# -*- coding: utf-8 -*- #+TITLE: Constructing and transforming formulas #+DESCRIPTION: Code example for constructing and transforming formulas in Spot #+SETUPFILE: setup.org #+HTML_LINK_UP: tut.html 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: #+BEGIN_SRC C++ // 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 #+END_SRC 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 [[https://spot.lrde.epita.fr/tl.pdf][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. #+BEGIN_SRC C++ :results verbatim :exports both #include #include #include 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; } #+END_SRC #+RESULTS: : 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: #+BEGIN_SRC python :results output :exports both 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") #+END_SRC #+RESULTS: : 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 [[file:tut02.org][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. #+BEGIN_SRC C++ :results verbatim :exports both #include #include #include #include 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; } #+END_SRC #+RESULTS: #+begin_example 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 === #+end_example 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: #+BEGIN_SRC C++ :results verbatim :exports both #include #include #include #include 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; } #+END_SRC #+RESULTS: : 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: #+BEGIN_SRC python :results output :exports both 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 ===") #+END_SRC #+RESULTS: #+begin_example 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 === #+end_example Here is the =F= and =G= exchange: #+BEGIN_SRC python :results output :exports both 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)) #+END_SRC #+RESULTS: : before: FGa -> (GFb & GF(b & c & d)) : after: GFa -> (FGb & FG(b & c & d))