# Constructing and transforming formulas

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

We will first describe how to build a formula from scratch, by using the constructors associated to each operators, and show the 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 straightforward 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::X(arg, min, max);     // X[min..max] arg
formula::F(arg);
formula::F(arg, min, max);     // F[min..max] arg
formula::G(arg);
formula::G(arg, min, max);     // G[min..max] arg
formula::Closure(arg);
formula::NegClosure(arg);
formula::first_match(arg);      // SVA's first match operator
// 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
// syntactic sugar built on top of previous operators
formula::sugar_goto(arg, min, max); // arg[->min..max]
formula::sugar_equal(arg, min, max); // arg[=min..max]
formula::sugar_delay(left, right, min, max); // left ##[min..max] right
```

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 operator 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';

// kindstr() 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 << ", right: " << f << '\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
<< (f.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
<< (f.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)

# kindstr() 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}, right: {f}".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, "is F" if f.kind() == spot.op_F else "is not F")
# "is" is keyword in Python, the so shortcut is called _is:
print(f, "is G" if f._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. Algorithms that "transform" formulas (for instance the relabeling function) actually recursively traverse 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 each subformulas of a given formula, including that starting 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 formula 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 assemble the result under the same top-level operator 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));
if (in.is(spot::op::G))
return spot::formula::F(xchg_fg(in));
// No need to transform subformulas without F or G
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))
```
• Additional tricks about `map` and `traverse` in C++

As seen above, the first argument of `map()` and `traverse()` is a function `fun()` (or actually any object that as an `operator()`) that will be applied to subformulas. If additional arguments are passed to `map()` or `traverse()`, those will be passed on to `fun()` after the formula.

For instance instead of having a lambda capturing the `gcount` variable in the first example, we could pass a reference to this variable:

```#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([](spot::formula f, int& count)
{
if (f.is(spot::op::G))
++count;
return f.is_sugar_free_ltl();
}, gcount);
std::cout << "=== " << gcount << " G seen ===\n";
return 0;
}
```
```=== 3 G seen ===
```

(Here we have removed the print statement inside the lambda to focus more on how `gcount` get passed as the `&count` reference. Here there is no real advantage to passing such reference by argument instead of capturing them in the lambda.

The possibility to pass additional arguments is however more useful in the case of `map`. Let's write a variant of our `xchg_fg()` example that counts the number of exchanges performed. First, we do it without lambda:

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

spot::formula xchg_fg(spot::formula in, int& count)
{
if (in.is(spot::op::F, spot::op::G))
++count;
if (in.is(spot::op::F))
return spot::formula::G(xchg_fg(in, count));
if (in.is(spot::op::G))
return spot::formula::F(xchg_fg(in, count));
// No need to transform subformulas without F or G
if (in.is_sugar_free_ltl())
return in;
// Apply xchg_fg recursively on any other operator's children
return in.map(xchg_fg, count);
}

int main()
{
spot::formula f = spot::parse_formula("FGa -> (GFb & GF(c & b & d))");
std::cout << "before: " << f << '\n';
int count = 0;
std::cout << "after:  " << xchg_fg(f, count) << '\n';
std::cout << "exchanges:  " << count << '\n';
return 0;
}
```
```before: FGa -> (GFb & GF(b & c & d))
after:  GFa -> (FGb & FG(b & c & d))
exchanges:  6
```

Now let's pretend that we want to define `xchg_fg` as a lambda, and `count` to by captured by reference. In order to call pass the lambda recursively to `map`, the lambda needs to know its address. Unfortunately, if the lambda is stored with type `auto`, it cannot capture itself. A solution is to use `std::function` but that has a large penalty cost. We can work around that by assuming that that address will be passed as an argument (`self`) to the lambda:

```#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))");
std::cout << "before: " << f << '\n';

int count = 0;
auto xchg_fg = [&count](spot::formula in, auto&& self) -> spot::formula
{
if (in.is(spot::op::F, spot::op::G))
++count;
if (in.is(spot::op::F))
return spot::formula::G(self(in, self));
if (in.is(spot::op::G))
return spot::formula::F(self(in, self));
// No need to transform subformulas without F or G
if (in.is_sugar_free_ltl())
return in;
// Apply xchg_fg recursively on any other operator's children
return in.map(self, self);
};
std::cout << "after:  " << xchg_fg(f, xchg_fg) << '\n';
std::cout << "exchanges:  " << count << '\n';
return 0;
}
```
```before: FGa -> (GFb & GF(b & c & d))
after:  GFa -> (FGb & FG(b & c & d))
exchanges:  6
```

### 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));
if i._is(spot.op_G):
return spot.formula.F(xchg_fg(i));
# No need to transform subformulas without F or G
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))
```

Like in C++, extra arguments to `map` and `traverse` are passed as additional to the function given in the first argument.