On Expressive Powers of Timed Logics: Comparing Boundedness, Non-punctuality, and Deterministic Freezing
Timed temporal logics exhibit a bewildering diversity of operators and the resulting decidability and expressiveness properties also vary considerably. We study the expressive power of timed logics and as well as of their several fragments. Extending the LTL EF games of Etessami and Wilke, we define Ehrenfeucht-Fraïssé games on a pair of timed words. Using the associated EF theorem, we show that, expressively, the timed logics , and (respectively incorporating the restrictions of boundedness, unary modalities and non-punctuality), are all pairwise incomparable. As our first main result, we show that is strictly contained within the freeze logic for both weakly and strictly monotonic timed words, thereby extending the result of Bouyer et al and completing the proof of the original conjecture of Alur and Henziger from 1990. We also relate the expressiveness of a recently proposed deterministic freeze logic (with NP-complete satisfiability) to . As our second main result, we show by an explicit reduction that lies strictly within the unary, non-punctual logic . This shows that deterministic freezing with punctuality is expressible in the non-punctual .
Temporal logics are well established formalisms for specifying qualitative ordering constraints on the sequence of observable events. Real-time temporal logics extend this vocabulary with specification of quantitative timing constraints between these events.
There are two well-established species of timed logics with linear time. The logic makes use of freeze quantification together with untimed temporal modalities and explicit constraints on frozen time values; the logic uses time interval constrained modalities and . For example,the formula and the formula both characterize the set of words that have a letter with time stamp 2 where this is preceded only by a string of letters . Timed logics may be defined over timed words (also called pointwise time models) or over signals (also called continuous time models). Weak monotonicity (as against strict monotonicity) allows a sequence of events to occur at the same time point. In this paper we confine ourselves to finite timed words with both weakly and strictly monotonic time, but the results straightforwardly carry over to infinite words too.
In their pioneering studies [1, 3, 4], Alur and Henzinger investigated the expressiveness and decidability properties of timed logics and . They showed that can be easily translated into . Further, they conjectured, giving an intuitive example, that is more expressive than (see  section 4.3). Fifteen years later, in a seminal paper, Bouyer et al  formally proved that the purely future time logic is strictly more expressive than and that is more expressive than , for both pointwise and continuous time. In this paper, we complete the picture by proving the original conjecture of Alur and Henzinger for the full logic with both future and past over pointwise time.
In their full generality, and are both undecidable even for finite timed words. Several restrictions have been proposed to get decidable sub-logics (see  for a recent survey). Thus, Bouyer et al.  introduced with “bounded” intervals and showed that its satisfiability is -complete. Alur and Henzinger argued, using reversal bounded 2-way deterministic timed automata , that the logic permitting only non-singular (or non-punctual) intervals was decidable with complexity [2, 5]. Unary modalities have played a special role in untimed logics , and we also consider unary fragments and in our study. Further sub-classes can be obtained by combining the restrictions of bounded or non singular intervals and unary modalities. Decidable fragments of are less studied but two such logics can be found in [15, 13].
In this paper, we mainly compare the expressive powers of various real-time temporal logics. As our main tool we define an -round MTL EF game with “until” and “since” moves on two given timed words. As usual, the EF theorem equates the inability of any formula with modal depth from distinguishing two timed words to the existence of a winning strategy for the duplicator in -round games. Our EF theorem is actually parametrized by a permitted set of time intervals, and it can be used for proving the lack of expressiveness of various fragments of .
Classically, the EF Theorem has been a useful tool for proving limitations in expressive power of first-order logic [11, 17]. In their well-known paper, Etessami and Wilke  adapted this to the LTL EF games to show the existence of the “until” hierarchy in LTL definable languages. Our EF theorem is a generalization of this to the timed setting. We find that the use of EF theorem often leads to simple game theoretic proofs of seemingly difficult questions about expressiveness of timed logics. The paper contains several examples of such proofs.
Our main expressiveness results are as follows. We show these results for finite timed words with weakly and strictly monotonic time. However, we remark that these results straightforwardly carry over to infinite timed words.
We show that logics , and are all pairwise incomparable. These results indicate that the restrictions of boundedness, non-punctuality, and unary modalities are all semantically “orthogonal” in context of .
As one of our main results, we show that the unary and future fragment of the freeze logic is not expressively contained within for both strictly monotonic and weakly monotonic timed words. Thus, is a strict subset of for pointwise time, as originally conjectured by Alur and Henzinger almost 20 years ago [1, 3, 6].
It is easy to show that for strictly monotonic timed words, logic can be translated to the unary fragment and for expressiveness the two logics coincide. For weakly monotonic time, we show that and are expressively incomparable.
In the second part of this paper, we explore the expressiveness of a recently proposed “deterministic” and “unary” fragment of called . This is an interesting logic with exact automaton characterization as partially ordered two way deterministic timed automata . Moreover, by exploiting the properties of these automata, the logic has been shown to have NP-complete satisfiability. The key feature of this logic is the “unique parsing” of each timed word against a given formula. Our main results on the expressiveness of are as follows.
By an explicit reduction, we show that is contained within the unary and non-punctual logic . The containment holds in spite of the fact that can have freeze quantification and punctual constraints (albeit only occurring deterministically).
Using the unique parsability of , we show that neither nor are expressively contained within .
Thus, the full logic is more expressive than . But its unary fragment with deterministic freezing, , lies strictly within the unary and non-punctual logic . In our recent work , we have also shown by explicit reduction that the bounded fragment is strictly contained within . Figure 1 provides a succinct pictorial representation of all the expressiveness results achieved.
The rest of the paper is organized as follows. Section 2 defines various timed logics. The EF games and the EF Theorem are given in Section 3. Section 4 explores the relative expressiveness of various fragments of and the subsequent section compares to . Section 6 studies the expressiveness of relative to sub logics of .
2 Timed Temporal Logics: Syntax and Semantics
We provide a brief introduction of the logics whose expressiveness is investigated in this paper.
Let and be the set of reals, rationals, integers, and natural numbers, respectively and be the set of non-negative reals. An interval is a convex subset of , bounded by non-negative integer constants or . The left and right ends of an interval may be open ( ”(” or ”)” ) or closed ( ”[” or ”]” ). We denote by a generic interval whose ends may be open or closed. An interval is said to be bounded if it does not extend to infinity. It is said to be singular if it is of the form for some constant , and non-singular (or non-punctual) otherwise. We denote by all the intervals (including singular intervals and unbounded intervals ), by the set of all non-punctual (or extended) intervals, and by the set of all bounded intervals. Given an alphabet , its elements are used also as atomic propositions in logic, i.e. the set of atomic propositions .
A finite timed word is a finite sequence , of event-time stamp pairs such that the sequence of time stamps is non-decreasing: . This gives weakly monotonic timed words. If time stamps are strictly increasing, i.e. , the word is strictly monotonic. The length of is denoted by , and . For convenience, we assume that as this simplifies the treatment of “freeze” logics. The timed word can alternately be represented as with and . Let . We shall use the two representations interchangeably. Let be the set of timed words over the alphabet .
2.2 Metric Temporal Logics
The logic MTL extends Linear Temporal Logic by adding timing constraints to the ”Until” and ”Since” modalities of LTL. We parametrize this logic by a permitted set of intervals and denote the resulting logic as . Let range over formulas, and . The syntax of is as follows:
Let be a timed word and let . The semantics of formulas is as below:
The language of an formula is given by . Note that we use the ”strict” semantics of and modalities. We can define unary ”future” and ”past” modalities as: and . The subset of using only these modalities is called . We can now define various well known variants of .
Unary , denoted = uses only unary modalities. It is a timed extension of the untimed unary temporal logic studied by .
Metric Interval Temporal Logic , denoted . In this logic, the timing constraints in the formulas are restricted to non-punctual (non-singular) intervals. is confined to the unary modalities and .
Bounded , denoted . Other logics can be obtained as intersections of the above logics. Specifically, the logics , , and are defined respectively as , , and .
Let denote the set of all intervals of the form or , with . Let denote the set of all bounded (i.e. non-infinite) intervals. Then and are respectively the logic and . Also, given an formula , let denote the maximum integer constant (apart from ) appearing in its interval constraints.
2.3 Freeze Logics
These logics specify timing constraints by conditions on special variables, called freeze variables which memorize the time stamp at which a subformula is evaluated. Let be a finite set of freeze variables. Let and let
be a valuation which assigns a non-negative real number to each freeze variable. Let be the initial valuation such that
denote the valuation such that and if .
A timing constraint in freeze logics has the form:
where and .
Let denote that the timing constraint evaluates to in valuation with assigned to the variable .
The semantics of formulas over a timed word with
and valuation is as follows. The boolean connectives have their usual meaning.
The language defined by a formula is given by . Also, is the unary sub logic of .
Deterministic Freeze Logic
is a sub logic of .
A guarded event over an alphabet and a finite set of freeze variables is a pair where is an event and is a timing constraint over as defined before.
Logic uses the deterministic modalities and which access the position with the next and previous occurrence of a guarded event, respectively. This is the timed extension of logic  using freeze quantification. The syntax of a formula is as follows:
The semantics of formulas over timed words is as given below. denotes the formula . This and the boolean operators have their usual meaning.
3 EF Games for
We extend the LTL EF games of  to timed logics, and use these to compare expressiveness of various instances of the generic logic . Let be a given set of intervals. A -round -EF game is played between two players, called and , on a pair of timed words and . A configuration of the game (after any number of rounds) is a pair of positions with and . A configuration is called partially isomorphic, denoted iff .
The game is defined inductively on from a starting configuration and results in either the or winning the game. The wins the -round game iff . The round game is played by first playing one round from the starting position. Either the spoiler wins in this round (and the game is terminated) or the game results into a new configuration . The game then proceeds inductively with -round play from the configuration . The wins the game only if it wins every round of the game. We now describe one round of play from a starting configuration .
At the start of the round, if then the wins the game and the game is terminated. Otherwise,
The chooses one of the words by choosing . Then gives the other word. The also chooses either an -move or a move, including an interval . The remaining round is played in two parts.
Part I: The chooses a position such that and .
2by choosing a position in the other word s.t. and . If the cannot find such a position, the wins the game. Otherwise the play continues to Part II.
Part II: chooses to play either -part or -part.
-part: the round ends with configuration .
-part: verifies that iff and wins the game if this does not hold. Otherwise checks whether . If yes, the round ends with configuration . If no, chooses a position in the other word such that . The responds by choosing such that . The round ends with the configuration .
Move This move is symmetric to where the chooses positions as well as in “past” and the also responds accordingly. In Part II, the will a have choice of -part or -part. We omit the details. This completes the description of the game.
Given two timed words and , we define
iff for every -round EF-game over the words and starting from the configuration , the always has a winning strategy.
iff for every formula of operator depth , . ∎
We shall now state the EF theorem. Its proof is a straight-forward extension of the proof of LTL EF theorem of . The only point of interest is that there is no a priori bound on the set of intervals that a modal depth formula can use and hence the set of isomorphism types seems potentially infinite. However, given timed words and , we can always restrict these intervals to not go beyond a constant where is the smallest integer larger than the biggest time stamps in and . This restricts the isomorphism types to a finite cardinality. The complete proof is given in detail in Appendix A.
if and only if ∎
When clear from context, we shall abbreviate by and by . As temporal logic formulas are anchored to initial position , define and . It follows from the EF Theorem that if and only if .
We can modify the EF game to match the sub logic . An game is obtained by the restricting game such that in PART II of any round, the always chooses an -part or a -part. The corresponding EF Theorem also holds.
4 Separating sub logics of
Each formula of a timed logic defines a timed language. Let denote the set of languages definable by the formulas of logic . A logic is at least as expressive as (or contains) logic if . This is written as . Similarly, we can define (strictly contained within), (not contained within), (incomparable), and (equally expressive).
We consider three sub logics of namely , and .These have fundamentally different restrictions and using their corresponding EF-games, we show that they are all incomparable with each other.
Consider the formula . Consider a family of words and . We have with the ’s occurring at integral time stamps in both words. In , the letter occurs at time and hence time distance between any and is more than . In , the occurs at time and the time distance between the and the preceding is in . Clearly, whereas for any .
We prove the theorem using an -round EF game on the words and where . We show that has a winning strategy. Note that in such a game the is allowed to choose intervals at every round with maximum upper bound of and hence can shift the pebble at most positions to the right. It is easy to see that the is never able to place a pebble on the last . Hence, the has a winning strategy where she exactly copies the moves. Using the EF theorem, we conclude that no modal depth formula of logic can separate the words and . Hence, there doesn’t exist a formula giving the language . ∎
Consider the formula . Consider a family of words such that . Let and . All the ’s are in the interval (0,1) at time stamps and all the ’s are in the interval , at time stamps for . Every has a paired , which is at a distance from it. Hence, . Let be a word identical to but with the middle shifted leftwards by , so that it is exactly at a distance of t.u. (time units) from the middle . Thus, .
We prove the theorem using the -round EF game on the words and where we can show that has a winning strategy. This proves that no modal depth formula of logic can separate and . Hence, there is no formula giving The full description of the strategy can be found in the Appendix B. ∎
over strict monotonic timed words (and hence also over weakly monotonic timed words).
over weakly monotonic timed words. ∎
These results follow by embedding untimed LTL into logics MTL as well as TPTL. The proof can be found in Appendix B.
5 TPTL and MTL
Consider the formula . Bouyer et al  showed that this formula cannot be expressed in for pointwise models. They also gave an formula equivalent to it thereby showing that is strictly more expressive than . Prior to this, Alur and Henzinger  considered the formula and they conjectured that this cannot be expressed within . Using a variant of this formula and the EF games, we now show that is indeed expressively incomparable with .
In Theorem 4.3 we showed that over weakly monotonic timed words. We now consider the converse.