spot/doc/org/tut03.org

Constructing and transforming formulas

• Constructing formulas
  • C++
  • Python
• Transforming formulas
  • C++
  • Python

This page explains how to build formulas and how to iterate over their syntax trees.

We'll first describe how to build a formula from scratch, by using the constructor functions associated to each operators, and show that basic accessor methods for formulas. We will do that for C++ first, and then Python. Once these basics are covered, we will show examples for traversing and transforming formulas (again in C++ then Python).

Constructing formulas

C++

The spot::formula class contains static methods that act as constructors for each supported operator.

The Boolean constants true and false are returned by formula::tt() and formula:ff(). Atomic propositions can be built with formula::ap("name"). Unary and binary operators use a straighforward syntax like formula::F(arg) or formula::U(first, second), while n-ary operators take an initializer list as argument as in formula::And({arg1, arg2, arg3}).

Here is the list of supported operators:

  // atomic proposition
  formula::ap(string)
  // constants
  formula::ff();
  formula::tt();
  formula::eword();               // empty word (for regular expressions)
  // unary operators
  formula::Not(arg);
  formula::X(arg);
  formula::F(arg);
  formula::G(arg);
  formula::Closure(arg);
  formula::NegClosure(arg);
  // binary operators
  formula::Xor(left, right);
  formula::Implies(left, right);
  formula::Equiv(left, right);
  formula::U(left, right);        // (strong) until
  formula::R(left, right);        // (weak) release
  formula::W(left, right);        // weak until
  formula::M(left, right);        // strong release
  formula::EConcat(left, right);  // Seq
  formula::UConcat(left, right);  // Triggers
  // n-ary operators
  formula::Or({args,...});        // omega-rational Or
  formula::OrRat({args,...});     // rational Or (for regular expressions)
  formula::And({args,...});       // omega-rational And
  formula::AndRat({args,...});    // rational And (for regular expressions)
  formula::AndNLM({args,...});    // non-length-matching rational And (for r.e.)
  formula::Concat({args,...});    // concatenation (for regular expressions)
  formula::Fusion({args,...});    // concatenation (for regular expressions)
  // star-like operators
  formula::Star(arg, min, max);   // Star (for a Kleene star, set min=0 and omit max)
  formula::FStar(arg, min, max);  // Fusion Star

These functions implement some very limited type of automatic simplifications called trivial identities. For instance formula::F(formula::X(formula::tt())) will return the same formula as formula::tt(). These simplifications are those that involve the true and false constants, impotence (F(F(e))=F(e)), involutions (Not(Not(e)=e), associativity (And({And({e1,e2},e3})=And({e1,e2,e3})). See tl.pdf for a list of these trivial identities.

In addition, the arguments of commutative operators (e.g. Xor(e1,e2)=Xor(e2,e1)) are always reordered. The order used always put the Boolean subformulas before the temporal subformulas, sorts the atomic propositions in alphabetic order, and otherwise order subformulas by their unique identifier (a constant incremented each time a new subformula is created). This reordering is useful to favor sharing of subformulas, but also helps algorithms that perform memoization.

Building a formula using these operators is quite straightforward. The second part of the following example shows how to print some detail of the top-level oeprator in the formula.

  #include <iostream>
  #include <spot/tl/formula.hh>
  #include <spot/tl/print.hh>

  int main()
  {
    // Build FGa -> (GFb & GFc)
    spot::formula fga = spot::formula::F(spot::formula::G(spot::formula::ap("a")));
    spot::formula gfb = spot::formula::G(spot::formula::F(spot::formula::ap("b")));
    spot::formula gfc = spot::formula::G(spot::formula::F(spot::formula::ap("c")));
    spot::formula f = spot::formula::Implies(fga, spot::formula::And({gfb, gfc}));

    std::cout << f << '\n';

    // kindstar() prints the name of the operator
    // size() return the number of operands of the operators
    std::cout << f.kindstr() << ", " << f.size() << " children\n";
    // operator[] accesses each operand
    std::cout << "left: " << f[0] << ", right: " << f[1] << '\n';
    // you can also iterate over all operands using a for loop
    for (auto child: f)
      std::cout << "  * " << child << '\n';
    // the type of the operator can be accessed with kind(), which
    // return an element of the spot::op enum.
    std::cout << f[1][0]
              << (f[1][0].kind() == spot::op::F ? " is F\n" : " is not F\n");
    // however because writing f.kind() == spot::op::XXX is quite common, there
    // is also a is() shortcut:
    std::cout << f[1][1]
              << (f[1][1].is(spot::op::G) ? " is G\n" : " is not G\n");
    return 0;
  }

FGa -> (GFb & GFc)
Implies, 2 children
left: FGa, right: GFb & GFc
  * FGa
  * GFb & GFc
GFb is not F
GFc is G

Python

The Python equivalent is similar:

  import spot

  # Build FGa -> (GFb & GFc)
  fga = spot.formula.F(spot.formula.G(spot.formula.ap("a")))
  gfb = spot.formula.G(spot.formula.F(spot.formula.ap("b")));
  gfc = spot.formula.G(spot.formula.F(spot.formula.ap("c")));
  f = spot.formula.Implies(fga, spot.formula.And([gfb, gfc]));

  print(f)

  # kindstar() prints the name of the operator
  # size() return the number of operands of the operators
  print("{}, {} children".format(f.kindstr(), f.size()))
  # [] accesses each operand
  print("left: {f[0]}, right: {f[1]}".format(f=f))
  # you can also iterate over all operands using a for loop
  for child in f:
     print("  *", child)
  # the type of the operator can be accessed with kind(), which returns
  # an op_XXX constant (corresponding the the spot::op enum of C++)
  print(f[1][0], "is F" if f[1][0].kind() == spot.op_F else "is not F")
  # "is" is keyword in Python, the so shortcut is called _is:
  print(f[1][1], "is G" if f[1][1]._is(spot.op_G) else "is not G")

FGa -> (GFb & GFc)
Implies, 2 children
left: FGa, right: GFb & GFc
  * FGa
  * GFb & GFc
GFb is not F
GFc is G

Transforming formulas

C++

In Spot, Formula objects are immutable: this allows identical subtrees to be shared among multiple formulas. Algorithm that "transform" formulas (for instance the relabeling function) actually recursively traverses the input formula to construct the output formula.

Using the operators described in the previous section is enough to write algorithms on formulas. However there are two special methods that makes it a lot easier: traverse and map.

traverse takes a function fun, and applies it to a subformulas of a formula, including the formula itself. The formula is explored in a DFS fashion (without skipping subformula that appear twice). The children of a formula are explored only if fun returns false. If fun returns true, that indicates to stop the recursion.

In the following we use a lambda function to count the number of G in the formula. We also print each subformula to show the recursion, and stop the recursion as soon as we encounter a subformula without sugar (the is_sugar_free_ltl() method is a constant-time operation, that tells whether a formulas contains a F or G operator) to save time time by not exploring further.

  #include <iostream>
  #include <spot/tl/formula.hh>
  #include <spot/tl/print.hh>
  #include <spot/tl/parse.hh>

  int main()
  {
    spot::formula f = spot::parse_formula("FGa -> (GFb & GF(c & b & d))");

    int gcount = 0;
    f.traverse([&gcount](spot::formula f)
               {
                 std::cout << f << '\n';
                 if (f.is(spot::op::G))
                   ++gcount;
                 return f.is_sugar_free_ltl();
               });
    std::cout << "=== " << gcount << " G seen ===\n";
    return 0;
  }

FGa -> (GFb & GF(b & c & d))
FGa
Ga
a
GFb & GF(b & c & d)
GFb
Fb
b
GF(b & c & d)
F(b & c & d)
b & c & d
=== 3 G seen ===

The other useful operation is map. This also takes a functional argument, but that function should input a formula and output a replacement formula. f.map(fun) applies fun to all children of f, and rebuild a same formula as f.

Here is a demonstration of how to exchange all F and G operators in a formula:

  #include <iostream>
  #include <spot/tl/formula.hh>
  #include <spot/tl/print.hh>
  #include <spot/tl/parse.hh>

  spot::formula xchg_fg(spot::formula in)
  {
     if (in.is(spot::op::F))
       return spot::formula::G(xchg_fg(in[0]));
     if (in.is(spot::op::G))
       return spot::formula::F(xchg_fg(in[0]));
     // No need to transform Boolean subformulas
     if (in.is_sugar_free_ltl())
       return in;
     // Apply xchg_fg recursively on any other operator's children
     return in.map(xchg_fg);
  }

  int main()
  {
    spot::formula f = spot::parse_formula("FGa -> (GFb & GF(c & b & d))");
    std::cout << "before: " << f << '\n';
    std::cout << "after:  " << xchg_fg(f) << '\n';
    return 0;
  }

before: FGa -> (GFb & GF(b & c & d))
after:  GFa -> (FGb & FG(b & c & d))

Python

The Python version of the above two examples uses a very similar syntax. Python only supports a very limited form of lambda expressions, so we have to write a standard function instead:

  import spot

  gcount = 0
  def countg(f):
      global gcount
      print(f)
      if f._is(spot.op_G):
          gcount += 1
      return f.is_sugar_free_ltl()

  f = spot.formula("FGa -> (GFb & GF(c & b & d))")
  f.traverse(countg)
  print("===", gcount, "G seen ===")

FGa -> (GFb & GF(b & c & d))
FGa
Ga
a
GFb & GF(b & c & d)
GFb
Fb
b
GF(b & c & d)
F(b & c & d)
b & c & d
=== 3 G seen ===

Here is the F and G exchange:

  import spot

  def xchg_fg(i):
     if i._is(spot.op_F):
        return spot.formula.G(xchg_fg(i[0]));
     if i._is(spot.op_G):
        return spot.formula.F(xchg_fg(i[0]));
     # No need to transform Boolean subformulas
     if i.is_sugar_free_ltl():
        return i;
     # Apply xchg_fg recursively on any other operator's children
     return i.map(xchg_fg);

  f = spot.formula("FGa -> (GFb & GF(c & b & d))")
  print("before:", f)
  print("after: ", xchg_fg(f))

before: FGa -> (GFb & GF(b & c & d))
after:  GFa -> (FGb & FG(b & c & d))