Handling Unbounded Temporal Operators

• Handling unbounded temporal operators is a set of techniques that define and process infinite-time modalities like always, eventually, and until within temporal logics.
• The methods utilize syntactic reductions, automata encodings, and grounding strategies to ensure tractability and domain independence in complex temporal logic programs.
• Advanced techniques such as event-freezing operators and operator splitting provide rigorous error analysis and efficient computational handling of infinite behaviors.

Handling unbounded temporal operators entails rigorous semantic, syntactic, and algorithmic techniques that render formulas involving modalities like "always," "eventually," and "until" tractable within both logic programming and specification contexts. This article surveys the central methods for coping with such operators, focusing on temporal logics, temporal equilibrium semantics, syntactic reductions, automata-based reasoning, and computational frameworks.

1. Formal Semantics of Unbounded Temporal Operators

Unbounded operators are those quantifying over potentially infinite time horizons. In Linear-time Temporal Logic (LTL), the operators $G$ ("always"), $F$ ("eventually"), and $U$ ("until") are interpreted as follows on an infinite trace $\sigma=(M,\tau,\mu)$:

• $G\varphi$: true iff for all $t' \geq t$, $\sigma, t' \models \varphi$.
• $F\varphi$: true iff there exists $t' \geq t$ such that $\sigma, t' \models \varphi$.
• $A U B$: true iff there exists $t' \geq t$ such that $\sigma, t' \models B$ and for all $t \leq u < t'$, $\sigma, u \models A$.

In modal extensions like Temporal Equilibrium Logic (TEL), unbounded operators are handled via minimality conditions over here-and-there traces, with "temporal stable models" corresponding to total equilibrium structures selecting minimal LTL models (Aguado et al., 2016). The satisfaction of unbounded modalities proceeds as in classical LTL for infinite traces, but equilibrium selection adds an extra minimization layer.

2. Syntactic Transformations and Negation-Free Reduction

Recent work demonstrates that universal temporal operators in fragments like Metric Temporal Logic (MTL)—notably negation-free MTL—can be eliminated in favor of strictly existential and binary primitives:

• Any instance of $G^{+}_I A$ ("always in future over interval $I$") is expressible via compositions of "once" ($\Diamond$) and "until" ($U$) operators, with equivalences such as $$G^{+}_I A \equiv \Diamond^{+}_{[i_1,i_1]}(A U_{[i_2-i_1,i_2-i_1]}\top)$$ for punctual intervals, or explicit overlap-based rewrites for non-punctual intervals to stay in the PSPACE-complete MITL fragment (Noort et al., 12 Sep 2025).
• "Once" operators themselves can be compiled away: $\Diamond^{+}_I A \equiv \top U_{I} A$.
• Thus, a negation-free temporal engine need only support $U$ and $S$ (since), vastly reducing syntactic and algorithmic complexity.

This transformation is particularly valuable in open systems without reliable negation as failure, as it preserves monotonicity and tractability.

3. Grounding and Domain Independence in Temporal Logic Programs

In rule-based settings (TEL, ASP extensions), unbounded operators pose challenges for grounding—the process of generating all possible rule instances over the domain:

• The subclass of "splittable temporal logic programs" restricts operators to $X$ ("next"), $G$, and $F$, and excludes general $U$ (Aguado et al., 2016).
• A syntactic safety condition on variables (every variable appears in some positive atom in rule body) guarantees domain independence: the set of temporal equilibrium models remains unchanged when the domain is arbitrarily enlarged.
• A finite ASP grounding is realized by collapsing rules beyond time $i=2$, computing a least model $\Delta$ via a non-temporal positive program over at most three time slices, and pruning the infinite grounding to only those rule instances relevant for any temporal stable model.

This method ensures that infinite behaviors induced by unbounded operators are handled via finite, compositional calculation—reinstating feasible model generation even in the presence of infinite time or domains.

4. Event-Freezing Operators and Elimination Techniques

Extensions of first-order LTL with "event freezing" functions introduce explicit mechanisms to refer to the value of a term at the next or previous occurrence of a formula, thus directly internalizing potentially unbounded dependencies:

• Operators $u@_{\scriptscriptstyle F}(\varphi)$ and $u@_{\scriptscriptstyle P}(\varphi)$ denote "value of $u$ at the next/previous point where $\varphi$ holds".
• These generalize Event-Clock modalities and allow encoding unbounded $U$/$S$ and metric variants (Tonetta, 2017).

A key algorithmic innovation is the prophecy variable elimination: all event-freezing expressions are replaced by fresh variables and LTL constraints, yielding a propositional LTLNext formula equivalent (equisatisfiable) to the original. Discretization methods further ensure correctness across dense and super-dense time models.

5. Deduction Systems for Unbounded "Until": The History Operator

The "until" operator is both existential and universal in nature—"exists a future point where $B$, for all points until then $A$ holds". This duality complicates deduction systems. The introduction of a unary "history" operator $\nabla$ solves this by packaging the universal component into $\nabla$ (0910.4500):

• $\nabla A$ is true on an interval if $A$ holds at every point in that interval.
• $A U B$ is re-expressed as $B \lor F(XB \wedge \nabla A)$, decoupling the existential and universal parts.
• Finitary labeled natural deduction is achieved, with uniform intro/elimination rules restoring the subformula property and enabling normalization proofs.

This methodology generalizes to other temporal logics, including branching-time and interval-based frameworks.

6. Automata-Based Evaluation of Unbounded Temporal Properties

For both finite and infinite traces, automata-theoretic methods are central in reasoning about formulas with unbounded operators:

• For classical LTL, unbounded $G/F/U$ are compiled into Büchi automata for infinite words and NFAs/alternating automata for finite words (Aguado et al., 2020).
• Temporal equilibrium models are realized by automata intersection: accepting only those models for which no strictly smaller assignment satisfies the formula.
• Second-order LTL and QLTL encodings establish the complexity bounds and form the basis for effective algorithms (ExpSpace for TELf).

Automata-based approaches are prevalent in temporal answer set programming toolchains (e.g., TELINGO).

7. Computational Methods: Splitting Unbounded Operators in Analysis

Unboundedness is also addressed in operator splitting for numerical computation in Banach spaces with unbounded generators:

• Classical Taylor expansions for splitting methods break down for unbounded operators due to insufficient domain regularity (Iserles et al., 22 Mar 2024).
• Duhamel's formula provides an exact error expression for splitting schemes involving unbounded components, requiring only bounded commutator conditions.
• The approach generalizes to arbitrary numbers of components and higher-order splittings through nested Duhamel integrals, while maintaining rigorous error bounds.

This technique preserves the validity and convergence of splitting schemes even in infinite-dimensional, unbounded contexts.

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In summary, handling unbounded temporal operators in logic programming, specification, and computational analysis encompasses semantic decoupling, syntactic elimination, safety and grounding paradigms, automata encodings, interval quantification strategies, and advanced error analysis—all of which together ensure tractability, correctness, and expressive completeness for infinite time behaviors, large domains, and complex system properties in temporal logic frameworks (Aguado et al., 2016, Tonetta, 2017, 0910.4500, Noort et al., 12 Sep 2025, 0806.2802, Iserles et al., 22 Mar 2024, Aguado et al., 2020).