Partially Punctual Metric Temporal Logic is Decidable
\IEEEoverridecommandlockoutsMetric Temporal Logic is one of the most studied real time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints . Henzinger et al., in their seminal paper showed that the nonpunctual fragment of called is decidable. In this paper, we sharpen this decidability result by showing that the partially punctual fragment of (denoted ) is decidable over strictly monotonic finite point wise time. In this fragment, we allow either punctual future modalities, or punctual past modalities, but never both together. We give two satisfiability preserving reductions from to the decidable logic . The first reduction uses simple projections, while the second reduction uses a novel technique of temporal projections with oversampling. We study the tradeoff between the two reductions: while the second reduction allows the introduction of extra action points in the underlying model, the equisatisfiable formula obtained is exponentially succinct than the one obtained via the first reduction, where no oversampling of the underlying model is needed. We also show that is strictly more expressive than the fragments and .
1 Introduction
Metric Temporal Logic is a well established logic useful for specifying quantitative properties of real time systems. The main modalities of are (read “until ”) and (read “since ”), where is a time interval with end points in . These formulae are interpreted over timed behaviours or timed words. A formula holds at a position of a timed word iff there is a position strictly in the future of where holds, and at all intermediate positions between and , holds good; moreover, the difference in the time stamps of and must lie in the interval [2,3]. Similarly, holds good at a point iff there is a position strictly in the past of where holds, and at all intermediate positions between and holds; further, the difference in the time stamps between and lie in the interval [2,3]. The intervals can be bounded of the form , or unbounded of the form , with , and represents left closed or left open, while represents right closed or right open intervals. The unary modalities (read “fut ”) and (read “past ”) are special cases of until and since: and .
The satisfiability question for various fragments of has evoked lot of interest and work over the past years. In their seminal paper, Alur and Henzinger showed that the satisfiability of is undecidable, while the satisfiability of the “nonpunctual” fragment of is decidable. As the name suggests, the nonpunctual fragment disallows punctual intervals : these are intervals of the form . The satisfiability of the future only fragment of , viz., was open for a long time, till Ouaknine and Worrell [12] showed its decidability via a reduction to 1clock alternating timed automata. Even though the logic is more expressive than , it was shown to be decidable [3] by an equisatisfiable reduction to . The decidability of the unary fragment has remained open for a long time, it was recently shown undecidable [7]. The only fragment whose decidability is unknown is thus, the “partially punctual fragment” of , where we allow punctualities only in the future or in the past modalities, but never in both. The main result of this paper is the decidability of the partially punctual fragment of for finite strictly monotonic timed words; our results can be adapted to work for weakly monotonic finite words.
2 Metric Temporal Logic
Let be a finite set of propositions. A finite timed word over is a tuple
where and are sequences and respectively, with , and
for . Let be the set of positions in the timed word.
Let . An example of a timed word is .
is strictly monotonic iff for all .
Otherwise, it is weakly monotonic.
Given , the formulae of are built from using boolean connectives and
time constrained versions of the modalities and as follows:
where is an open, halfopen or closed interval
with end points in .
Formulae of are interpreted over timed words over a chosen set of propositions. Let be an formula. If is interpreted over timed words over , then we say that is interpreted over . Note that this is different from saying is built from a set of propositions : this just means that the propositions in are taken from .
Given a finite timed word , and an formula , in the pointwise semantics, the temporal connectives of quantify over a finite set of positions in . For an alphabet , a timed word , a position , and an formula , the satisfaction of at a position of is denoted , and is defined as follows:
and
,
,
and
,
, ,
and
satisfies denoted iff . Let . The set of all timed words over is denoted .
A nonpunctual interval has the form with . We denote by the class of formulae with nonpunctual past modalities. Similarly, is the class of formulae with nonpunctual future modalities. The class of partially punctual formulae, consists of all formulae with nonpunctual future or nonpunctual past. .
Additional temporal connectives are defined in the standard way: we have the constrained future and past eventuality operators and , and their duals , . Weak versions of operators are defined as : , .
3 Temporal Projections
In this section, we discuss the notion of “temporal projections” that are central to this paper. We discuss two kinds of temporal projections: simple projections, and oversampling projections.
3.1 Simple Extensions and Projections
simple extensions:
Let be finite sets of propositions such that .
A simple extension is a timed word over such that at any point , .
For ,
is a simple extension. However,
is not a simple extension for
the same choice of , since for the position , .
Simple Projections:
Consider a simple extension .
We define the simple projection of with respect to , denoted as the word obtained by erasing the symbols of from each . Note that .
For example, if , , and
, then
.
is thus, a timed word over .
If the underlying word is not a simple extension, then
the simple projection of with respect to is undefined.
Equisatisfiability modulo Simple Projections: Given formulae and , we say that is equisatisfiable to modulo simple projections iff there exist disjoint sets such that

is interpreted over , and is interpreted over ,

For any timed word over ,
is a simple extension and 
For any timed word over such that , a simple extension such that , and
.
We denote by ,
the fact that is equisatisfiable to modulo simple projections.
Extended Normal Form(ENF):
Given a formula built from ,
the extended normal form of with respect to denoted
is the formula .
Lemma 1 (Boolean Closure Lemma).
Let be formulae built from . Let be formulae built from and respectively. Let for , and let . Then, and .
Proof.
The proof can be found in Appendix .1. ∎
3.2 Flattening
Let built from . Given any subformula of , and a fresh symbol , is called a temporal definition and is called a witness. Let be the formula obtained by replacing all occurrences of in , with the witness . Flattening is done recursively until we have replaced all future/past modalities of interest with witness variables, obtaining , where is the conjunction of all temporal definitions. Let be the set of all witness propositions. For example, consider the formula . Replacing the modalities with witness propositions and we get , along with the temporal definitions and . Hence, is obtained by flattening the modalities from . Here . Note that is a formula built from .
Given a timed word over , flattening marks precisely positions in satisfying with witnesses . This marked word over satisfies iff . Hence, we have . ensures that any timed word over that satisfies is indeed a simple extension. is the set of all those simple extensions satisfying such that .
3.3 Oversampled Behaviours and Projections
oversampled behaviours:
Let be finite sets of propositions such that .
A oversampled behaviour is a timed word over , such that
and .
For ,
is a oversampled behaviour, while
is not.
If is a oversampled behaviour,
then points where is not true are called
nonaction points. Hence, in any oversampled behaviour,
the first as well as the last points are action points.
Oversampled Projections:
Given a oversampled behaviour , we define
the oversampled projection of with respect to ,
denoted as the timed word obtained
by deleting points for which , and then
erasing the symbols of from the remaining points ().
The result of oversampling, = is a timed word over .
If , there exists a strictly increasing function
such that , , and

, , , and

, , , and

For , and iff

, and ,

, and ,

For all , .

For , a oversampled behaviour for , we have . We have with , and .
Equisatisfiability modulo Oversampled Projections: Given formulae and , we say that is equisatisfiable to modulo oversampled projections iff there exist disjoint sets such that

is interpreted over , and over ,

For any oversampled behaviour ,

For any timed word over such that , there exists a oversampled behaviour such that , and .
We denote by the fact that is equisatisfiable to modulo oversampled projections. The above conditions establish the existence of some oversampled behaviour corresponding to that satisfies , when satisfies . If condition 3 above holds for all possible oversampled behaviours, i.e,

if for any timed word over such that , all oversampled behaviours for which
satisfy ,
then we say that and are equivalent modulo oversampled projections and denote it by
Oversampled Normal Form (ONF): Let be a formula built from . Let denote . The oversampled normal form with respect to of denoted is obtained by replacing recursively

all subformulae of the form by ,

all subformulae of the form with
, 
all subformulae of the form with
. 
all subformulae of the form with
, and all subformulae of the form with .
and conjuncting the resultant formulae with .
Let ,
and for .
Then
where
denotes .
Proofs of Lemmas 2, 3 and 4
can be found in Appendices .2, .3 and
.4.
Lemma 2 (Oversampling Closure Lemma).
Let be a formula built from . Then .
Lemma 3.
Let be a formula built from and let . Then, .
Lemma 4.
Consider formulae built from .
Let be formulae built from and respectively.
Let , for , and .
Let and . Then,
and
.
Lemma 5.
Let be built from , and be the set of witness variables obtained while flattening . Then .
4 Decidability of
In this section, we show that the class is decidable, by giving a satisfiability preserving reduction to . Given a timed word , and a nonsingular past modality of the form , Lemma 6 establishes a relationship between time stamps of the points in where holds and the time stamps of points where holds in with respect to .
Lemma 6.
Given a timed word and a point . Let and denote respectively the first and last occurrences of in . iff

, where is when is , and is when is , or

, where is when is , and is when is ,or

for all points where holds consecutively (that is there does not exist any point , where holds). Note that in this case .
Proof.
We prove the lemma for intervals of the form . The proof can be extended for other type of intervals also. Assume that . We then show that and and for consecutive points where holds.

Let . implies that there is a point such that , such that . Then, , contradicting that is the first point where holds.

Let . Again, implies that there is a point such that such that . We then have , contradicting that is the last point where holds.

Assume that there exist consecutive points where holds. Also, let . implies that there exists a point such that and . Also, and . This gives contradicting the assumption that are consecutive points where holds.
The converse can be found in Appendix .5. Figure 1 illustrates regions for cases (a) and (b), while Figure 2 illustrates the region for case (c). In the rest of the paper, we refer to regions in case(a) as Region I, regions in case(b) as Region II and regions in case (c) as Region III. ∎
In the rest of this section, we show the decidability of by reducing any formula to a formula . We have two techniques for this proof: one using oversampling projections, and the other, using simple projections.
4.1 Elimination of Past with Oversampled Projections
In this section, given a formula in built from , we synthesize a formula built from equisatisfiable to modulo oversampled projections, whose size is linear in . Starting with a timed word over , we synthesize an oversampled behaviour such that iff .

Start with a formula built from , and a timed word over ,

Flatten obtaining . Let be the witness propositions used. is a formula built from , with .

Let be the conjunction of all temporal definitions in . Each has the form , with , and is built from . , with . We know from Lemma 5 that .

For , let , where are a set of fresh propositions, such that for . Synthesize a formula over such that .

Using Lemma 4, is such that
, for .
Lemma 7 and Lemma 8 show how to synthesize an equisatisfiable formula in corresponding to . Lemma 7 shows step 4 for intervals of the form , while Lemma 8 shows step 4 for bounded intervals of the form . The results of these lemmas can be extended to work for any interval . If all the past modalities involved have unbounded intervals, then we get an equivalent formula, as shown by Lemma 7.
Lemma 7.
Consider a temporal definition built from . Then we can synthesize a formula built from equivalent to .
Proof.
It can be shown that is equivalent to , for . Details in Appendix .6. ∎
Lemma 8.
Consider a temporal definition , built from . Then we can synthesize a formula built from linear in the size of , such that .
Proof.
We start with and a oversampled behaviour . Let . If there exists a point marked , then we want to ensure that all points in marked such that are marked . This is enforced by the following formula:
enforces the direction of . Marking points of with is considerably more involved. We use Lemma 6 to characterize the points where holds, and use this to ensure that such points are marked . Recall that by Lemma 6, such points can be classified into three regions.
Region I consists of all those points to the left of .
In any model, these points are described by the formula
Let us now discuss how to mark points lying in region III with . Recall that these are the points in for any two consecutive points such that , but . Consider as two consecutive points where holds. If , then clearly, there are no points in to be marked . Assume now that . We need to mark exactly the points falling in with . It is quite possible that, we dont have the points in such that and . Here, we use the idea of oversampled projections, to obtain a behaviour from , by adding extra points to . Corresponding to every pair of consecutive points, such that , we add points to , such that and . We mark these new points with fresh propositions and respectively. We then say that between and , no can occur. To pindown the points correctly, we mark the points respectively with fresh propositions and .
To summarize the marking scheme, given a oversampled behaviour