![](bg5.png)
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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