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Definability of Functional Properties in the Basic Modal-Temporal Language over Ordered Frames

This paper analyzes the expressive power of a basic modal-temporal language over various ordered frames, demonstrating that while the language struggles to define functional properties in general multiflow settings due to uncontrolled functional multiplicity, restricting semantics to minimal functional frames or uniform domains significantly enhances definability, though the lack of connectivity remains a fundamental obstacle in non-linear orders.

Original authors: Alfredo Burrieza

Published 2026-07-02

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CC BY 4.0

Original authors: Alfredo Burrieza

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to figure out the rules of a mysterious game played by invisible messengers. These messengers travel between different "worlds" (or points in time and space), carrying messages. Your goal is to write a single, perfect rulebook (a logical formula) that can describe exactly how these messengers behave.

The paper by Alfredo Burrieza is an investigation into how good our rulebook is at describing specific behaviors of these messengers. The behaviors we care about are things like:

Totality: Does every starting point have a messenger?
Injectivity: Do two different starting points ever send messengers to the same destination? (No duplicates allowed).
Surjectivity: Does every destination get at least one messenger?
Monotonicity: Do messengers always move forward in a consistent direction?
Constancy: Do all messengers from a specific spot go to the exact same place?

The paper tests our rulebook in two main scenarios: the "Chaotic City" and the "Quiet Village."

1. The Chaotic City (The Original Setting)

Imagine a huge, crowded city where thousands of messengers are running around at once. You can see them all, but you can't tell which messenger belongs to which route. They all mix together in a big pile.

The Problem: In this chaotic city, our rulebook is very weak. It's like trying to describe the behavior of a single ant in a massive anthill just by looking at the whole hill.
The Result: The paper finds that in this setting, we can only successfully describe two things: Totality (is the hill full?) and Surjectivity (are all the exits covered?).
The Failure: We cannot describe if the messengers are unique (Injectivity), if they move in a straight line (Monotonicity), or if they stay in one spot (Constancy). The chaos of having too many messengers at once "blurs" the picture so much that the specific rules get lost. It doesn't matter if the city is a straight line or a messy web; the noise is too loud.

2. The Quiet Village (Minimal Frames)

Now, imagine we shrink the city down to a tiny, quiet village with only two houses and exactly one messenger running between them. We remove all the noise and confusion.

The Improvement: Suddenly, our rulebook becomes much sharper. Because there is only one messenger, we can finally see their specific habits.
The New Success: In this quiet village, we can now define Monotonicity (do they move forward?) and Antitonicity (do they move backward?) in almost any type of village layout. We can also define Constancy (do they always go to the same spot?) if the village is laid out in a straight line.
The "Strict" Glasses: The paper also tests wearing "strict glasses" (ignoring the current moment and only looking at the future/past). With these glasses in the quiet village, we can even define Injectivity (uniqueness) in straight-line villages. It's like the strict glasses help us ignore the "self" and focus purely on the path ahead.

3. The "Hard Core" Mystery

Even in the quiet village, there is a limit. The paper discovers a "Hard Core" of behaviors that remain impossible to define if the village layout is messy (non-linear).

The Obstacle: If the village has branches or dead ends (like a tree or a web) rather than a single straight road, we still cannot define Totality, Surjectivity, Injectivity, or Constancy.
The Reason: The rulebook relies on "connectivity." It needs a straight line to trace a path. If the path splits or stops, the rulebook gets confused. The lack of a single, continuous line is the fundamental wall that stops the rulebook from working, no matter how quiet the village is.

4. The "Uniform Domain" Shortcut

The paper also checks a third scenario: a city where, even though there are many messengers, they all start from the exact same set of houses. This is called the "Uniform Domain."

The Surprise: This setup behaves exactly like the "Quiet Village." Even though there are many messengers, because they all start from the same place, the rulebook can "see" them as if they were just one. This proves that the problem in the "Chaotic City" wasn't the messengers themselves, but the fact that they started from different, confusing places.

The Big Takeaway

The paper concludes that our logical language is actually quite powerful, but it gets blinded by structural noise.

Too many paths (Multiflow): If you have too many messengers starting from different places, you can't describe their specific rules.
Too much branching (Non-linear): Even if you simplify the messengers, if the map itself is a messy web instead of a straight line, you still can't describe the most basic rules (like "is everyone covered?" or "is everyone unique?").

The paper essentially maps out exactly where our logical tools work and where they hit a wall, showing that the wall is caused by the shape of the world (the order) and the confusion of having too many actors, not by the tools themselves being weak.

Technical Summary: Definability of Functional Properties in the Basic Modal-Temporal Language over Ordered Frames

Problem Statement
This paper investigates the expressive power of the basic modal-temporal language $LT \times W$ , formed by adding Prior's temporal operators $G$ (always in the future) and $H$ (always in the past) to the basic modal language with $\Box$ (in all functional images). The study focuses on the definability of nine specific functional properties—totality, non-totality, injectivity, surjectivity, monotonicity, strict monotonicity, antitonicity, strict antitonicity, and constancy—across five distinct order types: preorders (PRE), strict preorders (sPRE), partial orders (PO), linear orders (LO), and strict linear orders (sLO).

A central challenge addressed is the "multiflow" setting, where multiple accessibility functions coexist, and modal operators quantify indiscriminately over their combined images. Previous work (e.g., [1, 3]) had restricted analysis to strict linear orders and global constraints like the Uniform Domain (U-Dom) property. This paper aims to determine whether these properties are definable in isolation and how their definability varies across a broader spectrum of order structures, specifically examining the impact of two semantic readings: the standard interpretation (including the current instant) and the strict interpretation (excluding the current instant).

Methodology
The authors employ a two-stage analytical approach to map the expressive boundaries of $LT \times W$ :

Arbitrary Ordered Multiflow Setting: The study first establishes definability limits in the general setting where frames consist of arbitrary families of functions. Here, modal operators act as second-order quantifiers over the union of all function images. The authors utilize surjective p-morphisms (which induce functional bisimulations) to prove undefinability. If a p-morphism maps a frame satisfying a property to one that does not, the property is undefinable.
Structural Refinement (Minimal Frames): To isolate the source of expressive limitations, the authors introduce "minimal functional frames" (the $O_2$ family), restricting frames to at most two flows and a single accessibility function. This reduction collapses the second-order complexity of the multiflow setting into a first-order relational reading.
Comparative Analysis: The study compares the results of minimal frames against two other frameworks:
- Indexed Languages: Where operators are explicitly labeled to distinguish specific functions.
- Uniform Domain (U-Dom) Frames: Where all functions in a frame share a common domain.
Semantic Variations: The analysis is conducted under both standard ( $G, H$ ) and strict ( $G^*, H^*$ ) interpretations to determine if excluding the current instant alters the definability landscape.

Key Contributions and Results

Expressive Limits in the Multiflow Setting:
In the general multiflow setting, the language is significantly weak. The study proves that only totality and surjectivity are definable, and only within linear orders (LO and sLO). All other properties (injectivity, monotonicity, constancy, etc.) are undefinable in any order type, including linear ones. This is attributed to the "indiscriminate quantification" over multiple functions, which obscures individual functional behaviors.
The Impact of Minimal Frames:
By restricting the semantics to minimal frames (single function), the definability landscape expands dramatically:
- Monotonicity and Antitonicity: These become definable in all order types (PRE, sPRE, PO, LO, sLO).
- Injectivity: Becomes definable in strict linear orders (sLO) under the standard interpretation.
- Strict Monotonicity/Antitonicity: Become definable in strict preorders and strict linear orders.
- Constancy: Remains undefinable in non-linear orders (PRE, sPRE, PO) even in minimal frames, requiring a distinct semantic equivalence argument (as it is preserved by p-morphisms).
The Role of the Strict Interpretation:
In the general multiflow setting, the choice between standard and strict interpretations yields identical results. However, in minimal frames, the strict interpretation ( $G^*, H^*$ ) provides crucial gains:
- It enables the definition of injectivity in linear orders (LO) where the standard interpretation fails.
- It allows the definition of strict monotonicity and strict antitonicity in all order types, including non-strict ones (PRE, PO, LO), by effectively treating them as their strict counterparts.
Equivalence of Frameworks:
The paper demonstrates that the definability patterns obtained in minimal frames are identical to those in indexed languages (where functions are syntactically distinguished) and Uniform Domain (U-Dom) frames. This indicates that the expressive limitations of the original framework stem not from the weakness of the operators themselves, but from the uncontrolled multiplicity of functions with divergent domains. Once functional multiplicity is controlled (via structural simplification, domain uniformity, or syntactic indexing), the basic operators recover their full capacity.
Fundamental Obstacles:
Even after controlling functional multiplicity, a "hard core" of properties—totality, non-totality, injectivity, surjectivity, and constancy—remains undefinable in all non-linear orders (PRE, sPRE, PO). The authors conclude that the lack of connectivity (incomparability of points) is a fundamental obstacle that temporal modalities cannot overcome, regardless of the interpretation or structural refinement.

Significance
The paper provides a systematic classification of the expressive limits of the basic modal-temporal language over ordered frames. Its primary significance lies in identifying that the inability to define fundamental functional properties in the original multiflow setting is a structural artifact of uncontrolled functional multiplicity, not an inherent flaw in the temporal operators.

By showing that minimal frames, indexed languages, and U-Dom frames yield identical definability patterns, the work clarifies the precise conditions under which functional properties become definable. It establishes that while structural simplification can recover the definability of monotonicity and antitonicity across all orders, the lack of linear connectivity remains a persistent barrier for properties requiring global domain comparisons (like totality and injectivity) in non-linear structures. The study also highlights the special status of constancy, which resists standard undefinability proofs via p-morphisms in minimal frames, requiring a unique semantic equivalence argument.