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The Complexity of Regularity in Grammar Logics and
Related Modal Logics
Stéphane Demri
To cite this version:
Stéphane Demri. The Complexity of Regularity in Grammar Logics and Related Modal Logics. Jour-
nal of Logic and Computation, Oxford University Press (OUP), 2001, 11 (6), pp.933-960. �10.1093/log-
com/11.6.933�. �hal-03189648�
The Complexity of Regularity in
Grammar Logics
(long version)
St´ephane Demri
∗
Lab. Sp´ecification et V´erification
ENS de Cachan & CNRS UMR 8643
61 Av. Pdt. Wilson
94235 Cachan Cedex, France
email: demri@lsv.ens-cachan.fr
Abstract
A modal reduction principle of the form [i
1
] . . . [i
n
]p ⇒ [j
1
] . . . [j
n
0
]p can be
viewed as a production rule i
1
· . . . · i
n
→ j
1
· . . . · j
n
0
in a formal grammar.
We study the extensions of the multimodal logic K
m
with m independent K
modal connectives by finite addition of axiom schemes of the above form such
that the associated finite set of production rules forms a regular grammar. We
show that given a regular grammar G and a modal formula φ, deciding whether
the formula is satisfiable in the extension of K
m
with axiom schemes from G
can be done in deterministic exponential-time in the size of G and φ, and this
problem is complete for this complexity class. Such an extension of K
m
is called
a regular grammar logic. The proof of the exponential-time upper bound is
extended to PDL-like extensions of K
m
and to global logical consequence and
global satisfiability problems. Using an equational characterization of context-
free languages, we show that by replacing the regular grammars by linear ones,
the above problem becomes undecidable. The last part of the paper presents
non-trivial classes of exponential time complete regular grammar logics.
This is a long version of a paper to appear in Journal of Logic
and Computation.
Keywords: computational complexity, modal logic, formal gram-
mar, regular language, finite automaton.
∗
On leave from Laboratoire LEIBNIZ, Grenoble, France.
1
1 Introduction
Capturing the decidability/complexity status of modal logics.
A nowadays popular approach to establish the decidability of modal log-
ics consists in studying the decidability status of fragments of first-order
logic [Var97, ANB98] (see also [Gab81]). Sometime, these fragments are
augmented by features that are not present in the standard first-order
language allowing more expressive power, often at the cost of losing de-
cidability (see e.g. [GOR97]). By contrast, the guarded fixed point logic
µLGF [GW99] is a decidable fragment of fixed-point first-order logic in
which can be naturally embedded the modal µ-calculus (see e.g. [Koz83]).
Once an interesting decidable fragment is identified, the design of deci-
sion procedures that meet the best worst-case complexity upper bounds
is often the next step (see e.g. [Sch97, Niv98, Hus99]). However, even if
your favorite modal logic can be embedded in a known decidable frag-
ment of second-order logic, the characterization of the worst-case com-
plexity of your logic is not straightforward from the complexity of the
second-order fragment. For instance, the standard modal logic K can
be embedded into FO2, the fragment of classical predicate logic with
only two individual variables and without function symbols but FO2-
satisfiability is NEXPTIME-complete whereas K-satisfiability is “only”
PSPACE-complete [Lad77] (see e.g. [Pap94] for a thorough introduction
to computational complexity). Therefore, there is a need to develop gen-
eral metho ds dedicated to the computational complexity of modal logics.
Spaan’s thesis [Spa93] can be considered as an important step towards
this direction. Indeed, the worst-case complexity of independent fusion
of modal logics is studied there (see also [Hem94]). The study of the com-
plexity of PDL-like logics (e.g. the modal µ-calculus, Combinatory PDL)
can be understood as the modal counterpart for the study of decidable
fragments of second-order logic. Indeed, many modal logics can be natu-
rally embedded into PDL-like logics (see e.g. [Tuo90, Sch91, Gia95]) and
therefore the design of efficient decision procedures for PDL-like logics is
another way to study the complexity of modal logics in a uniform frame-
work. Typically, there is some natural transformation from satisfiability
for the standard modal logics B, S4, S5 into PDL with converse (see e.g.
[FL79]). However, such translations have not been studied in a system-
atic way. In the present paper, the main object of study is a countably
infinite class of polymodal logics such that the satisfiability problem can
be embedded in linear-time to first-order logic. Unfortunately, the target
2
fragment is not known to belong to identified decidable fragments and
therefore decidability and complexity shall be established by modal-like
techniques partly based on formal language theory.
Grammar logics. An important idea in logic programming is to trans-
late the special purpose formalism of formal (context-free) grammars into
a general purpose one, namely first-order predicate logics. In [FdCP88],
a similar approach is suggested where the analysis or generation of a
sentence is transformed to theorem proving for modal logics. The modal
logics (called “grammar logics”) introduced in [FdCP88] are closely re-
lated to formal grammars. Namely, with each production rule i
1
·. . .·i
n
→
j
1
· . . . · j
n
0
in the grammar is associated a modal axiom [i
1
] . . . [i
n
]p ⇒
[j
1
] . . . [j
n
0
]p. Such axioms are called reduction principles in [Ben76, CS94]
and they are a special type of Sahlqvist formulae [Sah75] and primitive
formulae [Kra96]. They are typical in modal logic. In this paper, we
study the extensions of the multimodal logic K
m
with m independent K
modal connectives by finite addition of axiom schemes of the above form
such that the associated finite set of production rules forms a regular
formal grammar.
Having in mind the initial motivation to introduce the grammar logics
in [FdCP88], it is worth observing that the regular grammar logics are
too express ive to encode the generation of strings by regular grammars.
Indeed, whether a string b elongs to a context-free language (defined by a
context-free grammar) is a P-complete problem (see e.g. [JL76, Corollary
11]) whereas the satisfiability problem of any regular grammar logic shall
be shown to be PSPACE-hard. In a sense, introducing regular gram-
mar logics to analyze regular languages is not very efficient. However,
we claim that it is more interesting to take advantage of the wealth of
knowledge about regular languages in order to analyze the computational
complexity of regular grammar logics.
Related modal logics. Although the grammar logics may seem ar-
tificial, many polymodal logics containing fragments that are regular
grammar logics can be found in the literature (see e.g. [FL79, Cat89,
HM92, G as94, FdCH95, Hem96, HM97] to quote a few examples). More
importantly, Description Logics (DLs) that are used to represent termi-
nological knowledge (see e.g. [SSS91]) are strongly related to grammar
logics. A current line of research in DLs community consists in studying
more and more expressive description logics as soon as they are mean-
3
ingful for knowledge representation languages (see e.g. [Wol99, HST00]).
Typically, in order to obtain expressive roles, one can either add role
constructors or constraint the interpretation of roles. Inverse roles, tran-
sitive roles, role value inclusions and role hierarchies are features of DLs
that allow a gain of expressive power for roles (see e.g. [HS99, HM00,
HST00, Are00]). The grammatical constraints on relations considered in
the present paper can be viewed as role inclusion axioms for roles built
from atomic roles and finite composition.
That is why understanding the complexity of grammar logics can help
understanding the complexity of other related modal logics, including
DLs.
Our contribution. We show that every regular grammar logic is de-
cidable. Up to now, it is only known that every right linear grammar
logic is decidable [Bal98, BGM98] and this solves an open problem men-
tioned in [Bal98, BGM98]. It is worth observing that although the right
linear grammars generate the same class of languages as the left lin-
ear grammars, this correspondence is not relevant at the level of regular
grammar logics. It is fair to mention that the initial motivation for this
work was to understand why the decidability proof in [Bal98, BGM98]
cannot be naturally extended to left linear grammars. Our decidability
proof consists in defining transformations into the satisfiability problem
for PDL. This allows us to prove that the satisfiability problem f or any
regular grammar logic is in EXPTIME and this result can be extended
to PDL-like logics, to description logics with inclusion axioms (some re-
strictions are made here), to the global logical consequence problem and
to the global satisfiability problem. Up to now, it is only known that
the satisfiability problem for any right linear grammar logics is in NEX-
PTIME [Bal98, BGM98]. Unfortunately, our transformation is not in
polynomial-time in the size of the regular grammars. We then show that
given a regular grammar G and a modal formula φ, deciding whether the
formula is satisfiable in the extension of K
m
with axiom schemes from
G can be done in deterministic exponential-time in the size of G and φ.
We refer to this problem as the general satisfiability problem for regular
grammar logics. The complexity upper bound is established by defin-
ing a polynomial time transformations into the satisfiability problem for
PDL with finite automata. PDL with automata is a succinct variant
of PDL where the regular expressions are replaced by finite automata.
Up to now, it is only known that the general satisfiability problem re-
4