Weakly Continuation Closed Homomorphisms on

Automata

Thierry Nicola and Ulrich Ultes-Nitsche

Department of Computer Science, University of Fribourg

Boulevard de P

erolles 90, CH-1700 Fribourg, Switzerland

Abstract. A major limitation of system and program veriﬁcation is the state

space explosion problem. To avoid this problem, there exists several approaches

to reduce the size of the system state space. Some methods try to keep the state

space small during the veriﬁcation run, other methods reduce the original state

space prior to the veriﬁcation. One of the later are abstraction homomorphisms.

Weakly Continuation Closed homomorphisms are abstraction homomorphisms

preserving exactly those properties of the original behaviour which are satisﬁed

inherently fair [1]. However, the practical use of WCC homomorphisms is lim-

ited by the lack of a reasonably efﬁcient algorithm, checking whether or not a

homomorphism is WCC and performing reasonably well. This paper presents a

result which allows to develop algorithms for WCC on automata.

1 Introduction

There exists several methods to tackle the state space explosion problem for system

veriﬁcation [2–4]. An optimal solution is to use combinations of known methods. Ab-

straction homomorphisms try to reduce the original system behaviour state space prior

to the veriﬁcation. As the abstract system model is used for the veriﬁcation, the ab-

stract model should satisfy exactly those properties the original system model satisﬁes.

Weakly Continuation Closed(WCC) homomorphisms preserve exactly those properties

which are satisﬁed inherently fair [1]. This means the conclusion whether or not the

original system satisﬁes a property follows from the veriﬁcation of the property on the

abstract model.

A problem of abstraction homomorphisms is that the complete state space has to

be computed before the abstraction homomorphism can be applied. This leads to a

dilemma, it is not possible to apply the abstraction onto the original state space due to

its vast state space size. This problem is resolved for WCC homomorphisms, as these

abstraction homomorphisms work as well on abstract-compatible trace reductions [1].

The motivation for using WCC homomorphims for model checking is two-folded.

Firstly, WCC homomorphims preserve the original properties on the abstract model

and secondly, WCC homomorphims can be applied on trace reductions of the original

system.

Supported by the Swiss National Science Foundation under grant # 200021-103985/1

Nicola T. and Ultes-Nitsche U. (2008).

Weakly Continuation Closed Homomorphisms on Automata.

In Proceedings of the 6th International Workshop on Modelling, Simulation, Veriﬁcation and Validation of Enterprise Information Systems, pages 39-48

DOI: 10.5220/0001736500390048

 SciTePress

We consider system behaviours which are represented by a deterministic B

uchi

automaton. The abstraction homomorphisms are applied onto the original system au-

tomata. The result is again a deterministic B

uchi Automaton, representing the abstract

system behaviour.

The goal of this paper is to present a new starting point for the development of more

efﬁcient algorithms for weakly continuation closed homomorphisms.

This paper is structured as follows. After some preliminary deﬁnitions, we explain

how homomorphisms are applied onto automata. In section 4 the main results of this

paper is established. Section 5 gives a rough outline of what a possible algorithm might

look like, Section 6 gives an example and Section 7 concludes the paper.

2 Preliminaries

We assume the reader is familiar with the common notions of formal language, ω-

languages and automata theory as presented in [5, 6].

For an alphabet Σ, let Σ

∗

be the set of all ﬁnitely long sequences on Σ, let Σ

the set of all inﬁnitely long sequences, and, let Σ

∞

= Σ

∗

∪Σ

. A set L ⊆ Σ

∗

is called

a (ﬁnitary) language and a set M ⊆ Σ

is called an ω-language.

Let L be language. Then the set of all ﬁnite preﬁxes of L is pre(L) = {w ∈

∗

|∃x ∈ Σ

∞

: wx ∈ L}. The continuation of a preﬁx w ∈ pre(L) in the language L

is cont(w, L) = {x ∈ Σ

∞

|wx ∈ L}.

The behaviour of a system is a set of sequences of actions. Such a sequence is then

called a behaviour. It is therefore justiﬁed to say that system behaviours are languages.

We consider systems which do not terminate and therefore the sequences are inﬁnitely

long. We will use B

uchi Automaton for representing these system behaviours.

Let A = (Q, Σ, δ, q

, F ) be a B

uchi automaton, where Q is a ﬁnite set of states,

Σ is an alphabet, q

∈ Q is the initial state, F ⊆ Q is a set of accepting states and

δ : Q × Σ → 2

is a transition relation. If p ∈ δ(q, a) then we call (q, a, p) a transition

from state q to state p with symbol a. For the rest of this paper, we assume the transition

relation to be extended to δ : 2

× Σ

∗

→ 2

in the usual way.

Let w = a

. . . ∈ Σ

be an inﬁnite sequence. A run of the automaton A on

w is a sequence of states ρ(w) = s

. . ., such that s

i+1

∈ δ(s

, a

i+1

) for i ≥ 0. A

run is called successful if s

= q

and inf (ρ(w)) ∩ F 6= ∅ where inf (ρ(w)) is the

set of states that occur inﬁnitely often in ρ(w). The automaton A B

uchi-accepts w if

and only there is a successful run of A on w. The inﬁnite-string language accepted by

uchi automaton A is L(A ) = {w ∈ Σ

|A B

uchi-accepts w}.

A state q ∈ Q of automaton A is called reachable if there exists a path from

the initial state q

to q. A state q ∈ Q of A is called co-reachable if there exists a

run ρ(w) = q

. . . of A on an inﬁnite sequence w ∈ Σ

such that q

= q and

inf(ρ(w)) ∩ F 6= ∅. Let q ∈ Q, the set r(q, A ) ⊆ Q is the set of all reachable states

from the state q in the automaton A . We assume that each state is reachable from itself

(at least with the empty string), so for all state q ∈ Q, we have that q ∈ r(q, A ). This

notation is extended to r({q

, . . . , q

}, A ) = r(q

, A ) ∪ . . . ∪ r(q

, A ).

Let S ⊆ Q be a Strongly Connected Bottom Component(SCBC). A SCBC is a set of

states such that from each state within S there exists a path to every other state within

S. Further no state within S has a path to a state p such that p 6∈ S and we require

that S is maximal, no more states can be added to S without destroying the property of

connectivity of S.

Let q ∈ Q be a state of the automaton A , we denote by c(q, A ) the language

accepted by the automaton A

= (Q, Σ, δ, q, F ).(The automaton A with q as initial

state) It holds that c(q

, A ) = L(A ). Again, we extend the notation to applied to sets

of states, let {q

, . . . , q

} ⊆ Q, then we deﬁne the set c({q

, . . . , q

}, A ) = c(q

, A )∪

. . . ∪ c(q

, A ).

Let q ∈ Q be a state from automaton A and let b ∈ pre(L(A )) be a preﬁx leading

to state q, in general we have that c(q, A ) ⊆ cont(b, L(A )). If the automaton A is

deterministic then it holds that cont(b, L(A )) = c(q, A ), as each preﬁx b leads to

exactly one state q. δ(b, q

) is the set of states reached by the preﬁx b in A , then we

have that c(δ(b, q

), A ) = cont(b, L(A ))

Throughout this paper the automaton B = (Q

, Σ, δ

, q

, F

) represents the

original system behaviour.

We will consider system behaviours where every state of

the automaton is accepting, F

= Q

. It can therefore be assumed that each automaton

representing a system behaviour is deterministic, because each B

uchi Automaton A =

(Q, Σ, δ, q

, Q) can be transformed into a deterministic B

uchi automaton accepting the

same language [7]. Further, we require that each state is reachable and co-reachable,

every automaton has no useless states.

Let h be a homomorphism on alphabet Σ, and w = a

. . . a

a sequence of sym-

bols from Σ, then we have that h(w) = h(a

)h(a

) . . . h(a

). An homomorphism h

can be applied on a language L, by applying h to each string w ∈ L, h(L) = {h(w)|w ∈

L}.

A special class of homomorphisms are abstraction homomorphism. Abstraction Ho-

momorphisms are deﬁned as follows:

Deﬁnition 1. h : Σ

∞

→ Σ

0∞

is an abstraction homomorphism if and only if the

following conditions hold:

– h(Σ) ⊆ Σ

∪ {},

– ∀v, w ∈ Σ

∗

, x ∈ Σ

: h(vw) = h(v)h(w) and h(vx) = h(v)h(x),

– h(Σ

) ⊆ Σ

0ω

Abstraction homomorphisms are partial mappings since they are not deﬁned for

ω-words that would be taken to ﬁnitely long words, and that they do not increase the

length of the words, i.e. |h(w)| ≤ |w|.

Weakly Continuation Closed homomorphisms are abstraction homomorphisms. In

general abstraction homomorphisms do not preserve the properties of the original sys-

tem. Weakly Continuation Closed homomorphism have been shown to preserve exactly

those properties which are satisﬁed inherently fair [8]. Regarding a model checking

algorithm, weakly continuation closed homomorphisms work best with inherently fair

linear time veriﬁcation(IFLTV) [9] and together they provide a promising tool for model

checking purposes when fairness [10] is enabled.

The deﬁnition of weakly continuation closed homomorphisms is as follows

By referring to the original behaviour, it might also be the system behaviour’s trace reduction.

Deﬁnition 2. The homomorphism h is weakly continuation closed on a language L if

and only if, for all w ∈ Σ

∗

, there exists v ∈ cont(h(w), h(L)) such that cont(v, cont

(h(w), h(L))) = cont(v, h(cont(w, L))).

In general, we know that h(cont(v, L)) ⊆ cont(h(w), h(L)) holds, which im-

plies that cont(v,h(cont(w, L))) ⊆ cont(v, cont(h(w), h(L))) for all v ∈ cont( h(

w), h(L)) always holds.

3 Homomorphisms on Automata

Let B = (Q

, Σ, δ

, q

, Q

) be an automaton representing a system behaviour and

let h : Σ → Σ

be an abstraction homomorphism. Applying the homomorphism h onto

the automaton B, results in a new automaton. We will refer to this resulting automaton

by H = (Q

, Σ

, δ

, q

, Q

The homomorphism h applied on the automaton B, translates the automaton’s al-

phabet Σ to another alphabet Σ

. The homomorphism changes the symbol a ∈ Σ of

the transition (q, a, p) of automaton B to a new symbol α ∈ Σ

, h(a) = α. The tran-

sition becomes (q, α, p) in the automaton H . The automaton H is identical to the

automaton B, except for the transition symbols and the alphabet used. In other words

the state set, the initial state and the set of accepting states are equal for both automata,

= Q

and q

= q

. The transition relation δ

contains the transition (q, α, p) if

and only if the transition (q, a, p) is in the original automaton’s transition relation δ

and h(a) = α.

In general, the automaton H might be non-deterministic (and might have -transitions).

The automaton H is therefore not suited as an automaton representing a system be-

haviour, as automata representing system behaviours are deterministic. To obtain an

automaton representing a system behaviour, H needs to be determinized. By applying

the powerset construction the resulting deterministic automaton accepts the same lan-

guage as the automaton H , as all state of H are accepting [7]. We will refer to the

resulting automaton by D = (Q

, Σ

, ∆, q

, Q

As we will work on these automata, it is important to specify certain observations

about the relation between these automata. A ﬁrst observation is that h(L(B)) =

L(H ) = L(D). This guarantees that the ﬁnal resulting automaton D, corresponds

again to a system behaviour, as it is deterministic, has only accepting states and is the

abstracted automaton to B, i.e. h(L(B)) = L(D ).

For the rest of this paper, let B = (Q

, Σ, δ

, q

, Q

) be the automaton rep-

resenting the original system behaviour, refered to as the original automaton, and let

H = (Q

, Σ, δ

, q

, Q

) and D = (Q

, Σ

, ∆, q

, Q

) be the automaton as de-

scribed above.

4 Weakly Continuation Closed Homomorphisms on Automata

In Deﬁntion 2, weakly continuation closed homomorphisms are deﬁned for languages.

The deﬁnition has to be slightly modiﬁed for automata. Let A = (Q, Σ, δ, q, F ) be a

uchi automaton then we deﬁne WCC homomorphisms on automata as follows

Deﬁnition 3. The homomorphism h is weakly continuation closed on A if and only if

h is weakly continuation closed on L(A ).

We say that a homomorphism is WCC on the automaton A if and only if h is WCC

on the language accepted by this automaton, i.e. L(A ).

This deﬁnition’s na

ıve algorithm, would need to verify all preﬁxes of L(A ). By ver-

ifying preﬁxes, we understand taking a preﬁx w and comparing it to all v ∈ cont(h(w),

h(L(A )), whether or not cont(v, cont(h(w), h(L(A )))) = cont(v, h(cont(w, L(A ))))

holds. But as we are working with ω-languages and there are inﬁnitely many preﬁxes,

this procedure cannot be regarded as efﬁcient. At best it is possible to reduce the veriﬁ-

cation to the number of states in A . Our approach reduces the number of such veriﬁca-

tion and at the same time tries to reduce the amount of possible continuations to choose

from, the number of v’s.

The ﬁrst lemma makes the switch from preﬁxes to states. This step decreases the

veriﬁcation steps from an inﬁnite amount of preﬁxes to a ﬁnite amount of states. But

this number may still be too large for an efﬁcient practical application.

Lemma 1. The homomorphism h is weakly continuation closed on B if and only if for

all q

∈ Q

and for all q ∈ Q

, where q

∈ q, there exists α ∈ c(q, D ) such that

cont(α, c(q, D ) = cont(α, h(c(q

, B))).

Proof. ’⇒’: Let’s assume h is weakly continuation closed on B. This means, for all b ∈

pre(L(B)), there exists α ∈ c(h(b), L(D )) such that cont(α, cont(h(b), L( D ))) =

cont(α, h(cont(b, L(B)))).

As B is deterministic, we know that we can replace cont(b, L(B)) by c(q

, B),

where q

∈ Q

is the state reached with preﬁx b in B, and, as D is deterministic,

we can replace cont(h(b), L(D )) by c(q, D ), where q ∈ Q

is the state reached with

preﬁx h(b) ∈ pre(L(D)) in D .

This holds for all b ∈ pre(L(B)) and cont(b, L(B)) = c(q

, B), where q

is the

state reached by b, it follows that it holds for all states which are reachable. In D , h(b)

then leads to a state q such that q

∈ q. Therefore, for a state q

∈ Q

, to get all all

the corresponing h(b), one needs all states q ∈ Q

, such that q

∈ q. This then ﬁnally

makes

For all q

∈ Q

, and for all q ∈ Q

, where q

∈ q, there exists α ∈ c(q, D ) such

that cont(α, c(q, D )) = cont(α, h(c(q

, B))).

’⇐’: Let’s assume h is not WCC on B. This implies that there exists b ∈ pre(L(B)),

such that for all α ∈ cont(h(b), L(D )) we have that cont(α, cont(h(b), L(D ))) 6=

cont(α, h(cont(b, L(B)))).

Let q

∈ Q

be the state in B reached with the preﬁx b and let q ∈ Q

be the state

reached with h(b) in D . In this case we know that q

∈ q. For all α ∈ c(q, D ), we get

cont(α, c(q, D )) 6= cont(α, h(c(q

, B))).

This is a contradiction, therefore h is weakly continuation closed on B if for all

∈ Q

and for all q ∈ Q

, where q

∈ q, there exists α ∈ c(q, D )) such that

cont(α, c(q, D )) = cont(α, h(c(q

, B))).

The following lemma reduces the number of states from the original automaton

which needs to be veriﬁed to the number of Strongly Connected Bottom Components.

Lemma 2. For all q

∈ Q

and for all q ∈ Q

, where q

∈ q, there exists α ∈

c(q, D ) such that cont(α, c(q, D ) = cont(α, h(c(q

, B))) if and only if for all Strongly

Connected Bottom Components S ⊆ Q

of B, there exists p

∈ S, and for all states

p ∈ Q

where p

∈ p there exists β ∈ c(p, D ) such that

cont(β, c(p, D )) = cont(β, h(c(p

, B)))

Proof. ’⇒’: Let’s assume that for all q

∈ Q

and for all q ∈ Q

there exists an

α ∈ c(q, D ) such that cont(α, c(q, D )) = cont(α, h(c(q

, B))).

This implies, as the above holds for all states q

∈ Q

, it holds for all q

∈ Q

, as

⊆ Q

, where Q

contains all states within SCBC. Therefore it holds for all states

within a SCBC, which implies that for each SCBC S there exists a state q

∈ S such

that the hypothesis holds.

’⇐’: Let’s assume there exists a state q

∈ Q

and a state q ∈ Q

, where q

∈ q,

such that for all α ∈ c(q, D ) we have cont(α, c(q, D )) 6= cont(α, h(c(q

, B))). This

implies that we have cont(α, h(c(q

, B))) ⊂ cont(α, c(q, D )).

Let x

∈ r(q

, B), be the state reached by a ∈ c(q

, B). This makes then that

h(c(x

, B)) ⊆ cont(h(a), h(c(c

, B))), as we have cont(a, c(q

, B)) = c(x

, B) and,

in general, h(cont(a, B)) ⊆ cont(h(a), h(c(c

, B))).

We know that cont(α, h(c(q

, B))) ⊆ cont(α, c(q, D )), we get h(c(x

, B)) ⊂

cont(α, c(q, D )), where α = h(a). The sequence α leads from state q ∈ Q

, to a state

x ∈ r(q, D ), where x

∈ x and c(x, D) = cont(α, c(q, D )).

Finally, there exists q

∈ Q

and q ∈ Q

, where q

∈ q such that for all x

∈

r(q

, B) and for all x ∈ r(q, D ), where x

∈ x, we have that h(c(x

, B)) ⊂ c(x, D ),

and therefore h(c(x

, B)) 6= c(x, D ).

This implies, there exists x

∈ r(q

, B) and x ∈ r(q, D ), where x

∈ x and x

is a state within a Strongly Connected Bottom Component S, such that h(c(x

, B)) 6=

c(x, D ).

By repeating the same procedure for x

as before for state q

, we ﬁnd that there

exists a Strongly Connected Bottom Component S, where for all states q

∈ S and

q ∈ Q

, q

∈ q such that for all α ∈ c(q, D ), we have that cont(α, c(q, D ) 6=

cont(α, h(c(q

, B))).

This is a contradiction therefore it follows that h is weakly continuation closed on B

if for all Strongly Connected Bottom Component S of B, there exists a q

∈ S, and for

all states q ∈ Q

where q

∈ q it holds that ∃α ∈ c(q, D ) such that cont(α, c(q, D )) =

cont(α, c(q

, B

)).

Combining Lemma 1 with Lemma 2 leads us to the main result of this paper, namely

Theorem 1. The theorem states that a homomorphism h is weakly continuation closed

on the automaton B if and only if for all SCBCs, there exists a state q

within this com-

ponent such that for all occurences of q

within a macrostate q of D , there exists a con-

tinuation α ∈ c(q, D ) such that eventually cont(α, c(q, D )) = cont(α, h(c(q

, B))),

note that h(c(q

, B)) = c(q

, H ).

Theorem 1. The homomorphism h is weakly continuation closed on B if and only if

for all Strongly Connected Bottom Component S ⊆ Q

, there exists q

∈ S and for all

q ∈ Q

, where q

∈ q, there exists α ∈ c(q, D ) such that

cont(α, c(q, D )) = cont(α, h(c(q

, B)))

Proof. The proof of the theorem is immediate with Lemma 1 and Lemma 2.

5 Draft of an Algorithm for WCC Homomorphisms

Theorem 1 reduces the number of states that needs to be veriﬁed to the number of

Strongly Connected Bottom Components within the original automaton B. This section

sketches an algorithm implementing the above result.

By applying the homomorphism h, the automaton B is ﬁrst transformed into the

automaton H , by changing the transition symbols according to the homomorphism

h. Then in a second step, by determinizing the automaton H , the result will be the

automaton D describing the abstract system behaviour. The check whether h is WCC

or not, is done in a third step.

For each SCBC S of the original automaton B, we select one state q

∈ S and

identify all macrostates q of D where q

∈ q. The next step is to gain access to the

preﬁxes of c(q

, H ), the SCBC S is regarded as an automaton with q

as initial state.

In general this automaton is non-deterministic. A determinization of this automaton

results in an automaton called S and it holds that L(S ) = c(q

, H ).

This means by taking state r of S and α the sequence leading to state r in S , it

holds that α ∈ c(q

, H ). Then α ∈ c(q, D ), for all q where q

∈ q and let α lead from

q to state t in D . Then the following lemma follows

Lemma 3. for all Strongly Connected Bottom Component S ⊆ Q

, there exists q

∈ S

and for all q ∈ Q

, where q

∈ q, there exists α ∈ c(q, D ) such that cont(α, c(q, D )) =

cont(α, h(c(q

, B))) if and only if c(r, S ) = c(t, D ), where state r is reached by α in

S and α leads from state q to state t in D .

Proof. As both S and D are deterministic,cont(α, c(q

, H )) = c(r, S ) and cont(α, c

(q, D )) = c(t, D ), because α leads from state q

to r in H and from q to t in D .

By reducing the number of possible states t in D, the number of necessary language

comparisons, c(r, S ) = c(t, D ), is reduced too. Such that c(r, S ) = c(t, D ) might

hold, it must be true that r ⊆ t, where r a state of S . This fact must hold as we know

that D is the determinized automaton of H and that q

∈ q and α leads from q

to state

r in S

and α leads from q to t in D .

For checking whether or not c(t, D ) = c(r, S ), we will rely on standard algorithms,

as language comparison is a quite common task for automata. In fact we only need to

verify whether or not c(t, D )) ⊆ c(r, S ), as we know that r ⊆ t, it always holds that

c(r, S )) ⊆ c(t, D ).

6 Example

Let B be the automaton in Figure 1a. Every state is an accepting state, therefore we

omit the usual accepting conditions of the states in the ﬁgures. The homomorphism

Which means α leads from state q

to states r in H .

h hides

all the symbols except req, rej, res, which are mapped to their identity. By

applying the homomorphism h to B, the ﬁnal result will then be the automaton D , ﬁg

1b.

h hides

all the symbols except req, rej, res, which are mapped to their identity. By

applying the homomorphism h to B, the ﬁnal result will then be the automaton D , ﬁg

1b.

req

rej

req

yes

res

lock

free

lock

free

lock

free

(a)

, q

}

, q

}

req

rej

res

(b)

Fig. 1: The automaton B representing the original system behaviour (a) and the automa-

ton D representing the abstracted system behaviour (b) reduced by homomorphism h.

To verify that h is weakly continuation closed on B, we will apply the presented

procedure. The automaton B consists of exactly one SCBC. So we select one state of

B and identify all occurences of this state within a macrostate in D . As the automaton

B is a SCBC, we can select any state of it. We choose state q

and there is exactly one

occurence of the state q

within macrostate {q

, q

As the automaton B is composed of one SCBC, determinization of the SCBC con-

taining state q

results in the automaton S , which is identical to D, the determinization

of B. This is why a homomorphism applied on an automaton which is strongly con-

nected, will always be weakly continuation closed.

The next step is then to choose a state r of S and then for each state t ∈ r(q, D )

we verify if c(t, D) = c(r, S ) holds. For our example, we choose the state {q

, q

}

as state r of S . Such that c(t, D ) = c({q

, q

}, S ) holds, {q

, q

} ⊆ t must be true.

The only state satisfying this property and being reachable from state q = {q

, q

}, is

the state {q

, q

}. And we know that c({q

, q

}, D ) = c({q

, q

}, S ) is always true,

D = S .

The homomorphism h is therefore weakly continuation closed on B and the abstract

system automaton D satisﬁes the same property as the B inherently fair.

Let B

be automaton describing a different system, see ﬁg 2a. The same homomor-

phism h will be applied to B

, which results in the automaton is D

, ﬁg 2b.

The automaton B

has one SCBC, so we select for example state q

. There is one

macrostate of D

containing an occurence of q

. By determinization of the SCBC con-

taining state q

, an automaton S is obtained such that L(S ) = (req · rej)

. The

automaton S = (Q

, Σ

, δ

, q

, Q

), where Q

contains two states, namely, q

and

, q

} and δ

has two transitions, (q

, req, {q

, q

}) and ({q

, q

}, rej, q

The state {q

, q

} of D

contains the only occurence of q

. It can easily be seen

that c({q

, q

}, D

) 6= c(q

, S ), because c({q

, q

}, D

) contains the sequence (req ·

res)

and c(q

, S ) does not. Then for state {q

, q

} of S , there exists state t =

Set a symbol to the empty string 

Fig. 1. The automaton B representing the original system behaviour (a) and the automaton D

representing the abstracted system behaviour (b) reduced by homomorphism h.

To verify that h is weakly continuation closed on B, we will apply the presented

procedure. The automaton B consists of exactly one SCBC. So we select one state of

B and identify all occurences of this state within a macrostate in D. As the automaton

B is a SCBC, we can select any state of it. We choose state q

and there is exactly one

occurence of the state q

within macrostate {q

, q

As the automaton B is composed of one SCBC, determinization of the SCBC con-

taining state q

results in the automaton S , which is identical to D, the determinization

of B. This is why a homomorphism applied on an automaton which is strongly con-

nected, will always be weakly continuation closed.

The next step is then to choose a state r of S and then for each state t ∈ r(q, D )

we verify if c(t, D ) = c(r, S ) holds. For our example, we choose the state {q

, q

}

as state r of S . Such that c(t, D ) = c({q

, q

}, S ) holds, {q

, q

} ⊆ t must be true.

The only state satisfying this property and being reachable from state q = {q

, q

}, is

the state {q

, q

}. And we know that c({q

, q

}, D ) = c({q

, q

}, S ) is always true,

D = S .

The homomorphism h is therefore weakly continuation closed on B and the abstract

system automaton D satisﬁes the same property as the B inherently fair.

Let B

be automaton describing a different system, see ﬁg 1a. The same homomor-

phism h will be applied to B

, which results in the automaton is D

, ﬁg 1b.

The automaton B

has one SCBC, so we select for example state q

. There is one

macrostate of D

containing an occurence of q

. By determinization of the SCBC con-

taining state q

, an automaton S is obtained such that L(S ) = (req · rej)

. The

automaton S = (Q

, Σ

, δ

, q

, Q

), where Q

contains two states, namely, q

and

, q

} and δ

has two transitions, (q

, req, {q

, q

}) and ({q

, q

}, rej, q

The state {q

, q

} of D

contains the only occurence of q

. It can easily be seen

that c({q

, q

}, D

) 6= c(q

, S ), because c({q

, q

}, D

) contains the sequence (req ·

res)

and c(q

, S ) does not. Then for state {q

, q

} of S , there exists state t =

, q

}, where {q

, q

} ⊆ t. But again c(t, D

) 6= c({q

, q

}, S ) because

c(t, D

) contains the sequence (res · req)

and c({q

, q

}, S ) does not.

Set a symbol to the empty string 

This implies that for all states r of S it holds that c(r, S ) 6= c(t, D

), where t

is reachable from state q and r ⊆ t, therefore the homomorphism h is not weakly

continuation closed on B

, q

}, where {q

, q

} ⊆ t. But again c(t, D

) 6= c({q

, q

}, S ) because

c(t, D

) contains the sequence (res · req)

and c({q

, q

}, S ) does not.

This implies that for all states r of S it holds that c(r, S ) 6= c(t, D

), where t

is reachable from state q and r ⊆ t, therefore the homomorphism h is not weakly

continuation closed on B

req

yes

res

lock

rej

(a)

req

rej

res

, q

}

, q

}

(b)

Fig. 2: The automaton B

representing the original system behaviour (a) and the au-

tomaton D

representing the abstracted system behaviour (b) reduced by homomor-

phism h.

7 Conclusions

This paper is a ﬁrst step towards an optimized algorithm for checking whether or not a

homomorphism is weakly continuation closed on a given automaton. The approach of

this paper is to reduce the number of language comparison needed to check for WCC

homomorphisms. The task of language comparison is a well-known task, so the authors

assume this task to be optimized and therefore the interest to loose complexity of a pro-

cedure in a different step. As ω-language comparison presents a complex task, reducing

its number results in a great optimization beneﬁt.

We assume that the beneﬁt is sufﬁcient enough to think of a practical implementa-

tion of the procedure. As the presented procedure is only a sketch of a possible algo-

rithm, we believe that there is still room for improvements.

Actual work in progress includes he development and implementation of an opti-

mized algorithm for verifying whether or not a homomorphism is weakly continuation

closed.

References

1. Ultes-Nitsche, U., James, S.S.: Improved veriﬁcation of linear-time properties within fair-

ness: weakly contiuation-closed behaviour abstractions computed from trace reductions.

Software testing, Veriﬁcation and Reliability 13 (2003) 241–255

2. Clarke, E.M., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Transac-

tions on Programming Languages and Systems 16 (1994) 1512–1542

Fig. 2. The automaton B

representing the original system behaviour (a) and the automaton D

representing the abstracted system behaviour (b) reduced by homomorphism h.

7 Conclusions

This paper is a ﬁrst step towards an optimized algorithm for checking whether or not a

homomorphism is weakly continuation closed on a given automaton. The approach of

this paper is to reduce the number of language comparison needed to check for WCC

homomorphisms. The task of language comparison is a well-known task, so the authors

assume this task to be optimized and therefore the interest to loose complexity of a pro-

cedure in a different step. As ω-language comparison presents a complex task, reducing

its number results in a great optimization beneﬁt.

We assume that the beneﬁt is sufﬁcient enough to think of a practical implementa-

tion of the procedure. As the presented procedure is only a sketch of a possible algo-

rithm, we believe that there is still room for improvements.

Actual work in progress includes he development and implementation of an opti-

mized algorithm for verifying whether or not a homomorphism is weakly continuation

closed.

References

1. Ultes-Nitsche, U., James, S.S.: Improved veriﬁcation of linear-time properties within fair-

ness: weakly contiuation-closed behaviour abstractions computed from trace reductions.

Software testing, Veriﬁcation and Reliability 13 (2003) 241–255

2. Clarke, E.M., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Transac-

tions on Programming Languages and Systems 16 (1994) 1512–1542

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37–56

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191

7. Ultes-Nitsche, U.: A power-set construction for reducing B

uchi automata to non-

determinism degree two. Information Processing Letters (IPL) 101 (2007) 107–111

8. Nitsche, U., Ochsenschl

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9. Nicola, T., Niessner, F., Ultes-Nitsche, U.: Model-checking inherently fair linear-time prop-

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