First-Order Definability of Transition Structures

The transition semantics presented in Rumberg (J Log Lang Inf 25(1):77–108, 2016a) constitutes a fine-grained framework for modeling the interrelation of modality and time in branching time structures. In that framework, sentences of the transition language Lt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_\mathsf{t}$$\end{document} are evaluated on transition structures at pairs consisting of a moment and a set of transitions. In this paper, we provide a class of first-order definable Kripke structures that preserves Lt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_\mathsf{t}$$\end{document}-validity w.r.t. transition structures. As a consequence, for a certain fragment of Lt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_\mathsf{t}$$\end{document}, validity w.r.t. transition structures turns out to be axiomatizable. The result is then extended to the entire language Lt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_\mathsf{t}$$\end{document} by means of a quite natural ‘Henkin move’, i.e. by relaxing the notion of validity to bundled structures.


Introduction
The Prior-Thomason theory of branching time provides a perspicuous representation of the idea that the past is fixed while the future may be open.In a branching time structure, the interrelation of modality and time is depicted as a tree of moments that branches toward the future (cf.Prior 1967;Thomason 1970).At any given moment, there may be alternative possibilities for the future, and which of the alternative future possibilities will be realized depends on how the future unfolds.
Branching time structures allow for different parameters of truth, resulting in different descriptions of the interrelation of modality and time.There are two traditional semantic approaches to branching time, which Prior (1967) refers to as Peirceanism and Ockhamism.While Peirceanism makes use of a moment parameter only, in the Ockhamist semantics, truth at a moment is relativized to a history, which represents a complete possible course of events.The distinctive feature of the transition semantics presented in Rumberg (2016a) is that it builds on local future possibilities, viz.transitions, rather than on histories.Possible courses of events are modeled by sets of transitions, where each transition specifies an immediate possible future continuation at a branching point.As a consequence, incomplete possible courses of events become available as well, which can then be extended toward the future.The Ockhamist history parameter is replaced by a dynamic transition parameter.In addition to modal and temporal operators, the transition language L t is equipped with a stability operator S, which is interpreted as a universal quantifier over the possible future extensions of a given transition set and brings to the fore the idea that contingencies about the future dissolve as time progresses.The semantics on transition structures developed along those lines enables a fine-grained picture of the interrelation of modality and time and comprises both Peirceanism and Ockhamism as limiting cases, exceeding both accounts in terms of expressive strength.
In this paper, we establish axiomatizability results for the transition framework by showing that transition structures are first-order definable.Axiomatizations of branching time logics are an intricate issue.Complications arise, first and foremost, from the fact that branching time logics may involve higher-order quantification over possible courses of events (cf.Zanardo 2006b).The literature on branching time offers several results on the axiomatizability of Peirceanism and Ockhamism, whose Priorean semantics makes use of second-order quantification over histories.In Burgess (1980), Peirceanism is proven to be axiomatizable and a finite axiomatization is provided. 1he proof rests on a so-called 'Henkin move': branching time structures are endowed with a primitive set of histories, a so-called bundle; and it is shown that, in the Peircean case, validity w.r.t.bundled trees coincides with validity w.r.t.branching time structures.When it comes to Ockhamism, the situation is more complex: in the Ockhamist semantics, histories are employed as a second parameter of truth, which creates the need to unravel the structure between the indices of evaluation.The resulting Ockhamist frames are first-order definable Kripke structures with a genuine Kripe-style semantics that preserve Ockhamist validity w.r.t.bundled trees (cf.Zanardo 1985Zanardo , 1996)), which is strictly weaker than Ockhamist validity w.r.t.branching time structures (cf.Burgess 1978;Reynolds 2002).It follows that Ockhamist validity w.r.t.bundled trees is axiomatizable.A finite axiomatization of Ockhamist bundled tree validity is given in Zanardo (1985).Finding a complete axiomatization of Ockhamist validity w.r.t.branching time structures, which is known to be axiomatizable as well (cf.Burgess 1979;Gurevich and Shelah 1985), seems still an open problem.
The result we establish in the present paper parallels the Peircean and the Ockhamist cases in a certain respect.Just as Ockhamism, the transition semantics is not a genuine Kripke-style semantics: the semantics makes use of a second parameter of truth next to the moment parameter, and the language L t is equipped with intensional operators that are interpreted as quantifiers over that second parameter.The crucial difference with Ockhamism then consists in the fact that, in the transition semantics, the structural elements that are employed as a second parameter of truth are sets of transitions rather than histories; histories play only a secondary, Peircean-like role.Sets of transitions are set-theoretically rather complex, however, much more complex than histories: they are sets of pairs whose second component is a set of histories.In this paper, we provide a class of genuine Kripke structures that preserves L t -validity w.r.t.transition structures, viz. the class of so-called index structures.They are the transition-theoretic analogue of Ockhamist frames.One merit of index structures is that they evade the set-theoretic complexity of transition sets.In fact, index structures are first-order definable, which then naturally leads to axiomatizability results.
Our definition of an index structure draws on the fact that transition sets correspond one-to-one to certain substructures of branching time structures, which we call prunings.The first-order definability of index structures is essentially a result of that correspondence: the prunings of a given branching time structure are first-order definable substructures.In index structures, the semantics of all but one of the intensional operators of the transition language L t dissolves into first-order quantification over the points of the structure in accordance with the respective accessibility relations.The only exception is given by the strong future operator F, whose index semantics is a Peircean-like semantics that involves second-order quantification over histories.Then, since index structures are first-order definable, validity of F-free L t -formulas w.r.t.transition structures is axiomatizable.
Similar to the Peircean case, the complication arising from the strong future operator can be dealt with by means of a quite natural 'Henkin move', i.e. by relaxing the notion of validity to bundled structures.Both transition structures and index structure are endowed with a primitive set of histories, or a so-called bundle.We show that bundled index structures are again first-order definable and that L t -validity w.r.t.bundled transition structures is equivalent to L t -validity w.r.t.bundled index structures.As a consequence, L t -validity w.r.t.bundled transition structures turns out to be axiomatizable.
The paper is structured as follows: in Sect.2, we introduce the framework of branching time and provide a brief overview of the transition semantics presented in Rumberg (2016a).In Sect.3, we establish a bijective correspondence between transition sets and prunings.On the basis of that result, in Sect.4, we then put forth our definition of the notion of an index structure, and, in Sect.5, we show that there is a one-to-one correspondence up to isomorphism between index structures and transition structures that preserves L t -validity.Section 6 is devoted to a proof of the first-order definability of index structures and its implications with regard to axiomatizability.Finally, in Sect.7, we deal with the complication posed by the strong future operator F and generalize the results of Sects.5 and 6 to bundled structures.

Preliminaries
In this section, we introduce the Prior-Thomason theory of branching time and review the core ideas of the transition semantics.For a detailed discussion of the transition framework and its philosophical motivation, we refer the reader to Rumberg (2016a).
As said, in the theory of branching time, the modal-temporal structure of the world is represented as a tree of moments that branches toward the future.Formally, a branching time structure (or short: BT structure) is defined as a non-empty strict partial ordering of moments M = M, < that is (BT1) left-linear, (BT2) jointed and (BT3) serial.Every maximal <-linear set of moments in M is called a history.
Definition 1 (BT structure) A BT structure M = M, < is a non-empty strict partial order (i.e. a set M = ∅ together with a relation < that is irreflexive, asymmetric and transitive) s.t.
Transitions provide a local alternative to histories: whereas histories represent complete possible courses of events, a transition specifies one of the alternative immediate future possibilities open at a branching point.The definition of a transition is based on the relation of undividedness, which captures the local branching behavior of histories at a given moment.
Definition 3 (Undividedness and Branching) Given a BT structure M = M, < and a moment m ∈ M, two histories h, h ∈ hist(M) are undivided-at-m, in symbols: h ≡ m h , iff there is some m ∈ h ∩ h s.t.m > m.In case m ∈ M is the greatest element in h ∩ h , we say that h and h branch at m (in M), in symbols: h ⊥ m h (in M), and we call the moment m a branching point.
Conditions (BT1) and (BT2) of Definition 1 jointly ensure that any two distinct histories branch at some moment.Also observe that h ⊥ m h implies h = h .We can then extend the notation h ⊥ m h to moments in a natural way: we write m ⊥ m m (in M) to indicate that m > m < m is a branching triangle in M, i.e. m and m are <-incomparable and m is their greatest common lower <-bound in M.
For any moment m ∈ M, the relation of undividedness-at-m is an equivalence relation on the set H m of histories containing m and hence yields a partition of that set.We denote the ≡ m -equivalence class of a history h ∈ H m by [h] m .Note that the relation of undividedness is downward entailing: any two histories that are undivided at some moment m, are undivided at any moment m < m.In particular, for all A transition is defined as an initial-outcome pair m H consisting of a branching point m ∈ M and an equivalence class Fig. 1 The possible course of events represented by a transition set.Transitions are indicated by arrows.The possible course of events is an incomplete one, viz.one that allows for two possible future continuations.
The set of histories allowed by the transition set is {h 2 , h 3 }.
Definition 4 (Transition) For M = M, < a BT structure, a transition is a pair m, H , also written m H , where m ∈ M is a branching point and H ∈ H m / ≡ m .
Let trans (M) be the set of all transitions in M.
Sets of transitions provide a perspicuous means to represent possible courses of events in a branching time structure, as indicated in Fig. 1.We can define a natural ordering ≺ between the transitions in trans (M) in terms of proper set inclusion on the outcomes.Note that Every downward closed ≺-chain of transitions represents a possible course of events, a complete or an initial partial one.It allows certain histories to occur while it may excludes others.The set of histories allowed by a transition set is thereby nothing but the intersection over the outcomes of those transitions.There are two limiting cases: on the one hand, there is the empty set of transitions, which allows any history in M to occur; on the other hand, there are maximal ≺-linear sets of transitions, each of which excludes all but a single history.A set of transitions is called consistent iff it allows at least one history to occur.In Rumberg (2016a, Prop. 1) it is shown that consistency is equivalent to the requirement that the transition set be linearly ordered via ≺.
Definition 6 (The set of histories allowed by T) Given a BT structure M = M, < and a (possibly empty) set of transitions T ⊆ trans (M), the set of histories allowed by T is given by Given a BT structure M, the set of all consistent, ≺-downward closed transition sets in M is denoted by dcts(M).Definition 8 (The set dcts) For M = M, < a BT structure, the set of consistent, ≺-downward closed sets of transitions in M is given by: As a possibly non-maximal ≺-chain that is closed toward the past, every transition set T ∈ dcts(M) uniquely specifies a possible course of events that stretches linearly all the way from the past toward a possibly open future.We show that for all T ∈ dcts(M), the correspondence T → H(T ) is injective and reverses the inclusion relation.3 Lemma 1 Let M = M, < be a BT structure.For all T , T ∈ dcts(M), we have Proof Obviously, T = T implies H(T ) = H(T ).We prove that the converse holds as well.
The transition semantics, in its most general form, is based on transition structures.A transition structure is a BT structure M together with a non-empty set of transition sets ts ⊆ dcts(M) that covers the entire BT structure.
Definition 9 (Transition structure) A transition structure is a triple M ts = M, <, ts where M = M, < is a BT structure and ts ⊆ dcts(M) a non-empty set of transition sets s.t. for every m ∈ M, there is some Transition structures allow for great generality.Both Peirceanism and Ockhamism can be viewed as restrictions on the class of transition structures: restricting the set ts to the empty transition set yields Peirceanism, while a restriction to all maximal ≺linear transition sets yields Ockhamism (cf.Rumberg 2016a, Sect.4).The definition of a transition structure involves a so-called 'Henkin move', a standard technique in branching time logics. 4In Sect.7, we will perform a second 'Henkin move': transition structures will be endowed with a primitive set of histories.
The language L t of the transition semantics extends the standard propositional language L (with propositional variables and the usual Boolean connectives) by a past operator, P, a weak and a strong future operator, f and F, an inevitability operator and a stability operator S. The stability operator is specific to the transition approach, which allows for the relativization to incomplete possible courses of events.In the transition semantics, the truth value of a sentence at a moment can change if the transition set is extended so that it stretches further into the future.As a universal quantifier over the possible future extensions of a given transition set, the stability operator enables us to capture the peculiar behavior of the truth value of a sentence at a moment in the course of time: its changing from contingent to stably-true or stably-false.The stability operator gains its significance in the context of future contingents, whose truth values only stabilize as the future unfolds.
Sentences of L t are evaluated on a transition structure M ts = M, <, ts at pairs m/T , where m ∈ M, T ∈ ts and H(T ) ∩ H m = ∅.Given a transition structure M ts , we denote the set of indices of evaluation by Ind(M ts ).

Definition 10 (Transition
The following semantic clauses extend the valuation v t on the propositional variables p ∈ At in a transition model M ts to any arbitrary sentence φ

Transition Sets and Prunings
The transition sets that are employed as a second parameter of truth in the transition semantics are set-theoretically rather complex.From a logical point of view, however, a 4 By endowing BT structures with a primitive set of transition sets ts ⊆ dcts(M), we obtain a variation of the semantics in which unrestricted higher-order quantification over transition sets is replaced by restricted first-order quantification over the set ts.The idea goes back to Henkin (1950).The technique is prominent in branching time logics, where it forms the basis of the notion of a bundled tree (cf.Burgess 1978Burgess , 1980)).
In bundled trees, unrestricted second-order quantification over histories dissolves into restricted first-order quantification over a primitive set of histories, the so-called bundle.We will come back to the notion of a bundled tree in Sect.7 below.
set of transitions is nothing over and above the set of histories it admits.In this section, we show that sets of transitions correspond one-to-one to certain substructures of BT structures, which we call prunings.In a nutshell, a pruning of a given BT structure is a substructure of the latter that is obtained by cutting out certain branches, namely all those branches that are excluded by the corresponding transition set. 5 pruning of a BT structure M = M, < is a substructure M ⊆ M that (i) shares at least one history with M and is such that (ii) if it contains two <-incomparable moments, it contains all moments in the past and future of their greatest common lower <-bound in M as well.For any moment m ∈ M, we denote the set {m Let prun(M) be the set of all prunings of M.
In the sequel, whenever X ⊆ X and R is an order relation on X , we use X , R as an abbreviation for X , R| X .The substructure M ⊆ M in the previous definition, for instance, will then be written M , < .
Note that condition (i) of Definition 11, as stated above, is a second-order condition as it implicitly involves quantification over histories.The condition is triggered by the need to cover the case in which M consists of just a single history.There is no need, however, to formulate the condition as a second-order condition.Since the first condition is implied by the second one whenever M comprises more than one history, we can replace clause (i) in Definition 11 by the following condition without loss of generality: Conditions (i) [resp.(i')] and (ii) of Definition 11 jointly ensure that every pruning M of a BT structure M is spanned by a non-empty set of histories in M. As a consequence, prunings are themselves BT structures.
Lemma 2 Let M = M, < be a BT structure, and let M = M , < be a pruning of M. Then hist(M ) ⊆ hist(M).
Proof Straightforward by Definition 11.
Even though it is not the case in general that the intersection hist(M) of all histories in a BT structure M is non-empty, for every proper pruning M of M, we have hist(M ) = ∅.This is due to the fact that, by Definition 11, all histories in M need to be undivided at every moment at which a branch has dropped off (cf.Lemma 3).Given a BT structure M, we call the (possibly empty) intersection hist(M) the trunk of M, and we denote it by Trunk(M).
Lemma 3 Let M = M, < be a BT structure, and let M = M , < be a proper Proof Under the given assumptions, we can consider some m ∈ h \ M s.t.m < m .Assume for reductio that there is some h ∈ hist(M ) \ [h ] m .This implies that there is some m ∈ M s.t.m ≤ m and h ⊥ m h .By Definition 11 (ii) we then have It is readily verified that the pruning relation between BT structures is reflexive, antisymmetric and transitive, i.e. a partial order.When restricted to the set prun(M) of prunings of a given BT structure M, the pruning relation coincides with the substructure relation.In the remainder of this section, we show that, for any given BT structure M, there is a one-to-one correspondence between the set of transition sets dcts(M) and the set of prunings prun(M), as indicated in Fig. 2. The connecting link between sets of transitions and prunings is the set of histories admitted by a transition set.
Let M be a BT structure.We prove that for every set of transitions T ∈ dcts(M), the structure H(T ), < spanned by the set H(T ) of histories allowed by T is a pruning of M with hist( H(T ), < ) = H(T ).The correspondence between transition sets and prunings obtained along those lines is injective and reverses the inclusion relation: whenever T is a proper extension of T , the structure H(T ), < is a proper pruning of H(T ), < , and vice versa.
Proposition 1 Let M = M, < be a BT structure, and consider the function χ with Dom(χ ) = dcts(M) defined by The following holds: Proof (i) Take any T ∈ dcts(M).We first show that hist(χ (T )) = H(T ).Obviously, H(T ) ⊆ hist(χ (T )).We prove that hist(χ (T )) ⊆ H(T ) holds as well.Assume for reductio that there is some h ∈ hist(χ (T )) \ H(T ).This implies that there is some , which shows that condition (ii) of Definition 11 is satisfied.Conditions (ii) and (iii) are immediate consequences of Lemma 1 given the equality hist(χ (T )) = H(T ) established in (i).
We now turn to the converse correspondence: we show that every pruning M of M in turn determines a transition set T in dcts(M) with H(T ) = hist(M ), viz. the set of all transitions whose outcomes include the set hist(M ).
Proposition 2 Let M = M, < be a BT structure, and consider the function ξ with Dom(ξ ) = prun(M) defined by (3.2) The following holds: Proof (i) Take any M ∈ prun(M).It is straightforward that ξ(M ) ∈ dcts(M), and obviously hist(M ) ⊆ H(ξ(M )).We prove that H(ξ(M )) ⊆ hist(M ) holds as well.Two cases can be considered.Case (i): Consequently, ξ(M ) = ∅ because there is no transition m H ∈ trans (M) s.t.hist(M) ⊆ H .Then, since H(∅) = hist(M), we have H(ξ(M )) = hist(M ).Case (ii): Suppose M = M, and assume for reductio that there is some h ∈ H(ξ(M )) \ hist(M ).Let h ∈ hist(M ), and suppose h ⊥ m h in M. By Lemma 3 it follows that hist(M ) ⊆ [h ] m .This implies that m [h ] m ∈ ξ(M ), which contradicts the assumption that h ∈ H(ξ(M )).Conditions (ii) and (iii) follow immediately from Lemma 1 on the basis of the equality By combining the results established in Propositions 1 and 2 above, we obtain a one-to-one correspondence between the set of transition sets dcts(M) and the set of prunings prun(M) in a given BT structure M.
Proposition 3 For any BT structure M = M, < , the functions χ and ξ are bijections and inverses of each other.

Index Structures
Drawing on the correspondence between transition sets and prunings established in Sect.3, we now put forth the definition of an index structure.Unlike transition structures, index structures are genuine Kripke structures: in an index structure, sentences of the transition language L t are evaluated at the points of the structure, and quantification over transition sets dissolves into restricted first-order quantification over that set.
An index structure is a Kripke structure W = W , , ∼, consisting of a nonempty set W and three primitive relations: , ∼ and .A fourth relation ≈ on W can be defined along the following lines: for all w, w ∈ W , w ≈ w iff there is some z ∈ W s.t.w z w .⊆ ∼; (vi) for all w, w ∈ W , there is some x ∈ [w] ≈ and there is some x ∈ [w ] ≈ s.t.(i) x w and x w , (ii) x ∼ x and (iii) for all y, y ∈ W s.t.y w, y w and y ∼ y , we have y x and y x ; (vii) for all w, w ∈ W , the intersection ] and (iii) for all x, x , x ∈ [w] ≈ , x ⊥ x x implies x / ∈ Dom( f w,w ), and for all y, y , y ∈ [w ] ≈ , y ⊥ y y implies y / ∈ Im( f w,w ).
In an abstract sense, an index structure is but a tree of trees: it is a tree-like arrangement of disjoint BT structures, as shown in  In the following lemmas, we state some results about index structures that will become important in the remainder of the paper.Lemma 5 For W = W , , ∼, an index structure and w, w ∈ W , we have either Then w ∈ Dom( f w,w ).Assume for reductio that [w] ∼ ∩ [w ] ≈ ⊇ {v, u}.This implies that f w,w (w) = u and f w,w (w) = v, which contradicts the fact that f w,w is a function.
Lemma 6 Let W = W , , ∼, be an index structure.For every w ∈ W , the structure [w] ≈ , is a BT structure.
Proof By Definition 12 (ii), the relation is a left-linear and serial strict partial order.The jointedness of follows from Definition 12 (vi) on the basis of Lemma 5.
Lemma 7 For W = W , , ∼, an index structure and w, w ∈ W , the following holds: Proof (i) Assume w w, and let x ∈ [w] ∼ .Then w ∈ Dom( f w,x ) and f w,x (w) = x.By Definition 12 (vii), the function f w,x is an -isomorphism whose domain is downward closed.Consequently, w ∈ Dom( f w,x ) and f w,x (w ) x. Let x be f w,x (w ).
(ii) The case w = w is trivial: let x be x.Now assume w w, and let x ∈ [w] ≈ .Then, by Definition 12 (vii.a), In an index structure W = W , , ∼, , sentences of the transition language L t are evaluated at the elements of W .The temporal operators P, f and F shift the index of evaluation along the relation , the inevitability operator shifts the index of evaluation along the relation ∼ and the stability operator S along the relation .While the operators P, f, and S are genuine Kripke-style operators, the strong future operator F is not: its index semantics is a Peircean-like semantics that involves secondorder quantification over histories.This is to say, the semantics of the strong future operator F does not reduce to the relations , ∼ and on W .We will deal with that complication in Sect.7 below. 9 Definition 13 (Index model) An index model is a quintuple W = W , , ∼, , v i where W = W , , ∼, is an index structure and v i : At × W → {0, 1} a valuation function.
The following semantic clauses extend the valuation v i on the propositional variables p ∈ At in an index model W to any arbitrary sentence φ ∈ L t : (At) W, w i p iff v i ( p, w) = 1; 9 Unlike the strong future operator F, the operator F considered in Rumberg (2016a, Sect. 3.2) is a genuine Kripke-style operator in the index semantics.In the case of the F -operator, quantification over histories is replaced by quantification over the possible future extensions of the given transition set.We have W, w i F φ iff for all w ∈ W s.t.w w, there is some x w s.t. for all x ∈ W s.t.x x and x w, W, x i φ.
(¬) W, w i ¬φ iff W, w i φ; (∧) W, w i φ ∧ ψ iff W, w i φ and W, w i ψ; (P) W, w i Pφ iff there is some w ∈ W s.t.w w and W, w i φ; (f) W, w i fφ iff there is some w ∈ W s.t.w w and W, w i φ; (F) W, w i Fφ iff for all h ∈ hist( [w] ≈ , ) s.t.w ∈ h, there is some w ∈ h s.t.w w and W, w i φ; ( ) W, w i φ iff for all w ∼ w, W, w i φ; (S) W, w i Sφ iff for all w w, W, w i φ.

Transition Structures and Index Structures
In this section, we show that there is a one-to-one correspondence up to isomorphism between transition structures and index structures that preserves L t -validity. 10The proof proceeds in several steps: we first show that to every transition structure, there naturally corresponds an index structure (cf.Definition 14 and Theorem 1).We then establish the converse correspondence (cf.Definition 15 and Theorem 2).Finally, we show that the two correspondences are inverses of each other up to isomorphism (cf. Theorem 3) and that L t -validity is preserved (Proposition 4 and Theorem 4).
Let M ts = M, <, ts be a transition structure.We show that M ts determines an index structure Ind(M ts ), , ∼, on the set Ind(M ts ) of indices of evaluation in M ts , where the relations , ∼ and are defined in terms of the relations < on M and ⊆ on ts between moments and transition sets, respectively.(5.1) A verification of that claim will be provided within the proof of following theorem.
Theorem 1 For every transition structure M ts = M, <, ts , the structure λ(M ts ) is an index structure.
Proof It is readily verified that the structure λ(M ts ) = Ind(M ts ), , ∼, fulfills conditions (i)-(vi) of Definition 12. Condition (i) is guaranteed by the definition of a transition structure (Definition 9).Condition (ii) follows immediately from the corresponding properties of the relation < on M (Definition 1), while condition (iv) is a consequence of the properties of the inclusion relation ⊆ on ts.Conditions (iii) and (v) are straightforward by definition.Condition (vi) follows from the jointedness of the relation < on M (Definition 1), taking into account that for all T ∈ ts, if m < m and m/T ∈ Ind(M ts ), then m /T ∈ Ind(M ts ).It remains to be shown that λ(M ts ) satisfies condition (vii) of Definition 12.
As an intermediate step, we establish some results about the defined relation ≈ on Ind(M ts ).In the present context, (4.1) reads: m/T ≈ m /T iff there is some m /T ∈ Ind(M ts ) s.
) is an injective and -preserving function, and we denote it by f m/T , m /T .The first part of condition (vii) thus holds for λ(M ts ).
We now turn to condition (vii.a).Assume

and for all h
Concerning the converse correspondence, we show that, given an index structure W = W , , ∼, , we can lift a BT structure W /∼, from the ∼-equivalence classes in W and extract a set of transition sets ts(W /≈) in that BT structure from the ≈-equivalence classes in W .
Definition 15 For any index structure and where ξ is the function defined in (3.2) and σ : W → W /∼ is defined by (5.4) For every [w] ≈ ∈ W /≈, the structure σ ([w] ≈ ), will be denoted by T [w] ≈ and the corresponding element ξ( σ Being defined in terms of quantification over the elements of ∼-equivalence classes, the relation is clearly well-defined.It will be shown below that the structure W /∼, is in fact a BT structure [cf.(5.6)] and that for every ) [cf. (5.7)].This warrants the application of the function ξ , which then for every [w] ≈ ∈ W /≈, yields a transition set [cf. (5.11)].
An auxiliary relation on W /≈ can be defined along the following lines: (5.5) Note that by Definition 12, we have [w] ≈ [w ] ≈ iff there is some x ∈ [w] ≈ and some x ∈ [w ] ≈ s.t.x x .Just as the relation on W /∼, the relation on W /≈ is obviously well-defined.Whereas the relation on W /∼ represents the temporal ordering on the set of moments W /∼, the relation on W /≈ induces an ordering among the various transition sets in ts(W /≈), as we shall see.Recall that by Lemma 7, the following holds: Theorem 2 For every index structure W = W , , ∼, , the structure ζ(W) is a transition structure.
Proof We first show that W /∼, is a BT structure. (5.6) Obviously, W /∼ = ∅.That the relation is a left-linear and serial strict partial order on W /∼ follows from the corresponding properties of the relation on W [Definition 12 (ii)] on the basis of Lemmas 5 and 7 (i).The jointedness of the relation is a straightforward consequence of Definition 12 (vi) and Lemma 7 (i).
We now prove that ts(W /≈) yields a set of transition sets in the BT structure W /∼, , i.e. ts(W /≈) ⊆ dcts( W /∼, ).To this end, we show that for every (5.7) By Lemma 5, it is straightforward that σ | [w] ≈ is injective and order-preserving: for all x, x ∈ [w] ≈ , x x implies σ (x) σ (x ).Then, by Lemma 6, T [w] ≈ ⊆ W /∼, is a BT structure.Moreover, on the basis of Lemmas 5 and 7 (ii), from Definition 12 (iv) it follows that the relation on W /≈ is a left-linear partial order, and, by Definition 12 (vii), (5.8) In order to prove (5.7), we establish the following two claims.For every maximal (5.9) (5.10) Proof of (5.9).Consider any history h in T [w] ≈ , and let h be any history in  [z] ∼ .This shows that [x] ≈ ∈C T [x] ≈ and W /∼, also fulfill condition (ii) of Definition 11.From (5.9) and (5.10), the claim in (5.7) follows immediately by the transitivity of the pruning relation.On the basis of Proposition 2 (i), (5.7) implies that for all [w] ≈ ∈ W /≈, (5.11) The proof can now be concluded by observing that for every [w] ∼ ∈ W /∼, H(T [w] ≈ )∩ H [w] ∼ = ∅, which is a consequence of (5.11).
The correspondences λ and ζ established in Definitions 14 and 15 above are inverses of each other up to isomorphism.Before we turn to a proof of that claim, however, we show that the correspondence ζ between index structures and transition structures induces a bijection between the sets of indices of evaluation of the respective structures.
Lemma 8 For any index structure W = W , , ∼, , the function ν : is a bijection, and for all w, w ∈ W : Proof We show that the function ν is a bijection.The injectivity of ν is a straightforward consequence of Lemma 5 and the following claim: for all (5.13)By Theorem 3, every transition structure is isomorphic to the ζ -image of some index structure.On the basis of Lemma 8, the correspondence ζ between index structures and transition structures can be extended in a natural way to a correspondence between models.For every index model (5.17) We show that the correspondence ζ preserves L t -validity.
Proposition 4 Let W = W, v i be an index model on the index structure W = W , , ∼, .Then for every φ ∈ L t and every w ∈ W , Proof The proof runs by induction on the structure of φ and rests on the induction hypothesis that the claim holds for any proper subformula ψ of φ.Given the correspondence in (5.17), the base clause is straightforward.We restrict ourselves here to the case for the strong future operator F, all other cases being similar and simpler."⇒": Assume W, w i Fψ.
The following formula states that the domain of f w,w is given by [w] ≈ : In order to give expression to the idea that Im( f w,w ), is a pruning of [w ] ≈ , , we distinguish two cases, viz. the case in which [w] ≈ , consists of just a single history and the case in which [w] ≈ , contains a branching point.That is, clause (FO.6) corresponds to the modification (i') of condition (i) of Definition 11 and clause (FO.7) to condition (ii) of that definition.
Let us now consider clause (vii.b), which deals with the case in which ∩ ( In that case, f w,w is not only required to be not onto, but the domain of f w,w must also be properly included in [w] ≈ .What is more, both Dom( f w,w ) and Im( f w,w ) are required to be -downward closed subsets of the trunks of [w] ≈ , resp.
[w ] ≈ , that do not contain a branching point.The fact that Dom( f w,w ) is a proper subset of [w] ≈ can easily be expressed by The following formula expresses the idea that both the domain and the image of f w,w are subsets of the trunks of the respective -trees: Finally, neither Dom( f w,w ) nor Im( f w,w ) may contain a branching point in [w] ≈ , resp.

123
In this section, have proven that index structures are first-order definable.Recall that on index structures, the semantic clauses for the intensional operators of the transition language L t -with the exception of the one for F-involve only first-order quantification over the set W (cf. Definition 13).Moreover, by Theorem 4, L t -validity w.r.t.transition structures is equivalent to L t -validity w.r.t.index structures.As a consequence, validity of F-free L t -formulas w.r.t.transition structures is axiomatizable.
Theorem 5 The set of F-free L t -validities w.r.t.transition structures is recursively enumerable and hence, by Craig's theorem, axiomatizable.
Proof Follows from Definition 13 by Theorem 4 and Proposition 5.

Bundled Structures
In this section, we generalize the results established in Sects.5 and 6 via a 'Henkin move' to so-called bundled structures.Rather than considering plain transition structures, we consider transition structures that are endowed with a primitive set of histories that is such that it covers the entire BT structure.A set of histories that fulfills the given requirement is called a bundle (cf.Burgess 1978Burgess , 1980)).The corresponding index structures will then have to be bundled structures as well, in a sense to be specified below.
The need of generalization is triggered by the strong future operator F, which involves second-order quantification over histories.In bundled structures, quantification over histories dissolves into first-order quantification over the elements of the bundle.We show that L t -validity w.r.t.bundled transition structures is equivalent to L tvalidity w.r.t.bundled index structures and that bundled index structures are first-order definable.Hence, L t -validity w.r.t.bundled transition structures turns out to be axiomatizable.
When it comes to the definition of bundled transition structures, some care is needed in order to avoid conflicts between the primitive set of histories and the primitive set of transition sets.
Definition 17 (Bundled transition structure) A bundled transition structure is a quadruple M ts B = M, <, ts, B where M ts = M, <, ts is a transition structure and B is a bundle on M, < s.t.(i) for every h ∈ B, there is some T ∈ ts s.t.h ∈ H(T ) and (ii) for every T ∈ ts, H(T ) ∩ B = ∅.
The above definition guarantees that every history of the bundle B is compatible with at least one transition set in ts.Moreover, the definition rules out the possibility that the bundle delimits the set of histories allowed by the transition parameter to the empty set.In a bundled transition structure M Since moving to bundled structures does not affect the set of indices of evaluation, on the basis of Theorem 6, the correspondence ζ between index models and transition models provided in (5.17) straightforwardly extends to bundled structures.We show that the extension preserves L t -validity.As said, the need of generalization to bundled structures was triggered by the strong future operator F, whose index semantics involves second-order quantification over histories.By Proposition 9, in bundled index structures, the truth conditions for F can now be expressed as follows: We then have the following theorem: With those results in place, the obvious next step for future work is to provide an explicit list of axioms and derivation rules that is complete for validity w.r.t.(bundled) index structures.A chronicle construction on index structures that proceeds by the elimination of counterexamples is expected to prove successful to this end.
Another important issue concerns the question whether validity w.r.t.bundled transition structures is properly weaker than validity w.r.t.plain transition structures.The 'Henkin move' to bundled structures was triggered by the need to avoid the Peirceanlike second-order quantification over histories in the index semantics for the strong future operator F. In Burgess (1980) it is shown that in the Peircean case, bundled tree validity coincides with validity w.r.t.plain BT structures.Even though BT structures are much less complex than index structures, investigating the possibility of transferring Burgess' result to our case seems worthwhile.
Definition 11 is trivially fulfilled by M and M : by Lemma 2, every h ∈ hist(M ) is a history in M and hence belongs to hist(M ) because M ⊆ M ⊆ M. Now assume that there are m , m ∈ M s.t.m ⊥ m m in M. By Definition 11 (ii), M m ⊆ M , which implies M m ⊆ M because M ⊆ M. Then M and M satisfy condition (ii) of Definition 11 as well.

Fig. 2
Fig. 2 One-to-one correspondence between transition sets and prunings.(a) M with transition set T , (b) Pruning M of M. The BT structure M with hist(M ) = {h 2 , h 3 } is a pruning of M. It corresponds one-to-one to the transition set T in dcts(M) with H(T ) = {h 2 , h 3 }.We have M = H(T ), < and T = { m H ∈ trans (M) | hist(M ) ⊆ H }.
(4.1)For any w ∈ W , we denote the set {w ∈ W | w ≈ w } by [w] ≈ .The notation suggests that ≈ is an equivalence relation, and we will see below that this in fact the case [cf.(4.2)].We state the definition of an index structure outright and then discuss its implications.Definition 12 (Index structure) An index structure is a quadruple W = W , , ∼, s.t.(i) W = ∅; (ii) the relation is a left-linear and serial strict partial order on W ; (iii) the relation ∼ is an equivalence relation on W , and we denote the ∼-equivalence class of an element w ∈ W by [w] ∼ ; (iv) the relation is a left-linear partial order on W ;(v)

Fig. 3 .Fig. 3
Fig. 3 An index structure.Endowed with the order relation ⊆ ≈, each equivalence class in W /≈ forms a BT structure, and the internal relation ⊆ ∼ on the equivalence classes in W /∼ defines a tree-like ordering beween those BT structures.

7
Condition (vii.b.i) is needed to cover the limiting case in which [w] ≈ , or [w ] ≈ , consist of just a single history and hence are identical to their trunks.In case both structures consist of more than a single history, condition (vii.b.i) is implied by condition (vii.b.ii). 8If [w] ≈ , and [w ] ≈ , are -incomparable and there is some w ∈ W s.t.both the intersections ∩ ([w] ≈ × [w ] ≈ ) and ∩ ([w ] ≈ × [w ] ≈ ) are non-empty, condition (vii.b.iii) follows from condition (vii.a).

Definition 14
For any transition structure M ts = M, <, ts , let λ(M ts ) be the structure Ind(M ts ), , ∼, with ( ) m/T m /T iff T = T and m < m ; (∼) m/T ∼ m /T iff m = m ; ( ) m/T m /T [m/T = m /T ] iff m = m and T ⊂ T [T = T ].In the context of Definition 14, the relation ≈ defined in (4.1) captures sameness of transition set, i.e. for all m/T , m /T ∈ Ind(M ts ), m/T ≈ m /T iff T = T .

Proposition 8
Let W B = W B , v i be an index model on the bundled index structure W B = W , , ∼, , B .Then for every φ ∈ L t and every w ∈ W ,W B , w b i φ iff ζ(W B ), [w] ∼ /T [w] ≈ b t φ.obvious candidate for the history represented by an element b ∈ B is h b = {w ∈ W | w ε b}. 13 Proposition 9 Let W B = W , B, , ∼, , ε be a model of axioms (FO.0)-(FO.10)and (FO.B.0)-(FO.B.2), and let B be {h b | b ∈ B}.Then W B = W , , ∼, , B is a bundled index structure.Proof Straightforward.