Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

TLDR: AABB properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB of Allen's "begun by/begins" relations and propositional neighborhood logic AA, in its many variants (including metric ones).

Abstract: In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations "meets", "met by", "begun by", and "begins" of Allen's interval algebra (AABB logic). AABB properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB of Allen's "begun by/begins" relations and propositional neighborhood logic AA, in its many variants (including metric ones). We prove that the satisfiability problem for AABB, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AABB turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AABB is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, Q, and R.

Figures

Fig. 3. Decision algorithm for the satisfiability problem over finite linear orders.

Fig. 1. A compass structure.

Fig. 2. Contraction of a compass structure.

Full text

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Maximal Decidable Fragments of Halpern and Shoham’s
Modal Logic of Intervals
Angelo Montanari, Gabriele Puppis, Pietro Sala
To cite this version:
Angelo Montanari, Gabriele Puppis, Pietro Sala. Maximal Decidable Fragments of Halpern and
Shoham’s Modal Logic of Intervals. Proceedings of ICALP 2010, 2010, Bordeaux, France. pp.345-356,
�10.1007/978-3-642-14162-1_29�. �hal-00717788�

Maximal decidable fragments of Halpern
and Shoham’s modal logic of intervals
Angelo Montanari
1
, Gabriele Puppis
2
, and Pietro Sala
1
1
Department of Mathematics and Computer Science, Udine University, Italy
{angelo.montanari,pietro.sala}@dimi.uniud.it
2
Computing Laboratory, Oxford University, England
gabriele.puppis@comlab.ox.ac.uk
Abstract. In this paper, we focus our attention on the fragment of
Halpern and Shoham’s modal logic of intervals (HS) that features four
modal operators corresponding to the relations “meets”, “met by”, “be-
gun by”, and “begins” of Allen’s interval algebra (A
¯
AB
¯
B logic). A
¯
AB
¯
B
properly extends interesting interval temporal logics recently investigated
in the literature, such as the logic B
¯
B of Allen’s “begun by/begins” rela-
tions and propositional neighborhood logic A
¯
A, in its many variants (in-
cluding metric ones). We prove that the satisfiability problem for A
¯
AB
¯
B,
interpreted over finite linear orders, is decidable, but not primitive recur-
sive (as a matter of fact, A
¯
AB
¯
B turns out to be maximal with respect to
decidability). Then, we show that it becomes undecidable when A
¯
AB
¯
B is
interpreted over classes of linear orders that contains at least one linear
order with an infinitely ascending sequence, thus including the natural
time flows N, Z, and R.
1 Introduction
For a long time, the role of interval temporal logics in computer science has been
controversial. On the one hand, it is commonly recognized that they provide a
natural framework for representing and reasoning about temporal properties in
many computer science areas (quoting Kamp and Reyle [11], “truth, as it per-
tains to language in the way we use it, relates sentences not to instants but
to temporal intervals”), including specification and design of hardware compo-
nents, concurrent real-time processes, event modeling, temporal aggregation in
databases, temporal knowledge representation, systems for temporal planning
and maintenance, qualitative reasoning, and natural language semantics [9]. On
the other hand, the computational complexity of most interval temporal logics
proposed in the literature has been a barrier to their systematic investigation
and their extensive use in practical applications. This is the case with the modal
logic of time intervals HS introduced by Halpern and Shoham in [10]. HS makes
it possible to express all basic binary relations that may hold between any pair
of intervals (the so-called Allen’s relations [1]) by means of four unary modal-
ities, namely, B, E and their transposes 
¯
B, 
¯
E, corresponding to Allen’s
relations “begun by”, “ended by” and their inverses “begins”, “ends”, provided

that singleton intervals are included in the temporal structure [21]. HS turns
out to be highly undecidable under very weak assumptions on the class of linear
orders over which its formulas are interpreted [10]. In particular, undecidability
holds for any class of linear orders that contains at least one linear order with
an infinitely ascending or descending sequence, thus including the natural time
flows N, Z, Q, and R. In fact, undecidability occurs even without infinitely as-
cending/descending sequences: undecidability also holds for any class of linear
orders with unboundedly ascending sequences, that is, for any class such that
for every n, there is a structure in the class with an ascending sequence of length
at least n, e.g., for the class of all finite linear orders. In [12], Lodaya sharpens
the undecidability of HS showing that the two modalities B, E suffice for un-
decidability over dense linear orders (in fact, the result applies to the class of all
linear orders [9]).
The recent identification of expressive decidable fragments of HS, whose de-
cidability does not depend on simplifying semantic assumptions such as locality
and homogeneity [9], shows that such a trade-off between expressiveness and de-
cidability of interval temporal logics can actually be overcome. The most signifi-
cant ones are the logic B
¯
B (resp., E
¯
E) of Allen’s “begun by/begins” (resp., “ended
by/ends”) relations [9], the logic A
¯
A of temporal neighborhood, whose modal-
ities correspond to Allen’s “meets/met by” relations (it can be easily shown
that Allen’s “before/after” relations can be expressed in A
¯
A) [8], and the logic
D
¯
D of the subinterval/superinterval relations, whose modalities correspond to
Allen’s “contains/during” relations [14]. In this paper, we focus our attention
on the logic A
¯
AB
¯
B that joins B
¯
B and A
¯
A (the case of A
¯
AE
¯
E is fully symmet-
ric). The decidability of B
¯
B (resp., E
¯
E) can be proved by translating it into the
point-based propositional temporal logic of linear time with temporal modalities
F (sometime in the future) and P (sometime in the past), which has the finite
(pseudo-)model property and is decidable [9]. Unfortunately, such a reduction
to point-based temporal logics does not work for most interval temporal logics
as their propositional variables are evaluated over pairs of points and translate
into binary relations. This is the case with A
¯
A. Unlike the case of B
¯
B (resp.,
E
¯
E), when dealing with A
¯
A one cannot abstract away from the left (resp., right)
endpoint of intervals, as contradictory formulas may hold over intervals with the
same right (resp., left) endpoint and a different left (resp., right) one. The decid-
ability of A
¯
A, over various classes of linear orders, has been proved by Bresolin
et al. [3] by reducing its satisfiability problem to that of the two-variable frag-
ment of first-order logic over the same classes of linear orders [17]. An optimal
(NEXPTIME) tableau-based decision procedure for A
¯
A over the integers has
been given in [5] and later extended to the classes of all (resp., dense, discrete)
linear orders [6], while a decidable metric extension of the future fragment of
A
¯
A over the natural numbers has been proposed in [7] and later extended to
the full logic [4]. Finally, a number of undecidable extensions of A
¯
A have been
given in [2, 3].
In [16], Montanari et al. consider the effects of adding the modality A to
B
¯
B, interpreted over the natural numbers. They show that AB
¯
B retains the

simplicity of its constituents, but it improves a lot on their expressive power.
In particular, besides making it possible to easily encode the until operator of
point-based temporal logic (this is possible neither with B
¯
B nor with A), AB
¯
B
allows one to express accomplishment conditions as well as metric constraints.
Such an increase in expressiveness is achieved at the cost of an increase in com-
plexity: the satisfiability problem for AB
¯
B is EXPSPACE-complete (that for
A is NEXPTIME-complete). In this paper, we show that the addition of the
modality 
¯
A to AB
¯
B drastically changes the characteristics of the logic. First,
decidability is preserved (only) if A
¯
AB
¯
B is interpreted over finite linear orders,
but the satisfiability problem is not primitive recursive anymore. Moreover, we
show that the addition of any modality in the set {D, 
¯
D, E, 
¯
E, O, 
¯
O}
(modalities O, 
¯
O correspond to Allen’s “overlaps/overlapped by” relations)
to A
¯
AB
¯
B leads to undecidability. This allows us to conclude that A
¯
AB
¯
B, inter-
preted over finite linear orders, is maximal with respect to decidability. Next,
we prove that the satisfiability problem for A
¯
AB
¯
B becomes undecidable when
it is interpreted over any class of linear orders that contains at least one lin-
ear order with an infinitely ascending sequence, thus including the natural time
flows N, Z, Q, and R. As matter of fact, we prove that the addition of B to A
¯
A
suffices to yield undecidability (the proof can be easily adapted to the case of
¯
B). Paired with undecidability results in [2, 3], this shows the maximality of A
¯
A
with respect to decidability when interpreted over these classes of linear orders.
2 The interval temporal logic A
¯
AB
¯
B
In this section, we first give syntax and semantics of the logic A
¯
AB
¯
B. Then,
we introduce the basic notions of atom, type, and dependency. We conclude the
section by providing an alternative interpretation of A
¯
AB
¯
B over labeled grid-like
structures (such an interpretation is quite common in the interval temporal logic
setting).
2.1 Syntax and semantics
Given a set Prop of propositional variables, formulas of A
¯
AB
¯
B are built up from
Prop using the boolean connectives ¬ and ∨ and the unary modal operators
A, 
¯
A, B, 
¯
B. As usual, we shall take advantage of shorthands like ϕ
1
∧ ϕ
2
=
¬(¬ϕ
1
∨ ¬ϕ
2
), [A]ϕ = ¬A¬ϕ, [B]ϕ = ¬B¬ϕ, ⊺ = p ∨ ¬p, and  = p ∧ ¬p,
with p ∈ Prop. Hereafter, we denote by ϕ the size of ϕ.
We interpret formulas of A
¯
AB
¯
B in interval temporal structures over finite
linear orders with the relations “meets”, “met by”, “begins”, and “begun by”.
Precisely, given N ∈ N, we define I
N
as the set of all (non-singleton) closed
intervals [x, y], with 0 ≤ x < y ≤ N. For any pair of intervals [x, y], [x
′
, y
′
] ∈ I
N
,
Allen’s relations “meets” A, “met by”
¯
A, “begun by” B, and “begins”
¯
B are
defined as follows:

“meets”: [x, y] A [x
′
, y
′
] iff y = x
′
;

“met by”: [x, y]
¯
A [x
′
, y
′
] iff x = y
′
;


“begun by”: [x, y] B [x
′
, y
′
] iff x = x
′
and y
′
< y;

“begins”: [x, y]
¯
B [x
′
, y
′
] iff x = x
′
and y < y
′
.
Given an interval structure S = (I
N
, A,
¯
A, B,
¯
B, σ), where σ ∶ I
N
→ P(Prop) is
a labeling function that maps intervals in I
N
to sets of propositional variables,
and an initial interval I, we define the semantics of an A
¯
AB
¯
B formula as follows:

S, I ⊧ a iff a ∈ σ(I), for any a ∈ Prop;

S, I ⊧ ¬ϕ iff S, I ⊧ ϕ;

S, I ⊧ ϕ
1
∨ ϕ
2
iff S, I ⊧ ϕ
1
or S, I ⊧ ϕ
2
;

for every relation R ∈ {A,
¯
A, B,
¯
B}, S, I ⊧ Rϕ iff there is an interval J ∈ I
N
such that I R J and S, J ⊧ ϕ.
Given an interval structure S and a formula ϕ, we say that S satisfies ϕ if there
is an interval I in S such that S, I ⊧ ϕ. We say that ϕ is satisfiable if there
exists an interval structure that satisfies it. We define the satisfiability problem
for A
¯
AB
¯
B as the problem of establishing whether a given A
¯
AB
¯
B-formula ϕ is
satisfiable.
2.2 Atoms, types, and dependencies
Let S = (I
N
, A,
¯
A, B,
¯
B, σ) be an interval structure and ϕ be a formula of A
¯
AB
¯
B.
In the sequel, we shall compare intervals in S with respect to the set of subfor-
mulas of ϕ they satisfy. To do that, we introduce the key notions of ϕ-atom and
ϕ-type.
First of all, we define the closure Cl(ϕ) of ϕ as the set of all subformulas
of ϕ and of their negations (we identify ¬¬α with α, ¬Aα with [A]¬α, etc.).
For technical reasons, we also introduce the extended closure Cl
+
(ϕ), which is
defined as the set of all formulas in Cl(ϕ) plus all formulas of the forms Rα
and ¬Rα, with R ∈ {A,
¯
A, B,
¯
B} and α ∈ Cl(ϕ).
A ϕ-atom is any non-empty set F ⊆ Cl
+
(ϕ) such that (i) for every α ∈ Cl
+
(ϕ),
we have α ∈ F iff ¬α ∉ F and (ii) for every γ = α ∨ β ∈ Cl
+
(ϕ), we have γ ∈ F iff
α ∈ F or β ∈ F (intuitively, a ϕ-atom is a maximal locally consistent set of formulas
chosen from Cl
+
(ϕ)). Note that the cardinalities of both sets Cl(ϕ) and Cl
+
(ϕ)
are linear in the number ϕ of subformulas of ϕ, while the number of ϕ-atoms
is at most exponential in ϕ (precisely, we have Cl(ϕ) = 2ϕ, Cl
+
(ϕ) = 18ϕ,
and there are at most 2
9ϕ
distinct atoms).
We also associate with each interval I ∈ S the set of all formulas α ∈ Cl
+
(ϕ)
such that S, I ⊧ α. Such a set is called ϕ-type of I and it is denoted by Type
S
(I).
We have that every ϕ-type is a ϕ-atom, but not vice versa. Hereafter, we shall
omit the argument ϕ, thus calling a ϕ-atom (resp., a ϕ-type) simply an atom
(resp., a type).
Given an atom F, we denote by Obs(F) the set of all observables of F, namely,
the formulas α ∈ Cl(ϕ) such that α ∈ F. Similarly, given an atom F and a relation
R ∈ {A,
¯
A, B,
¯
B}, we denote by Req
R
(F) the set of all R-requests of F, namely,
the formulas α ∈ Cl(ϕ) such that Rα ∈ F. Note that, for every pair of in-
tervals I = (x, y) and J = (x
′
, y
′
) in S, if y = y
′
(resp., x = x
′
) holds, then

Frequently asked questions

Q1. What are the contributions in "Maximal decidable fragments of halpern and shoham’s modal logic of intervals" ?

In this paper, the authors focus their attention on the fragment of Halpern and Shoham ’ s modal logic of intervals ( HS ) that features four modal operators corresponding to the relations “ meets ”, “ met by ”, “ begun by ”, and “ begins ” of Allen ’ s interval algebra ( AĀBB̄ logic ). The authors prove that the satisfiability problem for AĀBB̄, interpreted over finite linear orders, is decidable, but not primitive recursive ( as a matter of fact, AĀBB̄ turns out to be maximal with respect to decidability ). Then, the authors show that it becomes undecidable when AĀBB̄ is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, and R. 

Q2. How is the decidability of ABB shown?

The decidability of AĀ, over various classes of linear orders, has been proved by Bresolin et al. [3] by reducing its satisfiability problem to that of the two-variable fragment of first-order logic over the same classes of linear orders [17]. 

Q3. What is the satisfiability problem for ABB?

The satisfiability problem for AĀB, and hence that for AĀBB̄, interpreted over finite linear orders, is not primitive recursive. 

Q4. What is the simplest way to describe a compass structure?

If there exist a cover C of G and two rows y0 and y1 in G, with 1 < y0 < y1 ≤N, such that i) ShadingG(y0) ⊆ ShadingG(y1), ii) CountG(y0) ≥ CountG∣C(y1), then there exists a compass structure G ′ of length N ′ <N that features ϕ. 

Q5. What is the proof of the satisfiability problem for AB?

The proof of Theorem 2 is given in the Appendix and it is based on a reduction from the reachability problem for lossy counter machines, which is known to have strictly non-primitive recursive complexity [19], to the satisfiability problem for AĀB. 

Q6. What is the simplest way to define a compass structure?

a cover of a compass structure G = (PN,L) is a subset C of PN that satisfies the following two conditions:if (x,y) ∈ C and x < y − 1, then (x,y − 1) ∈ C as well; for every point q = (y − 1,y) ∈ PN, the set ReqĀ(L(q)) coincides with theunion of the sets Obs(L(p)) for all p = (x,y − 1) ∈ C. Given a cover C of G, the authors extend the notations RowG(y), ShadingG(y), and CountG(y) respectively to RowG∣C(y), ShadingG∣C(y), and CountG∣C(y), having the obvious meaning (e.g., RowG∣C(y) is the set of all points of G along the rowy that also belong to C). 

Q7. What is the definition of the satisfiability problem for ABB?

In this section, the authors prove that the satisfiability problem for AĀBB̄ interpreted over finite linear orders is decidable, but not primitive recursive. 

Q8. What is the definition of the satisfiability problem for ABB?

In this section, the authors prove that the satisfiability problem for AĀBB̄ interpreted over finite linear orders is decidable, but not primitive recursive. 

Q9. What is the role of interval temporal logics in computer science?

On the one hand, it is commonly recognized that they provide a natural framework for representing and reasoning about temporal properties in many computer science areas (quoting Kamp and Reyle [11], “truth, as it pertains to language in the way the authors use it, relates sentences not to instants but to temporal intervals”), including specification and design of hardware components, concurrent real-time processes, event modeling, temporal aggregation in databases, temporal knowledge representation, systems for temporal planning and maintenance, qualitative reasoning, and natural language semantics [9]. 

Q10. What is the maximum value of A with respect to decidability?

Paired with undecidability results in [2, 3], this shows the maximality of AĀ with respect to decidability when interpreted over these classes of linear orders. 

Q11. What is the satisfiability problem for the logic AB?

The satisfiability problem for the logic AĀB, and hence that for the logic AĀBB̄, interpreted over over any class of linear orders that contains at least one linear order with an infinitely ascending sequence is undecidable.