Logics with a bounded number of variables have been extensively studied, in particular in the context of descriptive complexity [17, 18, 10, 21] and temporal logics [20, 7, 14, 16]. One recurring question of interest [7, 25, 19, 4, 26, 1] is to determine, in a given class of structures, whether all properties expressible in monadic first-order logic (FO) can be defined in the fragment consisting of formulas which use at most variables. (A same variable may be quantified over several times in a formula.) In fact, several non-equivalent versions of this question appear in the literature, many of which are compared in . We say that has the -variable property if every formula of FO with at most free variables is equivalent over to a formula of . Note that this is strictly stronger than requiring that all sentences (without free variables) of FO are equivalent to some formulas. Indeed, Hodkinson and Simon gave an example of a class of structures where no sentence requires more than 3 variables, but which does not have the -variable property for any .
The problem of whether a given class of structures has the -variable property is closely related to the question of the existence of an expressively complete temporal logic (with a finite set of FO-definable modalities). A temporal logic is called expressively complete if any first-order formula with a single free variable can be expressed in it. For instance, it is well-known that linear temporal logic (LTL) over Dedekind-complete time flows, or its extension with Stavi connectives over all time flows, are expressively complete for first-order logic [20, 8]. More recently, it was shown that over the real numbers equipped with binary relations for all , metric temporal logic (MTL) is expressively complete . However, the questions of having the -variable property for some or admitting an expressively complete temporal logic are incomparable in general: there exist a class of structures which admits a finite expressively complete set of temporal connectives but which does not have the -variable property for any , and one which has the -variable property but for which no temporal logic is expressively complete . However, Gabbay established that having the -variable property implies the existence of a multi-dimensional expressively complete temporal logic, with multiple reference points .
Another classical approach to proving or disproving that a class of structures has the -variable property is through Ehrenfeucht-Fraïssé games, with a bounded number of pebbles [12, 25, 19, 1]. This was applied by Immerman and Kozen to linear orders and bounded-degree trees , and by Antonopoulos et al. to real-time signals .
Natural candidates for classes which might have the -variable property are classes of linearly ordered structures. Indeed, a typical counter-example to unrestricted structures having the -variable property is a formula such as “there exists distinct elements which satisfy some predicate ”. It is in general not expressible in , but it is easily expressible in if all models are equipped with a linear order . For instance for , we take the formula . As mentioned before, Immerman and Kozen showed that the class of linear orders has the -variable property . However, adding a single binary relation suffices to obtain a class of linearly ordered structures which does not have the -variable property for any . Venema gave an example of a dense linear order with a single equivalence relation which does not have the -variable property for any ; this was adapted in  to give another example where the equivalence relation is replaced with a bijection. In fact, even for finite linear orders, Rossman  proved that the class of linearly ordered graphs does not have the -variable property for any , resolving a problem which had been open for more than 25 years . Therefore, adding binary relations to linear orders while keeping the -variable property requires some restrictions on the interpretation of the relation symbols.
On the positive side, Antonopoulos et al. proved that the class of structures over (or signals) has the -variable property . Such structures have been studied in the context of real-time verification. As a corollary, they also showed that has the -variable property for any linear function .
We consider the class of linearly ordered structures with an additional (finite or infinite) number of binary interval-preserving relations. These are binary relations such that, for all intervals , any point which is in between two points of and has a preimage by must have one in . (We also require a symmetric condition of the converse relation .) We show that FO over this class of structures also has the -variable property.
This generalizes results from  and  described above. Moreover, this answers some open questions mentioned in the conclusion of , which asked if the result could be extended from linear functions to polynomials over the reals, or other linear orders and families of monotone functions. In fact, all increasing or decreasing partial functions (over arbitrary linear orders) are special cases of interval-preserving relations, and thus covered by our result.
Our proof relies on different techniques than [19, 1], which were based on Ehrenfeucht-Fraïssé games. We give an effective translation from FO to which goes through a star-free variant of Propositional Dynamic Logic (PDL) with converse. Propositional dynamic logic was introduced by Fischer and Ladner 
to reason about program schemes, and has now found a large range of applications in artificial intelligence and verification[11, 5, 23, 22, 9]. It combines local formulas containing modal operators, and path formulas using the concatenation, union and Kleene star operations. Several extensions have been studied, including PDL with converse , intersection , or negation of atomic programs . The particular star-free variant of PDL we use here is in fact very similar to Tarski’s relation algebras , which was used as a basis for formalizing set theory. It also corresponds to a two-dimensional temporal logic in the sense of Gabbay .
We applied similar proof techniques in , where we introduced a star-free variant of PDL and proved that it is equivalent to FO over message sequence charts (MSCs) (and thus obtained a 3-variable property result for MSCs as a corollary). MSCs are discrete partial orders which represent behaviors of concurrent message passing systems. They consist of a fixed, finite number of linear orders called process orders (one for each process in the system), together with FIFO binary message relations connecting matching send and receive actions. Having a (fixed) finite number of total orders instead of a single one is not an important difference, as we could always put them one after the other to extend them into a single linear order. FIFO relations are a special case of interval-preserving relations, thus the result of the present paper can in fact be seen as a strict generalization of our previous result in . More importantly, a major difference between MSCs studied in  and the setting we consider here is that MSCs are discrete structures, whereas here we allow arbitrary linear orders. In fact,  relied on the definition of formulas describing the minimum or the maximum of some binary relations. As such, it is interesting to see that the same kind of techniques can still be applied to a priori very different linear orders.
2 Interval-preserving relations and first-order logic
In this section, we define the class of structures covered by our results, and recall the syntax of first-order logic.
Interval-preserving binary relations.
Let be a binary relation between sets and . We write if , and . For a subset , we also write . We define the converse of a relation as , and the composition of two binary relations and as . Finally, we write for the complement of . Note that we have the following identities:
A linear order over a set is a reflexive, transitive and antisymmetric relation such that for all , we have or . Let be a linearly ordered set. For , we will also denote by the restriction of to , so that is still a linearly ordered set. Moreover, for , we write if for all , , and if for all , .
An interval of is a set such that for all with , we have . For , we denote by the interval , and similarly for the intervals , , . We call a relation between two linearly ordered sets and interval-preserving if:
For all intervals of , is an interval of .
For all intervals of , is an interval of .
In other terms, for all and with , for all , if there exists some such that , then there exists one in (cf. Figure 1). Note that we do not require that all elements between and are in , but only those which are in the image of . The second condition is symmetric: for all and with , for all , if there exists some such that , then there exists one in .
For any linear order and partial function , if is increasing or decreasing then the relation is interval-preserving.
As another example, consider a temporal structure over a set of atomic propositions AP, where indicates the set of propositions which are true at a given point. For , we let . Then is interval-preserving.
The following lemma states some simple closure properties of interval-preserving relations.
Let , , be linearly ordered sets.
For all interval-preserving relation , is interval-preserving.
For all interval-preserving relations , is interval-preserving.
For all interval-preserving relations and , is interval-preserving.
Part 1 follows from the fact that .
Let us prove 2. Since , by symmetry, it suffices to prove that for all interval of , is an interval of . Let and such that and for some . If , then we are done. Otherwise, suppose for instance that (the other cases are similar). Since is interval-preserving, there exists such that . Then, since and is an interval of , we obtain . Similarly, . Hence .
Let us show that 2 implies 3. Again, by symmetry, it suffices to prove that for all interval of , is an interval of . Let denote the relation . It is an interval-preserving relation between and . Moreover, we have . Now, let be some interval of , and . Then is an interval of . Moreover, since is interval-preserving, we have , hence
Then, according to 2, is an interval of , i.e., an interval of . ∎
Let be an infinite set of monadic predicates, and be a finite or infinite set of binary relation symbols. Throughout the paper, will denote a structure where is a linear order over , is an interval-preserving relation for all , and for all .
Monadic first-order logic.
We assume an infinite supply of variables . The set of monadic first-order logic formulas over is defined as follows:
We assume that all formulas are interpreted over structures defined as above. Given an formula , we denote by its set of free variables. We define the satisfaction relation as usual, where and is an interpretation of the free variables of . We say that two formulas are equivalent, written , if for all and , we have if and only if .
For , we denote by the set of first-order formulas with at most variables. Note that a same variable may be quantified over several times in the formula.
Let be a polynomial function, and its local extrema (we suppose that ). Fix . For convenience, we will write instead of in formulas. We focus on models of the form where is the usual ordering of the reals, and . Let us describe an formula such that for all and , we have if and only if . First, we write for the formula
We can then define formulas and which state that is a local minimum (resp. maximum) of , for instance:
The formula then states that there exists at least local extrema before , alternating existential quantifications over and to identify them; for instance, is the formula
3 Star-free Propositional Dynamic Logic
Star-free Propositional Dynamic Logic.
Propositional dynamic logic (PDL)  consists of two sorts of formulas: state formulas which are evaluated at single elements, and path formulas which are evaluated at pairs of elements and allow to navigate inside the model. Here we consider a star-free variant of PDL (with converse). The syntax of star-free propositional dynamic logic over , written , is given below:
where and .
Compared to classical PDL, star-free PDL uses the operators of star-free expressions, instead of the rational operators .
Let . The semantics or of a state or path formula in is defined below. The state formula is true at a point in (that is, ) if there exists some such that satisfies and is true at . The path formula is stationary and tests if the state formula is true. The semantics of other formulas is straightforward:
For simplicity, we will often write or instead of and . We also write if , and if .
We will use the abbreviations , , , , and . For all formulas , we also define a state formula which holds at if and only if .
Suppose that , and that we consider only models over and with . Let and . The formula of metric temporal logic, which holds at time if there exists such that and for all , , can be expressed in as follows:
An interval-preserving fragment of star-free PDL.
We say that a path formula is interval-preserving if for all , is interval-preserving. Notice that and (for all ) are interval-preserving. By Lemma 2 (and assumption on ), all formulas constructed without the boolean operators and are interval-preserving. However, the complement or the union of interval-preserving relations are not in general interval-preserving. We define below a fragment of where all path formulas are interval-preserving, and which will turn out to be as expressive as (and in fact, ) when it comes to state formulas. To do so, we will introduce several restrictions of which will turn out to be interval-preserving, and which will suffice to characterize .
Let us first look at the different reasons for which we may have , assuming that is interval-preserving. To begin with, we focus on . One first sufficient condition for having is that . Now, suppose . If is interval-preserving, there are only three possible cases in which : , or , or . We define formulas and corresponding respectively to the first two cases. We let
Now, if we look at instead of , we can make the same observations, by symmetry: we have if and only if , and if is interval-preserving, there are again only four possible cases: , or , or , or .
Unfortunately, the formulas and are still not interval-preserving in general. However, if we take a more symmetric restriction of , where we look at all the possible positions of and relatively to and , we obtain four cases, illustrated in Figure 2, which we will later show correspond to interval-preserving restrictions of .
More precisely, let
Notice that .
Let be the following restriction of :
All formulas are interval-preserving.
We proceed by induction on the formula. By assumption, is interval-preserving for all . Moreover, and are interval-preserving. For , and , we apply Lemma 2.
Suppose that is interval-preserving. Let us show that is interval-preserving. Notice that . So we only need to show that for all intervals , for all and such that , there exists such that . Let such that . Let us show that we can in fact take . The proof is illustrated in the picture below.
First, we have . Moreover, (since ). Now, suppose towards a contradiction that . Let such that . Since , there exists such that . We then have and . Since is interval-preserving, we obtain , a contradiction with the fact that . Thus, .
Let us show that is also interval-preserving. Similarly to the previous case, we show that for all and such that , we have .
First, , and . Suppose towards a contradiction that . Let such that , and . We have , and . Since is interval-preserving, we obtain , a contradiction with the fact that . Symmetrically, let be an interval, , and such that . Then for any such that , we also have , hence .
Since , this also implies that is interval-preserving.
Finally, the case of is symmetric to the case of : for all and such that , we have . ∎
4 Star-free PDL is expressively equivalent to FO
Let be a state formula in , and an formula with a single free variable . We say that and are equivalent, written , if for all and elements in , we have if and only if . Similarly, for a path formula and an formula with exactly two free variables and , we write