1 Introduction
The definition and study of full combinations of modal [5] and intuitionistic [6, 23] logics can be quite challenging [30], and temporal logics, such as [28], are no exception. Some intuitionistic analogues of temporal logics have been proposed, including logics with ‘past’ and ‘future’ tenses [9] or with ‘next’ [7, 19], and ‘henceforth’ [17]. We proposed an alternative formulation in [4], where we defined the logics and using semantics similar to those of expanding and persistent products of modal logics, respectively [13], and the tenses (‘next’), (‘eventually’), and (‘henceforth’). in particular differs from previous proposals (e.g. [9, 27]) in that we consider minimal frame conditions that allow for all formulas to be upwardclosed under the intuitionistic preorder, which we denote . We then showed that with (‘next’), (‘eventually’), and (‘henceforth’) is decidable, thus obtaining the first intuitionistic analogue of which contains the three tenses, is conservative over propositional intuitionistic logic, is interpreted over unbounded time, and is known to be decidable.
Note that both and are taken as primitives, in contrast with the classical case, where may be defined by , whereas the latter equivalence is not intuitionistically valid. The same situation holds in the more expressive language with (‘until’): while the language with and is equally expressive to classical monadic firstorder logic with over [12], admits a firstorder definable intuitionistic dual, (‘release’), which cannot be defined in terms of using the classical definition. However, this is not enough to conclude that cannot be defined in a different way. Thus, while in [4] we explored the question of decidability, here we will focus on definability; which of the modal operators can be defined in terms of the others? As is wellknown, and ; these equivalences remain valid in the intuitionistic setting. Nevertheless, we will show that cannot be defined in terms of , and cannot be defined in terms of ; in order to prove this, we will develop a theory of bisimulations on models.
Following Simpson [30] and other authors, we interpret the language of using birelational structures, with a partial order to interpret intuitionistic implication, and a function or relation, which we denote , representing the passage of time. Alternatively, one may consider topological interpretations [8], but we will not discuss those here. Various intuitionistic temporal logics have been considered, using variants of these semantics and different formal languages. The main contributions include:

Logics with were axiomatized by Kamide and Wansing [17], where was interpreted over bounded time.

Nishimura [25] provided a sound and complete axiomatization for an intuitionistic variant of the propositional dynamic logic .

Balbiani and Diéguez axiomatized the hereandthere variant of with [2], here denoted .

FernándezDuque [10] proved the decidability of a logic based on topological semantics with and a universal modality.

The authors [4] proved that the logic with has the strong finite model property and hence is decidable, yet the logic , based on a more restrictive class of frames, does not enjoy the fmp.
In this paper, we extend to include (‘until’) and (‘release’). We will introduce different notions of bisimulation which preserve formulas with and each of , , and . With this, we will show that (or even ) may not be defined in terms of over the class of hereandthere models, while can be defined in terms of and can be defined in terms of over this class. However, we show that over the wider class of expanding models, cannot be defined in terms of .
2 Syntax and semantics
We will work in sublanguages of the language given by the following grammar:
where is an element of a countable set of propositional variables . All sublanguages we will consider include all Boolean operators and , hence we denote them by displaying the additional connectives as a subscript; for example, denotes the free, free fragment. As an exception to this general convention, denotes the fragment without or .
Given any formula , we define the length of (in symbols, ) recursively as follows:

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;

, with ;

, with .
Broadly speaking, the length of a formula corresponds to the number of connectives appearing in .
2.1 Dynamic posets
Formulas of are interpreted over dynamic posets. A dynamic poset is a tuple , where is a nonempty set of states, is a partial order, and is a function from to satisfying the forward confluence condition that for all if then An intuitionistic dynamic model, or simply model, is a tuple consisting of a dynamic poset equipped with a valuation function from to sets of propositional variables that is monotone, in the sense that for all if then In the standard way, we define and, for all , . Then we define the satisfaction relation inductively by:

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iff ;

;

iff and ;

iff or ;

iff ;

iff , if , then ;

iff there exists ;

iff for all , ;

iff there exists and , ;

iff for all , either , or s.t. .
As usual, a formula is satisfiable over a class of models if there is a model and a world of so that , and valid over if, for every world of every model , . Satisfiability (validity) over the class of models based on an arbitrary dynamic poset will be called satisfiability (validity) for , or expanding domain linear temporal logic.^{5}^{5}5Note that in [4] we used ‘’ to denote the fragment of this logic without .
The relation between dynamic posets and expanding products of modal logics is detailed in [4], where the following is also shown. Below, we use the notation
Lemma 1.
Let , where is a poset and is any function. Then, is a dynamic poset if and only if, for every valuation on and every formula , is monotone, i.e., if and , then .
The proof that all valuations on a dynamic poset are monotone proceeds by a standard structural induction on formulas, and the cases for are similar to those for in [4]. This suggests that dynamic posets provide suitable semantics for intuitionistic . Moreover, dynamic posets are convenient from a technical point of view:
Theorem 1 ([4]).
There exists a computable function such that any formula satisfiable (resp. falsifiable) on an arbitrary model is satisfiable (resp. falsifiable) on a model whose size is bounded by .
It follows that the fragment of is decidable. Moreover, as we will see below, many of the familiar axioms of classical are valid over the class of dynamic posets, making them a natural choice of semantics for intuitionistic .
2.2 Persistent posets
Despite the appeal of dynamic posets, in the literature one typically considers a more restrictive class of frames, similar to persistent frames, as we define them below.
Definition 1.
Let be a poset. If is such that, whenever , there is such that , we say that is backward confluent. If is both forward and backward confluent, we say that it is persistent. A tuple where is persistent is a persistent intuitionistic temporal frame, and the set of valid formulas over the class of persistent intuitionistic temporal frames is denoted , or persistent domain .
As we will see, persistent frames do have some technical advantages over arbitrary dynamic posets. Nevertheless, they have a crucial disadvantage:
Theorem 2 ([4]).
The logic does not have the finite model property, even for formulas in .
2.3 Temporal hereandthere models
An even smaller class of models which, nevertheless, has many applications is that of temporal hereandthere models [2]. Some of the results we will present here apply to this class, so it will be instructive to review it. Recall that the logic of hereandthere is the maximal logic strictly between classical and intuitionistic propositional logic, given by a frame with . The logic of hereandthere is obtained by adding to intuitionistic propositional logic the axiom
A temporal hereandthere frame is a persistent frame that is ‘locally’ based on this frame. To be precise:
Definition 2.
A temporal hereandthere frame is a persistent frame such that for some set , and there is a function such that for all and , if and only if and and .
The prototypical example is the frame , where , if and , and . Note, however, that our definition allows for other examples (see Figure 1). In [2], this logic is axiomatized, and it is shown that cannot be defined in terms of , a result we will strengthen here to show that cannot be defined even in terms of . It is also claimed in [2] that is not definable in terms of over the class of hereandthere models, but as we will see in Proposition 5, this claim is incorrect.
3 Some valid and nonvalid formulas
In this section we explore which axioms of classical are still valid in our setting. We start by showing that the intuitionistic version of the interaction and induction axioms used in [2] remain valid in our setting. However, not all FisherServi axioms [11], which are valid in the hereandthere of [2], are valid in .
Proposition 1.
The following formulas:

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;

;

;

;

;

;

;

;


.
are valid.
Proof.
Let us consider (10) and (11). For (10), let be any model and be such that . Let be arbitrary and assume that . Then, by induction on we obtain that for all ; since for all , it follows that for all as well. Hence an easy induction shows that for all , which means that . Since was arbitrary, we conclude that the formula (10) is valid.
For (11), let be as above and be such that . Let be such that . It follows that , so . Since were arbitrary, the formula (11) is valid as well.
The proofs for the rest of formulas are left to the reader. ∎
Some of the wellknown Fisher Servi axioms [11] are only valid on the class of persistent frames.
Proposition 2.
The formulas

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,

are not valid. However they are valid.
Proof.
Let be a set of propositional variables and let us consider the model defined as: ; , and ; ; . Clearly, , so . By definition, and ; however, it can easily be checked that and , so and . Let us check their validity over the class of persistent frames. For (1), let be an model and a world of such that . Suppose that satisfies . By backward confluence, there exists such that , so that and thus . But this means that , and since was arbitrary, , i.e. . Similarly, for (2) let us assume that is an model and a world of such that . Consider arbitrary , and suppose that is such that . Then, it is readily checked that the composition of backward confluent functions is backward confluent, so that in particular is backward confluent. This means that there is such that . But then, , hence , and . It follows that , and since was arbitrary, . ∎
We make a special mention of the schema , which characterises the class of weakly connected frames [14] in classical modal logic. We say that a frame is weakly connected iff it satisfies the following firstorder property: for every , if and , then either , , or .
Proposition 3.
The axiom schema is not valid.
Proof.
Let us consider the set of propositional variables and the model defined as: ; , , and ; and ; and . The reader can check that and . Consequently . ∎
Finally, we show that (resp. ) can be defined in terms of (resp. ) and the axioms involving and are also valid in our setting:
Proposition 4.
The following formulas are valid:

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;

;

;

;

;

;

;

.
Proof.
We consider some cases below. For (1), from left to right, let us assume that . Therefore there exists s.t. and for all satisfying , . If then while, if it follows that and . Therefore . From right to left, if then by definition. If then and so, due to the semantics, we conclude that . In any case, .
For (2), we work by contrapositive. From right to left, let us assume that . Therefore there exists s.t. and for all satisfying , . If then while, if it follows that and . In any case, . From left to right, if then by definition. If then and so, due to the semantics of , we conclude that . In any case, .
The remaining items are left to the reader. ∎
As in the classical case, over the class of persistent models we can ‘push down’ all occurrences of to the propositional level. Say that a formula is in normal form if all occurrences of are of the form , with a propositional variable.
Theorem 3.
Given , there exists in normal form such that is valid over the class of persistent models.
Proof.
We remark that the only reason that this argument does not apply to arbitrary models is the fact that is not valid in general (Proposition 2).
4 Bounded bisimulations for and
In this section we adapt the classical definition of bounded bisimulations for modal logic [3] to our case. To do so we combine the ordinary definition of bounded bisimulations with the work of [26] on bisimulations for propositional intuitionistic logic. Such work introduces extra conditions involving the partial order . In our setting, we combine both approaches in order to define bisimulation for a language involving , and as modal operators plus an intuitionistic . Since all languages we consider contain Booleans and , it is convenient to begin with a ‘basic’ notion of bisimulation for this language.
Definition 3.
Given and two models and , a sequence of binary relations is said to be a bounded bisimulation if for all and for all , the following conditions are satisfied:
Atoms. If then for all propositional variables , iff .
Forth . If then for all , if , there exists such that and .
Back . If then for all if then there exists such that and .
Forth . if then .
Note that there is not ‘back’ clause for ; this is simply because is a function, so its ‘forth’ and ‘back’ clauses are identical. Bounded bisimulations are useful because the preserve the truth of relatively small formulas.
Lemma 2.
Given two models and and a bounded bisimulation between them, for all and , if then for all satisfying ^{6}^{6}6Although not optimal, we use the length of the formula in this lemma to simplify its proof. More precise measures like counting the number of modalities and implications could be equally used., .
Proof.
We proceed by induction on . Let be such that for all the lemma holds. Let and be such that and let us consider such that . The cases where is an atom or of the forms , are as in the classical case and we omit them. Thus we focus on the following:
Case . We proceed by contrapositive to prove the lefttoright implication. Note that in this case we must have .
Assume that . Therefore there exists such that , , and . By the Back condition, it follows that there exists such that and . Since and , by the induction hypothesis, it follows that and . Consequently, . The converse direction is proved in a similar way but using the Forth .
Case . Once again we have that . Assume that , so that . By Forth , . Moreover, , so that by the induction hypothesis, , and . The righttoleft direction is analogous. ∎
Next, we will extend the notion of a bounded bisimulation to include other tenses. Let us begin with .
Definition 4.
Given and two models and , a bounded bisimulation is said to be a bounded bisimulation if for all and for all , if , then the following conditions are satisfied:
Forth . For all there exist and such that , and .
Back . For all there exist and such that , and .
As was the case of Lemma 2, if two worlds are related by a bounded bisimulation, then they satisfy the same formulas of small length.
Lemma 3.
Given two models and and a bounded bisimulation between them, for all and , if then for all^{7}^{7}7We remind the reader that, as per our convention, is the free fragment. A similar comment applies to other sublanguages of mentioned below. satisfying , .
Proof.
We proceed by induction on . Let be such that for all the lemma holds. Let and be such that and let us consider such that . We only consider the case where , as other cases are covered by Lemma 2.
From left to right, if then there exists such that . By Forth , there exists and such that , and . By monotonicity, . Then, by the induction hypothesis and the fact that , it follows that , thus by monotonicity once again, , so that . The converse direction is proved similarly by using Back . ∎
We can define bounded bisimulations in a similar way.
Definition 5.
A bounded bisimulation is said to be a bounded bisimulation if for all and for all , if , then:
Forth . For all there exist and s.t. , and .
Back . For all there exist and s.t. , and .
Lemma 4.
Given two models and and a bounded bisimulation between them, for all and , if then for all such that , then .
Proof.
We proceed by induction on . Let be such that for all the lemma holds. Let and be such that and let us consider such that . Note that the cases for atoms as well as propositional and connectives are proved as in Lemma 2, so we only consider .
For the lefttoright implication, we work by contrapositive, and assume that . Then, there exists such that . By Forth , there exist and s.t. , and . As in the proof of Lemma 3, by monotonicity, the induction hypothesis and the fact that , it follows that ; thus , and again by monotonicity . The converse direction follows a similar reasoning but using Back . ∎
5 Bounded bisimulations for and
In this section we adapt the bisimulations defined for a language with until and since [18] presented by Kurtonina and de Rijke [20] to our case. Let us begin with bounded bisimulations for .
Definition 6.
Given and two models and , a bounded bisimulation is said to be a bounded bisimulation iff for all and for all :
Forth . For all there exist and such that

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, and , and

for all there exist and such that , and .
Back . For all there exist and such that

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, and , and

for all there exist and such that , and .
As was the case before, the following lemma states that two bounded bisimilar models agree on small formulas.
Lemma 5.
Given two models and and a bounded bisimulation between them, for all and , if then for all such that , .
Proof.
Once again, proceed by induction on . Let be such that for all the lemma holds. Let and be such that and let us consider such that . As before, we only consider the ‘new’ case, where . From left to right, assume that . Then, there exists such that and for all satisfying , . By Forth , there exist and such that , and ; for all satisfying there exist and s. t.
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