Linear-Time Logic of Here-and-There

• Linear-time HT is the temporal extension of here-and-there logic, integrating a linear progression with a fixed two-world intuitionistic structure.
• The framework uses dynamic posets and persistent frames to ensure monotonic truth and synchronous temporal evolution in the 'here' and 'there' layers.
• It exhibits unique expressive behavior, demonstrating both operator interdefinability and limitations in until-based fragments compared to classical LTL.

Searching arXiv for recent and foundational papers on linear-time logic of here-and-there and related HT/QHT background. Linear-time logic of here-and-there is the temporal extension of the logic of here-and-there obtained by combining a linear temporal progression with the characteristic two-world intuitionistic structure of HT. In the notation of "Intuitionistic Linear Temporal Logics," it is the logic $\mathsf{ITL}_{HT}$, interpreted over temporal frames in which each time instant contains exactly two ordered worlds, “here” and “there,” and temporal evolution proceeds synchronously in both layers (Balbiani et al., 2019). As a temporal specialization of HT, it inherits the persistence-based semantics of intuitionistic Kripke models while adding the operators of linear temporal logic; as a specialization of intuitionistic temporal logic, it constrains the intuitionistic dimension to the fixed two-point HT frame. This places it at the intersection of temporal logic, intermediate logics, and the semantic foundations of equilibrium logic and answer-set programming, for which static HT is a central reference logic (Otten et al., 7 Jan 2026).

1. Static HT background and the temporal extension

The non-temporal logic of here-and-there is an intermediate logic between intuitionistic logic and classical logic. In the first-order setting, it has the same syntax as classical and intuitionistic first-order logic, with atomic formulas $P(t_1,\ldots,t_n)$, connectives $\wedge,\vee,\rightarrow,\neg$, and quantifiers $\forall x$, $\exists x$; semantically, it is given by an intuitionistic Kripke structure with two worlds $h$ and $t$, a constant individual domain, and a reflexive and transitive accessibility relation between those worlds (Otten et al., 7 Jan 2026). The characteristic semantic condition is that a formula true in $h$ is also true in $t$, while implication and negation are evaluated with respect to both worlds. HT is strictly weaker than classical logic, since it does not validate excluded middle $A\vee \neg A$, and strictly stronger than intuitionistic logic, since it validates the weak law of excluded middle $P(t_1,\ldots,t_n)$0 (Otten et al., 7 Jan 2026).

Linear-time HT preserves the local two-world structure but adds a temporal axis. The relevant temporal setting is not an arbitrary modal or branching-time semantics, but a deterministic temporal transition on a Kripke-style order. The general framework begins with dynamic posets $P(t_1,\ldots,t_n)$1, where $P(t_1,\ldots,t_n)$2 is a poset and $P(t_1,\ldots,t_n)$3 is an order-preserving successor function. Temporal here-and-there frames are then obtained as a very specific subclass in which each time slice is the two-point HT frame and time advances uniformly in both layers (Balbiani et al., 2019). In this sense, linear-time HT is the temporal reiteration of the classical HT frame $P(t_1,\ldots,t_n)$4 with $P(t_1,\ldots,t_n)$5, now indexed by time.

This construction is significant because it separates two dimensions that are often conflated: the intuitionistic order and the temporal transition. In static HT, the order has only the “here”/“there” distinction. In $P(t_1,\ldots,t_n)$6, that same distinction is preserved at every temporal stage, while the temporal component is handled by a function rather than by a branching accessibility relation. A plausible implication is that many static HT phenomena survive pointwise in time, but operator interaction introduces new expressive behavior that is not reducible to the non-temporal case.

2. Semantic structures, persistence, and temporal HT frames

The semantic hierarchy developed for intuitionistic temporal logic consists of dynamic posets, persistent frames, and temporal here-and-there frames (Balbiani et al., 2019). A dynamic poset is a structure

$P(t_1,\ldots,t_n)$7

with forward confluence: $P(t_1,\ldots,t_n)$8 Valuations are monotone in the sense that $P(t_1,\ldots,t_n)$9. Proposition 1 states the key equivalence: $\wedge,\vee,\rightarrow,\neg$0 is forward confluent if and only if truth of every formula is monotone with respect to $\wedge,\vee,\rightarrow,\neg$1 (Balbiani et al., 2019). This gives the semantic basis for persistence of truth in the temporal setting.

Persistent frames strengthen dynamic posets by imposing both forward and backward confluence. The backward condition is formulated as follows: if $\wedge,\vee,\rightarrow,\neg$2, there exists $\wedge,\vee,\rightarrow,\neg$3 such that $\wedge,\vee,\rightarrow,\neg$4 (Balbiani et al., 2019). Temporal here-and-there frames form a special subclass of persistent frames: $\wedge,\vee,\rightarrow,\neg$5 with order

$\wedge,\vee,\rightarrow,\neg$6

and temporal successor

$\wedge,\vee,\rightarrow,\neg$7

for some function $\wedge,\vee,\rightarrow,\neg$8 (Balbiani et al., 2019). Thus the order never compares different times; it only compares the “here” and “there” layers at the same time point.

This semantics makes explicit what “linear-time here-and-there” means. The temporal axis is linear in the sense that the successor is functional, and the intuitionistic axis is fixed to two levels. Because $\wedge,\vee,\rightarrow,\neg$9 acts layerwise,

$\forall x$0

time advances synchronously in both worlds (Balbiani et al., 2019). The result is a semantics in which intuitionistic persistence is local to each time slice, while temporal evolution is shared across slices. This sharply constrains the models compared with arbitrary persistent posets, and that constraint is central to the logic’s expressivity results.

3. Language and interpretation of temporal operators

The language of $\forall x$1 contains propositional variables, $\forall x$2, the intuitionistic connectives $\forall x$3, and the temporal operators next, eventually, henceforth, until, and release (Balbiani et al., 2019). Negation is an abbreviation: $\forall x$4 Satisfaction is defined inductively on dynamic models $\forall x$5. The clauses for conjunction and disjunction are standard. The clause for implication is intuitionistic: $\forall x$6 Thus implication quantifies over all worlds above the current one in the intuitionistic order, not merely over the current state.

The temporal operators are interpreted by iterating the successor function. For next,

$\forall x$7

For eventually and henceforth,

$\forall x$8

$\forall x$9

For until and release,

$\exists x$0

$\exists x$1

These clauses are intuitionistic-temporal rather than classical-temporal: implication is evaluated against the order, and temporal truth must respect persistence.

Several formulas that are familiar from classical temporal logic remain valid already on arbitrary dynamic posets: $\exists x$2 The paper also records two valid identities for $\exists x$3: $\exists x$4

$\exists x$5

(Balbiani et al., 2019). These equivalences are important because they show that, despite the failure of classical dualities in intuitionistic settings, some operator interdefinability persists.

4. Expressive behavior and definability phenomena

A distinguishing feature of linear-time HT is that temporal expressivity differs from both classical LTL and more general intuitionistic temporal logics. Over HT frames, $\exists x$6 is definable in terms of $\exists x$7 and implication. The defining formula uses

$\exists x$8

$\exists x$9

and Proposition $h$0 states that

$h$1

over here-and-there frames (Balbiani et al., 2019). This result corrects a prior claim attributed to Balbiani–Diéguez: the paper explicitly states that the earlier claim that $h$2 is not definable from $h$3 over HT models is false.

The definability pattern is asymmetric. Although $h$4 is definable from $h$5 over HT models, $h$6 is not definable in terms of $h$7, even over finite here-and-there models. The paper proves: $h$8 (Balbiani et al., 2019). The proof uses bounded bisimulations and a family of finite HT models in which states are indistinguishable by $h$9-formulas of bounded size but differ with respect to $t$0. This establishes a strict expressive limitation for until-based fragments.

The contrast with persistent models is also sharp. Over persistent frames, $t$1 is not definable from $t$2, even though it is definable over HT frames (Balbiani et al., 2019). This shows that the two-level HT constraint is not merely a simplification of general persistence semantics; it changes expressivity in substantive ways. A plausible implication is that the fixed local frame $t$3 permits formulas to discriminate between the lower and upper intuitionistic layers in ways unavailable on arbitrary persistent posets.

The paper further notes that some classical frame-characterizing formulas fail intuitionistically. For example,

$t$4

is not valid in the full intuitionistic temporal logic, even though classically it characterizes weakly connected frames (Balbiani et al., 2019). Linear-time HT therefore sits within an expressivity landscape in which classical correspondences cannot be transferred mechanically.

5. Relation to answer-set semantics and quantified HT

The logic of here-and-there has a well-established role in answer-set programming. In the propositional setting, strong equivalence of programs is exactly equivalence in HT: two formulas are strongly equivalent if and only if they are satisfied by the same HT-interpretations (Lifschitz, 2021). HT-interpretations are pairs $t$5 with $t$6, and the nonclassical clause for implication is the key semantic feature distinguishing HT. This connection underlies proof-theoretic methods for establishing strong equivalence by translating rules into formulas and deriving equivalence in HT.

The broader semantic framework for ASP equivalences extends from propositional HT to quantified HT. In the first-order setting, QHT-interpretations are triples

$t$7

where $t$8 interprets the domain and function symbols, and $t$9 are the “here” and “there” components (Fink, 2010). Equilibrium models and answer sets are then defined by minimality of total interpretations $h$0. The same paper shows that strong equivalence, uniform equivalence, answer-set equivalence, classical equivalence, and relativized hyperequivalence can be characterized via HT/QHT countermodels and equivalence interpretations, and that the framework lifts to first-order theories and non-ground programs (Fink, 2010).

This static and quantified background is not itself a temporal theory, but it is essential context for linear-time HT. The temporal logic $h$1 retains the same local “here/there” semantics while introducing the successor-based temporal dimension. This suggests continuity with the semantic role of HT in equilibrium logic, but the available results in the cited papers concern the non-temporal setting: strong equivalence, quantified countermodel characterizations, and arithmetic enrichments of HT, not a temporal equilibrium logic as such. Accordingly, linear-time HT is best understood as a temporal extension of a logic that already has deep semantic and proof-theoretic importance in ASP.

6. Metatheory, scope, and unresolved issues

The metatheoretic situation is uneven across the surrounding logics. For the full intuitionistic temporal logic over dynamic posets, denoted $h$2, the paper proves the effective finite model property and hence decidability (Balbiani et al., 2019). By contrast, $h$3, the logic over persistent posets, does not have the finite model property (Balbiani et al., 2019). Temporal here-and-there logic $h$4 is presented as a special case of persistent semantics, but the paper does not provide a full axiomatization or a completeness theorem for $h$5, and it does not establish decidability or a finite model property specifically for the HT fragment (Balbiani et al., 2019).

This boundary is important. The paper’s main positive technical results for the HT fragment are semantic placement within the larger hierarchy and precise definability and non-definability theorems. Its broader contribution is to show how the interaction between intuitionistic persistence and linear time changes operator behavior. What is preserved from static HT is the two-world order and the monotonicity of truth from “here” to “there.” What changes is that temporal progression is encoded by a deterministic function compatible with that order, so formulas may express constraints that mix local HT persistence with temporal iteration.

From the standpoint of proof theory, the static logic of HT is comparatively well developed. There is a sound and complete sequent calculus $h$6 for first-order HT, together with an axiomatic embedding into intuitionistic logic, native proof search, and implemented theorem provers (Otten et al., 7 Jan 2026). No corresponding proof-theoretic package is provided for linear-time HT in the cited temporal work. A plausible implication is that temporal HT remains less settled proof-theoretically than static HT: the semantics and expressivity are relatively explicit, while axiomatization and decision procedures for the specific HT temporal fragment remain open in the available account.

Overall, linear-time logic of here-and-there is best characterized as a constrained intuitionistic temporal logic whose models are persistent two-layer temporal frames. Its importance lies in the way it imports the two-world HT perspective into linear time, yielding a semantics that is more structured than general intuitionistic temporal models and expressive phenomena that differ from both classical LTL and general persistent-frame logics (Balbiani et al., 2019).