BRANCHING-TIME VERSUS LINEAR-TIME

A Cooperative and Feasible Approach

Norihiro Kamide

Waseda Institute for Advanced Study, 1-6-1 Nishi Waseda, Shinjuku-ku, Tokyo 169-8050, Japan

Keywords:

Temporal reasoning, Branching-time formalism, Linear-time formalism, Computation tree logic, Linear-time

temporal logic, Model checking.

Abstract:

A new temporal logic called linear-time computation tree logic (LCTL) is obtained from computation tree

logic (CTL) by adding some modiﬁed versions of the temporal operators of linear-time temporal logic (LTL).

A theorem for embedding LCTL into CTL is proved. The model-checking, validity and satisﬁability problems

of LCTL are shown to be deterministic PTIME-complete, EXPTIME-complete and deterministic EXPTIME-

complete, respectively.

1 INTRODUCTION

It is known that computation tree logic (CTL) (Clarke

and Emerson, 1981) and linear-time temporal logic

(LTL) (Pnueli, 1977) are the most useful tempo-

ral logics for verifying concurrent systems by model

checking (Clarke et al., 1999). CTL has some feasi-

ble model checking algorithms, which are determin-

istic PTIME-complete (Emerson and Clarke, 1982),

but CTL cannot express some important tempo-

ral properties such as strong fairness. LTL can ex-

press almost all important temporal properties, but

LTL has no feasible model-checking algorithms. The

model-checking problem of LTL is indeed PSPACE-

complete (Sistla and Clarke, 1985). Although CTL

and LTL have been rivaled each other (Vardi, 2001),

cooperating CTL and LTL is considered to be a

good choice to obtain a more useful model check-

ing tool. Full computation-tree logic (CTL

∗

) (Emer-

son and Sistla, 1984; Emerson and Halpern, 1986) is

known to be a result of cooperating CTL and LTL.

However, the model-checking problem of CTL

∗

PSPACE-complete. This paper tries to obtain a coop-

erative and feasible approach to the traditional issue

of “branching-time versus linear-time”. The proposed

logic in this paper includes CTL and subsumes some

versions of the linear-time temporal operators of LTL

(i.e., cooperative). The proposed logic also has the

By “feasible”, we mean “computable in practice”.

There is a widespread opinion that PTIME computability

is the correct mathematical model of feasible computation.

same complexity result as CTL model-checking (i.e.,

feasible).

The results of this paper are then summarized

as follows. A new computation tree logic called

linear-time computation tree logic (LCTL) is ob-

tained from CTL by adding some bounded ver-

sions of the linear-time temporal operators of LTL.

A theorem for embedding LCTL into CTL is

proved. The model-checking, validity and satisﬁabil-

ity problems of LCTL are shown to be determinis-

tic PTIME-complete, EXPTIME-complete and deter-

ministic EXPTIME-complete, respectively. The em-

bedding and decidability results indicate that we can

reuse the existing CTL-based algorithms for model-

checking, validity and satisﬁability. This fact is

regarded as an advantage of LCTL. The proposed

bounded linear-time temporal operators, which are re-

garded as ﬁnite approximations of the usual linear-

time temporal operators, have the central role for ob-

taining the complexity results. Although the stan-

dard LTL operators have an inﬁnite (unbounded) time

domain, i.e., the set ω of natural numbers, the pro-

posed bounded operators havea bounded time domain

which is restricted by a ﬁxed positive integer l, i.e.,

the set ω

:= {x ∈ ω | x ≤ l}. Despite this restriction,

the proposed bounded operators can derive almost all

the typical LTL axioms including the time induction

axiom.

522

Kamide N. (2010).

BRANCHING-TIME VERSUS LINEAR-TIME - A Cooperative and Feasible Approach.

In Proceedings of the 2nd International Conference on Agents and Artiﬁcial Intelligence - Artiﬁcial Intelligence, pages 522-526

DOI: 10.5220/0002709205220526

 SciTePress

2 LINEAR-TIME COMPUTATION

TREE LOGIC

Formulas of LCTL are constructed from countably

many atomic formulas, → (implication) ∧ (conjunc-

tion), ∨ (disjunction), ¬ (negation), X (next), G (glob-

ally), F (eventually), U (until), X

(linear next), G

(linear globally), F

(linear eventually), A (all com-

putation paths) and E (some computation path) where

, G

and F

are based on a bounded time domain.

The symbols X, G, F, U, X

, G

and F

are called

temporal operators, and the symbols A and E are

called path quantiﬁers. The symbol ATOM is used

to denote the set of atomic formulas. An expression

A ≡ B is used to denote the syntactical identity be-

tween A and B.

Deﬁnition 2.1 Formulas α are deﬁned by the follow-

ing grammar, assuming p ∈ ATOM:

α ::= p | α→α | α∧ α | α∨ α | ¬α |

α | G

α | F

α | AXα | EXα | AGα |

EGα | AFα | EFα | A(αUα) | E(αUα).

Note that pairs of symbols like AG and EU are in-

divisible, and that the symbols X,G,F and U cannot

occur without being preceded by an A or an E. Simi-

larly, every A or E must have one of X, G, F and U to

accompany it. Some operators are redundant as those

in CTL, because some operators can be obtained by

the other operators (e.g., AGα := ¬EF¬α).

The symbol ω is used to represent the set of nat-

ural numbers. Lower-case letters i, j, k, m and n are

sometimes used to denote any natural numbers. An

expression X

α for any m ∈ ω is deﬁned inductively

by X

α ≡ α and X

n+1

α ≡ X

α. The symbols ≤

and ≥ are used to represent a linear order on ω. The

symbol ω

is used to represent the set {i ∈ ω | i ≤ l}.

In the following discussion, the number l is ﬁxed as a

certain positive integer.

Deﬁnition 2.2 A structure hS,S

,R, {L

}

m∈ω

i is

called a time-indexed Kripke structure if:

1. S is the set of states,

2. S

is a set of initial states and S

⊆ S,

3. R is a binary relation on S which satisﬁes the con-

dition: ∀s ∈ S ∃s

′

∈ S [(s,s

′

) ∈ R],

4. L

(m ∈ ω) are functions from S to the power set

of a nonempty subset AT of ATOM.

A path in a time-indexed Kripke structure is an

inﬁnite sequence of states, π = s

,... such that

∀i ≥ 0 [(s

i+1

) ∈ R].

The logic LCTL is then deﬁned as a time-indexed

Kripke structure with satisfaction relations |=

(m ∈

ω).

Deﬁnition 2.3 Let AT be a nonempty subset of

ATOM. Satisfaction relations |=

(m ∈ ω) on a time-

indexed Kripke structure M = hS, S

,R, {L

}

m∈ω

i are

deﬁned inductively as follows (s represents a state in

S):

1. for any p ∈ AT, M,s |=

p iff p ∈ L

(s),

2. M,s |=

→α

iff M,s |=

implies M, s |=

3. M,s |=

∧ α

iff M,s |=

and M,s |=

4. M,s |=

∨ α

iff M,s |=

or M, s |=

5. M,s |=

¬α

iff not-[M,s |=

6. for any m ≤ l − 1, M, s |=

α iff M, s |=

m+1

α,

7. for any m ≥ l, M, s |=

α iff M,s |=

α,

8. for any n ∈ ω, M, s |=

l+n

α iff M,s |=

α,

9. M,s |=

α iff M, s |=

m+n

α for all n ∈ ω

10. M,s |=

α iff M,s |=

m+n

α for some n ∈ ω

11. M,s |=

AXα iff ∀s

∈ S [(s, s

) ∈ R implies

M,s

α],

12. M,s |=

EXα iff ∃s

∈ S [(s,s

) ∈ R and

M,s

α],

13. M,s |=

AGα iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

M,s

α,

14. M,s |=

EGα iff there is a path π ≡ s

,...,

where s≡ s

, and for all states s

along π, we have

M,s

α,

15. M,s |=

AFα iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

M,s

α,

16. M,s |=

EFα iff there is a path π ≡ s

,...,

where s ≡ s

, and for some state s

along π, we

have M, s

α,

17. M,s |=

A(α

Uα

) iff for all paths π ≡

,..., where s ≡ s

, there is a state s

along

π such that [(M,s

) and ∀ j (0 ≤ j < k im-

plies M, s

)],

18. M,s |=

E(α

Uα

) iff there is a path π ≡

,..., where s ≡ s

, and for some state s

along π, we have [(M,s

) and ∀ j (0 ≤ j <

k implies M, s

)].

We can naturally consider the unbounded version

LCTL

which is obtained from LCTL by deleting the

conditions 7 and 8 and replacing the conditions 6, 9

and 10 by the standard conditions:

′

. M,s |=

α iff M, s |=

m+1

α,

′

. M,s |=

α iff M, s |=

m+n

α for all n ∈ ω,

′

. M,s |=

α iff M, s |=

m+n

α for some n ∈ ω.

BRANCHING-TIME VERSUS LINEAR-TIME - A Cooperative and Feasible Approach

523

However, the decidability of validity, satisﬁability

and model-checking problems for LCTL

cannot be

shown using the proposed embedding-based method.

The logic LCTL

is embeddable into the inﬁnitary

version CTL

which is obtained from CTL by adding

the inﬁnitary conjunction and disjunction connectives

and

. But, logics with

and

are known to

be undecidable, and hence such an embedding result

cannot imply the decidability.

Deﬁnition 2.4 A formula α is valid (satisﬁable)

in LCTL if and only if M, s |=

α holds for

any (some) time-indexed Kripke structure M =

hS, S

,R, {L

}

m∈ω

i, any (some) s ∈ S, and any (some)

satisfaction relations |=

(m ∈ ω) on M.

Deﬁnition 2.5 Let M be a time-indexed Kripke struc-

ture hS, S

,R, {L

}

m∈ω

i for LCTL, and |=

(m ∈ ω)

be satisfaction relations on M. Then, the model

checking problem of LCTL is deﬁned by: for any for-

mula α, ﬁnd the set {s ∈ S | M,s |=

α}.

Let C be a ﬁnite set of formulas. Then, expres-

sions

C and

C represent the conjunction and dis-

junction of all elements of C, respectively. An expres-

sion α ↔ β is used to represent (α→β) ∧ (β→α).

Proposition 2.6 The following formulas are valid in

LCTL: for any formulas α and β,

1. X

(α ◦ β) ↔ X

α ◦ X

β where ◦ ∈ {→,∧, ∨},

2. X

(¬α) ↔ ¬(X

α),

3. G

α→α,

4. G

α→X

α,

5. G

α→X

α,

6. G

α→G

α,

7. α∧ G

(α→X

α)→G

α (time induction),

8. for any n ∈ ω, X

l+n

α ↔ X

α,

9. G

α ↔

α | n ∈ ω

10. F

α ↔

α | n ∈ ω

Note that the formula 8 in in Proposition 2.6

means that the nesting of X is bounded by l. Note also

that the formulas 9 and 10 in Proposition 2.6 mean

that G

and F

are ﬁnite approximations of the stan-

dard linear-time temporal operators.

Deﬁnition 2.7 A Kripke structure for CTL is a struc-

ture hS,S

,R, Li such that

1. S is the set of states,

2. S

is a set of initial states and S

⊆ S,

3. R is a binary relation on S which satisﬁes the con-

dition: ∀s ∈ S ∃s

′

∈ S [(s,s

′

) ∈ R],

4. L is a function from S to the power set of a

nonempty subset AT of ATOM.

A satisfaction relation |= on a Kripke structure M =

hS, S

,R, Li for CTL is deﬁned by the same conditions

1–5 and 9–16 as in Deﬁnition 2.3 by deleting the su-

perscript “m”. The validity, satisﬁability and model-

checking problems for CTL are deﬁned similarly as

those for LCTL.

Remark that |=

of LCTL includes |= of CTL, and

hence LCTL is an extension of CTL.

3 EMBEDDING AND

COMPLEXITY

Deﬁnition 3.1 Let AT be a non-empty subset of

ATOM, and AT

(m ∈ ω) be the sets {p

| p ∈ AT

}

of atomic formulas where p

:= p (i.e., AT

:= AT).

The language L

(the set of formulas) of LCTL is de-

ﬁned using AT, X

, G

, F

, ¬,→, ∧, ∨, X, F, G, U, A

and E. The language L of CTL is obtained from L

by adding

[

m∈ω

and deleting {X

A mapping f from L

to L is deﬁned inductively

by:

1. for any p ∈ AT, f(X

p) := p

∈ AT

, esp.,

f(p) := p,

2. f(X

(α ♯ β)) := f(X

α) ♯ f(X

β) where ♯ ∈

{∧,∨,→},

3. f(X

♯α) := ♯ f(X

α) where ♯ ∈

{¬,AX, EX,AG, EG,AF, EF},

4. f(X

♯(αUβ))) := ♯( f(X

α)Uf (X

β)) where

♯ ∈ {A,E},

5. f(X

α) :=

{ f(X

m+n

α) | n ∈ ω

6. f(X

α) :=

{ f(X

m+n

α) | n ∈ ω

Lemma 3.2 Let f be the mapping deﬁned in Def-

inition 3.1. For any time-indexed Kripke structure

M := hS,S

,R, {L

}

m∈ω

i for LCTL, and any satisfac-

tion relations |=

(m ∈ ω) on M, we can construct a

Kripke structure N := hS,S

,R, Li for CTL and a sat-

isfaction relation |= on N such that for any formula α

in L

and any state s in S,

M,s |=

α iff N,s |= f (X

α).

Proof. Let AT be a nonempty subset of ATOM, and

be the sets {p

| p ∈ AT} of atomic formulas.

Suppose that M is a time-indexed Kripke structure

hS, S

,R, {L

}

m∈ω

i such that

(m ∈ ω) are functions from S to the power

set of AT.

Suppose that N is a Kripke structure hS,S

,R, Li such

that

ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence

524

L is a function from S to the power set of

[

m∈ω

Suppose moreover that for any s ∈ S and any p ∈ AT,

p ∈ L

(s) iff p

∈ L(s).

The lemma is then proved by induction on the

complexity of α.

• Base step:

Case α ≡ p ∈ AT: We obtain: M, s |=

p iff p ∈

(s) iff p

∈ L(s) iff N,s |= p

iff N, s |= f (X

(by the deﬁnition of f).

• Induction step:

Case α ≡ β ∧ γ: We obtain: M, s |=

β ∧ γ iff

M,s |=

β and M,s |=

γ iff N,s |= f(X

β) and

N,s |= f(X

γ) (by induction hypothesis) iff N,s |=

f(X

β)∧ f(X

γ) iff N,s |= f(X

(β∧γ)) (by the def-

inition of f).

Case α ≡ β∨ γ: Similar to Case α ≡ β∧ γ.

Case α ≡ β→γ: We obtain: M, s |=

β→γ iff

M,s |=

β implies M,s |=

γ iff N,s |= f(X

β) im-

plies N, s |= f(X

γ) (by induction hypothesis) iff

N,s |= f(X

β)→ f(X

γ) iff N,s |= f (X

(β→γ)) (by

the deﬁnition of f).

Case α ≡ ¬β: We obtain: M,s |=

¬β iff not-

[M,s |=

β] iff not-[N,s |= f(X

β)] (by induction hy-

pothesis) iff N,s |= ¬ f(X

β) iff N, s |= f(X

¬β) (by

the deﬁnition of f).

Case α ≡ X

β:

Subcase m ≤ l − 1: We obtain: M, s |=

β iff

M,s |=

m+1

β iff N,s |= f(X

m+1

β) (by induction hy-

pothesis).

Subcase m ≥ l: We obtain: M, s |=

β iff

M,s |=

β iff M,s |=

m+1

β iff N,s |= f (X

m+1

β) (by

induction hypothesis).

Case α ≡ G

β: We obtain: M,s |=

β iff

M,s |=

m+n

β for any n ∈ ω

iff N,s |= f(X

m+n

β)

for any n ∈ ω

(by induction hypothesis) iff N,s |=

{ f(X

m+n

β) | n ∈ ω

} iff N,s |= f(X

β) (by the

deﬁnition of f).

Case α ≡ F

β: Similar to Case α ≡ G

β.

Case α ≡ AXβ: We obtain: M, s |=

AXβ iff

∀s

∈ S [(s,s

) ∈ R implies M, s

β] iff ∀s

∈

S [(s,s

) ∈ R implies N,s

|= f(X

β)] (by induc-

tion hypothesis) iff N,s |= AXf(X

β) iff N,s |=

f(X

AXβ) (by the deﬁnition of f).

Case α ≡ EXβ: Similar to Case α ≡ AXβ.

Case α ≡ AGβ: We obtain:

M,s |=

AGβ

iff for all paths π ≡ s

,..., where s ≡ s

, and all

states s

along π, we have M,s

iff for all paths π ≡ s

,..., where s ≡ s

, and

all states s

along π, we have N,s

|= f(X

β) (by

induction hypothesis)

iff N,s |= AGf(X

β)

iff N,s |= f(X

AGβ) (by the deﬁnition of f).

Cases α ≡ EGβ, α ≡ AFβ and α ≡ EFβ: Similar

to Case α ≡ AGβ.

Case α ≡ A(βUγ): We obtain:

M,s |=

A(βUγ)

iff for all paths π ≡ s

,..., where s ≡ s

, there is

a state s

along π such that [M,s

γ and ∀ j[i ≤

j < k implies M, s

β]

iff for all paths π ≡ s

,..., where s ≡ s

, there

is a state s

along π such that [N, s

|= f(X

γ) and

∀ j[i ≤ j < k implies N,s

|= f(X

β)] (by induc-

tion hypothesis)

iff N,s |= A( f(X

β)Uf(X

γ))

iff N,s |= f(X

A(βUγ)) (by the deﬁnition of f).

Case α ≡ E(βUγ): Similar to Case α ≡ A(βUγ).

Lemma 3.3 Let f be the mapping deﬁned in Deﬁni-

tion 3.1. For any Kripke structure N := hS, S

,R, Li

for CTL, and any satisfaction relation |= on N, we

can construct a time-indexed Kripke structure M :=

hS, S

,R, {L

}

m∈ω

i for LCTL and satisfaction rela-

tions |=

(m ∈ ω) on M such that for any formula α

in L

and any state s in S,

N,s |= f(X

α) iff M,s |=

α.

Proof. Similar to the proof of Lemma 3.2.

Theorem 3.4 (Embedding) Let f be the mapping

deﬁned in Deﬁnition 3.1. For any formula α, α is

valid (satisﬁable) in LCTL iff f(α) is valid (satisﬁ-

able) in CTL.

Proof. By Lemmas 3.2 and 3.3.

We then obtain the main theorem of this paper.

Theorem 3.5 (Complexity) The model-checking,

validity and satisﬁability problems for LCTL are

deterministic PTIME-complete, EXPTIME-complete

and deterministic EXPTIME-complete, respectively.

Proof. By the mapping f deﬁned in Deﬁnition 3.1, a

formula α of LCTL can ﬁnitely be transformed into

the corresponding formula f(α) of CTL. By Lem-

mas 3.2 and 3.3 and Theorem 3.4, the model check-

ing, validity and satisﬁability problems for LCTL can

be transformed into those of CTL. Since the model

checking, validity and satisﬁability problems for CTL

are decidable, the problems for LCTL are also de-

cidable. Since the mapping f from LCTL into CTL

is a polynomial-time reduction, the complexity re-

sults for LCTL become the same results as CTL, i.e.,

BRANCHING-TIME VERSUS LINEAR-TIME - A Cooperative and Feasible Approach

525

the model-checking, validity and satisﬁability prob-

lems for LCTL are deterministic PTIME-complete,

EXPTIME-complete and deterministic EXPTIME-

complete, respectively.

4 CONCLUDING REMARKS

In this paper, a new logic, linear-time computation

tree logic (LCTL), was introduced by “cooperat-

ing” CTL and LTL, and the deterministic PTIME-

completeness (i.e., the existence of “feasible” algo-

rithms) of the LCTL model-checking problem was

shown. It was thus shown that there is a coopera-

tive and feasible approach to the traditional issue of

“branching-time versus linear-time”.

In the following, we give some remarks on the

idea of bounding time and on the concept of combin-

ing logics.

To restrict the time domain of the LTL operators

is not a new idea. Such an idea was discussed in

(Biere et al., 2003; Cerrito et al., 1999; Cerrito and

Mayer, 1998; Hodkinson et al., 2000). For exam-

ple, by using and introducing a bounded time domain

and the notion of bounded validity in a semantics,

bounded tableaux calculi (with temporal constraints)

for propositional and ﬁrst-order LTLs were intro-

duced by Cerrito, Mayer and Prand (Cerrito et al.,

1999; Cerrito and Mayer, 1998). It is also known that

to restrict the time domain is a technique to obtain

a decidable or efﬁcient fragment of ﬁrst-order LTL

(Hodkinson et al., 2000). Restricting the time domain

implies not only some purely theoretical merits dis-

cussed above, but also some practical merits for de-

scribing temporal databases and planning speciﬁca-

tions (Cerrito et al., 1999; Cerrito and Mayer, 1998),

and for implementing an efﬁcient model checking al-

gorithm called bounded model checking (Biere et al.,

2003). Such practical merits are due to the fact that

there are problems in computer science and artiﬁcial

intelligence where only a ﬁnite fragment of the time

sequence is of interest (Cerrito et al., 1999).

As mentioned in (Sernadas and Sernadas, 2003),

there are some general theories for various combined

modal logics (Sernadas and Sernadas, 2003), includ-

ing the theories of fusion, product and ﬁbring. Vari-

ous combined modal logics have been studied based

on these theories. The proposed logic LCTL may be

categorized by a fusion of CTL and a bounded-time

version of LTL.

ACKNOWLEDGEMENTS

This research was supported by the Alexander von

Humboldt Foundation and by the Japanese Ministry

of Education, Culture, Sports, Science and Technol-

ogy, Grant-in-Aid for Young Scientists (B) 20700015.

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