Exploring the Jungle of Intuitionistic Temporal Logics 3

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arXiv:1912.12895v2 [cs.LO] 12 Mar 2021 tsol o esrrsn htcmiain fitiinsi oi n lin and logic cr intuitionistic a of play combinations phenomena that surprising these be which nat not in a should it processes it making dynamic topology, about or information reason computation, on based tions nutoitclgc( logic Intuitionistic ne osdrto o ulcto nTer n Practice and Theory in publication for consideration Under n,uigorsudesrsls hwta h axiomatical Practic and the Theory that in show consideration Under results, relations. b soundness order on our based using logics and, semantically-defined classic the the between of structu variants i relation intuitionistic of distinct natural class two are provide a which We for of systems. sound topological is dynamic systems intui on these or for of systems each pro that axiomatic logic show several for persp consider semantics semantics We several the framework. and from systems while dynamical theory, about the type f From extend via years. to features possible few new it last made have the logics in temporal itionistic clear Comput increasingly in logics become temporal has intuitionistic of importance The ∗ hsrsac a upre yteSca cecsadHuman and Sciences Social the by supported was research This xlrn h ugeo nutoitcTemporal Intuitionistic of Jungle the Exploring umte aur 03 eie aur 03 cetd1 accepted 2003; January 1 revised 2003; January 1 submitted eateto hlspy nvriyo oot,Toronto, Toronto, of University Philosophy, of Department eateto ahmtc,GetUiest.Get Belgi Ghent, University. Ghent Mathematics, of Department IL ..(etn 90 it 00 nosamra finterpreta- of myriad a enjoys 2000) Mints 1930; (Heyting e.g. ) ( e-mail: EI.Uiest fAgr.Agr,France. Angers, Angers. of University LERIA. RT olueUiest,Tuos,France. Toulouse, University, Toulouse IRIT, ( e-mail: ( AI FERN DAVID ( e-mail: e-mail: [email protected] [email protected] MART OEHBOUDOU JOSEPH HLPKREMER PHILIP [email protected] Introduction 1 [email protected] Logics Abstract ´ NDI IN ANDEZ-DUQUE ´ EGUEZ ´ loe ecmltl sals h order the establish completely We one. al fLgcProgramming Logic of ntoa rgamn agae with languages programming unctional ∗ fLgcPormig(TPLP). Programming Logic of e rSineadAtfiilIntelligence Artificial and Science er t nepeain f‘henceforth’, of interpretations oth trrttoso hneot’ both ‘henceforth’, of nterpretations insi iertmoa oi and logic temporal linear tionistic te eerhCuclo Canada. of Council Research ities rmighv hi ot nthis in roots their have gramming e ae ihro rpeframes Kripke on either based res ciesvrllgc o reasoning for logics several ective ydfie oisejytesame the enjoy logics ly-defined ro-hoypito iw intu- view, of point proof-theory ) ) ) ) aur 2003 January Canada. um. rlfaeokto framework ural ca oe Thus, role. ucial a temporal ear 1 2 Boudou et al. logic (LTL) (Pnueli 1977) have been proposed for applications within several different contexts:

Types for functional programming languages. The Curry-Howard correspondence identi- ﬁes intuitionistic proofs with the λ-terms of functional programming (Howard 1980). Sev- eral extensions of the λ-calculus with operators from LTL have been proposed in order to introduce new features to functional programming languages: (Davies 1996; Davies 2017) has suggested adding a ‘next’ (#) operator to IL in order to deﬁne the type system λ#, which allows extending functional programming languages with staged computa- tion1 (Ershov 1977). (Davies and Pfenning 2001) proposed the functional programming language Mini-ML2 which is supported by intuitionistic S4 and allows capturing complex forms of staged computation as well as runtime code generation. (Yuse and Igarashi 2006) later extended λ# to λ2 by incorporating the ‘henceforth’ operator (2), useful for modelling persistent code that can be executed at any subsequent state.

Semantics for dynamical processes. Intuitionistic temporal logics have been proposed as a tool for modelling semantically-given processes. (Maier 2004) observed that an intuitionistic temporal logic with ‘henceforth’ and ‘eventually’2 (3) could be used for reasoning about safety and liveness conditions in possibly-terminating reactive systems, and (Fern´andez-Duque 2018) has suggested that a logic with ‘eventually’ can be used to provide a decidable framework in which to reason about topological dynamics.

Temporal answer set programming. In the areas of nonmonotonic reasoning, knowledge representation (KR), and artificial intelligence, intuitionistic and intermediate logics have played an important role within the successful answer set programming (ASP) (Brewka et al. 2011) paradigm for KR, leading to several extensions of modal ASP (Cabalar and Pérez Vega 2007) that are supported by intuitionistic-based modal logics like temporal here and there (Balbiani and Diéguez 2016).

Despite interest in the above applications, there is a large gap to be filled regarding our understanding of the computational behaviour of intuitionistic temporal logics. We have successfuly employed semantical methods to show the decidability of the logic ITLe defined by a natural class of Kripke frames (Boudou et al. 2017) and shown that these semantics correspond to a natural calculus over the 2-free fragment (Diéguez and Fernández-Duque 2018). However, as we will see, in the presence of 2, new validities arise which may be undesirable from the point of view of an extended Curry-Howard isomorphism. Thus, our goal is to provide semantics for weaker axiomatically-defined intuitionistic temporal logics in order to provide tools for understanding their computational behaviour. We demonstrate the power of our semantics by separating several natural axiomatically-given calculi, which in particular answers in the negative a conjecture of (Yuse and Igarashi 2006) that the Gentzen-style and the Hilbert-style calculi presented there prove the same set of formulas.

1 Staged computation is a technique that allows dividing the computation in order to exploit the early availability of some arguments. 2 In this paper, ‘eventually’ should be understood as ‘occurring at least once, either now or in the future,’ while ‘henceforth’ should be understood as ‘from now on.’ Exploring the Jungle of Intuitionistic Temporal Logics 3

There have already been some notable efforts towards a semantical study of intuitionistic temporal logics. (Kojima and Igarashi 2011) endowed Davies’s logic with Kripke semantics and provided a complete deductive system. Bounded-time versions of logics with henceforth were later studied by (Kamide and Wansing 2010). Both use semantics based on Simpson’s bi-relational models for intuitionistic modal logic (Simpson 1994). Since then, (Balbiani and Diéguez 2016) have shown that temporal here-and-there is decidable and enjoys a natural axiomatization. Topological semantics for intuitionistic modal and tense logics have also been studied by (Davoren 2009; Davoren et al. 2002), and (Kremer 2004) suggested a topologically-defined intuitionistic variant of LTL with # and an intuitionistic variant of ‘henceforth’, which we will denote by . Kremer’s pro- posal was never published, and is presented from the axiomatic perspective in Section 2 and its semantics in Section 3. The decidability of the logic of the weak semantics remains open, but (Fernández-Duque 2018) has shown that a similar logic with ‘eventually’ 3 instead of 2 is decidable.

In this paper we lay the groundwork for an axiomatic treatment of intuitionistic linear temporal logics. We will introduce a ‘basic’ intuitionistic temporal logic, ITL32, defined by adding standard axioms of LTL to intuitionistic modal logic (see Section 2 for details). We also consider additional Fischer Servi axioms (FS), a ‘constant domain’ axiom CD := 2(p∨q) → 2p∨3q, and a ‘conditional excluded middle’ axiom CEM := (¬#p∧#¬¬p) → (#q ∨¬#q). Combining these, we obtain seven intuitionistic temporal logics. Logics with the constant domain axiom are sound for their Kripke semantics, given by the class of dynamical systems based on a poset, also called expanding posets. In the setting of Kripke semantics, the Fischer Servi axioms correspond to backwards-confluence of the transition function. The constant domain axiom is not derivable from the others, and to show this, we will consider topological semantics for intuitionistic temporal logic. This is in contrast to the setup in (Balbiani et al. 2019), where all logics are based on expanding posets and hence validate CD. The crucial difference between working with expanding posets vs. topological semantics is that the standard interpretation of ‘henceforth’ in classical LTL may readily be applied to the setting of expanding posets, whereas in the topological setting, it requires some modification. Recall that topological spaces are pairs (X, T ), where T is a collection of subsets of X closed under unions and finite intersections (see Section 4). Elements of T are called open sets. In the topological semantics of intuitionistic logic, each proposition ϕ must be interpreted as an open set ϕ ⊆ X. In order to interpret tenses, we equip (X, T ) with a continuous function S : XJ K→ X. The classical semantics for next and eventually yield well-defined operations in this setting: for example, we define #ϕ = S−1 ϕ , which amounts to the standard definition where x ∈ #ϕ iff S(x) ∈ ϕJ . TheK continuityJ K of S ensures that #ϕ is an open set whenever ϕJ isK (recall thatJ byK definition, S is continuous iff preimagesJ K of open sets are open). Similarly,J K setting x ∈ 3ϕ iff there is n ≥ 0 such that Sn(x) ∈ ϕ ensures that 3ϕ will always be open. J K However, the classicalJ definitionK of 2ϕJ wouldK have that x ∈ 2ϕ iff Sn(x) ∈ ϕ 2 J K J K J K for all n ≥ 0 or, equivalently, ϕ = n≥0 ϕ . The problem is that open sets need not be closed under infinite intersections,J K so anJ intuitionisticK interpretation for 2ϕ must T modify the classical semantics in a way that only open sets are produced. There are at 4 Boudou et al. least two ways to achieve this. We call these the ‘weak’ and ‘strong’ interpretations of 2. The first was originally proposed by (Kremer 2004) in an unpublished note, and is treated similarly to the universal quantifier in the context of intuitionistic semantics of first order logic. In order to distinguish it from the strong interpretation, we will denote it by .3 As we will see, the operator does not satisfy some key LTL validities, namely p →# p, #p → # p, and p → p. Consequently, some of the standard LTL axioms are not sound for this interpretation. We thus propose a logic ITL3, where the axiom p → # p is replaced by the weaker p → #p. Nevertheless, 2p → #2p is arguably one of the defining axioms for henceforth, so it is convenient to have semantics that validate it. In order to obtain semantics for ITL32, we propose a new interpretation for 2. Our approach is natural from an algebraic perspective, as we define the interpretation of 2ϕ via a greatest fixed point in the Heyting algebra of open sets.4 We will show that dynamic topological systems provide semantics for the logics without the constant domain axiom, from which we conclude the independence of the latter. Moreover, we show that the Fischer Servi axioms are valid for the class of open dynamical topological systems, and that in this setting, the semantics for and 2 coincide. The constant domain axiom shows that the {3, 2}-logic of expanding posets is different from that of dynamic topological systems. We show via an alternative axiom that the {#, 2}-logics are also different. We also consider the special case where topological semantics are based on Euclidean spaces. We show that this leads to logics strictly between that of all spaces and that of expanding posets. In the special case of the real line, we can prove that every formula falsified on a persistent poset is falsifiable on the real line. Layout. Section 2 introduces the syntax and the axiomatic systems as well as its weak counterparts that we propose for intuitionistic temporal logic. Section 3 reviews dynamic topological systems, which are used in Section 4 to provide semantics for our formal language. Section 5 shows that four of our logics and their weak companions are each sound for a class of dynamical systems, and Section 6 shows that the remaining logics are sound for Euclidean spaces. In Section 7 we focus on ITL32 interpreted on persistent posets and its connection with the real line. In Section 8 we show that several of the logics we consider are pairwise distinct. Finally, Section 9 lists some open questions.

2 Syntax and axiomatics In this section we will introduce several natural intuitionistic temporal logics. Most of the axioms we consider have appeared either in the intuitionistic logic, the temporal logic, or the intuitionistic modal logic literature. They will be based on the language of linear temporal logic, as deﬁned next. Fix a countably inﬁnite set P of propositional variables. The full language L32 of intuitionistic (linear) temporal logic ITL is given by the grammar in Backus-Naur form ϕ, ψ := ⊥ | p | ϕ ∧ ψ | ϕ ∨ ψ | ϕ → ψ | #ϕ | 3ϕ | 2ϕ | ϕ, where p ∈ P. As usual, we use ¬ϕ as a shorthand for ϕ →⊥ and ϕ ↔ ψ as a shorthand

3 (Kremer 2004) instead uses ∗. 4 We will not discuss Heyting algebras in this text, but see e.g. (Heyting 1930; Mints 2000). Exploring the Jungle of Intuitionistic Temporal Logics 5

for (ϕ → ψ) ∧ (ψ → ϕ). We read # as ‘next’, 3 as ‘eventually’, 2 as ‘strong henceforth’ and as ‘weak henceforth’. The intuition is that formulas are evaluated at moments of discrete time. The formula #ϕ indicates that ϕ will hold at the next moment, 3ϕ that it will hold in some subsequent moment, and 2ϕ and ϕ both indicate that ϕ will hold in every subsequent moment, including the current moment. However, as we will see, making sense of the latter notion in intuitionistic semantics is not straightforward, thus giving rise to two natural, but distinct, interpretations. Given any formula ϕ, we denote the set of subformulas of ϕ by sub(ϕ). For Θ ⊆ {3, 2, }, the language LΘ is the sub-language of L32 whose only tenses are # and those in Θ; we will not consider languages without #. So, for example, L3 only has tenses # and 3. We will write L◦ instead of L∅. The tenses , 2 represent two possible intuitionistic readings of ‘henceforth’ and thus we will rarely consider logics with both. In order to compare logics based on with logics based on 2, we introduce the translations t, where t(ϕ) ∈ L3 is the formula obtained by replacing every occurrence of 2 in ϕ by , and similarly define t2, which replaces every occurrence of by 2. The semantics for first appeared in the unpublished note (Kremer 2004), while those for 2 were first introduced in a preliminary version of this paper (Boudou et al. 2019). We begin by establishing our basic axiomatization for logics over L32. It is obtained by adapting the standard axioms and inference rules of LTL (Lichtenstein and Pnueli 2000), as well as their dual versions.

Deﬁnition 2.1 The logic ITL32 is the least set of L32-formulas closed under the following axioms and rules.

(i) All intuitionistic tautologies; (ix) 2ϕ → #2ϕ; (ii) ¬#⊥; (x) ϕ → 3ϕ; (iii) # (ϕ ∧ ψ) ↔ (#ϕ ∧ #ψ); (xi) #3ϕ → 3ϕ; (iv) # (ϕ ∨ ψ) ↔ (#ϕ ∨ #ψ); (xii) 2(ϕ → #ϕ) → (ϕ → 2ϕ); (v) # (ϕ → ψ) → (#ϕ → #ψ); (xiii) 2(#ϕ → ϕ) → (3ϕ → ϕ); ϕ ϕ → ψ (vi) 2 (ϕ → ψ) → (2ϕ → 2ψ); (xiv) ; ψ (vii) 2 (ϕ → ψ) → (3ϕ → 3ψ); ϕ ϕ (xv) , . (viii) 2ϕ → ϕ; #ϕ 2ϕ

Axioms (v) and (vi) hold in any normal modal logic and (vii) is a dual version of (vi); such dual axioms are often needed in intuitionistic modal logic, since 3 and 2 are not typically inter-deﬁnable. The axioms (ii)-(iv) have to do with the passage of time being deterministic in linear temporal logic, and are related to a functional modality, i.e. a modality that is interpreted using a function rather than a relation. The axioms (viii) and (x) have to do with future tenses being interpreted reﬂexively, i.e. ϕ is considered to hold eventually if it holds now. The axiom (xi) states that if something will henceforth be the case, then in the next moment, it will still henceforth be the case, and (xii) is successor induction, as time is interpreted over the natural numbers. Axioms (xi) and (xiii) are their duals. All rules are standard in any normal modal logic. Each axiom is either included in the axiomatization of Goldblatt (Goldblatt 1992, page 6 Boudou et al.

87) or is a variant of one of them (e.g., a contrapositive); this is standard in intuitionistic modal logic, as such variants are needed to account for the independence of the basic connectives. We do not consider ‘until’ and ‘release’ in this paper, but these operators have previously studied within an intuitionistic context in (Balbiani et al. 2019). Next we deﬁne our base logic for weak henceforth. It is convenient to present it as a logic over L32 and then translate to L3. The main reason for this is that we view the weak and strong semantics of henceforth as two possible interpretations of intuitionistic temporal logic, rather than two independent tenses. From this point of view, the notation should be seen as an indication that weak semantics are being used. Moreover, we are interested in comparing logics based on with those based on 2, and uniform notation will be helpful for this. In particular, we will see that the weak semantics give rise to weaker logics, partially motivating the terminology. We will then use the translation t (which, recall, replaces 2 by ) when we wish to indicate that we are working with weak semantics. 0 We deﬁne ITL32 as ITL32, but replacing axiom (ix) by

def WH = 2ϕ → 2#ϕ. Here, WH stands for ‘weak henceforth’. This terminology is justiﬁed by the following.

Lemma 2.2 Every instance of WH is derivable in ITL32.

Proof From (viii), (xv) and (vi) we obtain #2ϕ → #ϕ, which combined with axiom (ix) allows us to conclude ⊢ 2ϕ → #ϕ. Thanks to (xv) and axiom (vi) we get ⊢ 22ϕ → 2#ϕ. (1)

From axiom (xv), necessitation and (xii) we obtain ⊢ 2 (2ϕ → #2ϕ) → (2ϕ → 22ϕ) . (2)

From (1) and (2) we conclude ⊢ 2ϕ → 2#ϕ.

0 Thus ITL32 ⊆ ITL32. We will later see that the inclusion is strict, since ϕ → # ϕ is not valid in general for our semantics, but ϕ → #ϕ and 2ϕ → #2ϕ are. We can 0 then deﬁne ITL3 = t(ITL32). Modal intuitionistic logics often involve additional axioms, and in particular (Fischer Servi 1984) includes the schema def FS3(ϕ, ψ) = (3ϕ → 2ψ) → 2 (ϕ → ψ) .

We also deﬁne def FS◦(ϕ, ψ) = (#ϕ → #ψ) → # (ϕ → ψ) . This notation is justiﬁed in view of the fact that # is self-dual in linear temporal logic: since time is modeled deterministically, ‘in some next moment’ is equivalent to ‘in every next moment,’ so # may be regarded as a ‘box’ or a ‘diamond.’ It is further motivated by the following. Exploring the Jungle of Intuitionistic Temporal Logics 7

Proposition 2.3

The formula FS◦(3p, 2q) → FS3(p, q) is derivable in ITL32.

Proof

We reason within ITL32. Assume 1) FS◦(3p, 2q) and 2) (3p → 2q). Notice that #3p → 3p and 2p → #2p are instances of axioms (ix) and (xi). From this and assumption 2) we conclude #3p → #2p. Thanks to assumption 1) and Modus Ponens we conclude # (3p → 2q). Therefore, (3p → 2p) → # (3p → 2q). By Rule (xv), we obtain 2 ((3p → 2p) → # (3p → 2q)). By the induction axiom (xii) we derive (3p → 2p) → 2 (3p → 2q). From the assumption 2) and Modus Ponens it follows that 2 (3p → 2q). Note that (3p → 2q) → (p → q) is derivable in ITL32. From rule (xv) and axiom (vi) we obtain 2 (3p → 2q) → 2 (p → q). From this and 2 (3p → 2q) it follows that 2 (p → q), as required.

Later we will show that these schemas lead to logics strictly stronger than ITL32. Next we consider additional axioms reminiscent of the constant domain axiom in ﬁrst- order intuitionistic logic, namely ∀x(ϕ(x) ∨ ψ(x)) → ∃xϕ(x) ∨ ∀xψ(x); we maintain the terminology ‘constant domain’ due to this similarity, although they do not retain this meaning in our logics. As we will see, in the context of intuitionistic temporal logics, these axioms separate Kripke semantics from the more general topological semantics.

def CD(ϕ, ψ) = 2(ϕ ∨ ψ) → 2ϕ ∨ 3ψ def BI(ϕ, ψ) = 2(ϕ ∨ ψ) ∧ 2(#ψ → ψ) → 2ϕ ∨ ψ. Here, CD stands for ‘constant domain’ and BI for ‘backward induction’. We also deﬁne as a special case CD−(ϕ) = CD(¬ϕ, ϕ). The axiom CD might not be desirable from a constructive perspective, as from 2(ϕ∨ψ) one cannot in general extract an upper bound for a witness for 3ψ.5 The axiom BI is a 3-free version of CD, as witnessed by the following.

Proposition 2.4 0 The following formulas are derivable in ITL32:

1. CD(p, q) → BI(p, q); 2. BI(p, 3q) → CD(p, q).

Proof 0 We reason within ITL32. For the ﬁrst claim, assume that 1) CD(p, q), 2) 2(#q → q), and 3) 2(p ∨ q). From 1) and 3) we obtain 2p ∨ 3q, which together with 2) and axiom (xiii) gives us 2p ∨ q, as needed. For the second, assume 1) BI(p, 3q) and 2) 2(p ∨ q). From 2(p ∨ q), axiom (xi) and some modal reasoning, 2(p ∨ 3q). Also from axiom (xi) and rule (xv), 2(#3q → 3q). From BI(p, 3q) we obtain 2p ∨ 3q, as needed.

5 For example, if ϕ represents the ‘active’ states and ψ the ‘halting’ states of a program, then CD would require us to decide whether the program halts, which is not possible to do constructively. 8 Boudou et al.

Finally, we introduce the conditional excluded middle axiom

def CEM(p, q) = (¬#p ∧ #¬¬p) → (#q ∨ ¬#q).

This axiom states that a certain instance of excluded middle holds, provided some as- sumptions are satisfied. It is less familiar than others we have considered, but its role will become clear when we consider semantics based on the real line. With this, we define a handful of logics, listed in Table 2.1, along with definitions of the ‘optional’ axioms. The inclusions between these logics are summarized in Figure 2.1; as we will show in this paper, these are the only inclusions that hold between these logics.6

WH(ϕ) = 2ϕ → 2#ϕ FS◦(ϕ, ψ) = (#ϕ → #ψ) → # (ϕ → ψ) FS3(ϕ, ψ) = (3ϕ → 2ψ) → 2 (ϕ → ψ) CD(ϕ, ψ) = 2(ϕ ∨ ψ) → 2ϕ ∨ 3ψ CD−(ϕ) = 2(¬ϕ ∨ ϕ) → 2¬ϕ ∨ 3ϕ BI(ϕ, ψ) = 2(ϕ ∨ ψ) ∧ 2(#ψ → ψ) → 2ϕ ∨ ψ CEM(ϕ, ψ) = (¬#ϕ ∧ #¬¬ϕ) → (#ψ ∨ ¬#ψ)

ITL32 = (See Deﬁnition 2.1) 0 0 ITL32 = ITL32 − (ix) + WH ITL3 = t(ITL32) − 0 − ETL32 = ITL32 + CD ETL3 = t(ITL32 + CD ) − 0 − RTL32 = ITL32 + CD + CEM RTL3 = t(ITL32 + CD + CEM) CDTL32 = ITL32 + CD CDTL3 = t(CDTL32) + + + ITL32 = ITL32 + FS◦ ITL3 = t(ITL32) + − + + ETL32 = ITL32 + FS◦ + CD ETL3 = t(ETL32) + + + CDTL32 = ITL32 + FS◦ + CD CDTL3 = t(CDTL32)

Table 2.1: Axioms not listed in Deﬁnition 2.1 (above) and logics based on strong and weak henceforth (below). In the right-hand column, notice that only ITL3, ETL3 and 0 RTL3 are based on ITL32.

Here, RTL32 stands for ‘real temporal logic’, ETL32 for ‘Euclidean temporal logic’ 0 and CDTL32 for ‘constant domain temporal logic’. For a logic Λ in the above list, Λ is 0 deﬁned analogously but replacing ITL32 by ITL32. Logics over L3 are deﬁned in Table 2.1. Note that logics with either CD or FS◦ use the strong axiom, ϕ → # ϕ. This has to do with our semantics and will become clear later.

6 Note that our notation for logics has been modiﬁed from that in (Boudou et al. 2019), in order to 0 + accommodate the larger family we now consider. Speciﬁcally, ITL32 was denoted ITL , ITL32 was FS CD + 1 0 denoted ITL , CDTL32 was denoted ITL , and CDTL32 was denoted ITL . Note that ITL32 is weaker than ITL0. Exploring the Jungle of Intuitionistic Temporal Logics 9

+ + CDTL32 CDTL3

+ + ETL32 ETL3

+ + CDTL32 RTL32 ITL32 CDTL3 RTL3 ITL3

ETL32 ETL3

ITL32 ITL3

Fig. 2.1: Inclusions between the logics based on strong or weak henceforth we have deﬁned; an arrow Λ1 → Λ2 means that every theorem of Λ1 is a theorem of Λ2.

Our list is not meant to exhaust all combinations of axioms; rather, we only consider logics that arise from natural classes of models. Before discussing semantics, we establish the only non-trivial inclusion between these logics. Lemma 2.5 + Every instance of CEM is derivable in ITL32.

Proof + It is not hard to check that ¬(¬#p ∧ #¬¬p) is derivable in ITL32, hence so is CEM. + This immediately yields that RTL32 ⊆ ETL32: Proposition 2.6 + Every formula derivable in RTL32 is derivable in ETL32.

We are also interested in logics over sublanguages of L32 or L3. For any logic Λ defined above, let Λ2 be defined by restricting similarly all rules and axioms to L2, except that when CD is an axiom of Λ, we add the axiom BI to Λ2. In these cases, Λ is similarly defined using t(BI). The logic ITL2 is similar to a Hilbert calculus for the ∧, ∨-free fragment considered by Yuse and Igarashi (Yuse and Igarashi 2006), although they do not include induction but include the axioms 2ϕ → 22ϕ and 2#ϕ ↔ #2ϕ. It is not difficult to check that the latter are derivable from our basic axioms, and hence their logic is contained in ITL2. We also define Λ3 to be the logic obtained by restricting all rules and axioms to ϕ→ψ #ϕ→ϕ L3, and adding the rules 3ϕ→3ψ and 3ϕ→ϕ . Note that these rules correspond to axioms (vii), (xiii), respectively, but do not involve 2. In this paper we are mostly con- cerned with logics including ‘henceforth’, but 2-free logics are studied in detail by (Diéguez and Fernández-Duque 2018).

3 Dynamic topological systems The logics deﬁned above are pairwise distinct. We will show this by introducing semantics for each of them. They will be based on dynamic topological systems (or dynamical systems for short), which, as was observed in (Fern´andez-Duque 2018), generalize their 10 Boudou et al.

Kripke semantics (Boudou et al. 2017). In this section, we review the basic notions of topological dynamics needed in the rest of the text. Let us first recall the definition of a topological space, as in e.g. (Dugundji 1975): Definition 3.1 A topological space is a pair (X, T ) , where X is a set and T a family of subsets of X satisfying a) ∅,X ∈ T ; b) if U, V ∈ T then U ∩ V ∈ T , and c) if O ⊆ T then O ∈ T . The elements of T are called open sets. S If x ∈ X, a neighbourhood of x is an open set U ⊆ X such that x ∈ U. Given a set A ⊆ X, its interior, denoted A◦, is the largest open set contained in A. It is defined formally by A◦ = {U ∈ T : U ⊆ A} . (3) Dually, we define the closure A as X[\ (X \ A)◦; this is the smallest closed set containing A. If (X, T ) is a topological space, a function S : X → X is continuous if, whenever U ⊆ X is open, it follows that S−1[U] is open. The function S is open if, whenever V ⊆ X is open, then so is S[V ]. An open, continuous function is an interior map, and a bijective interior map is a homeomorphism; equivalently, S is a homeomorphism if it is invertible and both S and S−1 are continuous. A dynamical system is then a topological space equipped with a continuous function: Definition 3.2 A dynamical (topological) system is a triple X = (X, T ,S) such that (X, T ) is a topological space and S : X → X is continuous. We say that X is open if S is an interior map and invertible if S is a homeomorphism. Topological spaces generalize posets in the following way. Let F = (W, 4) be a poset; that is, W is any set and 4 is a transitive, reflexive, antisymmetric relation on W . To see F as a topological space, define ↑w = {v : w 4 v} . Then consider the topology T4 on W given by setting U ⊆ W to be open if and only if, whenever w ∈ U, we have ↑w ⊆ U. A topology of this form is an up-set topology (Aleksandroff 1937). The interior operator on such a topological space is given by A◦ = {w ∈ W : ↑w ⊆ A}; (4) i.e., w lies on the interior of A if whenever v < w, it follows that v ∈ A. Throughout this text we will often identify partial orders with their corresponding topologies, and many times do so tacitly. In particular, a dynamical system generated by a poset is called a dynamical or expanding poset, due to its relation to expanding products of modal logics (Gabelaia et al. 2006). It will be useful to characterize the continuous and open functions on posets (see Figure 3.1): Lemma 3.3 Consider a poset (W, 4) and a function S : W → W . Then, 1. the function S is continuous with respect to the up-set topology if and only if, whenever w 4 w′, it follows that S(w) 4 S(w′), and 2. the function S is open with respect to the up-set topology if whenever S(w) 4 v, there is w′ ∈ W such that w 4 w′ and S(w′)= v. Exploring the Jungle of Intuitionistic Temporal Logics 11

These are properties common in multi-modal logics and we refer to them as ‘conﬂuence properties.’ A persistent function is an open, continuous map on a poset.

S S w′ w′ v

4 4 4 4 S S w w

(a) Continuity (b) Openness

Fig. 3.1: On a dynamic poset, the above diagrams can always be completed if S is continuous or open, respectively.

4 Semantics In this section we will see how dynamical systems can be used to provide a natural intuitionistic semantics for the language of linear temporal logic. Classicall LTL may be interpreted over structures (X,S) where X is a set and S : X → X. In this setting, #ϕ is true on a point x if ϕ is true ‘at the next moment,’ i.e., on S(x); 3ϕ is true on a point x if ϕ is ‘eventually’ true, i.e. there is n ≥ 0 such that ϕ is true on Sn(x), and 2ϕ is true on a point x if ϕ is ‘henceforth’ true, i.e. for all n ≥ 0, ϕ is true on Sn(x). Meanwhile, topological spaces provide semantics for intuitionistic logic, where each formula is assigned an open set. Under this semantics, ϕ → ψ is true on x if there is a neighborhood U of x (i.e., an open set U with x ∈ U) such that every y ∈ U satisfying ϕ also satisfies ψ. Thus it is natural to interpret intuitionistic temporal logic on dynamical systems, which are endowed with both a topology and a transition function. In this setting, the classical definitions of # and 3 readily adapt to the topological setting without modification. On the other hand, the classical definition of 2 does not necessarily yield open sets, and to this end we consider two variants of henceforth, the weak variant, , and the strong variant, which we simply denote 2. Definition 4.1 Given a dynamical system X = (X, T ,S), a valuation on X is a function · : L32 → T such that J K

⊥ = ∅ 3ϕ = S−n ϕ Jϕ K∧ ψ = ϕ ∩ ψ J K n≥0 J K S Jϕ ∨ ψK = JϕK ∪ JψK 2ϕ = U ∈ T : S[U] ⊆ U ⊆ ϕ ◦ J K J K Jϕ → ψK =J (KX \J ϕK ) ∪ ψ ◦ ϕ = S n S−n ϕ o J#ϕ =KS−1 ϕ J K J K J K n≥0 J K J K J K T A tuple M = (X, T , S, · ) consisting of a dynamical system with a valuation is a dynamic topological model, JandK if T is generated by a partial order, we will say that M is a dynamic poset model. All of the semantic clauses are standard from either intuitionistic or temporal logic, with the exception of those for ϕ and 2ϕ, which we discuss in greater detail in the 12 Boudou et al. remainder of this section. It is not hard to check by structural induction on ϕ that ϕ is uniquely defined given any assignment of the propositional variables to open sets, andJ K that ϕ is always open. We define validity in the standard way, and with this introduce additionalJ K semantically-defined logics, two of which were studied in (Boudou et al. 2017). Definition 4.2 If M = (X, T , S, · ) is any dynamic topological model and ϕ ∈ L is any formula, we write M |= ϕ if JϕK = X. Similarly, if S = (X, T ,S) is a dynamical system, we write S |= ϕ if for any valuationJ K · on X , we have that (S, · ) |= ϕ; and, if X = (X, T ) is any topological space, then X |=J Kϕ if, for any continuous SJ :K X → X, (X, T ,S) |= ϕ. Finally, if Ω is a class of structures (either topological spaces, dynamical systems, or models), we write Ω |= ϕ if for every A ∈ Ω, A |= ϕ, in which case we say that ϕ is valid on Ω. Ω If Ω is either a structure or a class of structures and Θ ⊆{3, 2, }, we write ITLΘ for the set of LΘ formulas valid on Ω. We maintain the convention that # is assumed to be Ω in all languages, and we write ITL◦ when Θ = ∅. The main classes of dynamical systems we are interested in are listed in Table 4.1. c For example, ITL32 denotes the set of L32-formulas valid over the class of all dynamical systems.

c all dynamical systems (with a continuous function) e expanding posets p persistent posets o open dynamical systems Rn systems based on n-dimensional Euclidean space

Table 4.1: The main classes of dynamical systems appearing in the text.

In practice, it is convenient to have a ‘pointwise’ characterization of the semantic clauses of Deﬁnition 4.1. For a model M = (X, T , S, · ), x ∈ X and ϕ ∈ L, we write M, x |= ϕ if x ∈ ϕ , and M |= ϕ if ϕ = X. Then,J K in view of (3), given formulas ϕ and ψ, we haveJ thatK M, x |= ϕ → ψJifK and only if there is a neighbourhood U of x such that for all y ∈ U, if M,y |= ϕ then M,y |= ψ; note that this is a special case of neighbourhood semantics (Pacuit 2017). The following simple observation will be useful. Lemma 4.3 If M = (X, T , S, · ) is any model and ϕ, ψ ∈ L32, then M |= ϕ → ψ if and only if ϕ ⊆ ψ . J K J K J K Proof ◦ If ϕ ⊆ ψ then (X \ ϕ ) ∪ ψ = X, so ϕ → ψ = (X \ ϕ ) ∪ ψ = X◦ = X. ◦ Otherwise,J K J thereK is z ∈ JϕK suchJ K that z∈ / ψJ , so thatK z∈ / (XJ \K ϕ J) ∪K ψ , i.e. z∈ / ϕ → ψ . J K J K J K J K J K Using (4), this can be simpliﬁed somewhat in the case that T is generated by a partial order 4: Exploring the Jungle of Intuitionistic Temporal Logics 13

Proposition 4.4 If (X, 4, S, · ) is a dynamic poset model, x ∈ X, and ϕ, ψ are formulas, then M, x |= ϕ → ψ if andJ K only if from y < x and M,y |= ϕ, it follows that M,y |= ψ.

This is the standard relational interpretation of implication, and thus topological semantics are a generalization of the usual Kripke semantics. The semantics for were originally introduced by (Kremer 2004) as an intuitionistic reading of ‘henceforth’. By analogy with 3, one might try to interpret ϕ as −n J K n≥0 S ϕ . But this does not quite work since, on this interpretation, there would be no guaranteeJ K that ϕ is open. Instead, we consider interpreting ϕ as the interior T −n J K J K of n≥0 S ϕ . In other words, M, x |= ϕ if and only if there is a neighbourhood U of x so that forJ everyK y ∈ U and every n ∈N, one has that M,Sn(y) |= ϕ. T This interpretation of ‘henceforth’ is analogous to the interpretation in (Rasiowa and Sikorski 1963), and going back to (Mostowski 1948), of ∀x in the topological semantics for quantified intuitionistic logic. We may interpret variables as ranging over some non-empty domain D, and truth values as open sets in some topological space (X, T ). The semantic clauses for ∃x and ∀x are, essentially, the following: ∃xϕ = ϕ[d/x] J K d∈D J K ∀xϕ = (S ϕ[d/x] )◦ J K d∈D J K T Note that if D is infinite then the intersection in the definition of ∀xϕ may also be infinite and hence the application of the interior operator is necessaryJ in orderK to obtain an open truth value. The semantics for 2ϕ are also an intuitionistic interpretation of ‘henceforth’, but from a more algebraic perspective. In classical temporal logic, 2ϕ is the largest set contained in ϕ which is closed under S. In our semantics, 2ϕ is theJ greatestK open set which is closed underJ K S. If M, x |= 2ϕ, this fact is witnessedJ by anK open, S-invariant neighbourhood of x, where U ⊆ X is S-invariant if S[U] ⊆ U.

Proposition 4.5 If (X, T , S, · ) is a dynamic topological model, x ∈ X, and ϕ is any formula, then M, x |= 2ϕJifK and only if there is an S-invariant neighbourhood U of x such that for all y ∈ U, M,y |= ϕ.

In fact, the open, S-invariant sets form a topology; that is, the family of S-invariant open sets is closed under ﬁnite intersections and arbitrary unions, and both the empty set and the full space are open and S-invariant (this follows readily from the fact that the topology T already has these properties, as does the family of S-invariant sets). This topology is coarser than T , in the sense that every S-invariant open set is (tautologically) open. Thus 2 can itself be seen as an interior operator based on a coarsening of T , and 2ϕ is always an S-invariant open set. J K Example 4.6 As usual, the real number line is denoted by R and we assume that it is equipped with the standard topology, where U ⊆ R is open if and only if it is a union of intervals of the form (a,b). Consider a dynamical system based on R with S : R → R given by S(x)=2x. 14 Boudou et al.

We claim that for any model M based on (R,S) and any formula ϕ, M, 0 |= 2ϕ if and only if M |= ϕ. To see this, note that one implication is obvious since R is open and S-invariant, so if ϕ = R it follows that M, 0 |= 2ϕ. For the other implication, assume that M, 0 |= 2ϕ, soJ K that there is an S-invariant, open U ⊆ ϕ with 0 ∈ U. It follows from U being open that for some ε > 0, (−ε,ε) ⊆ U. Now, letJ Kx ∈ R, and let n be large enough so that |2−nx| <ε. Then, 2−nx ∈ U, and since U is S-invariant, x = Sn(2−nx) ∈ U. Since x was arbitrary, U = R, and it follows that M |= ϕ. On the other hand, suppose that 0

a) w ∈ 2ϕ ; b) w ∈ ϕ ; c) ∀n ∈ N Sn(w) ∈ ϕ . J K J K J K Proof −n ◦ By Lemma 4.7, a) implies b). That b) implies c) is immediate from n∈N S ϕ ⊆ S−n ϕ , so it remains to show that c) implies a). Suppose that for all Jn K∈ N, n∈N T M,Sn(w) J|=Kϕ, and let U = ↑Sn(w). That the set U is open follows from each T n∈N ↑Sn(w) being open and unions of opens being open. If v ∈ U, then v < Sn(w) for some S n ∈ N and hence by upwards persistence, from M,Sn(w) |= ϕ we obtain M, v |= ϕ; moreover, S(v) < Sn+1(w) so S(v) ∈ U. Since v ∈ U was arbitrary, we conclude that U is S-invariant and U ⊆ ϕ . Thus U witnesses that M, w |= 2ϕ. J K As we will see later, Proposition 4.8 fails over the class of general dynamical systems, but a weaker version holds over the class of open dynamical systems. Proposition 4.9 The formula p ↔ 2p is valid over the class of open dynamical systems. Exploring the Jungle of Intuitionistic Temporal Logics 15

Proof One implication is Lemma 4.7, so we focus on the other. Let (X, T , S, · ) be a dynamical −n ◦ J K model. Assume that w ∈ p , and let U = n∈N S p . Clearly U is open; we claim that it is S-invariant. WeJ haveK that J K ◦ T S[U]= S S−n p ⊆ S S−n p ⊆ S−n p , n N J K! ! n N J K! n N J K \∈ \∈ \∈ where the latter inclusion is obtained by distributing S over the intersection. Moreover, S[U] is open, since S is assumed to be an open function. Thus ◦ S[U] ⊆ S−n p = U, n N J K! \∈ witnessing that w ∈ 2p . J K

5 Soundness In this section we will show that several of the logics we have considered are sound for their semantics based on diﬀerent classes of dynamic topological systems. First we show that our basic logics (as given in Deﬁnition 2.1 and Table 2.1) are sound for the class of all dynamical systems. Below, recall that c denotes the class of all dynamical systems (see Table 4.1). Theorem 5.1 The logics ITL32 and ITL3 are sound for the class of dynamical systems; that is, c c ITL32 ⊆ ITL32 and ITL3 ⊆ ITL3.

Proof Let M = (X, T , S, · ) be any dynamical topological model; we must check that all the axioms (i)-(xiii)J areK valid on M and the rules (xiv), (xv) preserve validity. Note that all intuitionistic tautologies are valid due to the soundness for topological semantics (Mints 2000). Many of the other axioms can be checked routinely, so we focus only on those axioms involving the continuity of S or the semantics for 2 or . Axiom (v): #(ϕ → ψ) → (#ϕ → #ψ). Suppose that x ∈ #(ϕ → ψ) . Then, S(x) ∈ ϕ → ψ . Since S is continuous and ϕ → ψ is open, U =J S−1 ϕ →K ψ is a neigh- bourhoodJ K of x. Then, for y ∈ U, if yJ ∈ #ϕK , it follows that S(yJ) ∈ ϕK ∩ ϕ → ψ , so that S(y) ∈ ψ and y ∈ #ψ . Since Jy ∈KU was arbitrary, x ∈ #Jϕ →K #Jψ , thusK #(ϕ → ψ) ⊆ J#ϕK → #ψ , andJ byK Lemma 4.3 (which we will henceforthJ use withoutK Jmention), axiomK J (v) is validK on M. Axiom (vi): 2(ϕ → ψ) → (2ϕ → 2ψ). Observe that 2(ϕ → ψ) is an S-invariant open subset of ϕ → ψ . Similarly, 2ϕ is an S-invariantJ open subsetK of ϕ . Let U = 2(ϕ → ψ) ∩ J2ϕ . SinceK U is open,J K it suffices to prove that U ⊆ 2ψJ .K Moreover, UJ is S-invariant,K J thereforeK it suffices to prove that U ⊆ ψ , which isJ directK because U ⊆ ϕ → ψ ∩ ϕ and ϕ → ψ ⊆ (X \ ϕ ) ∪ ψ . J K J K J K J K J K J K Axiom (vii): 2(ϕ → ψ) → (3ϕ → 3ψ). As before, suppose that x ∈ 2(ϕ → ψ) , and let U be an S-invariant neighbourhood of x such that U ⊆ ϕ → ψ .J If y ∈ U ∩ K3ϕ , J K J K 16 Boudou et al. then Sn(y) ∈ ϕ for some n; since U is S-invariant, Sn(y) ∈ U, hence Sn(y) ∈ ψ and y ∈ 3ψ . WeJ concludeK that x ∈ 3ϕ → 3ψ . J K J K J K Axioms (viii), (ix): 2ϕ → ϕ ∧ #2ϕ. Suppose that x ∈ 2ϕ , and let U ⊆ ϕ be an S-invariant neighbourhood of x. Then, x ∈ U, so x ∈ ϕJ . Moreover,K U is alsoJ K an S-invariant neighbourhood of S(x), so S(x) ∈ 2ϕ and thusJ Kx ∈ #2ϕ . We conclude that x ∈ ϕ ∩ #2ϕ . J K J K J K J K Axiom (xii): 2(ϕ → #ϕ) → (ϕ → 2ϕ). Suppose that x ∈ 2(ϕ → #ϕ) . If x ∈ ϕ , then U = ϕ ∩ 2(ϕ → #ϕ) is open (by the intuitionistic semantics)J andK S-invariant,J K since if y J∈ KU,J from y ∈ ϕK → #ϕ we obtain S(y) ∈ ϕ . It follows that U is an S-invariant neighbourhood ofJ x, so xK∈ 2ϕ . J K J K Axiom (xiii): 2(#ϕ → ϕ) → (3ϕ → ϕ). Suppose that x ∈ 2(#ϕ → ϕ) ∩ 3ϕ . Let U ⊆ #ϕ → ϕ be an S-invariant neighbourhood of x. Let n beJ least so thatKSnJ(x) ∈K ϕ ; if n>J 0, sinceKU is S-invariant we see that Sn−1(x) ∈ U ⊆ #ϕ → ϕ , hence Sn−1(xJ) K∈ ϕ , contradicting the minimality of n. Thus n = 0 and x ∈J ϕ . K J K J K Axiom t(vi): (ϕ → ψ) → ( ϕ → ψ). Assume that x ∈ (ϕ → ψ) . We claim that if y ∈ (ϕ→ ψ) ∩ ϕ then y ∈ ψ , from which we obtainJ x ∈ K ϕ → ψ . If y ∈ (ϕJ→ ψ) ∩ K ϕ JthenK note thatJ forK all z ∈ (ϕ → ψ) ∩ ϕ , byJ definition K of theJ semantics,K weJ haveK that for all k ≥ 0, Sk(z) J∈ ϕ → ψK andJSKk(z) ∈ ϕ , so Sk(z) ∈ ψ . Since U := (ϕ → ψ) ∩ ϕ is a neighbourhoodJ K of y and z ∈JUKwas arbitrary,J yK∈ ψ . SinceJO := (Kϕ →Jψ)K is a neighbourhood of x and y ∈ O was arbitrary, this witnessesJ K that x ∈ Jϕ → ψ K. J K Axiom t(WH): ϕ → #ϕ. Suppose that x ∈ ϕ . Let U := ϕ be a neighbourhood of x such that, for all y ∈ U and n ∈ N, Sn(Jy) ∈K ϕ . Therefore,J K for all y ∈ U and n ∈ N, Sn+1(y) ∈ ϕ , i.e., y ∈ #ϕ , so that U witnessesJ K that x ∈ #ϕ . J K J K J K Axiom t(xii): (ϕ → #ϕ) → (ϕ → ϕ). Suppose that x ∈ (ϕ → #ϕ) . If x ∈ ϕ , then U = ϕ ∩ (ϕ → #ϕ) is a neighbourhood of x. It can beJ proved byK inductionJ onK i that for allJ Ki ≥J0 and y ∈ UK, Si(y) ∈ ψ , so that U witnesses that x ∈ ψ . J K J K Axioms t(vii), t(xiii): These are treated as their analogues for 2. The additional axioms we have considered are valid over specific classes of dynamical systems. Specifically, the constant domain axiom is valid for the class of expanding posets, while the Fischer Servi axioms are valid for the class of open systems. Let us begin by discussing the former in more detail. In the next few results, recall that the relevant definitions are summarized in Tables 2.1 and 4.1. Theorem 5.2 The logics CDTL32 and CDTL2 are sound for the class of expanding posets; that is, e e CDTL32 ⊆ ITL32 and CDTL2 ⊆ ITL2.

Proof Let M = (X, 4, S, · ) be a dynamic poset model; in view of Theorem 5.1, it only remains to check that CD andJ K BI are valid on M. However, by Proposition 2.4, BI is a consequence of CD, so we only check the latter. Suppose that x ∈ 2(ϕ ∨ ψ) , but x 6∈ 2ϕ . Then, in view of Proposition 4.8, for some n ≥ 0, Sn(x) 6∈ ϕJ . It followsK that Sn(Jx) ∈K ψ , so that x ∈ 3ψ . J K J K J K Exploring the Jungle of Intuitionistic Temporal Logics 17

Note that the relational (rather than topological) semantics are used in an essential way in the above proof, since Proposition 4.8 is not available in the topological setting, and indeed we will show in Proposition 8.2 that these axioms are not topologically valid. But before that, let’s turn our attention to the Fischer Servi axioms (see Table 2.1). Recall that o denotes the class of dynamical systems with a continuous and open map.

Theorem 5.3 + o + ITL32 ⊆ ITL32, i.e., the logic ITL32 is sound for the class of open dynamical systems.

Proof Let M = (X, T , S, · ) be a dynamical topological model where S is an interior map. J K We check that axiom FS◦ is valid on M. Suppose that x ∈ #ϕ → #ψ , and let U = #ϕ → #ψ . Since S is open, V = S[U] is a neighbourhoodJ of S(x). LetK y ∈ V ∩ ϕ , Jand chooseKz ∈ U so that y = S(z). Then, z ∈ U ∩ #ϕ , so that S(z)= y ∈ #ψ . SinceJ K y ∈ V was arbitrary, S(x) ∈ ϕ → ψ , and x ∈ #(Jϕ →Kψ) . J K J K J K As an easy consequence, we mention the following combination of Theorems 5.2 and 5.3. Recall that dynamic posets with an interior map are also called persistent, and the class of persistent posets is denoted p.

Corollary 5.4 + + The logics CDTL32 and CDTL2 are sound for the class of persistent posets, that is, + p + p CDTL32 ⊆ ITL32 and CDTL2 ⊆ ITL2.

6 Euclidean spaces

The celebrated McKinsey-Tarski theorem states that intuitionistic propositional logic is complete for the real line, and more generally for a wide class of metric spaces which includes every Euclidean space Rn (Tarski 1938). Thus, it is natural to ask if a similar result holds for intuitionistic temporal logics, which could lead to applications in spatio- temporal reasoning. As we will see in this section, the answer to this question is negative; however, we identify some principles which could lead to an axiomatization for Euclidean systems. Let us consider the real line. The conditional excluded middle axiom shows that even the #-logic of the real line is diﬀerent from the logic of all dynamical systems.

Lemma 6.1 The formula CEM(p, q) = (¬#p ∧ #¬¬p) → (#q ∨ ¬#q) is valid on R.

Proof Suppose that (R, S, · ) is a model based on R and that x ∈ ¬#p ∧ #¬¬p . From x ∈ #¬¬p and the semanticsJ K of double negation (discussed in (Fernández-DJ uqueK 2018)) we seeJ thatK there is a neighbourhood V of S(x) such that V ⊆ p . It follows from the intermediate value theorem that if U is a neighbourhood of x andJSK[U] is not a singleton, then S[U]∩V contains an open set and hence S[U]∩ p 6= ∅. Meanwhile, from x ∈ ¬#p J K J K 18 Boudou et al. we see that x has a neighbourhood U∗ such that S[U∗] ∩ p = ∅, hence for such a U∗ J K the set S[U∗] is the singleton {S(x)}. But then either S(x) ∈ q and x ∈ #q , or else J K J K S(x) 6∈ q , which means that U∗ ∩ #q = ∅ and thus U∗ witnesses that x ∈ ¬#q . In either case,J K x ∈ #q ∨ ¬#q , as required.J K J K J K Remark 6.2 It is tempting to conjecture that CEM axiomatizes the L#-logic of the real line, but there is a possibility that additional axioms are required. CEM is an intuitionistic variant of similar formulas in (Kremer and Mints 2005; Slavnov 2003) showing that the dynamic topological logic (DTL) of the real line is different from the dynamic topological logic of arbitrary spaces. We will not review DTL here, but it is a classical cousin of intuitionistic temporal logic; see (Fernández-Duque 2018; Diéguez and Fernández-Duque 2018). The problem of axiomatizing DTL over the real line has long remained open, and (Nogin and Nogin 2008) give further examples of valid formulas not derivable from the classical analogue of CEM. We do not know if these formulas also have intuitionistic counterparts, and leave this line of inquiry open. Since CEM is not valid for Rn in general, it cannot be used to show that our base logic is incomplete for Euclidean spaces. However, this can be shown using CD−, which is valid on any locally connected space. Recall that a subset C of a topological space X is connected if, whenever A, B are disjoint open sets such that C ⊆ A ∪ B, it follows that C ⊆ A or C ⊆ B. The space X is locally connected if whenever U is open and x ∈ U, there is a connected neighbourhood V ⊆ U of x. It is well-known that Rn is locally connected for all n. The following properties regarding connectedness will be useful below. Suppose that X is a topological space and S : X → X is continuous. Then, for every C ⊆ X, if C is connected, then so is S[C]. Moreover, any A ⊆ X can be partitioned into a family of maximal connected sets called the connected components of A. If X is locally connected and A is open, then the connected components of A are also open (see e.g. (Dugundji 1975)). Lemma 6.3 − − The formulas CD (p) and t(CD (p)) are valid on the class of locally connected spaces.

Proof Let (X, T , S, · ) be a model based on a locally connected space, and x ∈ X. First we J K − − show that x ∈ qt(CD (p))y. Recall that CD (p) = 2(p ∨ ¬p) → 2¬p ∨ 3p, so that − t(CD (p)) = (p ∨ ¬p) → ¬p ∨ 3p. We may thus assume that x ∈ (p ∨ ¬p) , so that using the local connectedness of X, there is a connected neighbourhoodJ U of Kx −n −n with U ⊆ n∈N S p ∨ ¬p . This means that, for each n ∈ N, U ⊆ S p ∨ ¬p = S−n p ∪ S−n ¬p and,J fromK the connectedness of U, U ⊆ S−n p or U ⊆ S−J n ¬p K, as T theseJ twoK sets areJ K disjoint and open. If x ∈ 3p there is nothingJ toK prove, so weJ assumeK otherwise. This means that for all n, Sn(x)J6∈ Kp , which implies that U 6⊆ S−n p , and hence U ⊆ S−n ¬p . But then, U witnesses thatJ Kx ∈ ¬p . J K To see that xJ ∈ KqCD−(p)y, suppose that x ∈ 2(Jp ∨ ¬pK) ; we must show that x ∈ J K 2¬p ∨ 3p . Let U ⊆ p ∨ ¬p be an S-invariant neighbourhood of x. For n ∈ N, let Vn J K J K n be the connected component of U containing S (x), and set V = n∈N Vn. Since each Vn ⊆ U ⊆ p ∪ ¬p and the latter are disjoint and open, it follows that either Vn ⊆ p J K J K S J K Exploring the Jungle of Intuitionistic Temporal Logics 19 or Vn ⊆ ¬p . If Vn ⊆ p for some n, it immediately follows that x ∈ 3p , and we J K J K J K are done. Otherwise, Vn ⊆ ¬p for all n. Clearly V is open; we claim that it is also J K S-invariant. To see this, note that S[Vn] ⊆ U. Note that U can be written as the disjoint union of two open sets as U = Vn+1 ∪(U \Vn+1); since Vn is connected, so is S[Vn], hence S[Vn] ⊆ Vn+1 or S[Vn] ⊆ U \ Vn+1. However, S[Vn] ∩ Vn+1 is non-empty, so we must have S[Vn] ⊆ Vn+1, and since n was arbitrary, S[V ] ⊆ V , as claimed. Hence V witnesses that x ∈ 2¬p , as needed. J K In conclusion, we obtain the following. Theorem 6.4

1. RTL32 and RTL3 are sound for R. n 2. ETL32 and ETL3 are sound for {R : n> 0}. + + n 3. ETL32 and ETL3 are sound for the class of invertible systems based on {R : n> 0}.

R2 e R In the remainder of this section we show that ITL32 ⊆ ITL32∩ITL32. We show this using results from (Fern´andez-Duque 2007) originally developed for dynamic topological logic, but applicable to ITL as well. We begin with the notion of dynamic morphism. Deﬁnition 6.5 Let X = (X, TX ,SX ) and Y = (Y, TY ,SY ) be dynamic topological systems. Let U ⊆ X be open and SX -invariant. A dynamic morphism from X to Y is an interior map f : U → Y such that for all x ∈ X, fSX (x)= SY f(x).

Proposition 6.6 Let X = (X, TX ,SX ) and Y = (Y, TY ,SY ) be dynamic topological systems. Let U ⊂ X be open and SX -invariant and f : U → Y be a dynamic morphism, and let · Y be J K any valuation on Y. Then, there is a valuation · X on X such that for every formula −1 J K ϕ ∈ L32, ϕ X ∩ U = f ϕ Y . J K J K Proof

Let · X be the unique valuation such that for any propositional variable p, p X = −1 J K −1 J K f p Y . We prove by induction on ϕ that ϕ X ∩U = f ϕ Y . Most cases are standard, saveJ theK cases for ϕ = ψ and ϕ = 2Jψ,K so we focusJ onK those. First assume that −n x ∈ U ∩ ψ X . Let V ⊆ U be a neighbourhood of x such that V ⊆ n≥0 SX ψ X . Then f[VJ] isK open, and since fS = S f, J K X Y T ih −n −n −n f[V ] ⊆ f SX ψ X ⊆ SX f ψ X ⊆ SY ψ Y . n 0 J K n 0 J K n 0 J K h \≥ i \≥ \≥ So, f[V ] witnesses that f(x) ∈ ψ Y . J K ′ ′ −n If f(x) ∈ ψ Y , we instead let V be a neighbourhood of f(x) so that V ⊆ n≥0 SY ψ Y . Then, J K J K T ih −1 ′ −1 −n −n −1 −n f [V ] ⊆ f SY ψ Y = SY f ψ Y ⊆ SX ψ X . n 0 J K n 0 J K n 0 J K h \≥ i \≥ \≥ Since f −1[V ′] is open, this witnesses that x ∈ ψ . J K 20 Boudou et al.

2 2 The arguments for ψ are similar. As above, if x ∈ ψ X , we let V ⊆ ψ X be J K J K an open, SX -invariant neighbourhood of x. Since U is open and SX -invariant, V ∩ U is also open and SX -invariant, so we may assume that V ⊆ U. Then, f[V ] is open and f[V ] ⊆ ψ Y by the induction hypothesis. It remains to check that f[V ] is SY -invariant. J K But if y ∈ f[V ] then y = f(z) for some z ∈ V , hence SY (y) = SY f(z) = fSX (z) and SX (z) ∈ V by SX -invariance, so SY (y) ∈ f[V ], as needed. ′ 2 −1 ′ 2 Similarly, if V witnesses that f(x) ∈ ψ Y , then f [V ] witnesses that x ∈ ψ X . J K J K

From here, we easily obtain that the logics based on R2 are contained in those based on R:

Theorem 6.7 2 2 R R Every formula of L32 valid on R is valid on R; that is, ITL32 ⊆ ITL32.

Proof Suppose that ϕ is not valid on R. Let S : R → R and · be such that (R, S, · ) 6|= ϕ. Define f : R2 → R2 to be given by f(x, y) = x and S′(x,J K y) = (S(x),y). Then,J itK is not hard to see that f is a surjective dynamic morphism from (R2,S′) onto (R,S). It follows 2 2 ′ −1 R that (R ,S ,f · ) 6|= ϕ, and hence ϕ 6∈ ITL32. J K R2 e In order to show that ITL32 ⊆ ITL32, it would suffice to construct a dynamic morphism from R2 onto a given dynamic poset (W, 4,S). We may assume that W is finite in view of the following result, which is established by (Balbiani et al. 2019). Theorem 6.8 Any formula satisfiable (falsifiable) on a dynamic poset is satisfiable (falsifiable) on a finite dynamic poset. We may then use the following result, essentially proven in (Fernández-Duque 2007). Theorem 6.9 2 2 If (W, 4,S) is a finite dynamic poset and w0 ∈ W , there exist continuous T : R → R , an open, T -invariant set U ⊆ R2, and a dynamic morphism f : R2 → W , such that 0 ∈ U and f(0) = w0.

Proof sketch. (Fernández-Duque 2007) shows the result for any dynamic preorder with limits that commute with S. Recall that 4 is a preorder if it is transitive and reflexive (but not necessarily antisymmetric). We say that (W, 4,S) is a dynamic preorder if S : W → W is 4-monotone, so that dynamic posets are a special case of dynamic preorders. We say that (W, 4,S) is a dynamic preorder with limits if every monotone sequence (wi)i∈N (i.e., a sequence such that wi 4 wi+1 for all i) is assigned a limit limi→∞ wi ∈ W with the property that wn 4 limi→∞ wi for all n and limi→∞ wi 4 wn for n large enough. The limits commute with S if S (limi→∞ wi) = limi→∞ S(wi). However, if W is finite and 4 is a partial order then every monotone sequence can trivially be assigned a limit, as in this case we have that wi is constant for i large enough, and we can define limi→∞ wi to be this unique constant. It is easy to see that this is a limit assignment that commutes with S, hence the desired U, T and f exist by (Fernández-Duque 2007). Exploring the Jungle of Intuitionistic Temporal Logics 21

Theorem 6.10 2 Every formula of L32 valid on R is valid on the class of expanding posets; that is, R2 e ITL32 ⊆ ITL32.

Proof e We prove the claim by contrapositive. If ϕ 6∈ ITL32, then by Theorem 6.8 there are a

ﬁnite dynamic poset model M = (W, 4,T, · W ) and w0 ∈ W \ ϕ W . By Theorem 6.9, there exist T : R2 → R2, an open, T -invariantJ K set U ⊆ R2, andJ aK dynamic morphism f : U → W , such that 0 ∈ U and f(0) = w0. By Proposition 6.6, there is a valuation 2 −1 · R2 on R such that ϕ R2 ∩ U = f ϕ W ; in particular, since w0 6∈ ϕ W , we have 2 J K J K2 J K R J K that 0 6∈ ϕ R2 , so that R 6|= ϕ and hence ϕ 6∈ ITL32. J K

7 Persistent posets and the real line

R2 e We have seen that ITL32 ⊆ ITL32. The question naturally arises whether a similar result holds when restricting to open systems: is every formula falsifiable on a persistent poset (i.e., a poset equipped with a continuous, open map) falsifiable on some Euclidean space, also equipped with a continuous, open map? Surprisingly, not only is the answer affirmative, but in this case, any falsifiable formula is falsifiable on the real line. In this p section, we will prove this fact, along with some properties of ITL32 which may be interesting on their own right. We remark that in this context 2 and coincide, so we restrict our attention to languages with the former. One challenge is that we do not have the finite model property in this setting. This is already proven in (Balbiani et al. 2019) for L32, but in fact, the finite model property already fails over L2. Proposition 7.1 The formula ϕ = 2¬¬p → ¬¬2p is valid over the class of all finite persistent posets, but not over the class of all persistent posets.

Proof First we show that ϕ is indeed valid over any finite persistent poset. Let M = (W, 4 , S, · ) be any model based on a finite persistent poset, and let w ∈ 2¬¬p . We show thatJ Kw ∈ ¬¬2p . It suffices to show that if v < w is maximal, thenJ v ∈K 2p .7 So, let n ∈ N.J Then, KSn(w) ∈ ¬¬p , and Sn(v) is maximal (as order-preservingJ persistentK functions preserve maximality),J K so Sn(v) ∈ p . It follows that v ∈ 2p . To see that ϕ is not valid over the classJ ofK persistent posets, letJ WK = Z, where 4 is the usual order and S(x) = x − 1. Let p = N. Then, every point satisfies ¬¬p (as every large-enough point satisfies p), so thatJ inK particular 0 ∈ 2¬¬p . However, no point satisfies 2p, since Sx+1(x) 6∈ p . Hence, in particular, 0 6∈ ϕJ . K J K J K Thus, we cannot avoid working with infinite models. However, we can still work with p models that have some ‘nice’ properties. For starters, ITL32 is complete for the class of product models (Kurucz et al. 2003). The products we consider will have a rather

7 In fact, the only property we use of M is that for every w there is a maximal v < w, so ϕ is valid over any persistent model with this property. 22 Boudou et al. particular form. We will need the following general deﬁnition, which applies to arbitrary topological spaces.

Deﬁnition 7.2

Let X = (X, T ) be any topological space. We deﬁne a new dynamical system X∞ = (X∞, T∞,S∞), where

• X∞ = X × N, • U ⊆ X∞ is open if and only if for every n ∈ N, {x ∈ X : (x, n) ∈ U} is open, and • S∞(x, n) = (x, n + 1).

We say that a dynamical system Y is a product system if Y = X∞ for some space X . If X was a poset, then Y is a product poset.

Products have the property that they ‘lift’ interior maps to dynamic morphisms, in the following sense:

Lemma 7.3

Let X = (X, TX ) be a topological space, Y = (Y, TY ,SY ) be an open dynamical system, and suppose that f : X → Y is an interior map. Then, there exists a dynamic morphism n f∞ : X∞ → Y whose range is n∈N SY f[X]. S Proof n Deﬁne f∞ : X ×N → Y by f∞(x, n)= SY f(x). First we must check that f∞ is continuous and open. If (x, n) ∈ X∞ and U is a neighbourhood of f∞(x, n), then since both f and n −1 −n −1 SY are continuous, f SY [U] ×{n} is a neighbourhood of (x, n) contained in f∞ [U]. Since (x, n) was arbitrary, f∞ is continuous. Similarly, if O is a neighbourhood of (x, n), n then SY f[O ∩ (X ×{n})] is a neighbourhood of f∞(x, n) contained in f∞[O], which since (x, n) was arbitrary shows that f∞[O] is open. Hence, f∞ is an open map. n n+1 Next we note SY f∞(x, n) = SY ◦ SY f(x) = SY f(x)= f∞(x, n +1) = f∞S∞(x, n). Finally, the range of f∞ is

n f∞[X × N]= f∞[X ×{n}]= SY f[X]. n N n N [∈ [∈

As a corollary, we immediately obtain that any formula satisfiable (falsifiable) on a persistent poset is satisfiable (falsifiable) on a product poset, since we can take X = Y and f to be the identity. With this, we can easily check that we can restrict our attention to the class of countable models. Below, if M = (W, 4, S, · ) is a dynamical poset model and Z ⊆ W , then M ↾ Z = (Z, 4↾ Z,S ↾ Z, · ↾ Z) is theJ K sub-structure with domain W and such that each of 4, S and · are restrictedJ K to Z. If Z is S-invariant but not necessarily open, then M ↾ Z is alsoJ basedK on a dynamic poset, although · ↾ Z may not be a valuation. However, this will indeed be the case if we choose Z appropriately.J K

Lemma 7.4 If ϕ ∈ L32 is falsiﬁable on a persistent poset, it is falsiﬁable on a countable model. Exploring the Jungle of Intuitionistic Temporal Logics 23

Proof We proceed as in a standard proof of the downward Löwenheim-Skolem theorem. Let M = (W, 4, S, · ) be a model based on a persistent poset such that M 6|= ϕ. In view of J K Lemma 7.3, we may assume that M is a product model, so that W = U∞ for some U, and there is w0 ∈ U such that (w0, 0) 6∈ ϕ . We define a sequence V0 ⊆ V1 ⊆ ... ⊆ U of J K countable sets, so that for V = n∈N Vn we have that M ↾ V falsifies ϕ. The construction is straightforward: V0 = {w0}× N, and if we are given V , define S n ψ→θ Vn+1 := Vn ∪{(vw,k ,m) : (w, k) ∈ Vn, ψ,θ ∈ L32, and m ∈ N},

ψ→θ ψ→θ where vw,k = w if (w, k) ∈ ψ → θ , and otherwise v = vw,k is chosen to satisfy v < w, J K ψ→θ (v, k) ∈ ψ and (v, k) 6∈ θ . Note that at each stage we add (vw,k ,m) for all m, which J K J K ensures that the resulting set is S-invariant and that S is open on each Vn, hence on V . One can then easily check that w0 satisﬁes the same formulas on M ↾ V as it did on M.

Given the lack of the finite model property for persistent posets we need to consider infinite posets, but in view of Lemma 7.4, it suffices to work with countable posets. Fortunately, we may work with a single, ‘universal’ countable poset. Definition 7.5 Define 2

Theorem 7.9 There exists a surjective interior map f : (0, 1) → 2≤N. This immediately gives us a dynamic morphism from an open subset of R to the ≤N

Proof ≤N Let S denote the map on 2∞ as given in Deﬁnition 7.2. By Theorem 7.9, there is a surjective interior map f ′ : (0, 1) → 2≤N, which can be viewed as an interior map f ′′ : (0, 1) → 2≤N × N with range 2≤N ×{0}. Hence Lemma 7.3 tells us that there is a ≤N n ≤N ≤N dynamic morphism f : (0, 1)∞ → 2∞ with range n∈N S [2 ×{0}]=2 × N. Deﬁne S : R → R by S(x)= x + 1, and let U = (0, ∞) \ N (i.e., the set of positive reals that are S not integers). Then it is easily seen that ι: (0, 1)∞ → U given by (x, n) 7→ x + n is an −1 ≤N isomorphism. Thus fι : U → 2∞ is also a surjective dynamic morphism.

Remark 7.11 In fact (Kremer 2013) proves Theorem 7.9 for any complete metric space without isolated points, and shows how the completeness assumption can be dropped using algebraic semantics. Using Kremer’s result, we could generalize Lemma 7.10 to many more spaces, including the rational numbers and the Cantor space. We require only the existence of suitable U and S, so as to be able to adapt the proof.

Proof

′ f is order-preserving: If U ⊆ U ∈ T< and a ∈ f(U) there is ﬁnite b ∈ U such that b ⊑ a, but then b ∈ U ′ and a ∈ f(U ′).

′ g is order-preserving: If V ⊆ V ∈ T≤, then clearly every ﬁnite element of V is a ﬁnite element of V ′, so g(V ) ⊆ g(V ′).

−1 −1

−1 −1 ≤N −1 g ◦ S≤ = S< ◦ g: Take V ∈ T≤ and a ∈ 2 × N. If a ∈ g ◦ S≤ [V ], then a is finite −1 and S≤(a) ∈ V ; but S≤(a) is also finite so S<(a)= S≤(a) ∈ g(V ), hence a ∈ S< ◦ g(V ). −1 −1 Conversely, if a ∈ S< ◦ g(V ), then S<(a) ∈ g(V ), so that S<(a) ∈ V and a ∈ S< [V ], −1 which since a is finite implies a ∈ g ◦ S< [V ]. f preserves invariant sets: If U ∈ T< is S<-invariant and (a,n) ∈ f(U) then there is finite b ⊑ a so that (b,n) ∈ U, hence (b,n + 1) ∈ U, witnessing that S≤(a,n) = (a,n + 1) ∈ f(U), and f(U) is S≤-invariant. g preserves invariant sets: If V ∈ T≤ is S≤-invariant and a ∈ g(U), then a is finite and S<(a) = S≤(a) ∈ V , which since a is finite implies that S≤(a) is finite so that S≤(a) ∈ g(V ).

Putting our results together, we obtain the following.

Theorem 7.13 Every formula of L32 valid over R equipped with an interior map is valid over the class o∩R p of persistent posets; that is, ITL32 ⊆ ITL32. 26 Boudou et al.

Proof If ϕ is falsiﬁable on some persitent poset, then by Corollary 7.7, it is falsiﬁable on 2

Remark 7.14 Compare the situation to that of DTL, where (Fern´andez-Duque 2011) showed that there is a formula that is sound for the class of complete metric spaces with an open map, but not sound for the class of persistent posets. The latter is due to the Baire category theorem; in view of Theorem 7.13, the Baire category theorem does not seem to aﬀect the intuitionistic temporal logic of R.

Figure 7.1 summarizes the inclusions between the semantically-deﬁned logics we have deﬁned. As we will see in the remainder of this paper, these are the only inclusions that hold between said logics.

p ITL32

o∩R ITL32

e R o ITL32 ITL32 ITL32

R2 ITL32

c ITL32

Fig. 7.1: Inclusions between the semantically deﬁned logics; arrows point from the smaller logic to the larger one. See Table 4.1 for the deﬁnitions of the relevant classes of dynamical systems.

8 Independence

In this section we will use our soundness results to show that many of the logics we have considered are pairwise distinct. We begin by showing that the weak logics (based on ) are in fact weaker than their strong counterparts. Indeed, certain key LTL principles are not valid for logics with . Recall that the semantics for and 2 are given in Deﬁnition 4.1.

Proposition 8.1 The following are not valid over R.

1. p → # p, 2. #p → # p, and 3. p → p. Exploring the Jungle of Intuitionistic Temporal Logics 27

Proof Let M = (R, S, · ), where S is deﬁned as follows: J K

0, if x ≤ 0 S(x) = (2x if x> 0 and p = (−∞, 1). Note that S−n( p ) = (−∞, 1/2n), for n ≥ 0. Thus, J K J K S−n p = S−n p = (−∞, 0]. n≥0 J K n≥1 J K T T So, p = #p = (−∞, 0). Hence, # p = p = ∅, and J K J K J K J K p → # p = #p → # p = p → p = (0, ∞) 6= R. J K J K J K

Next, we note that the formulas CD and BI separate Kripke semantics from the general topological semantics.

Proposition 8.2 The formulas CD(p, q) and BI(p, q) are not valid over the class of invertible dynamical + + systems based on R, hence ETL32 6⊢ CD(p, q) and ETL32 6⊢ BI(p, q).

Proof Deﬁne a model M = (R, S, · ) on R, with S(x)=2x, p = (−∞, 1) and q = (0, ∞). Clearly p ∨ q = R, so thatJ 2K (p ∨ q) = R as well. J K J K Let usJ seeK that M, 0 6|= CD(J p, q).K Since M, 0 |= 2(p ∨ q), it suﬃces to show that M, 0 6|= 2p ∨ 3q. It is clear that M, 0 6|= 3q simply because Sn(0) = 0 6∈ q for all n. Meanwhile, by Example 4.6, M, 0 |= 2p if and only if p = R, which is notJ theK case. We conclude that M, 0 6|= CD(p, q). J K To see that M, 0 6|= BI(p, q) we proceed similarly, where the only new ingredient is the observation that M, 0 |= 2(#q → q). But this follows easily from the fact that if M, x |= #q, then x> 0 so that M, x |= q, hence #q → q = R. J K

Remark 8.3

Proposition 8.2 also holds for t(CD(p, q)) and t(BI(p, q)). However, by Proposition 4.9, these are equivalent to their counterparts with 2 over the class of invertible systems, so we do not need to mention them separately. A similar comment holds when working over the class of dynamic posets.

The Fischer Servi axioms are also not valid in general, as already shown in (Balbiani et al. 2019). + From this and the soundness of ITL32 (Theorem 5.3), we immediately obtain that they are not derivable in ITL32. 28 Boudou et al.

p S

4 S S

Fig. 8.1: An expanding poset model falsifying both Fischer Servi axioms. Propositional variables that are true on a point are displayed; only one point satisﬁes p and no point satisﬁes q. It can readily be checked that FS◦(p, q) and FS3(p, q) fail on the highlighted point on the left. Note that S is continuous but not open, as can easily be seen by comparing to Figure 3.1.

Proposition 8.4

1. FS3(p, q) is not valid over the class of expanding posets, hence CDTL32 6⊢ FS3(p, q) and CDTL32 6⊢ FS◦(p, q). 2. FS3(p, q) and t(FS3(p, q)) are not valid over R, hence RTL32 6⊢ FS3(p, q) and RTL32 6⊢ FS◦(p, q).

Proof For the first claim, let us consider the model M = (W, 4,S,V ) defined by 1) W = {w,v,u}; 2) S(w)= v, S(v) = v and S(u) = u; 3) v 4 u; 4) p = {u}, and 5) q = ∅ (see Figure 8.1). Clearly, M,u 6|= p → q, so M, v 6|= p → Jq.K By definition, MJ, wK 6|= 2 (p → q); however, M, w |= 3p → 2q, since the negation of each antecedent holds, so M, w 6|= (3p → 2q) → 2 (p → q). For the second we let (R, S, · ) be a model based on R with S : R → R given by S(x) = 0 for all x, p = (0, ∞),J andK q = ∅. Then we have that 3p = (0, ∞), so that −1 ∈ ¬3p and henceJ K −1 ∈ 3p → 2J qK . However, if U is an S-invariantJ K neighbourhood of −1J thenK 0 = S(−1) ∈ U,J but 0 6∈ Kp → q = (−∞, 0), hence −1 6∈ 2(p → q) . It J K J K follows that −1 6∈ FS3(p, q) . Similar reasoning shows that −1 6∈ t(FS3(p, q)) . J K J K

Remark 8.5 As mentioned previously, (Yuse and Igarashi 2006) present a Hilbert-calculus which yields a sub-logic of ITL2. They also present a Gentzen-style calculus and conjecture that their two calculi prove the same set of formulas. However, (Kojima and Igarashi 2011) show that the formula FS◦(p, q) is derivable in this Gentzen calculus, while Proposition 8.4 shows that it is not derivable in ITL2. Hence, the two calculi are not equivalent.

Now we show that our axioms for Euclidean spaces are not valid in general. In particular, CEM is valid for R, but it is not valid for higher-dimensional spaces. In view of Theorem 6.10, it suﬃces to show that it is not valid over the class of expanding posets.

Lemma 8.6 The formula CEM is not valid on the class of expanding posets, hence CDTL32 6⊢ CEM. Exploring the Jungle of Intuitionistic Temporal Logics 29

Proof Consider the model M = (W, 4, S, · ), where W = {w0, w1, v0, v1, v2}, p = {v2} and J K J K q = {v1, v2}, and for xi,yj ∈ W , xi 4 yj if and only if x = y and i ≤ j, and S(xi)= vi J(seeK Figure 8.2). Then, it is not hard to check that CEM(p, q) = (¬#p ∧ #¬¬p) → (#q ∨ ¬#q) fails at w0.

p, q S

4 S q S

4 4 S w0 S

Fig. 8.2: Model falsifying CEM(p, q) at w0

Similarly, CD−(p), which is valid on all Euclidean spaces, is not valid on all dynamical systems, even those based on Q. Lemma 8.7 The formula CD−(p) is not valid on the class of invertible systems based on Q, hence + − ITL32 6⊢ CD (p). Proof Recall that CD−(ϕ)ψ = 2(p ∨ ¬p) → 3p ∨ 2¬p. Let S be given by S(x)= x + 1. Define a set 1 1 D = Q ∩ n − n+π ,n + n+π n∈N [ 1 and let p = Q \ D. It is readily verified that n+π 6∈ Q for any n ∈ N, and hence J K 1 1 1 1 Q ∩ n − n+π ,n + n+π = Q ∩ n − n+π ,n + n+π , so that D is both closed and open in Q. It follows thath ¬p = D, and hencei p ∨ ¬p = Q; but Q is open and S-invariant, so 2(p ∨ ¬p) = Q as well.J K In particular, 0 ∈J 2(p ∨K ¬p) . Moreover, we claim that J K J K (a) 0 6∈ 3p , but n (b) if x J∈ (0K, 1/2) and n> 1/x, then S (x) ∈ p . J K Indeed, for (a) we see that any n ∈ N, Sn(0) = n ∈ D, while for (b), if x ∈ (0, 1/2) and n> 1/x, then 1 1 1 1 Sn(x)= n + x ∈ n + ,n + ⊆ n + , (n + 1) − , n + π 2 n + π n + π so Sn(x) 6∈ D. If U is an S-closed neighbourhood of 0, U contains some x ∈ (0, 1/2). From (b) it follows that Sn(x) 6∈ ¬p , hence U 6⊆ ¬p ; since U was arbitrary, 0 6∈ 2¬ϕ . J K J K J K The above independence results are sufficient to see that the only non-trivial inclusions between our axiomatic systems are given by Proposition 2.6. 30 Boudou et al.

Theorem 8.8 For each of the following families of axiomatically defined logics (see Table 2.1) or semantically defined logics (see Table 4.1) has pairwise distinct elements, and all subset relations are as indicated in Figures 2.1 or 7.1. + + + 1. ITL32, ITL32, CDTL32, RTL32, ETL32, ETL32, and CDTL32; + + + 2. ITL3, ITL3, CDTL3, RTL3, ETL3, ETL3, and CDTL3; and 2 c o e R R o∩R p 3. ITL32, ITL32, ITL32, ITL32, ITL32, ITL32, and ITL32. The logics in the last item may be replaced by their fragments with only one of or 2. Proof For the first item, each arrow from Λ1 to Λ2 in Figure 8.3 is labelled by a formula which we have previously shown to belong to Λ2 \ Λ1. The same formulas may be used to separate the respective logics in the other two items. The non-trivial subset relations between the logics have been established in Propositions 2.6 and Theorems 6.7, 6.10, and 7.13. We may also classify 3-free logics. Theorem 8.9 For each of the following families of logics, their elements are pairwise distinct, and all subset relations are as indicated in Figures 8.4. + + 1. ITL2, ITL2 , CDTL2, RTL2, and CDTL2 ; + + 2. ITL, ITL, CDTL, RTL, and CDTL, or c o e R p 3. ITL2, ITL2, ITL2, ITL2, and ITL2. The logics in the last item may be replaced by their fragments with only one of or 2. Proof Similar to Theorem 8.8, Figure 8.3 displays formulas separating these logics, except that instances of CD should be replaced by BI.

Remark 8.10 Note that logics characterized by CD− are not included in the statement of Theorem 8.9. In particular, the formula BI(¬p,p) is already valid over the class of all dyamical systems. We do not know if the 3-free logic of dynamical systems based on R2 is diﬀerent from that of all dynamical systems.

9 Concluding Remarks and Future Perspectives

We have proposed a natural ‘basic’ intuitionistic temporal logic, ITL32, along with possible extensions including Fischer Servi or constant domain axioms, and weakened versions obtained by modifying the ﬁxed-point axioms for ‘henceforth’. We have seen that relational semantics validate the constant domain axiom, leading us to consider a wider class of models based on topological spaces, with two possible interpretations for ‘henceforth’: the weak henceforth, , and the strong henceforth, 2. With this, we have shown that + + the logics ITL32, CDTL32, ITL32 and CDTL32 are sound for the class of all dynamical Exploring the Jungle of Intuitionistic Temporal Logics 31

+ CDTL32

◦ FS + ETL32

CD FS◦ ◦ FS CD −

FS◦ CD FS◦ + CDTL32 RTL32 ITL32 CEM CD− CD

CEM FS◦ CD

− CD

◦ ETL32 FS

CD −

ITL32

Fig. 8.3: Graph displaying the dependences among the different logics studied in this paper. Nodes corresponds to different logic while edges mean two different kinds of relation. Edges of the form Λ1 – ϕ → Λ2 mean that Λ1 ⊆ Λ2 and, moreover, ϕ ∈ Λ2 \ Λ1. Edges of the form Λ1 -- ϕ 99K Λ2 mean that Λ2 6⊆ Λ1 and ϕ ∈ Λ2 \ Λ1.

+ + p CDTL2 CDTL ITL2

+ + e R o CDTL2 RTL2 ITL2 CDTL RTL ITL ITL2 ITL2 ITL2

c ITL2 ITL ITL2

Fig. 8.4: Inclusions between 3-free logics. 32 Boudou et al. systems, of all dynamical posets, of all open dynamical systems, and of all persistent dynamical posets, respectively, which we have used in order to prove that the logics + are pairwise distinct. We have also shown that the logics RTL32, ETL32, and ETL32, based on Euclidean spaces, are distinct from any of the above-mentioned logics. We have performed a similar analysis for logics using instead of 2. Of course this immediately raises the question of completeness, which we have not addressed. Speciﬁcally, the following are left open.

Question 1 Are the logics:

(a) ITL32, ITL3, and ITL2 complete for the class of dynamical systems? (b) CDTL32 and CDTL2 complete for the class of expanding posets? + o + (c) ITL32, ITL3 and ITL2 complete for the class of open dynamical systems? R (d) ITL◦ complete for the class of systems based on R? (e) ETL32, ETL3 complete for the class of systems based on Euclidean spaces? + (f) ETL32 complete for the class of systems based on Euclidean spaces with a homeomorphism? + p + (g) CDTL32, ITL3 and CDTL2 complete for the class of persistent posets?

We already know that ITL3 is sound and complete for the class of expanding posets and for Euclidean spaces (Diéguez and Fernández-Duque 2018). However, the completeness + p of ITL3 and ITL3 is likely to be a more difficult problem than that of ITL3, as in these cases it is not even known if the set of valid formulas is computably enumerable, let alone decidable.

Question 2 p o Are any of the logics Λ, Λ3,orΛ2 with Λ ∈{ITL32, ITL32} decidable and/or computably enumerable?

A negative answer is possible for any of these logics, since that is the case for their classical counterparts (Konev et al. 2006) and these logics do not have the ﬁnite model property (Boudou et al. 2017). Nevertheless, the proofs of non-axiomatizability in the classical case do not carry over to the intuitionistic setting in an obvious way, and these remain challenging open problems. o∩R Note that the semantic counterpart for ETL32 used in Theorem 8.8 is ITL32 . We n o∩{R :n≥1} could have used ITL32 instead, as ETL32 is also sound for this class. This raises the following.

Question 3 Is every formula falsiﬁable on some Rn with a homeomorphism also falsiﬁable on R?

Note that we have not considered weak logics with CD or FS. However, this is only due to the fact that the topological semantics we have considered do not yield semantically- deﬁned logics which satisfy the latter axioms without also satisfying ITL32. It may yet be that semantics for such logics may be deﬁned using other classes of dynamical systems. In particular, our techniques do not show whether the weak and standard logics coincide in these cases. Exploring the Jungle of Intuitionistic Temporal Logics 33

Question 4 + 0 Is the logic CDTL32 distinct from ITL32 +CD+FS? We conjecture that an affirmative answer could be given using more general algebraic semantics, but we leave this for future work. Finally, we remark that while we have not considered logics over the full language, it is possible to study logics which combine 2 and . Over dynamic posets or over open dynamical systems such an extension would be uninteresting since both operators are equivalent, but over the class of all dynamical systems, Lemma 4.7 suggests defining def ITL32 = ITL32 + ITL3 + 2p → p. This leaves us with one final question. Question 5 c Is the logic ITL32 decidable, and does it enjoy a natural axiomatization?

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