Maximal Subset of Allen's Relations

• Maximal Subset of Allen’s Relations is the largest set of interval relations that maintain essential properties such as decidability and tractability in temporal reasoning systems.
• It is characterized using complexity-theoretic analysis, automata-theoretic constructions, and algebraic reductions to balance expressiveness with efficient computational handling.
• This concept has practical implications in automated planning, qualitative CSPs, and temporal query processing, guiding robust system design in various domains.

The maximal subset of Allen's relations refers to the largest possible set of interval relations from Allen’s interval algebra that preserves a desirable property within a given computational or logical framework, most commonly decidability or efficient algorithmic handling. The structure and boundaries of this maximal subset differ depending on the semantic context—modal interval logic, constraint satisfaction, automated planning, or temporal query processing. Across these domains, precise characterizations have been obtained via complexity-theoretic analysis, automata-theoretic constructions, and algebraic reductions, leading to a nuanced taxonomy of maximal (and maximally tractable) fragments.

1. Fundamentals of Allen’s Interval Algebra and Maximal Subsets

Allen’s interval algebra consists of 13 atomic binary relations defining all possible qualitative temporal relationships between pairs of closed intervals, including "before" ($\mathbf{p}$), "meets" ($\mathbf{m}$), "overlaps" ($\mathbf{o}$), "during" ($\mathbf{d}$), "starts" ($\mathbf{s}$), "finishes" ($\mathbf{f}$), "equals" ($\equiv$), and their respective inverses. The combinatorial explosion of possible subsets (fragments) enables fine-grained control over expressiveness and complexity in qualitative temporal reasoning systems.

A "maximal subset" in this context is formally a set of Allen’s relations closed under definability (i.e., additional relations are not first-order definable in the subset) for which a targeted algorithmic or logical property (such as decidability of satisfiability) still holds, and for which adding any further relation strictly increases expressiveness at the cost of losing this property (Bresolin et al., 2010).

2. Decidability Borderlines in Interval Temporal Logics

Maximal Decidable Fragments

In interval temporal logic, particularly Halpern–Shoham logic (HS), which provides one modal operator per Allen relation, unrestricted use of all 13 relations leads to undecidability over most linear orders (Bresolin et al., 2010). The prominent result is that the fragment generated by the modalities "after" ($A$), "begins" ($B$), "begun-by" ($\bar{B}$), and "later" ($L$; a form of "before") constitutes a maximal decidable fragment over strongly discrete linear orders (i.e., finite, $\mathbb{N}$, and $\mathbb{Z}$), as adding any other non-definable relation provably leads to undecidability except in finite domains. The decision procedure operates within EXPSPACE, and the lower bound is tight.

Table 1. Maximal Decidable Subset of Allen Relations for Interval Temporal Logic

Relation	Symbol	Included in Maximal Decidable Fragment?
after	$A$	Yes
begins	$B$	Yes
begun-by	$\bar{B}$	Yes
before/later	$L$	Yes
ends/overlaps/during/etc.	–	No

This suggests that careful combinatorial selection of fragment operators enables a sharp balance between expressiveness and decidability.

Expressiveness-Complexity Trade-offs

Adding further relations, such as "meets", rapidly escalates complexity. For instance, equipping the "begins" and "during" fragment with the "meets" operator yields an EXPSPACE-complete fragment (Bozzelli et al., 2021), which is strictly harder than the PSPACE-completeness of "begins" and "during" alone. Thus, maximality is tightly related to detailed complexity-theoretic classifications.

3. Maximal Tractability in Constraint Satisfaction

ORD-Horn and Maximal Tractable Subsets

Within the framework of qualitative CSPs, Allen’s algebra gives rise to a vast taxonomy of tractable and intractable subclasses. Using primitive positive interpretations and pp-homotopy, the ORD-Horn class is established as a maximally tractable subset: any first-order definable extension that adds non-ORD-Horn relations results in NP-completeness of the corresponding CSP (Martin et al., 2018).

Table 2. Maximal Tractable Classes in CSPs

Subset	Algorithms Feasibility	Tractability
ORD-Horn	Polynomial-time	Maximal tractable
General Union	NP-complete	Not tractable

This demonstrates that algebraic–logical closure properties of relation sets play a decisive role in the identification of maximal tractable subsets.

4. Maximal Subsets for Deterministic Timeline-based Planning

Recent research isolates a "maximal subset" of Allen’s relations that allows for the synthesis of plans via deterministic finite automata (DFA) in timeline-based planning models (Acampora et al., 30 Oct 2024, Monica et al., 23 Jul 2025). The relevant property, termed the "eager" condition, requires that the semantics of synchronization rules be checkable on-the-fly by a DFA.

The resulting maximal deterministic subset—i.e., the largest collection of Allen’s relations for which synthesis and plan-existence are both settled by a DFA of singly exponential size—includes precisely those relations where at least one token can serve as an unambiguous "trigger." The following Allen relations (up to inversion) compose this deterministic subset:

• before ($a \ibefore b$): $\tokend(a) < \tokstart(b)$
• meets ($a \meets b$): $\tokend(a) = \tokstart(b)$
• ends ($a \ends b$): $\tokstart(b) < \tokstart(a)$, $\tokend(a) = \tokend(b)$
• starts ($a \starts b$): $\tokstart(a) = \tokstart(b)$, $\tokend(a) < \tokend(b)$
• contains ($a \icontains b$) (inverse of during)
• equals ($a \isequal b$)

Relations such as overlaps and during, by contrast, inherently induce ambiguity and preclude direct DFA-based synthesis.

Table 3. Maximal Eager (Deterministic) Subset for Timeline-based Planning

Allen Relation	Deterministic DFA Synthesis Feasible?
before	Yes
meets	Yes
starts	Yes
ends	Yes
contains	Yes (by inversion)
equals	Yes
overlaps	No
during	No

A plausible implication is that for practical planning requirements under determinism constraints, these relations suffice for expressing a significant range of temporal interaction while avoiding nondeterminism and exponential determinization overhead.

5. Maximal Subsets in Temporal Query Processing and Databases

Query evaluation over intervals in temporal (or genomic) databases exploits the fact that many interval relations—especially those expressing intersection or containment—can be rewritten as multidimensional range queries in $(x, y)$ space, allowing efficient representation and lookup via augmented range trees with fractional cascading (Mao et al., 2018, Piatov et al., 2020). The maximal subset tractable in this paradigm consists of the intersection relations, including:

• overlaps ($o$), overlapped-by ($oi$), during ($d$), contains ($di$), starts ($s$), started-by ($si$), finishes ($f$), finished-by ($fi$), meets ($m$), met-by ($mi$), equals ($=$)

These relations are efficiently supported by geometric data structures for scalable query answering. This maximal subset is significant for high-throughput applications where query time is critical (e.g., population-scale genome annotation).

6. Connections to Parameterized and Multivariate Complexity

In CSPs, the tractability border can also be analyzed via multivariate parameterization, most notably by bounding the maximum number $k$ of overlapping intervals per interval in a network (Eriksson et al., 2022). The subset of Allen’s relations for which this overlap remains bounded yields instances admitting single-exponential algorithms (in the XE class; time roughly $f(k)^n$), whereas in the unrestricted case (full Allen algebra), such tractability is elusive. Thus, maximality here is linked to combinatorial properties enforceable by the chosen relations, as well as by the corresponding tractable parameter regimes.

7. Practical and Theoretical Implications

Comprehensive classification of maximal subsets of Allen’s relations deconstructs the apparent intractability of full interval logics and CSPs into regimes of feasible automated reasoning, verification, constraint satisfaction, and planning. The choice of maximal fragment is essentially dictated by the targeted computational property:

• Decidability (under strong expressiveness) requires tightly controlled fragments like AB$\overline{B}$L (Bresolin et al., 2010).
• Polynomial/CSP tractability is characterized by ORD-Horn (Martin et al., 2018).
• Syntactic determinism for planning strategy synthesis corresponds to unambiguous, eager-fragment Allen relations (Acampora et al., 30 Oct 2024, Monica et al., 23 Jul 2025).
• High-throughput querying exploits the geometric maximal subset (intersection relations) (Mao et al., 2018, Piatov et al., 2020).
• Parameterized complexity delineates maximal subsets with respect to bounded overlap and fixed-parameter tractability (Eriksson et al., 2022, Dabrowski et al., 2023).

The precise delineation of these maximal subsets directly impacts algorithmic design, theoretical limits, and the practical selection of formalism for real-world temporal reasoning systems.