# -*- coding: utf-8 -*-
#+TITLE: Creating an alternating automaton by adding states and transitions
#+DESCRIPTION: Code example for constructing alternating ω-automata in Spot
#+SETUPFILE: setup.org
#+HTML_LINK_UP: tut.html

This example demonstrates how to create the following alternating
co-Büchi automaton (recognizing =GFa=) and then print it.

#+NAME: tut23-dot
#+BEGIN_SRC sh :results verbatim :exports none :var txt=tut23-cpp
autfilt --dot=.a <<EOF
$txt
EOF
#+END_SRC

#+BEGIN_SRC dot :file tut23-aut.svg :var txt=tut23-dot :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:tut23-aut.svg]]


Note that the code is very similar to the [[file:tut22.org][previous example]]: in Spot an
alternating automaton is just an automaton that uses a mix of standard
edges (declared with =new_edge()=) and universal edges (declared with
=new_univ_edge()=).

* C++
  :PROPERTIES:
  :CUSTOM_ID: cpp
  :END:

#+NAME: tut23-cpp
#+BEGIN_SRC C++ :results verbatim :exports both :wrap SRC hoa
  #include <iostream>
  #include <spot/twaalgos/hoa.hh>
  #include <spot/twa/twagraph.hh>

  int main(void)
  {
    // The bdd_dict is used to maintain the correspondence between the
    // atomic propositions and the BDD variables that label the edges of
    // the automaton.
    spot::bdd_dict_ptr dict = spot::make_bdd_dict();
    // This creates an empty automaton that we have yet to fill.
    spot::twa_graph_ptr aut = make_twa_graph(dict);

    // Since a BDD is associated to every atomic proposition, the
    // register_ap() function returns a BDD variable number that can be
    // converted into a BDD using bdd_ithvar().
    bdd a = bdd_ithvar(aut->register_ap("a"));

    // Set the acceptance condition of the automaton to co-Büchi
    aut->set_acceptance(1, "Fin(0)");

    // States are numbered from 0.
    aut->new_states(3);
    // The default initial state is 0, but it is always better to
    // specify it explicitely.
    aut->set_init_state(0U);

    // new_edge() takes 3 mandatory parameters: source state,
    // destination state, and label.  A last optional parameter can be
    // used to specify membership to acceptance sets.
    //
    // new_univ_edge() is similar, but the destination is a set of
    // states.
    aut->new_edge(0, 0, a);
    aut->new_univ_edge(0, {0, 1}, !a);
    aut->new_edge(1, 1, !a, {0});
    aut->new_edge(1, 2, a);
    aut->new_edge(2, 2, bddtrue);

    // Print the resulting automaton.
    print_hoa(std::cout, aut);
    return 0;
  }
#+END_SRC

#+RESULTS: tut23-cpp
#+BEGIN_SRC hoa
HOA: v1
States: 3
Start: 0
AP: 1 "a"
acc-name: co-Buchi
Acceptance: 1 Fin(0)
properties: univ-branch trans-labels explicit-labels trans-acc complete
properties: deterministic
--BODY--
State: 0
[0] 0
[!0] 0&1
State: 1
[!0] 1 {0}
[0] 2
State: 2
[t] 2
--END--
#+END_SRC

* Python

#+BEGIN_SRC python :results output :exports both :wrap SRC hoa
  import spot
  import buddy

  # The bdd_dict is used to maintain the correspondence between the
  # atomic propositions and the BDD variables that label the edges of
  # the automaton.
  bdict = spot.make_bdd_dict();
  # This creates an empty automaton that we have yet to fill.
  aut = spot.make_twa_graph(bdict)

  # Since a BDD is associated to every atomic proposition, the register_ap()
  # function returns a BDD variable number that can be converted into a BDD
  # using bdd_ithvar() from the BuDDy library.
  a = buddy.bdd_ithvar(aut.register_ap("a"))

  # Set the acceptance condition of the automaton to co-Büchi
  aut.set_acceptance(1, "Fin(0)")

  # States are numbered from 0.
  aut.new_states(3)
  # The default initial state is 0, but it is always better to
  # specify it explicitely.
  aut.set_init_state(0);

  # new_edge() takes 3 mandatory parameters: source state, destination state,
  # and label.  A last optional parameter can be used to specify membership
  # to acceptance sets.  In the Python version, the list of acceptance sets
  # the transition belongs to should be specified as a list.
  #
  # new_univ_edge() is similar, but the destination is a list of states.
  aut.new_edge(0, 0, a);
  aut.new_univ_edge(0, [0, 1], -a);
  aut.new_edge(1, 1, -a, [0]);
  aut.new_edge(1, 2, a);
  aut.new_edge(2, 2, buddy.bddtrue);

  # Print the resulting automaton.
  print(aut.to_str('hoa'))
#+END_SRC

#+RESULTS:
#+BEGIN_SRC hoa
HOA: v1
States: 3
Start: 0
AP: 1 "a"
acc-name: co-Buchi
Acceptance: 1 Fin(0)
properties: univ-branch trans-labels explicit-labels trans-acc complete
properties: deterministic
--BODY--
State: 0
[0] 0
[!0] 0&1
State: 1
[!0] 1 {0}
[0] 2
State: 2
[t] 2
--END--
#+END_SRC

* Additional comments

Alternating automata in Spot can also have a universal initial state:
e.g, an automaton may start in =0&1&2=.  Use =set_univ_init_state()=
to declare such as state.

We have a [[file:tut24.org][separate page]] describing how to explore the edges of an
alternating automaton.

Once you have built an alternating automaton, you can [[file:tut31.org][remove the
alternation]] to obtain a non-deterministic Büchi or generalized Büchi
automaton.
