x2y1 = k->new_state((!odd_x) & odd_y);
    unsigned x2y2 = k->new_state((!odd_x) & !odd_y);

    k->set_init_state(x0y0);

    k->new_edge(x0y0, x0y1); k->new_edge(x0y0, x1y0);
    k->new_edge(x0y1, x0y2); k->new_edge(x0y1, x1y1);
    k->new_edge(x0y2, x0y0); k->new_edge(x0y2, x1y2);
    k->new_edge(x1y0, x1y1); k->new_edge(x1y0, x2y0);
    k->new_edge(x1y1, x1y2); k->new_edge(x1y1, x2y1);
    k->new_edge(x1y2, x1y0); k->new_edge(x1y2, x2y2);
    k->new_edge(x2y0, x2y1); k->new_edge(x2y0, x0y0);
    k->new_edge(x2y1, x2y2); k->new_edge(x2y1, x0y1);
    k->new_edge(x2y2, x2y0); k->new_edge(x2y2, x0y2);

    if (named_states)
      {
        auto names = new std::vector<std::string>
          { "x0y0", "x0y1", "x0y2",
            "x1y0", "x1y1", "x1y2",
            "x2y0", "x2y1", "x2y2" };
        k->set_named_prop("state-names", names);
      }
    return k;
  }
#+END_SRC

To display this Kripke structure, we can call the =print_dot()=
function:

#+NAME: print-dot
#+BEGIN_SRC C++ :noweb strip-export :results verbatim :export code
  #include <spot/twaalgos/dot.hh>
  <<create-kripke>>
  int main()
  {
    spot::bdd_dict_ptr dict = spot::make_bdd_dict();
    spot::print_dot(std::cout, create_kripke(dict));
  }
#+END_SRC

=print_dot()= will output the graph in [[https://graphviz.gitlab.io/][GraphViz's syntax]], and you just
have to pass it to GraphViz's =dot= command to render it:

#+BEGIN_SRC dot :file kripke-52.svg :cmd circo :var txt=print-dot :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:kripke-52.svg]]

By default, states are just numbered.  If you want to name them, for
instance for debugging, you should define the ="state-names"=
properties to a vector of names.  Our =create_kripke()= function does
that when passed a true argument.

#+NAME: print-dot-2
#+BEGIN_SRC C++ :noweb strip-export :results verbatim :export code
  #include <spot/twaalgos/dot.hh>
  <<create-kripke>>
  int main()
  {
    spot::bdd_dict_ptr dict = spot::make_bdd_dict();
    spot::print_dot(std::cout, create_kripke(dict, true));
  }
#+END_SRC

#+BEGIN_SRC dot :file kripke-52b.svg :cmd circo :var txt=print-dot-2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:kripke-52b.svg]]

** Checking a property on this state space

Let us pretend that we want to verify the following property: if
=odd_x= infinitely often holds, then =odd_y= infinitely often holds.

In LTL, that would be =GF(odd_x) -> GF(odd_y)=.

To check this formula, we translate its negation into an automaton,
build the product of this automaton with our Kripke structure, and
check whether the output is empty.  If it is not, that means we have
found a counterexample.  Here is some code that would show this
counterexample:

#+NAME: demo-2
#+BEGIN_SRC C++ -r :exports code :noweb strip-export :results verbatim
#include <spot/tl/parse.hh>
#include <spot/twaalgos/translate.hh>
#include <spot/twaalgos/emptiness.hh>
<<create-kripke>>
int main()
{
   auto d = spot::make_bdd_dict();

   // Parse the input formula.
   spot::parsed_formula pf = spot::parse_infix_psl("GF(odd_x) -> GF(odd_y)");
   if (pf.format_errors(std::cerr))
     return 1;

   // Translate its negation.
   spot::formula f = spot::formula::Not(pf.f);
   spot::twa_graph_ptr af = spot::translator(d).run(f);

   // Find a run of our Kripke structure that intersects af.
   auto k = create_kripke(d, true);            (ref:ck)
   if (auto run = k->intersecting_run(af))
     std::cout << "formula is violated by the following run:\n" << *run;
   else
     std::cout << "formula is verified\n";
}
#+END_SRC

# temporary fix for an issue in Org 9.2, see
# http://lists.gnu.org/archive/html/emacs-orgmode/2019-01/msg00226.html
#+BEGIN_SRC text :noweb yes
<<demo-2()>>
#+END_SRC

Note that this main function is similar to the main function we used
for [[file:tut51.org::#check-prop][the on-the-fly version]] except for [[(ck)][the line that creates the Kripke
structure]].  You can modify it to display the counterexample similarly.

* Python implementation
** Building the state space

#+NAME: create-kripke-py
#+BEGIN_SRC python -r
  import spot
  from buddy import bdd_ithvar

  def create_kripke(bdddict, name_states=False):
    k = spot.make_kripke_graph(bdddict)
    odd_x = bdd_ithvar(k.register_ap("odd_x"))
    odd_y = bdd_ithvar(k.register_ap("odd_y"))

    x0y0 = k.new_state(-odd_x & -odd_y);
    x0y1 = k.new_state(-odd_x & odd_y);
    x0y2 = k.new_state(-odd_x & -odd_y);
    x1y0 = k.new_state(odd_x & -odd_y);
    x1y1 = k.new_state(odd_x & odd_y);
    x1y2 = k.new_state(odd_x & -odd_y);
    x2y0 = k.new_state(-odd_x & -odd_y);
    x2y1 = k.new_state(-odd_x & odd_y);
    x2y2 = k.new_state(-odd_x & -odd_y);

    k.set_init_state(x0y0);

    k.new_edge(x0y0, x0y1); k.new_edge(x0y0, x1y0);
    k.new_edge(x0y1, x0y2); k.new_edge(x0y1, x1y1);
    k.new_edge(x0y2, x0y0); k.new_edge(x0y2, x1y2);
    k.new_edge(x1y0, x1y1); k.new_edge(x1y0, x2y0);
    k.new_edge(x1y1, x1y2); k.new_edge(x1y1, x2y1);
    k.new_edge(x1y2, x1y0); k.new_edge(x1y2, x2y2);
    k.new_edge(x2y0, x2y1); k.new_edge(x2y0, x0y0);
    k.new_edge(x2y1, x2y2); k.new_edge(x2y1, x0y1);
    k.new_edge(x2y2, x2y0); k.new_edge(x2y2, x0y2);

    if name_states:
      k.set_state_names(["x0y0", "x0y1", "x0y2",   (ref:ns)
                         "x1y0", "x1y1", "x1y2",
                         "x2y0", "x2y1", "x2y2"])
    return k;
#+END_SRC

To display this structure, we would just call =to_str('dot')= (and
convert the resulting textual output into graphical form using the
=dot= command).

#+NAME: print-dot-py
#+BEGIN_SRC python :noweb strip-export :results output :export code
<<create-kripke-py>>
k = create_kripke(spot.make_bdd_dict())
print(k.to_str('dot'))
#+END_SRC

=print_dot()= will output the graph in [[https://graphviz.gitlab.io/][GraphViz's syntax]], and you just
have to pass it to GraphViz's =dot= command to render it:

#+BEGIN_SRC dot :file kripke-52c.svg :cmd circo :var txt=print-dot-py :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:kripke-52c.svg]]

Again, states may be named if that can help.  This is done by passing
a vector of names (indexed by state numbers) to the =set_name_states()= method,
[[(ns)][as done conditionally in our =create_kripke()= function]].


#+NAME: print-dot-py-2
#+BEGIN_SRC python :noweb strip-export :results output :export code
<<create-kripke-py>>
k = create_kripke(spot.make_bdd_dict(), True)
print(k.to_str('dot'))
#+END_SRC

#+BEGIN_SRC dot :file kripke-52d.svg :cmd circo :var txt=print-dot-py-2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:kripke-52d.svg]]


** Checking a property on this state space

Here is the Python code equivalent to our C++ check.  Please
read the C++ description for details.

#+NAME: demo-2py
#+BEGIN_SRC python -r :exports code :noweb strip-export :results output
  <<create-kripke-py>>

  d = spot.make_bdd_dict()

  # Translate the negation of the formula
  f = spot.formula.Not("GF(odd_x) -> GF(odd_y)")
  af = spot.translate(f, dict=d)

  # Find a run of our Kripke structure that intersects af.
  k = create_kripke(d, True)
  run = k.intersecting_run(af)
  if run:
    print("formula is violated by the following run:\n", run)
  else:
    print("formula is verified")
#+END_SRC

# temporary fix for an issue in Org 9.2, see
# http://lists.gnu.org/archive/html/emacs-orgmode/2019-01/msg00226.html
#+BEGIN_SRC text :noweb yes
<<demo-2py()>>
#+END_SRC

#  LocalWords:  utf html kripke SRC bool bdd ithvar ap init noweb svg
#  LocalWords:  GraphViz's cmd circo txt GF af py bdddict ns str
