Two-sorted Point-Interval Temporal Logics.

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Two-sorted Point-Interval Temporal Logics
Philippe Balbiani1
Institut de Recherche
en Informatique de Toulouse
Universit´ e Paul Sabatier,
Toulouse, France
Valentin Goranko2
Department of Informatics and Mathematical Modeling
Technical University of Denmark,
Lyngby, Denmark
Guido Sciavicco3
Department of Computer Engineering
Middle East Technical University, Northern Cyprus Campus, and
Department of Information Engineering and Communications
University of Murcia,
Murcia, Spain
Abstract
There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and
interval-based. Usually, interval-based temporal reasoning deals with points as particular, duration-less
intervals. Here we develop explicitly two-sorted point-interval temporal logical framework whereby time
instants (points) and time periods (intervals) are considered on a par, and the perspective can shift between
them within the formal discourse. We focus on fragments involving only modal operators that correspond
to the inter-sort relations between points and intervals. We analyze their expressiveness, comparative to
interval-based logics, and the complexity of their satisfiability problems. In particular, we identify some
previously not studied and potentially interesting interval logics.
Keywords: point and interval temporal logics, decidability, complexity.
1 Introduction
The predominant approach in the studies of temporal reasoning and logics has been
based on the assumption of time points (instants) as the primary temporal on-
1Email: philippe.balbiani@irit.fr
2Email: vfgo@imm.dtu.dk
3Email: guido@um.es
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Electronic Notes in Theoretical Computer Science 278 (2011) 31–45
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tological entities. However, there have also been active studies of interval-based
temporal reasoning and logics over the past 2 decades, starting with the seminal
work of Halpern and Shoham [11] introducing the multi-modal logic, that we will
call HS, comprising modal operators for all possible relations (known as Allen’s
relations [1]) between two intervals in a linear order, and followed by a series of
publications studying expressiveness and decidability/undecidability and complex-
ity of the fragments of HS, e.g.,
both approaches – point-based and interval-based – see [17]. Many studies on in-
terval logics have considered the so-called ’non-strict’ interval semantics, allowing
point-intervals (with coinciding endpoints) along with proper ones, and thus encom-
passing the instant-based approach, too; see e.g., [11,10,4]. Yet, little has been done
so far on formal treatment of both temporal primitives, points and intervals, on a
par, in a common, two-sorted framework. The present paper purports to provide a
systematic such treatment. Our work is motivated by several observations:
[10,4]. For a detailed philosophical study of
• Natural languages incorporate both ontologies on a par, without assuming
the primacy of one over the other, and have the capacity to shift smoothly the
perspective from instants to intervals and vice versa within the same discourse.
• There are various temporal scenarios which neither of the two ontologies alone
can grasp properly. In particular, sometimes neither the treatment of intervals
as sets of their internal points, nor the treatment of points as ‘instantaneous’
intervals, is really adequate. For example, a sentence like ‘Ever since he met
her for the first time, he could not stop thinking about her and kept calling her
several times every night until she would give him a brush-off, and then after
being silent for a while he would phone again...’ cannot be properly repre-
sented in either instant-only or interval-only framework. As another example,
consider a typical safety requirement of an intelligent systems that controls a
rail crossing:’At the exact moment in which the train passes over the sensor,
the rail crossing bar starts to close; the bar will start to open again a while
after the train passes over the second sensor’.
• The technical identification of intervals with sets of their internal points, or
of points as instantaneous intervals leads also to conceptual problems, e.g. of
confusing events and fluents. Instantaneous events are represented by time
intervals and should be distinguished from instantaneous holding of fluents,
which are evaluated at time points. Formally, the point a should be distin-
guished from the interval [a,a] and the truths in these should not necessarily
imply each other.
• Moreover, the area of artificial intelligence is concerned with purely practi-
cal problems related to the formal representation and reasoning of intelligent
agents on various temporal and spatial aspects such as position, motion, ac-
tions, processes, events, fluents, etc. Some of these can be adequately repre-
sented in either of the rival ontologies, while others become awkward, if not
meaningless, in one or the other of them.
• Finally, we note that, while differences in the expressiveness have been found
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between the strict and non-strict semantics for some interval logics (see [7],
for example), so far no distinction in the decidability of the satisfiability has
been found. Therefore, we believe that an attempt to systemize the role of
points, intervals, and their interaction, would make good sense not only from a
purely ontological point of view, but also from algorithmic and computational
perspectives.
There have been several logical studies of the relationship between instants and
intervals, including [12,17,15]. One of the conceptual precursors of our present study
is [8] where Galton introduces a two-sorted ‘aspectual calculus’ involving points and
events on a par. Further, in [9] Galton argues that Allen’s interval-based theory
of time and action is inadequate for representing continuous change and advocates
the necessity of adding time instants in their own capacity to it. Other explicit
two-sorted point-interval formal studies of time of which we are aware include the
system IP from [18,16], based on the first-order theory of point-interval structures,
and the system LNint from [6], where, however, the interval type plays a secondary
role since formulae are always evaluated at points, and the time line is assumed to
be discrete (the set of integers).
Here we develop explicitly two-sorted modal approach to the point-interval tem-
poral reasoning whereby time instants (points) and time periods (intervals) are
considered on a par, and the perspective can shift between them within the formal
discourse. We compare that language with those interval logics with non-strict se-
mantics, that we already know to be right on the border between decidable and
undecidable. One of the most important examples on the decidable side is that of
Propositional Neighborhood Logic (PNL) with non-strict semantics [4], that corre-
sponds to the fragment of HS with the modal operator for meets and met-by only,
plus the modal constant π for point-interval; In [14] it has been shown that PNL is
almost maximal (amongst the fragments of HS) w.r.t. decidability on the class of
all linear orders, while the pair of operators corresponding to Allen’s relations ends
and ended by constitutes the most interesting exception, as they can be added to
PNL without losing the decidability when interpreted over finite models or mod-
els based on the set of natural numbers N. Here, we first introduce the logic PI
comprising modal operators for all possible binary relations between points and in-
tervals. It is easy to see that PI is at l east as expressive as HS, and therefore
it is undecidable under the same assumptions where the latter is. Then, we focus
on the fragment PImixinvolving only the modal operators that correspond to the
inter-sort relations between points and intervals. We analyze the expressiveness and
the complexity of the satisfiability problem of PImixand some of its fragments. In
particular, we identify some previously not studied, in the context of fragments
of HS, and potentially interesting interval logics whose decidability/undecidability
status can be deduced from already known results for fragments of HS and at least
one of them has an unexpectedly low complexity.
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2 Preliminaries: the Logic HS
In the classical interval setting, given a linearly ordered set D = ?D,<?, an interval
(also called non-strict interval) is defined as a pair [a,b], where a,b ∈ D and a ≤ b.
The logic HS, introduced in [11], is defined over a set of propositional letters AP,
denoted by p,q,..., by using the classical propositional operators ¬,∨ (whereas the
remaining ones can be considered as shortcuts), and a modal operator for each of
the 12 Allen’s relation, that is, each possible binary relation between two intervals
on linear orders (excluding equality, the modal operator for which is trivial). The
standard notation for such modal operators is as follows: ?A? (in the non-strict
semantics, it is usually denoted by 3r) for the relation meets, ?B? for begins, ?E?
for ends; moreover and, for each ?X? ∈ {?A?,?B?,?E?}, ?X? denotes its inverse (in
the non-strict semantics, 3ldenotes the inverse of 3r), and the modal operators
for the remaining six operators can easily be defined in terms of the above ones. In
this setting, a HS-model M is defined as M = ?D,I(D)+,V?, where I(D)+is the set
of all non-strict intervals over D, and V : I(D)+→ 2APis a labeling function. The
semantics of HS-formulae ϕ is as follows:
• M,[a,b] ? p iff p ∈ V([a,b]);
• M,[a,b] ? π iff a = b;
• M,[a,b] ? ?A?ψ (resp., 3rψ) iff there exists c > b (resp., c ≥ b) such that
M,[b,c] ? ψ;
• M,[a,b] ? ?B?ψ iff there exists a ≤ c < b such that M,[a,c] ? ψ;
• M,[a,b] ? ?E?ψ iff there exists a < c ≤ b such that M,[c,b] ? ψ;
• M,[a,b] ? ?D?ψ iff there exist a < c < d < b such that M,[c,d] ? ψ;
• M,[a,b] ? ?O?ψ iff there exist a < c ≤ b < d such that M,[c,d] ? ψ;
• M,[a,b] ? ?L?ψ iff there exist a < b < c ≤ d such that M,[c,d] ? ψ.
where the semantics of classical operators is as standard. The semantics of the
inverse operators can be easily deduced from the above clauses; for example, we
have:
• M,[a,b] ? ?A?ψ (resp., 3lψ) iff there exists c < b (resp., c ≤ b) such that
M,[c,b] ? ψ;
• M,[a,b] ? ?B?ψ iff there exists c > b such that M,[a,c] ? ψ;
• M,[a,b] ? ?E?ψ iff there exists c < a such that M,[c,a] ? ψ.
With every subset {?X1?,...,?Xk?} of HS modal operators, we associate the
fragment X1X2...Xkof the logic HS that features all and only those modal op-
erators, possibly extended with the modal constant π, denoted in this case by
X1X2...Xkπ, if π is not definable in that particular fragment. In the recent lit-
erature, PNL (resp., PNLπ) is also used to denote the fragment AA (resp., 3r3π
Recent results concerning decidability, undecidability, and expressive power of frag-
ments of HS include [7,14,2].
l).
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In [11], the satisfiability problem for the logic HS has been proved to be unde-
cidable over all interesting classes of linearly ordered sets. In the classification of
decidability/undecidability for most of the interesting fragments of HS for both the
strict and non-strict semantics (i.e., classes of models that exclude, resp., include
point-intervals), the case of PNL/PNLπ, is probably the most interesting one on the
decidable side. Its satisfiability problem is decidable in finite, discrete, and dense
case, as well as in the class of all linearly ordered sets, among others; in all these
cases, it is NEXPTIME-complete . More recently, the fragment AAEE has been
shown to be decidable in the finite case and over the set of natural numbers [14], but
its complexity is non-elementary. EXPSPACE-completeness holds, among others,
for the fragment ABB in the same cases as PNL/PNLπ(see [5] for a complete survey
of these decidable fragments). Finally, in the view of our classification of the ex-
pressive power of some of the languages that we introduce in the following sections,
we also note that the fragment AAD (and, therefore, AADπand 3r3rDπ) is an im-
portant case of undecidable fragment, very close to the decidability/undecidability
border [2].
3Syntax and Semantics of PI and its Fragments
Point and Interval Relations3.1
Given a linearly ordered set D = ?D,<?, we call the elements of D points and
define an interval as an ordered pair [a,b] of points in D, where a < b. Now, as we
have mentioned above, there are 13 possible relations, including equality, between
any two intervals. From now on, we call these interval-interval relations. Besides,
there are 3 different relations that may hold between any two points (before, equal,
and after), called hereafter point-point relations, and 5 different relations that may
hold between a point and an interval and vice-versa, namely before, beginning point,
during, ending point, and after, called hereafter point-interval relations. Intuitively,
our language will follow the same principle as the logic HS, discussed in the previous
section: one modal operator for each one of the 19 relations, excluding the equalities
between points and between intervals. To provide a uniform and simple notation,
we first distinguish among two main types of modal operators: those evaluated
at points, denoted by single square brackets ??, and those evaluated at intervals,
denoted by double square brackets ????. Now, consider an interval [b,c]: it generates
a partition of the set D into five regions (see [13]): the region 0, of those points
before b, the region 1 that contains b only, the region 2 (between b and c), 3 (only
c), and 4 (the remaining regions). Using that notation, for k ∈ {0,1,2,3,4}, a point
modality may belong to one of two categories: ‘point to point’, of the type ?k?, that
refers to any point in the relation k with the current one – in which case the regions
1,2,3 coincide and we use the number 2 to indicate any of them – and ‘point to
interval’, of the type ?kk??, that refers to any interval such that its beginning point
is in the region k, and its ending point in the region k?, with respect to the current
point. Likewise, an interval modality can be of ‘interval to point’ type ??k?? referring
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to any point in the area k w.r.t. the current interval, or of ‘interval to interval’ type
??kk???, in which case it becomes a syntactic variation of the respective HS modality
in the strict semantics.
3.2 Syntax and Semantics of the Point-Interval Logic PI
The language of the Point-Interval Logic PI comprises the classical connectives ¬
and ∨ (the rest are considered definable), two sorts of propositional letters, namely
the set of point propositional letters APpoand the set of interval ones APint, and
unary modalities of each of the types specified above. For technical convenience we
will assume that each of APpoand APintis a copy of the set of propositional letters
AP from the language of HS. We will denote typical elements of AP by p,q,...,
respective typical elements of APpoby ppo,qpo,..., and respective typical elements
of APintby pint,qint,....
The logic PI has two sorts of formulae: point formulae and interval formulae.
Point formulae are obtained by the following grammar:
ϕpo::= ppo| ¬ϕpo| ϕpo∨ ψpo| ?pp?ϕpo| ?pi?ϕint,
where ?pp? represent any point-to-point modality, and ?pi? is any point-to-interval
modality. Similarly, interval formulae are formed by the following grammar:
ϕint::= pint| ¬ϕint| ϕint∨ ψint| ??ii??ϕint| ??ip??ϕpo,
where the modal operators are interval-to-point or interval-to-interval. The formulae
of the type ?pp?ϕpo, ?pi?ϕint, ??ip??ϕpo, and ??ii??ϕintare called (respectively, point
or interval) diamond formulae. The respective box formulae are defined, as usual,
as their duals, e.g. [ij]ψ := ¬?ij?¬ψ4. Lastly, a PI formula is a point-formula or
an interval-formula.
A point-interval structure is a pair F = ?D,I(D)? where D = ?D,<? is a linear
order and I(D) is the set of all strict intervals in D. A PI-model is a tuple M =
(D,I(D),Vpo,Vint) where (D,I(D) is a point-interval structure and Vpo: D → 2APpo
and Vint : I(D) → 2APintare valuations5assigning to each point (respectively,
interval) the set of point (respectively, interval) propositional letters that are true of
it. The truth (satisfaction) relation is defined in a PI-model by a mutual recursion
on point and interval formulae as follows (the clauses for the classical connectives
are standard):
• M,a ? ppoiff ppo∈ Vpo(a);
• M,a ? ?0?ϕ iff there exists b < a such that M,b ? ϕ;
• M,a ? ?4?ϕ iff there exists b > a such that M,b ? ϕ;
4The typographic similarity between a box [ij] and an interval [a,b] is unfortunate, but should not cause
confusion.
5Usually, in modal logic valuations are functions assigning sets of possible worlds to propositional letters,
and the functions defined here are called ‘labeling functions’, but in this paper we will use the term ’labeling
function’ for another purpose.
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• M,a ? ?00?ϕ iff there exist b,c such that b < c < a, and that M,[b,c] ? ϕ;
• M,a ? ?02?ϕ iff there exist b,c such that b < c = a, and that M,[b,c] ? ϕ;
• M,a ? ?04?ϕ iff there exist b,c such that b < a < c, and that M,[b,c] ? ϕ;
• M,a ? ?24?ϕ iff there exist b,c such that b = a < c, and that M,[b,c] ? ϕ;
• M,a ? ?44?ϕ iff there exist b,c such that a < b < c, and that M,[b,c] ? ϕ,
and respectively:
• M,[a,b] ? pintiff pint∈ Vint([a,b]);
• M,[a,b] ? ??0??ϕ iff there exists c < a such that M,c ? ϕ;
• M,[a,b] ? ??1??ϕ iff M,a ? ϕ;
• M,[a,b] ? ??2??ϕ iff there exist c such that a < c < b, and that M,c ? ϕ;
• M,[a,b] ? ??3??ϕ iff M,b ? ϕ;
• M,[a,b] ? ??4??ϕ iff there exist c such that b < c, and that M,c ? ϕ,
An interval- (resp., point-) PI-formula φ is satisfiable if there exists a PI-model
and an interval (resp., a point) in it that satisfies φ. Note that the clauses for the
‘interval-to-interval’ modalities are identical to those for the HS modalities in the
strict semantics.
4 Expressiveness of fragments of PI
In this section we systematically compare the expressiveness of PI and its fragments
to that of HS and its fragments.
4.1Transformations between models of PI and HS
In order for us to compare the expressiveness of fragments of PI and HS we need to
specify transformations of two-sorted PI models into HS-models, and vice-versa.
First, let M = (D,I(D),Vpo,Vint) be a point-interval model based on some lin-
early ordered domain D = ?D,<?. The corresponding non-strict HS-model τ(M) is
obtained by taking the set of all non-strict intervals over D: I(D)+= {[a,b] | a,b ∈
D,a ≤ b} and defining V as follows. For each proper interval [a,b], where a < b,
we put V([a,b]) = {p ∈ AP | pint∈ Vint([a,b]). Likewise, for each point interval
[a,a] we put V([a,a]) = {p ∈ AP | ppo∈ Vpo(a). Conversely, given any HS-model
M = ?D,I(D)+,V?, the corresponding PI-model σ(M) is obtained as follows. First,
we consider the set of all strict intervals over D, I(D) = {[a,b] | a,b ∈ D,a < b}.
Then, with every propositional letter p ∈ AP we associate two distinct new propo-
sitional letters: ppo and pint.Now, for each p ∈ AP and all points a ∈ D
such that p ∈ V([a,a]), we put ppo ∈ Vpo(a); respectively, for all strict intervals
[a,b] ∈ I(D) such that p ∈ V([a,b]), we put pint∈ Vint([a,b]). Finally, we define
σ(M) = ?D,I(D),Vpo,Vint?.
Further, with a slight abuse of notation we will use τ and σ to denote respec-
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tive syntactic translations of formulae of the one language to the other, such that
the translation τ maps each the propositional letters ppoand pintto p, while the
translation σ splits into two parts: σp, producing point formulae by mapping each
propositional letter p to ppoand σi, producing interval formulae by mapping each
propositional letter p to pint. Note that τ and σ are so far just notation; further we
will define different such translations, to match different fragments of HS and PI.
4.2Comparing expressiveness of some fragments of PI and HS
Now, we can compare the expressive power of standard interval logics and two-
sorted ones, by following the general terminology for comparing expressiveness of
logics with respect to the transformations τ and σ, as follows.
Let L be any fragment of PI, and L?any fragment of HS; we say that L?is
at least as expressive as L, denoted by L ? L?, iff τ can be extended to a truth
preserving syntactic translation from L to L?, that is, one that maps every L formula
to L?formula, such that:
(i) for every PI-model M, point a ∈ M and a point formula ψ of L:
M,a |= ψ iff τ(M),[a,a] |= τ(ψ),
(ii) for every PI-model M, interval [a,b] ∈ M and an interval formula ψ of L:
M,[a,b] |= ψ iff τ(M),[a,b] |= τ(ψ).
Likewise, let L be a fragment of HS and L?a fragment of PI. We say that
L?is at least as expressive as L, denoted L ? L?, iff σ can be extended to a truth
preserving syntactic translation from L to L?, that is, one that maps every L formula
to L?formula, such that:
(i) for every HS-model M, point interval [a,a] ∈ M and a formula ψ of L:
M,[a,a] |= ψ iff σ(M),a |= σp(ψ),
(ii) for every HS-model M, strict interval [a,b] ∈ M and a formula ψ of L:
M,[a,b] |= ψ iff σ(M),[a,b] |= σi(ψ).
In each of the cases above we say that L?is more expressive than L, denoted
L ≺ L?if L ? L?, and not L?? L. Respectively, we say that L and L?are expressively
incomparable6if neither L ? L?nor L?? L, and they are expressively equivalent,
denoted by L ≡ L?, if and only if L ? L?and L?? L.
Hereafter we only consider in detail the fragment PImixof PI that comprises
the inter-sort modalities, that is, the ‘point-to-interval’ and the ‘interval-to-point’
ones, and some notable fragments of it. We consider PImixto be the most inter-
esting of the fragments of PI because it captures precisely the expressiveness of the
interaction between points and intervals. We also denote by PImix−??k??the frag-
ment of PImixdevoid of the interval-to-point modal operator ??k??, and by PImix
−?kk??the fragment of PImixdevoid of the point-to-interval modal operator ?kk??.
Moreover, for any two-sorted language L, we write L>to denote the sub-language
6Note that these definitions are given in terms of the specific transformations τ and σ. However, it is easy
to see that these definitions are as general as possible.
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HS
AAEE
PImix
PImix−??2??
PImix−?04?
PImix−??2??,?04?≡PNLπ
3r3lDπ
AB
PImix−?04?>
Figure 1. Some interesting fragments of PI and their expressiveness relation with fragments of HS.
of L that only retains the ‘future’ operators of L, that is, those that refer to the
regions to the right of the current point or interval. In Fig. 1 we have indicated
expressiveness relations between each pair of fragments of HS and PI connected
by a line, where the relation ? applies between the lower to the higher fragment.
With a mild abuse of notation, we will use ≺ and ? to compare two fragments of
PI or of HS, in the obvious way (only syntactic translation needed).
Theorem 4.1 All expressiveness relations ? between fragments of HS and PI,
represented in Fig. 1 hold true.
Proof All relations ? on pairs of fragments of HS and on pairs of fragments of
PI are trivial. As for the remaining cases, it is sufficient to define suitable truth-
preserving translations τ and σ, as discussed above. We do that below for each case.
Notice that, in the non-strict semantics π is definable in a HS-fragment in presence
of ?B? (π ≡ [B]⊥) or ?E? (π ≡ [E]⊥). Also, observe that 3rϕ ≡ ?E?(ϕ∧π)∨?A?ϕ
and 3lϕ ≡ ?A?(ϕ ∨ ?E?(π ∧ ϕ)) (a similar definition can be devised in presence of
?B? instead of ?E?).
– PImix?AAEE:
The following translation τ works (the easy details are left to the reader).
• τ(?00?ψ) = π ∧ 3l(¬π ∧ 3l(¬π ∧ τ(ψ)));
• τ(?02?ψ) = π ∧ 3l(¬π ∧ τ(ψ));
• τ(?04?ψ) = π ∧ 3r(¬π ∧ ?E?τ(ψ));
• τ(?24?ψ) = π ∧ 3r(¬π ∧ τ(ψ));
• τ(?44?ψ) = π ∧ 3r(¬π ∧ 3r(¬π ∧ τ(ψ)));
• τ(??0??ψ) = ¬π ∧ 3l(¬π ∧ 3l(π ∧ τ(ψ)));
• τ(??1??ψ) = ¬π ∧ 3l(π ∧ τ(ψ));
• τ(??2??ψ) = ¬π ∧ ?E?(¬π ∧ 3l(π ∧ τ(ψ)));
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• τ(??3??ψ) = ¬π ∧ 3r(π ∧ τ(ψ));
• τ(??4??ψ) = ¬π ∧ 3r(¬π ∧ 3r(π ∧ τ(ψ))),
Note that the relationship PImix?AABB holds, too, by symmetry; nevertheless,
using AAEE gives us the decidability of PImix over the natural numbers - the
relationship PImix?AABB allows us to say that PImixis decidable over the (less
interesting) set of negative natural numbers [14].
– PImix−?04??3r3lDπ:
For this claim it suffices to modify the definition of τ from the previous case as
follows, taking into account that ?04? is no longer part of the language:
• τ(??2??ψ) = ¬π ∧ ?D?(π ∧ τ(ψ));
– PImix−?04?>?AB:
Again, by modifying τ above:
• τ(??1??ψ) = ¬π ∧ ?B?(π ∧ τ(ψ));
• τ(??2??ψ) = ¬π ∧ ?B?(¬π ∧ ?A?(π ∧ τ(ψ))).
– PImix−??2??,?04?≡PNLπ:
To show PImix−?04?,??2??? PNLπwe define τ as in the translation to AAEE above.
To show PNLπ? PImix−?04?,??2??we define σ as follows:
• σp(3rψ) = ?24?σi(ψ) ∨ σp(ψ);
• σp(3lψ) = ?02?σi(ψ) ∨ σp(ψ);
• σi(3rψ) = ??3??(?24?σi(ψ) ∨ σp(ψ));
• σi(3lψ) = ??1??(?02?σi(ψ) ∨ σp(ψ))
• σp(π) = ?,σi(π) = ⊥.
2
We note that the expressive embeddings above are not claimed here to be strict.
Proving their strictness requires proving respective non-expressibility results (which,
in general, depend on the particular class of structures in which the languages are
interpreted) for which there is no space here. However, related classification of
the expressive power of fragments of HS has recently appeared in [7], and some of
the expressive embedding results above can be proven strict by using the model-
theoretic techniques applied there.
5Decidability and complexity of fragments of PI
Clearly PI is at least as expressive as HS, and therefore its satisfiability problem
is undecidable over most interesting classes of linearly ordered sets. Also, the more
expressive fragments of PI are at least as expressive as some known undecidable
fragments of HS, and thus are undecidable themselves. For example, any fragment
of PI including at least the pair of modalities ??34??,??22?? (resp., the modality
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??22??, or the pair of modalities ??14??,??03??) is undecidable in every interesting
class of models, as it includes the HS-fragment AD (resp., O, BE). On the other
hand, a number of fragments of PI are readily embeddable in already known de-
cidable fragments of HS. Still, several fragments of PI give rise to essentially new
decidability and complexity problems. Because of space constraints, we will consider
in detail only one such case.
5.1 Some complexity bounds for PImixand its fragments
Here we use the comparative expressiveness results from the previous section to im-
mediately obtain results on decidability and complexity upper bounds of fragments
of PI, by using respective known results for fragments of HS.
Proposition 5.1
(i) The satisfiability problem for PImixinterpreted in the class of all finite models
and in the class of models based on the set of natural numbers N is decidable,
but with a non-elementary time complexity upper bound [14].
(ii) The satisfiability problem for PImix
of finite, discrete, dense, all linearly ordered models, as well as over models
based on N, is decidable in EXPSPACE [5];
(iii) The satisfiability problem for PImix
finite, discrete, dense, and all linearly ordered models, as well as over models
based on N, is NEXPTIME-complete [4].
−?04?>, interpreted in each of the classes
−??2??,?04?, in each of the classes of all
Note that matching lower bounds for most of these cases are not known yet, be-
cause PImixand its fragments do not have precise expressively matching fragments
of HS.
In the rest of this section we will adapt model-theoretic arguments used in [3,4]
to obtain a new decidability and complexity result for the satisfiability problem
for PImix−??2??interpreted over finite models by proving bounded-model property
with respect to models of exponential size. For that we will use the more general
notion of fulfilling labeling structures.
5.2Fulfilling labeling structures for PImix−??2??
To begin with, note that satisfiability of an interval formula ϕ is equivalent to
satisfiability of the point formula ?24?ϕ; therefore it suffices to consider satisfiability
of point formulas. Also, notice that two of the interval-to-point modalities are
definable in terms of the others:
??4??ψ ≡ ??3???24???3??ψ,
??0??ψ ≡ ??1???02???1??ψ,
and, therefore, we will treat them as shortcuts. Similarly, two of the point-to-interval
modalities are definable, too:
?44?ψ ≡ ?24???3???24?ψ,
?00?ψ ≡ ?02???1???02?ψ.
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Definition 5.2 The closure of ϕ is the set CL(ϕ) of all subformulae of ϕ and their
negations, where we identify every ¬¬ψ with ψ. We denote by CLpo(ϕ) the subset of
point formulae of CL(ϕ) and by CLint(ϕ) the subset of interval formulae of CL(ϕ).
Definition 5.3 A point ϕ-atom is a set A ⊆ CLpo(ϕ) such that for every ψ ∈
CLpo(ϕ), ψ ∈ A iff ¬ψ ?∈ A and for every ψ1∨ ψ2 ∈ CLpo(ϕ), ψ1∨ ψ2 ∈ A iff
ψ1∈ A or ψ2∈ A. An interval ϕ-atom is defined likewise, using CLint(ϕ) instead
of CLpo(ϕ).
We denote the set of point ϕ-atoms by Aϕ
int.
po and the set of interval ϕ-atoms by
Aϕ
Definition 5.4 Let ϕ be a PImix
ϕ is a tuple L = (D,I(D),Lpo,Lint), where (D,I(D)) is a point-interval structure
and Lpoand Lintare labeling functions defined respectively as Lpo: D → Aϕ
Lint: I(D) → Aϕ
• For every [b,c] ∈ I(D) and formula [24]ψ, if [24]ψ ∈ Lpo(b) then ψ ∈ Lint([b,c]);
• For every a ∈ D, [b,c] ∈ I(D) such that b < a < c, and formula [04]ψ, if
[04]ψ ∈ Lpo(a) then ψ ∈ Lint([b,c]);
• For every [a,b] ∈ I(D) and formula [02]ψ, if [02]ψ ∈ Lpo(b) then ψ ∈ Lint([a,b]),
• For every [a,b] ∈ I(D) and formula [[3]]ψ, if [[3]]ψ ∈ Lint([a,b]) then ψ ∈ Lpo(b);
• For every [a,b] ∈ I(D) and formula [[1]]ψ, if [[1]]ψ ∈ Lint([a,b]) then ψ ∈ Lpo(a).
Hereafter, by ‘labeling structure’ we will mean a labeling structure for some formula
ϕ.
−??2??-formula. A labeling structure (LS) for
po and
intand satisfying the following properties:
Note that every interval model M induces a LS, with labeling functions:
ψ ∈ Lpo(a) iff M,a ? ψandψ ∈ Lint([a,b]) iff M,[a,b] ? ψ.
Labeling structures can be thought of as quasi-models, in which the truth of
formulae containing no modal operators is determined by the labeling functions.
Furthermore, the labeling functions respect the semantics of the box operators. To
obtain ‘true models’, we must also guarantee that the labeling is in accordance
with the semantics of the diamond operators, too. To this end, we introduce the
following notion.
Definition 5.5 An LS L = (D,I(D),Lpo,Lint) for a formula ϕ is fulfilling (a FLS)
iff:
• For every formula ?24?ψ in CLpo(ϕ) and point a ∈ D, if ?24?ψ ∈ Lpo(a), then
there exists an interval [a,b] ∈ I(D) such that ψ ∈ Lint([a,b]);
• For every formula ?02?ψ in CLpo(ϕ) and point a ∈ D, if ?02?ψ ∈ Lpo(a), then
there exists an interval [b,a] ∈ I(D) such that ψ ∈ Lint([b,a]);
• For every formula ?04?ψ and point a ∈ D, if ?04?ψ ∈ Lpo(a), then there exists
an interval [b,c] ∈ I(D) such that b < a < c and ψ ∈ Lint([b,c]);
• For every formula ??1??ψ in CLint(ϕ) and interval [a,b] ∈ I(D), if ??1??ψ ∈
Lint([a,b]), then ψ ∈ Lpo(a);
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• For every formula ??3??ψ in CLint(ϕ) and interval [a,b] ∈ I(D), if ??3??ψ ∈
Lint([a,b]), then ψ ∈ Lpo(b).
Definition 5.6 A point (resp. interval) formula ψ is satisfied by a point (resp.
interval) in a given FLS if it belongs to its label. A point (resp. interval) formula
ψ is satisfied by an LFS if it is satisfied by some point (resp. interval) in it.
Proposition 5.7 For every PImix
model is equivalent to satisfiability in some fulfilling labeling structure.
−??2??-formula, satisfiability in a point-interval
Definition 5.8 Given an LS L = (D,I(D),Lpo,Lint) and a point a ∈ D, the modal
type of a in L is the set R(a) consisting of all diamond point formulae in the label
Lpo(a). We will call the formulae in R(a) requests at the point a.
5.3Decidability and NEXPTIME-completeness of PImix−??2??in the finite
Now we will show that every finite FLS satisfying a given formula of PImix−??2??
can be cut down to size exponentially bounded above by the size of the formula.
Let us define m = |CL(ϕ)|, and observe what follows:
(i) If R(ϕ) is the set of modal types for points in L, then |R(ϕ)| ∈ O(2m).
(ii) The set of diamond formulas of the type ?02?ξ or ?24?ξ in a given modal type
R(a) can be satisfied by using at most m distinct endpoints of intervals.
For lack of space we have to omit the proof of the next, main technical result.
Lemma 5.9 Let L = (D,I(D),Lpo,Lint), where D = (D,<), be a finite FLS for
a point formula ϕ, satisfying ϕ at some point a. Suppose that there exists a point
e ?= a ∈ D with modal type R = R(e) such that there are at least m2+m+1 points
with modal type R before e, and at least m2+ m + 1 such points after e in (D,<).
Then, there exists a FLS¯L = (¯D,I(¯D),¯
Corollary 5.10 Satisfiability of a PImix
linear orderings is equivalent to its satisfiability on the class of finite linear orderings
of size at most 2m+1(m2+ m + 1), where m = |CL(ϕ)|.
Corollary 5.11 Satisfiability of PImix−??2??on the class of all finite linear order-
ings is decidable in NEXPTIME, and hence it is NEXPTIME-complete.
Lpo, ¯
Lint) satisfying ϕ such that¯D = D\{e}.
−??2??-formula ϕ on the class of finite
5.4 Small ultimately periodic model property of PImix−??2??on N
Here, we will extend the previous result to decidability of PImix−??2??on N. Again,
for lack of space, we can only provide a sketch.
Definition 5.12 An LS L = (N,I(N),Lpo,Lint) is ultimately periodic (UPLS) with
prefix length K ≥ 0 and period length P if Lpo(c+P) = Lpo(c) for every point c ≥ K
and for every interval [c,d]:
• If c ≥ K, then Lint([c + P,d + P]) = Lint([c,d]);
• If d ≥ K, then Lint([c,d + P]) = Lint([c,d]),
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Note that every finite LS can be regarded as an UPLS with a period length 0.
Also, note that every UPLS is finitely representable: it suffices to define its labeling
functions only on all points c < K+P and on all intervals [c,d] such that d < K+P
or d < K + 2P and K < c < K + P, and then it can be uniquely extended by
periodicity.
Claim 5.13 (Small Periodic Model Property) If ϕ is any formula of PImix
−??2??satisfiable in N, then there exists a (possibly finite) ultimately periodic FLS
satisfying ϕ with lengths of the prefix and of the period at most exponential in
m = |CL(ϕ)|.
Proof [sketch] Given an interval model based on N and satisfying ϕ, we take
its corresponding FLS and transform it to an ultimately periodic FLS for ϕ, by
identifying sufficiently long prefix and period in it, and then modifying the rest of
the model to make it ultimately periodic. That produces an LS which we make a
fulfilling one by applying defect-repairing technique similar to the one in the proof of
Lemma 5.9, but now fixing simultaneously all defects involving points on the same
periodic orbit, and in a uniform way. Once the ultimately periodic FLS for ϕ is
obtained, we reduce both the prefix and the period down to size, by applying again
the defect-repairing technique in a uniform way. Thus, the satisfiability problem for
PImix−??2??over N is NEXPTIME-complete, too.
2
6Concluding remarks
In this paper we have considered a new approach to interval logics where time in-
stants (points) and time intervals are treated as separate sorts, and have introduced
the two-sorted point-interval logic PI that formalizes that approach. We have then
focused on its fragment PImix, involving only the inter-sort modalities and its sub-
fragments, and have analyzed the expressiveness of its fragments with respect to
fragments of HS. Using results and adapting techniques for the latter, and we have
obtained some decidability and complexity results for the former.
A number of open problems remain, including: determining the decidability
status and exact complexity of the satisfiability for the fragments of PI on the
most natural classes of models (based on all, dense, discrete, finite, etc. linear
models) and obtaining complete classification of these fragments with respect to
their expressiveness, analogous to (and generalizing) the one for the fragments of
HS recently completed in [7]. The current work is intended as a beginning of their
systematic exploration.
Acknowledgement
Guido Sciavicco was partially supported by the Spanish MEC project TIN2009-
14372-C03-01, and the Spanish fellowship ‘Ramon y Cajal’ RYC-2011-07821.
Valentin Goranko has been supported by a grant 24/2011-CSU 8.2.1 through the
bilateral Programme for French-Danish scientific collaboration, and by the HYLO-
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CORE project, funded by the Danish Natural Science Research Council. We also
thank the reviewers for some corrections and useful remarks.
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