linear tense logics of increasing setsuros.m/logcom/hdb/volume_12/issue_04/pdf/1… · j. logic...

Linear Tense Logics of Increasing Sets

BERNHARD HEINEMANN, Fachbereich Informatik, FernUniversitatHagen, D–58084 Hagen, Germany.E-mail: [email protected]

AbstractWe provide an extension of the language of linear tense logic with future and past connectives � and � , respectively,by a modality that quantifies over the points of some set which is assumed to increase in the course of time. In thisway we obtain a general framework for modelling growthqualitatively. We develop an appropriate logical system,prove a corresponding completeness and decidability result and discuss the various kinds of flow of time in the newcontext. We also consider decreasing setsbriefly.

Keywords: Temporal reasoning, modal logics of time, modal logics of set spaces.

1 Introduction

The temporal operators � and � describing the future and the past, respectively, are includedin the very basic system of tense logic, �� [7, chapter 3]. The relevant semantic domainsare based on irreflexive and transitive flows of time, i.e. non-empty sets � endowed witha binary relation � on � satisfying these properties. Apart from this one can specify �to a greater extent by additional axioms if need be for concrete applications. For instance,linearity, density, or Dedekind completeness, can be expressed in this way; see the listing onp. 32 of the textbook [7]. Sometimes the general irreflexivity assumption on the relation � isdropped and reflexive flows of time are considered. This is done in the first part of the presentpaper as well.

We focus on linear tense logics here; i.e. we assume for the most part of the paper thatthe relation � is a linear ordering on � . Now, imagine that a certain set � increases alongsuch a time line and that there is an observer analysing the change of � at a given moment.Then, looking into the future the observer sees the set grow,but looking into the past he seesit shrink; see Figure 1.

Thus, taking into account the observer’s point of view in modelling growth in the courseof time leads to a different description of the situation with regard to the past and the future,respectively. In contrast to this it is true that several properties of growing and shrinking canbe reflected as a pair of ‘dual’ axiom schemata in the formal system that is to be given below.Actually, this system turns out to be a suitable extension of linear tense logic.

We now mention two topics fitting in the framework just indicated and motivating our ap-proach. Afterwards we explain some more details of the logic. First, consider an agent �involved in a multi-agent system, for instance a processor in a distributed system. A certainkind of knowledgecan be ascribed to�� which coincides, by definition, with the set of formu-las being valid in every state of the system the agent considers possible. In connection withdistributed systems, the relation of possibility between states is an equivalence relation repre-senting indistinguishability of the statesto the agent. Thus every equivalence class representssome knowledge stateof � about the system. A corresponding logic of knowledgehas beenevolved and diversely applied, which may be identified with the well-known multi-modal

J. Logic Computat., Vol. 12 No. 4, pp. 583–606 2002 c� Oxford University Press

584 Linear Tense Logics of Increasing Sets

� �� Æ��

�

�observer

��

��

��

��

�

FIGURE 1. A discretely increasing set

�� in case the system consists of agents [5, 10]. Now, combining modal and temporalconcepts enables one to treat interesting aspects of the development of knowledgeformally.For example, the notions of synchronous systems, perfect recall,and no learning,have beenstudied to a certain extent in the literature [11]. Roughly speaking, in ‘synchronous systems’the agents have access to a common clock. For such systems, ‘perfect recall’ means a succes-sive shrinkingof the agent’s knowledge state, while ‘no learning’ means increasing,or rathernon-decreasing,of this set in the course of time. The language of growing sets we introducebelow originates from this reasoning about knowledge context, and a knowledge operatorisretained in it. However, this connective is given a different semantics presently.

The second of the potential areas of application of our system is spatio-temporal modellingand reasoning, which currently makes up a very active branch of research. It is one of themajor goals of the efforts undertaken in this field to express adequately and handle formallythe growing (and shrinking, respectively) of various objects, and the temporal change ofgeometric shapes in particular. This is of interest to people working on (spatio-)temporaldatabases or corresponding reasoning formalisms, for instance. Enlargements of the systemdescribed below can provide the theoretical basis for such tasks.

In order to get to the desired tense logic of increasing sets, we briefly revisit a special logicof knowledge, which can be applied to simple topological reasoningand, therefore, gets thead hocdesignation TR. This system has been proposed and investigated in [3]. By meansof TR one is able to deal modally with spaces of decreasing sets. Thus access to our topicis offered in a sense. We mention the very basic features of TR here, for convenience ofthe reader. Roughly speaking, TR represents a poly-modal logic with a modified semantics.Restricted to the single-agent case it contains two modalities: the first one, the ‘knowledgeoperator’ �, quantifies ‘horizontally’ over the elements of some set � and the second one,the ‘progression operator’ �� quantifies ‘vertically’ over certain subsets of � expressing‘shrinking’ in this way. In particular, usual Kripke frames are replaced by so-called subsetframesnow, i.e. pairs �� consisting of a fixed non-empty set � and a distinguished set� of subsets of � .

One of the most interesting questions concerning TR is how the modalities � and � in-teract. This interaction depends on the class of semantic structures to be examined, i.e. onthe properties of � one is interested in. It is a demanding question in general to capture theinterplay of the two connectives axiomatically; see [3], [8], and [21], for some examples.

As we have said before this paper is concentrated on the growthof sets. It is our idea to stick

Linear Tense Logics of Increasing Sets 585

with interpreting formulas in models based on subset frames as in [3], but to ‘split up’ the dualof the �-operator into � and � respectively, and modify the semantics appropriately. Thus,abstracting away the connection with knowledge, only the general framework of TRremains.However, concerning methods we benefit from [8] very much; we can suit both Georgatos’canonical model and decomposition techniques to proving completeness and decidability,respectively, in the present case.

Subsequently, a modal logic of some kind of temporal change is presented. One can finddifferent, in particular (bi-)modal, logics dealing with dynamic aspects of sets in the literature,but those have other objectives [4]. Concerning the connections between topology and modallogic, and their utilization for qualitative spatial reasoning in AI, cf [18]. Finally, there doesexist some related work on combinations of S5 and linear temporal logic, for instance therecent article [20]. We will refer to this later on. However, we will not ‘cross the propositionalborder’ in the present paper; cf [16].

The outline of the paper is as follows. We first introduce the underlying logical languageprecisely, in Section 2. In the same section we define the linear tense logic of growth andprove a corresponding completeness result. In Section 3 we argue that this logic satisfiesthe finite model property. Thus the set of satisfiable formulas of the language is decidable.Afterwards, in Section 4, we ask whether the same results can be obtained for other classes offlows of time. We also touch on the problem of modelling decrease. Finally, some concludingremarks are given.

The fundamentals of modal and tense logic respectively are presupposed in this paper. Inparticular, we assume acquaintance of the reader with canonical models and filtrations. Allthat we need can be found in the textbooks [1, 2] or [9].

2 An extension of linear tense logic

From a technical point of view we essentially combine linear tense logic, i.e.

�� connectedness axioms�

with the basic logic of knowledge of a single agent, i.e. the modal system ��. The type ofthis combination is determined by our goals, viz. treating formally sets which grow little bylittle.

We use the syntactic conventions which are common for the respective logics. In particular,the formulas can contain a past operator � , a future operator � and a knowledge operator�. � and � are ‘existential’ modalities (‘there is a point in the past/future such that . . . ’)whereas � is a ‘universal’ one (‘for all points reachable from the present one we have . . . ’).In order to define the set �� of well-formed formulas of the logical language we let � �� be an enumerable set of propositional variables. Designating formulas by lowercase Greek letters the set �� is given in Backus–Naur form as follows:

� ��

Concerning dualsof the modal connectives let

�� and ��

as usual. �, � and � are read ‘henceforth’, ‘hitherto’ and ‘possibly’, respectively. TheBoolean operators � and � are treated as abbreviations.


We next define the semantics of our language. To this end we first single out the flowsof time we consider admissible in this paper. These are exactly linear orderings,i.e. sets �equipped either with a reflexive and transitive or an irreflexive and transitive, binary relation� such that every two different points of � are comparable with respect to � .

DEFINITION 2.1Let � be a non-empty set and � a linear ordering on � . Then the pair �� is called alinear flow of time.

Intending to describe growth we are confronted with a set at the moment, but we willfind several supersets of in future (and, correspondingly, have found certain subsets of in the past). All these sets have to be considered in the formal model. Consequently, taking auniverse which contains all of them, and the system of these sets, then we are led to a subsetframeas mentioned in the introduction.

Subsequently, let �� denote the powerset of a given set �.

DEFINITION 2.2Let a non-empty set � and a subset � � �� be given. Then the pair � �� iscalled a subset frame.

Connecting time to sets is done by indexing � with a linear flow of time. Note that setsmay be unchanged for a while. This case is included in the following definition.

DEFINITION 2.3Let �� be a linear flow of time, and let � �� be a subset frame.

1. Assume that there exists an order-preserving surjection

� � ��

Then the structure� � � �� is called a linear set frame. Following the case distinc-tion of Definition 2.1 we speak of a reflexiveand an irreflexive linear set frame respec-tively, if need be.

2. Let � � �� be a linear set frame, as just defined.(a) An �-valuationis a mapping � � � �� .(b) Let � be an �-valuation. Then the quadruple � � � �� is called a model

(based on�).

Given a linear set frame � as above, a valuation is intended to associate with each point� � �� the set of elementary propositions which are true at �. Note that we assume theelements of � to be true (or false) regardless of time (and therefore of sets as well), beingin accord with the TR tradition. This allows us to give a simple definition of the satisfactionrelation � following immediately, but contrasts with other approaches. For example, in [20]a ‘two-dimensional’ valuation is considered.

In TR, a pair �� such that � � � is called a situation, and � is definedas a binary relation between situations of a given model and formulas. In this way the twodimensions of the logic get linked such that the system becomes one-dimensional in a sense.This proceeding is brought into line with linear set frames now by the next definition. Ap-propriately, we let situations consist of a state � and a time point � such that � � ��.


DEFINITION 2.4Let � � � �� be a model and �� a situation. Then we let

�� where � � �� and �� and �� and �� if � � �� then ��

If �� we say that � holds in� at �� . If � holds in � at every situation, �is called valid in �. By quantification over all models based on a linear set frame � oneextends the notion of validity to frames, as usual.

We next want to provide a ‘classical’ axiomatization of the set of validities. In particular,we will not make use of Gabbay’s irreflexivity rulein this paper; concerning this rule see [7],p. 31 ff. So, we are led to the following assumption for the moment: we assume that therelation� is reflexive. This should hold up to the end of Section 3.

We divide the axioms into three groups. The first one concerns the modality� and consistsof the usual ��-schemata.

�� All instances of propositional tautologies��

As one can see below, the ��-modality � appears in combination with linear time in thispaper. Interestingly enough, an ��-modality is also present in certain systems of branchingtimetemporal logic [19].

Apart from the first and the last schema the second group of axioms originates from lineartense logic. Following [9], Section 6, we let �� .

��

It is well-known that the mirror image of ��, �� , is a theorem of tense logic.An analogous statement is valid for the mirror image of ��, �� .

The schema �� expresses reflexivity of the relation �, and the schema �� transitivity.The schemata �� and �� correspond to linearity to the effect that both are valid in

an arbitrary flow of time ��, iff is weakly future-connectedand weakly past-connected;i.e. both properties

� �� if � � � and � � �� then � � � or � � or � � ��


and� �� if � � � and � � �� then � � � or � � or � � ��

must hold for .�� is due to our particular definition of the satisfaction relation in case of a propositional

variable � � �.The final group of axioms describes the interaction of the knowledge operator and the time

operators. We actually have two ‘transposition relations’ in this group which are dual to eachother in a sense, and a third schema forcing the chain property for subsets in case of lookinginto the past:

��

Adding modus ponens as well as �-, �-, and �-necessitation as derivation rules yields acalculus �. Notice that the mirror image of �� is missing in the above list. In fact, we getas a consequence of the subsequent Theorem 2.5 that this schema can be derived in �.

Theorem 2.5 actually states that the just given logical system is sound and complete withrespect to the class of models based on reflexive linear set frames. This is the first main resultof this paper.

THEOREM 2.5A formula � � �� is derivable in �, iff it is valid in every model � based on a reflexivelinear set frame.

The soundness part of Theorem 2.5 is straightforward to see. Thus, corresponding argu-ments may be omitted.

We now point up the crucial steps of the completeness proof. Not surprisingly, we use thecanonical model �� of the logical system. Let

��

��

��

denote the accessibility relations of �� belonging to the connectives � , � and �, respec-tively. (This denotation is kept from TR.) Then, the above axioms provide for the requiredproperties of these relations.

The following features are well-known from basic modal and tense logic respectively:��

is an equivalence relation,�� and

�� are reflexive and transitive as well as weakly future-

connected and weakly past-connected respectively, and both inclusions

��

�� and �

��

��

are valid; thus

��

�� and �

��

��

holds in particular. It is perhaps less well-known that

for all points �� of the canonical model such that ��

�� (or �

��

��),

there exists a point � such that ��

�� (and �

��

�� respectively),


� ��

� � � �

� �

� �

�

��

�

�

�

�

� �

��

��

� � � �

FIGURE 2. Cross propertiescorresponding to �� and ��

because of �� (and �� respectively); see Figure 2. These cross propertiesare typical ofthe various logics of subset spaces; the right diagram of Figure 2 appears in the fundamentalpaper [3], the left one is implicit in [13]. In the present context, �� represents the real‘axiom of growth’, whereas �� corresponds to ‘shrinking’.

Let �� denote the��-equivalence class of a point � of ��. Define two binary relations

� and � on the set of all such equivalence classes:

�� there are �� such that ��

�� there are �� such that ��

Clearly, these relations are reflexive. By means of the transitivity of�� and

�� as well as

the respective cross property one can easily prove that they are transitive as well.

LEMMA 2.6The relations� and � are transitive.

PROOF. We only show transitivity of �. So, let �� be given such that

��

Then there are �� and �� such that

�� and ��

��

Because of �� we find �� satisfying

��

��

according to the first cross property of Figure 2. Now, transitivity of�� yields ��

��

showing that �� .

Less trivial characteristics of � and � will be obtained by the subsequent propositions.The schema �� has to be used in order to prove the following assertion, among other

things.


PROPOSITION 2.7Each of the relations � and � is the converse of the other one. Moreover, � is weaklyfuture-connected and � is weakly past-connected.

PROOF. The first statement follows immediately from the definition of � and �� and from

the fact that�� and

�� are converse to each other.

Next, suppose that�� and ��

holds. It follows that there exist �� and �� satisfying

�� and ��

��

According to the cross property associated with growing there exists �� such that

��

��

since we have ��. Now, by weak connectedness along with reflexivity of the relation

��

�� or ��

��

is yielded. This implies weak connectedness of the relation �.

Finally, we assume towards a contradiction that we have

�� and �� and neither �� nor ��

Fix any elements �� and �� such that

�� and ��

��

Because of �� there are formulas �� satisfying

� � �� and � � ��

for otherwise we would have

�� or ��

which implies that

�� Æ � Æ � �� or �� Æ � Æ � ��

is consistent; thus we would get an element �� of the canonical model such that

��

�� or ��

��

��

violating�� and ��

It follows that��

contradicting axiom ��. This proves that � is weakly past-connected.


It should be emphasized that the methods required for the proof of the weak connectednessproperty of � and � are different. In the first case this property is ‘subsumed’ under thecorresponding cross property in a sense, whilst in the second case a separate axiom is needed.

A further important property of the relations� and� is stated in the next proposition.

PROPOSITION 2.8The relations

�� and � as well as their mirror images are antisymmetric.

PROOF. Since�� is an equivalence relation,

�� is reflexive, transitive and weakly con-

nected, and the cross property holds for these accessibility relations, it follows from [8],

Proposition 3.4 and Proposition 3.5, that both relations�� and � are antisymmetric. As

�� and � are converse to

�� and � respectively, the same is true for these relations. This

proves Proposition 2.8.

Now we define a model refuting a given non-derivable formula. Let �� hold for � ��. Take a point � of the canonical model�� containing��, and let�� be the submodelof �� generated by �. The carrier set � � of �� consists of a union of certain equivalence

classes with respect to��:

� � ��

��

Letting� � �� or ��

we get that � �� represents a linear ordering, according to the above propositions.The desired linear set frame can be obtained next as the space of ‘threads’ through the

classes of which �� consists. (Notice that such threads may or may not have a ‘left end-point’.) To be more precise, we proceed in the following way:

For every � � � � we let

� � �� or �

��

� is called the thread determined by�.It is clearly possible that two different points of � � determine the same thread. In

fact, this is the case if and only if these points are connected with respect to�� or

��.We let the following set � be the domain of the model we are looking for:

� � � � � � � ��

see Figure 3 (where only somepoints of� are marked by horizontal boxes, for clarityreasons).The distinguished set of subsets of � , �, are given by the next definition, togetherwith a natural indexing surjection � � ��

� � � ��


�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�� !�� !��

��

�

�

�

�

�

�

�

�

�

�

�

�

�� !

�� !

�� !

�� !� � � � � ��

� � � � �

� � � � �

� � � �

� � � �

� � �

� � �

� �

� �

�

�

FIGURE 3. A part of the almost final model

for all �� . Notice that � � �� consists of a single point if it is non-empty,justifying the naming ‘thread’; in case of existence this point is designated � ��

��.(This fact, which is implicitly contained in Proposition 2.8, can be proved with the

aid of [8], Proposition 3.4(d), using the above stated properties of�� and

��

additionally.)

We immediately get that � � � � �� carries the right structure.

LEMMA 2.9The just defined structure � is a linear set frame.

Notice that, in particular, sets from �� increase in the course of time.An �-valuation � � � �� is given by

� � � ��

for all � � � and � � � . Although depending on the representative � of � this definitionis correct because of Axiom ��.

The resulting model� � � � �� fulfills the following Truth Lemma.

LEMMA 2.10For all � ��, � � � and �� such that � � �� we have that

�� iff � � � ��

��

PROOF. The proof of Lemma 2.10 proceeds by induction on the structure of �. Subsequentlywe do not mention use of Definition 2.4 explicitly. Moreover, we frequently apply that � ��actually exists according to the assumption � � � ��.

Case � � � �: In this case we have

��

��

where Axiom �� is required for the last equivalence.


The cases of negation and conjunction are straightforward.

Case � ��:

�� and ��

�� and � � � ��

��

Here the second equivalence holds because of the induction hypothesis, whereas propertiesof the canonical model are used for the last one.

In the case � �� one argues analogously, taking into account the slight modificationconcerning the semantics of � � see Definition 2.4.

Case � ��:

��

��

��

��

The second equivalence is again obtained by the induction hypothesis. For the third one wegive a separate argument:

‘�’: Let �� and � � � ��. Then � �� exists, and we have

� ��

��

��

Thus � � � �� follows from �� .

‘�’: Now suppose that �� holds. Then there exists an element � of the canonicalmodel such that

� ��

�� and � ��

As � � �� it follows that � � � �� and � �� , proving the left-to-right direction.

This finishes successfully the structural induction.

Theorem 2.5 follows easily from Lemma 2.10 now. In fact, we first observe that � �� furthermore, since � �� is contained in �we conclude that �� thus � is refutedin the model� at the situation ��

3 Finite model property

Decidability of the temporal system studied in [20] was proved by means of the so-calledmosaic method,which is typical of higher-dimensional modal logics; cf [17]. We have men-tioned above that our logic is one-dimensional in a sense. Thus it is not surprising that we areable to prove decidability with the aid of more elementary methods.

We show in this section that our logic satisfies the finite model property. (This is not truefor the system in [20].) First, we remark that the previously considered model � has notbeen obtained via filtration as is the case in ordinary linear tense logic. This is due to thepresence of the modality �, which destroys connectednessof generated submodels; so, weare forced to work more directly with the canonical model.


Correspondingly, the finite model property cannot be guaranteed by means of filtrationeither. We will use appropriate decompositiontechniques instead; these techniques are bor-rowed from [8], where they have been successfully applied to get decidability of the subsetspace logic of tree-like structures.

DEFINITION 3.1Let � �� be a linear flow of time.

1. A subset � � � � � is called a segment of , iff there is no � � � � � � strictly betweenany two elements of � �.

2. A partition of � into segments is called a segmentation of .

We will have to consider segmentations of such that the truth value of a given formularemains unaltered on every segment of a model � � �� based on a linear setframe.

DEFINITION 3.2Let � � �� be a formula, � � �� a model based on a linear set frame, ! anindexing set and � �� " � !� a segmentation of . Then � is called stable on , iff forall " � ! and � in the carrier set of �, we have

�� for all � � � such that � � �� or�� for all � � � such that � � ��

It turns out that we can always achieve a finitesegmentation of on which a given formulais stable.

PROPOSITION 3.3Let � � �� be a formula and � � �� a model based on a linear set frame.Then there exists a finite segmentation �� of such that � is stable on .Moreover, can be chosen such that it refines � for every subformula � of �.

PROOF. We construct the partition by induction on the structure of �. We start with thetrivial segmentation �� in case � equals a propositional variable �. Axiom �� actuallyguarantees that � is stable on ��.

Inspection of the cases � ��, � �� and � �� shows that we can choose �here. This is immediately clear in the case of a negation. Concerning the future operator fixany point �, segment �� , and time � � �� such that � � ��, and notice that

�� and ��

�� and ��

��

which shows stability of �� on �. In fact, the left-to-right direction of the second equiv-alence holds because of the stability of � on � and the reflexivity of time, among otherthings; for the right-to-left direction of the third, one distinguishes between the cases �� and � � ��; all the other transitions are obvious. The same observations apply to the pastoperator, too.

Only the cases � � � � and � �� contribute to a refinement of the segmentation ob-tained so far, in the following manner:


Letting� �� and ��

��

be segmentations which exist for � and � respectively according to the induction hypothesis,then the set

�� # � $� � % � �

induces in an obvious way a finite segmentation �� of on which � � � is stable, therebyrefining both � and .

In order to achieve a segmentation�� on which �� is stable each segment �� of� is splitinto at most two disjoint segments. The exact number depends on how the truth value of �evolves within �� ( � # � $). To be more precise, this number is two if and only if there arepoints �� such that

� � �� for all � � �� and

�� for some � � ��

then, letting�� for all � � �� and

�� for some � � ��

yields a corresponding segmentation of ��.

Let a formula � � �� and a model � � �� based on a linear set frame begiven. We consider a finite segmentation �� of according to Proposition3.3. We define

��

��

for all # � � � � � � � $�. Furthermore, we let

� � �� $��

where� means the usual ordering on natural numbers, and define a mapping

�� $� ��

by ��#� � �� for � # � $ � here � denotes the domain of �. Evidently the structure

�� where � � � ��

belongs to the class of models we are interested in. We get the following proposition byinduction on the structure of formulas.

PROPOSITION 3.4Let �, � and �� be as above. Then, for all subformulas � of �, � � � , # � � � � � � � $� and� � �� such that � � ��, we have

�� # ��


PROOF. We treat only the cases � �� and � ��. Each of the remaining ones is eitherobvious or similar. We use several times that �� is contained in ��#� if � � ��, among otherthings.

Case � ��:

�� and ��

�� and �� #� �� (where #� satisfies ��

�� #� � # � �� #�� and �� #� ��

�� # ��

Case � ��: First we prove the right-to-left direction:

�� # �� # �� for all �� #�

� �� for all ��

� ��

Proving the other direction the stability of the finite segmentation has to be used:

�� for all �� such that � � ��

� �� for all �� and ��

� �� for all �� and ��

� �� # �� for all �� #�

� �� # ��

As a consequence we immediately obtain that passing from � to �� preserves the satis-fiability of �.

COROLLARY 3.5Let � � �� be a model based on a linear set frame such that �� forsome situation �� of �, and let �� be derived from � as previouslydescribed. Then �� # �� for some situation �� # of ��.

Accordingly, we have restricted the problem to decide whether a given formula � is satis-fiable to the class of models of ‘finite depth’. By a standard procedure of TR(see [8], Lemma3.35) we can reduce this question further to models which have ‘finite width’ additionally.This eventually yields the finite model property of our system. Consequently, the logic isdecidable.THEOREM 3.6The bimodal logic of reflexive linear set frames satisfies the finite model property. Thus, theset of �-derivable formulas as well as the set of satisfiable formulas is decidable.

4 Other properties of time

In this section we return to irreflexive flows of time ��. We ask whether the resultsof the previous sections can be generalized correspondingly. It turns out that this is in factpossible, if we confine ourselves to the ‘future fragment’ of basic linear tense logic.


So, we do not consider the modality � in this section. The accordingly modified set offormulas is designated �� in this section, too. The semantic structures remain the same,but we assume irreflexive flows of time underlying our considerations; see Definition 2.1 ff.Concerning axioms the changes are as follows:

� ��, ��, ��, ��, �� and �� are skipped;

� �� and �� are replaced by the single schema

��

� �� now reads��

Adding�-necessitation and modus ponens as derivation rules yields a calculus �� determin-ing a modallogic of increasing sets and time.

We now explain the modifications of the completeness proof for the new system. We start

out with the relations�� and � of Section 2 and prove the analogue to Proposition 2.8.

Notice that the technical details of this proof are different from those of the proof of thatproposition (cf [8]) simply because we have to deal with other axioms. (Nevertheless, thefundamental ideas are due to Georgatos.) The proof of Proposition 4.1 will again demonstratethe power of the schema ��.

PROPOSITION 4.1The relations

�� and � are antisymmetric.

PROOF. The proof is divided into several steps. First, we prove an auxiliary claim, by struc-

tural induction. This enables us to establish the assertion for�� . Applying the axiom of

growth suitably yields the assertion for � afterwards. The claim reads as follows:

CLAIM 4.2For every formula � � �� and all points �� of the canonical model satisfying �

��

and ��

� � �� or �

��

�� and � � �

��

� � �� or �

��

��

��

PROOF. (Claim 4.2) We do not have to give reasons for the right-to-left direction since it isobvious in each case of the induction.

Case � � � �: Let � � � for some � such that � � �� or ��

��. Assume

� �� . Then �� . With the aid of Axiom �� we get �� . It follows that �� ;contradiction. Consequently, � � �, thus �� . This shows � � � for all � such that

� � �� or ��

��.

The other propositional cases are immediate from the induction hypothesis.

Case � ��: Let �� for some � such that � � �� or ��

��. Then � � �� for

some point �� satisfying ��.

We assume towards a contradiction that there is some �� such that �� or ��

�� , and �� . Because of Axiom �� we may additionally assume that �� ,


" ��

�

��

��

�

��

��

��

��

�

�

�

�

� �

�� "�� "#� �

��

"" &

��

��

�

�

"�"�

" ��

" �#

FIGURE 4. The case �� in the proof of Proposition 4.1

without loss of generality. It follows that �� .

Since the relation�� is transitive and weakly future-connected we infer

�� or �� or ��

��

As �� and � � ��, the third alternative is impossible. In the case of the first one theinduction hypothesis applies, yielding � � ��. This is produced by the second alternative aswell.

Because��

��

��

the schema �� implies the existence of some point �� such that

��

��

Consequently, the induction hypothesis is applicable to the pair �� . We conclude � � ��.However, we have also �� because of �� . This yields the desired contradiction.Our assumption, therefore, is wrong. Thus the claim is proved in the case � ��.

Case � ��: Let �� for some � such that � � �� or ��

��. Again, we

assume towards a contradiction that there is some �� such that �� or ��

��,

and �� . Then �� , hence �� for some point �� satisfying ��. We only

treat the case�

��

�� and �

��

��

since the case� � �� or ��

is similar and even simpler; the former case is illustrated by Figure 4. As we have

��

��


there exists �� such that��

��

��

Transitivity and symmetry of�� together imply ��

��. Consequently,

��

��

The axiom of growth guarantees the existence of a point & satisfying

��&

��

It turns out that the induction hypothesis is applicable to the pair �� &�. We obtain ��

in particular.

We also have��

��

��

This yields

��

��

for some �. Moreover,

��

��

holds, implying

��

��

for some point �. It follows that the induction hypothesis is applicable to the pair �� aswell. This produces � � ��, for � � � holds because of �� . So, we have arrived at acontradiction. This proves the claim in the case � ��.

All in all, the proof of the claim is finished.

As a consequence of the claim we get that

�� if �� are points of the canonical model such that �� and �

�� then � ��.

Antisymmetry of the relation�� follows immediately from the subsequent assertion, which

can be proved by means of the above claim.

CLAIM 4.3For every formula � �� and all points �� of the canonical model:

if ��

�� then � � � iff � � ��

PROOF. (Claim 4.3) We carry out a structural induction, as suggested by the formulation ofthe claim. Only the case � �� is non-trivial:

Suppose that some formula �� is contained in � and not in ��. Then �� . Hence there

exists �� satisfying �� and �� . According to the cross property corresponding to

the schema �� there is some � such that

��

��


Moreover, there is some �� such that

��

��

for the same reason. It follows that the pair �� of points satisfies the presupposition of

the previously considered claim. Thus �� holds because of �� and ��

��.

But we get � � �, too, since �� ; contradiction. Consequently, �� as desired. Forreasons of symmetry this suffices to prove the assertion in the case � ��.

The argument yielding antisymmetry of the relation� is fairly canonical now: Assuming

��

i.e.��

�� and ��

��

for some �� and �� , and applying the cross property appropriately results in

��

�� for some ��

By means of the above corollary �� to Claim 4.2 we conclude �� . Then antisymmetry

of the relation�� yields �� proving �� .

This ends the proof of the proposition.

Again, let�� be the submodel of the canonical model generated by a point � that containsthe negation of a given non-derivable formula. The carrier set � � of �� reads

� � ��

now. Let� � ��

and � ��. It may happen that is not irreflexive, thus cannot serve as an appropriatelinear flow of time automatically. However, a semantically equivalent irreflexive linear flowof time can be constructed by means of the well-known bulldozingtechnique of modal logic;see for instance Theorem 3.20 in [2]. In the actual situation bulldozing means substitutingevery ‘reflexive point’ �� by the set �� (ordered in the natural way). In this way some of

the��-equivalence classes of �� become infinite sequences of copies of themselves.

Let � � �� be the resulting strict linear ordering. We use � in place of duringthe present completeness proof. We define

��

� ��

and let ' � �� be the canonical surjection; in case � � �� consists of two components,

we denote the second one '�� ('�� ). We lift the relations�� and

�� to �� by

�� '��

��'��

��

�'��

��'�� and

'�� '�� (if applicable)�


�

��

�

�

�

�

�

�

�

�

��

��

��

�

��

$

%

&

'

��

��

��

FIGURE 5. A segment of some cone of ��for all �� . Obviously the relevant properties of

�� and

�� hold also for

� �� and

� �� respectively;

� ��-equivalence classes are designated by means of square brackets, too.

Though formed like threads (see Section 2), the sets

� � �� or ��

� ��

do not carry the linear structure of threads. Because of that we have to work with suitableequivalence classes instead of them; see Figure 5. To this end we define

� ! � �� ( � �� ( or �

� ��(� and �� ( or �

� ��(�

��

for all �� . It is clear that ! is reflexive and symmetrical. Transitivity of ! holds

because of transitivity and weak connectedness of� ��. The equivalence class of � � �� with

respect to ! is designated ��; �� may be called the cone determined by�.As it has just been indicated, the set �� may consist of more than one point for some

�� . Nevertheless, the following assertion is valid, which will become important inconnection with the Truth Lemma4.5 below.

PROPOSITION 4.4Let � �� and �� . Then

� � '�� for some � � �� '�� for all � � ��

PROOF. The proposition can be proved by structural induction. The case � �� requiresthe following not completely obvious strengthening of the cross property (needed in the proofof Lemma 4.5 as well):

Let �� be elements of �� a class satisfying �� , and

( � �� such that �� (. Then there exists an element (� � �� such

that �� (�.


This can be seen with the aid of the cross property and weak connectedness of the relation� �� among other things. All the other steps of the proof are rather routine. We may omit

further details therefore.

We are going to define the desired model. First, the carrier set is

� � �� The set of subsets � and the mapping � are given in the same way as in Section 2:

��

for all � � �� and �� . Finally, the distinguished valuation is defined by

�� '��

for all � � � and � � ��. Let� be the resulting model. Then the subsequent Truth Lemmaholds for �.

LEMMA 4.5For all � ��, � � �� and �� such that �� we have that

�� iff � � '�� for all � � ��

The lemma can be proved in the same manner as Lemma 2.10.Utilizing Proposition 4.4, completeness of the system follows as before; cf the proof of

Theorem 2.5 at the end of Section 2.

THEOREM 4.6A formula � �� is derivable in ��, iff it is valid in every model� based on an irreflex-ive linear set frame.

The methods of Section 3 for proving decidability cannot be adapted directly to the contextconsidered in this section because they depend on reflexivity. However, we are able to modifythem appropriately, as we want to sketch in the following.

To this end we revisit the generated submodel

��

��

of the canonical model�� of the actual system we started with just after the proof of Propo-sition 4.1. (� � denotes the distinguished valuation of �� restricted to ��.) The followingproperties are satisfied for ��:

�� is an equivalence relation,

�� is transitive and weakly connected,

� the cross property corresponding with the schema �� holds for�� and

��,

�� induces a transitive and antisymmetric relation � on the set of

�� -equivalence

classes of �� with respect to which every two distinct classes are comparable,


� for all �� such that �� it holds that

��

for every � � �� here � means the usual modal satisfaction relation.

We call bimodal Kripke models sharing these properties special. For special models thesubsequent soundness and completeness result can be proved.

PROPOSITION 4.7A formula � �� is derivable in ��, iff it is valid in every finite special Kripke model.

We give reasons for the validity of this assertion. The soundness part is easy to see. As tocompleteness, we consider the structure �� introduced above (together with ��).The fact that �� is a generatedsubmodel of the canonical model of a modalsystem givesrise to the following specialized notion of segmentation for relations of this type.

DEFINITION 4.8Let " be a transitive and antisymmetric binary relation on a set � . Furthermore, assume thatevery two distinct elements of � are comparable with respect to " . Taking reflexive pointsas non-endpoints a segmentation of # � ��"� is called faithful, iff every segment of either consists of a single point or does not have a right endpoint.

Now we transfer the notion of stability (see Definition 3.2) to special Kripke models. Thismeans to the structure ��, in particular, that the truth value of a given formula has to beconstant on the set of worlds

��

for all points � � � � and segments � of any segmentation of . (Clearly, it is presupposedhere that the equivalence relation ! is defined for elements of � � in the same way as above,and �� denotes the cone determined by �.) Then, applying the decomposition technique ofSection 3 we get the subsequent analogue to Proposition 3.3.

PROPOSITION 4.9For every � �� there exists a finite faithful segmentation of on which � is stable.

This yields a finite special model falsifying a given non-derivable formula, similarly tobefore. Thus we have shown Proposition 4.7, from which decidability can be concluded in awell-known manner.

THEOREM 4.10The set of formulas of the future fragment of our temporal language of sets which are vaildin every irreflexive linear set frame is decidable.

Concluding our discussion about irreflexive flows of time we mention that the basic systemof this section can be augmented by further properties of time without affecting the validityof both, respective semantic completeness and decidability. In particular, one can treat linearset frames � � �� based on the standard flows of time �� or �� in the new context. In each case we have to add the typical axioms of the respective modaltense logic to the � -list above; e.g. along with the schema

��


expressing seriality of time, i.e. the property

� � � � � ��

the schema��

has to be added in the first case.The method of constructing finite (intermediate) models in completeness or decidability

proofs makes all the difference between usual linear tense logic and the logics of sets and timeconsidered in this paper. The method of filtration, which is common in classical modal logicsof linearity, has to be substituted by the method of segmentation, or to be suited appropriatelyto the respective situation. The technical details are laborious, as can be seen in [13] for themodal logic of continuously growing sets. We may dispense with further details here.

5 Decreasing sets

The topic of this short section is to deal with shrinking. This is the very field of the modallogic of subset frames, TR. But we can also look on spaces of descending sets from the pointof view of linear tense logic. In this way we get a new logical language which is moreexpressive in comparison with TR.

Actually, the situation of decreasing proves to be perfectly dual to that of increasing, for weget a logic of decreaseby changing the axiom schemata ��–�� from the list of Section2 into

�)�� ) � �� )��

and retaining all the other schemata. Completeness and decidability of the resulting systemcan be obtained in the same manner as above. We state these issues explicitly for the basictense logic of shrinking sets.

THEOREM 5.1The logical system determined by the schemata �)��–�)�� in place of ��–�� is soundand complete with respect to the class of descending linear subset frames. Moreover, thislogic fulfills the finite model property and, therefore, the set of formulas which are valid inall such structures is decidable.

Note that the problem of axiomatizing descending chains of sets in a sufficiently expres-sive language, which was left open in the framework of TR (cf [12]), has been ‘solved’ byTheorem 5.1 in a sense. (See also [14] for a ‘solution’ in the framework of propositionallinear time temporal logic of concurrency.)

Notice, however, that Theorem 5.1 cannot be transmitted easily to the ‘modal’ case con-sidered in Section 4. In order to succeed in this task it is very likely that one has to add newaxioms. Thus, determining the modallogic of linearly decreasing sets really remains an openproblem of TR.

6 Concluding remarks

We have introduced a basic linear tense logic of increasing sets in order to be able to modelgrowth. We have proved completeness and the finite model property of this system, and


discussed extensions by further axioms as well as variations with regard to decrease.Apart from this a formalism is desirable expressing both increasing and shrinking of sets,

and their acting in combination. We have studied a corresponding temporal system, whichcontains the usual temporal operators nexttime, henceforthand until; cf [15]. Nevertheless,we are faced with serious problems in the purely modal case.

There are some current directions of research which have not been adequately related tothe issues of this paper so far, e.g. combining logics;cf [6]. Moreover, the question of inter-pretability of our logic, for instance in some systems of [16] or [20], should be answered byfuture research.

Acknowledgement

I would like to thank the anonymous referees very much for their valuable suggestions onhow to improve the paper.

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Received 9 January 2001

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