Linear temporal logic to Büchi automaton


In formal verification,
finite state model checking needs to find a Büchi automaton equivalent to a given Linear temporal logic formula, i.e., such that the LTL formula and the BA recognize the same ω-language. There are algorithms that translate an LTL formula to a BA. This transformation is normally done in two steps. The first step produces a generalized Büchi automaton from a LTL formula. The second step translates this GBA into a BA, which involves a relatively easy construction. Since LTL is strictly less expressive than BA, the reverse construction is not possible.
The algorithms for transforming LTL to GBA differ in their construction strategies but they all have common underlying principle, i.e., each state in the constructed automaton represents a set of LTL formulas that are expected to be satisfied by the remaining input word after occurrence of the state during a run.

Transformation from LTL to GBA

Here two algorithms are presented for the construction. The first one provides a declarative and easy to understand construction. The second one provides an algorithmic and efficient construction. Both the algorithms assume that the input formula f is constructed using the set of propositional variables AP and f is in negation normal form.
For each LTL formula f' without ¬ as top symbol, let neg = ¬f', neg = f'.
For a special case f'=true, let neg = false.

Declarative construction

Before describing the construction, we need to present a few auxiliary definitions.
For a LTL formula f, Let cl be a smallest set of formulas that satisfies the following conditions:
cl is closure of sub-formulas of f under negation.
Note that cl may contain formulas that are not in negation normal form.
The subsets of cl are going to serve as states of the equivalent GBA.
We aim to construct the GBA such that if a state corresponds to a subset M ⊂ cl then the GBA has an accepting run starting from the state for a word iff the word satisfies every formula in M and violates every formula in cl/M.
For this reason,
we will not consider each formula set M that is clearly inconsistent
or subsumed by a strictly super set M' such that M and M' are equiv-satisfiable.
A set M ⊂ cl is maximally consistent if it satisfies the following conditions:
Let cs be the set of maximally consistent subsets of cl.
We are going to use only cs as the states of GBA.
;GBA construction
An equivalent GBA to f is A=, where
The three conditions in definition of Δ1 ensure that any run of A does not violate semantics of the temporal operators.
Note that F is a set of sets of states.
The sets in F are defined to capture a property of operator U that can not be verified by comparing two consecutive states in a run, i.e.,
if f1 U f2 is true in some state then eventually f2 is true at some state later.
Let ω-word w= a1, a2,... over alphabet 2AP. Let wi = ai, ai+1,....
Let Mw =, which we call satisfying set.
Due to the definition of cs, Mw ∈ cs.
We can define a sequence
ρw = init, Mw1, Mw2,....
Due to the definition of A,
if w f then ρw must be an accepting run of A over w.
Conversely, lets assume A accepts w.
Let ρ = init, M1, M2,... be a run of A over w.
The following theorem completes the rest of the correctness proof.
Theorem 1: For all i > 0, Mwi = Mi.
Proof: The proof is by induction on the structure of f' ∈ cl.
Due to the above theorem, Mw1 = M1. Due to the definition of the transitions of A, f ∈ M1. Therefore, f ∈ Mw1 and w f.

Gerth et al. algorithm

The following algorithm is due to Gerth, Peled, Vardi, and Wolper.
A verified construction mechanism of this by Schimpf, Merz and Smaus is also available.
The previous construction creates exponentially many states upfront and many of those states may be unreachable.
The following algorithm avoids this upfront construction and has two steps.
In the first step, it incrementally constructs a directed graph.
In the second step, it builds a labeled generalized Büchi automaton by defining nodes of the graph as states and directed edges as transitions.
This algorithm takes reachability into account and may produce a smaller automaton but the worst-case complexity remains the same.
The nodes of the graph are labeled by sets of formulas and are obtained by decomposing formulas according to their Boolean structure, and by expanding the temporal operators in order to separate what has to be true immediately from what has to be true from the next state onwards. For example, let us assume that an LTL formula f1 U f2 appears in the label of a node.
f1 U f2 is equivalent to f2 ∨.
The equivalent expansion suggests that f1 U f2 is true in one of the following two conditions.
  1. f1 holds at the current time and holds at the next time step, or
  2. f2 holds at the current time step
The two cases can be encoded by creating two states of the automaton and the automaton may non-deterministically jump to either of them.
In the first case, we have offloaded a part of burden of proof in the next time step
therefore we also create another state that will carry the obligation for next time step in its label.
We also need to consider temporal operator R that may cause such case split.
f1 R f2 is equivalent to ∨
and this equivalent expansion suggests that f1 R f2 is true in one of the following two conditions.
  1. f2 holds at the current time and holds at the next time step, or
  2. holds at the current time step.
To avoid many cases in the following algorithm, let us define functions curr1, next1 and curr2 that encode the above equivalences in the following table.
fcurr1next1curr2
f1 U f2
f1 R f2
f1 ∨ f2∅

We have also added disjunction case in the above table since it also causes a case split in the automaton.
Following are the two steps of the algorithm.
;Step 1. create_graph
In the following box, we present the first part of the algorithm that builds a directed graph. create_graph is the entry function, which expects the input formula f in the negation normal form. It calls recursive function expand that builds the graph by populating global variables Nodes, Incoming, Now, and Next.
The set Nodes stores the set of nodes in the graph. The map Incoming maps each node of Nodes to a subset of Nodes ∪, which defines the set of incoming edges.
Incoming of a node may also contain a special symbol init that is used in the final automaton construction to decide the set of initial states.
Now and Next map each node of Nodes to a set of LTL formulas.
For a node q, Now denotes the set of formulas that must be satisfied by the rest of the input word if the automaton is currently at node q.
Next denotes the set of formulas that must be satisfied by the rest of the input word if the automaton is currently at the next node after q.
function expand
The code of expand is labelled with line numbers for easy reference.
Each call to expand aims to add a node and its successors nodes in the graph.
The parameters of the call describes a potential new node.
At line 1, the if condition checks if curr contains any formula to be expanded.
If the curr is empty then at line 2 the if condition checks if there already exists a state q' with same set of expanded formulas.
If that is the case, then we do not add a redundant node, but we add parameter incoming in Incoming at line 3.
Otherwise, we add a new node q using the parameters at lines 5–9 and we start expanding a successor node of q using next as its unexpanded set of formulas at line 10.
In the case curr is not empty, then we have more formulas to expand and control jumps from line 1 to 12.
At lines 12–14, a formula f from curr is selected and moved to old.
Depending on structure of f, the rest of the function executes.
For the recursive calls, curr and next are modified using functions curr1, next1, and curr2 that were defined in the above table.
; Step 2. LGBA construction
Let = create_graph.
An equivalent LGBA to f is A=, where
Note that node labels in the algorithmic construction do not contain negation of sub-formulas of f. In the declarative construction a node has the set of formulas that are expected to be true. The algorithmic construction ensures the correctness without the need of all the true formulas to be present in a node label.

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