Divisible Updating Rules

• Divisible updating rules are mechanisms that partition global updates into structured subupdates (across agents, times, or logical strata) while maintaining mathematical invariants.
• They encompass diverse methods such as agent-wise partitioning, time-wise mixing, path-independent conditioning, and stratified modular updates to facilitate controlled recomposition.
• Applications span evolutionary game theory, probabilistic belief updating, hybrid knowledge bases, and combinatorial arithmetic, offering insights into maintaining system coherence.

Divisible updating rules are update mechanisms in which a global change is decomposed into structured subupdates across agents, times, strata, events, or combinatorial components. In the literature, this includes fixed coexistence of multiple strategy-updating rules in evolutionary games, annealed versus quenched mixtures of local-selection rules, path-independent conditioning of ambiguous beliefs, splitting-sequence semantics for hybrid knowledge bases, randomized deletion for $k$-divisible partitions, and incremental modular updates in factoradic arithmetic. This suggests a broad technical notion of divisibility: update operations are partitioned into components whose recomposition is itself mathematically controlled (Takesue, 2019, Szolnoki et al., 2018, Slota et al., 2011, Gong et al., 2017, Oliver et al., 21 Feb 2025).

1. Formal senses of divisibility

In the cited literature, divisibility does not denote a single formal property. It refers instead to several mathematically distinct ways of partitioning an update process. Some papers divide update rules across agents, others across time, others across evidence sequences, logical strata, or combinatorial prefixes. The central question is whether the partition preserves a target object: a stationary cooperation level, a posterior belief, an MKNF model, a phase classification, a partition law, or a modular remainder.

Form of divisibility	Core mechanism	Representative formulation
Agent-wise partition	Fixed rule types across individuals	$p$ aspiration-based, $1-p$ imitators
Time-wise mixing	Rule chosen at each event	$\Pr(\mathrm{DB})=q$, $\Pr(\mathrm{imitation})=1-q$
Path-independent conditioning	Sequential update equals one-shot update	$U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$
Stratified modular update	Layer-wise reducts and intersections	$M=\bigcap_{\alpha<\mu} X_\alpha$
Incremental modular update	Prefix-wise remainder recursion	$R_j\equiv R_{j-1}+a_j w_j \pmod{k}$

Two recurrent axes are especially prominent. The first is whether divisibility is spatial or agent-wise, as in fixed rule assignments, grouped entities, or distributed local controllers. The second is whether divisibility is temporal or sequential, as in annealed mixing, update schemas for cellular automata, staged communication, and repeated conditioning. A third axis is semantic modularity: splitting the update problem into strata or event refinements and asking whether the recomposed result is equivalent to the direct update (Szolnoki et al., 2018, Reia et al., 2015, Slota et al., 2011, Gong et al., 2017, Crane et al., 2011, Oliver et al., 21 Feb 2025).

2. Fixed heterogeneity and time-wise rule mixing in evolutionary games

A canonical agent-wise notion of divisibility appears in evolutionary Prisoner’s Dilemma models with heterogeneous strategy-updating rules. On an $L\times L$ square lattice with periodic boundary conditions and Moore neighborhood $k_i=8$, agents play the weak Prisoner’s Dilemma with

$p$0

A fraction $p$1 of agents is assigned aspiration-based updating and the remaining $p$2 uses imitation-based updating; these assignments are permanent throughout a run, so heterogeneity is not a per-step mixture but a fixed partition of the population. Imitators adopt a randomly chosen neighbor’s strategy with Fermi probability

$p$3

whereas aspiration-based agents switch to the opposite strategy with probability

$p$4

Mutation acts independently with probability $p$5 per agent per period and overwrites the strategy by a random choice in $p$6 with equal probability (Takesue, 2019).

In that model, divisibility is quantified by the aspiration share $p$7 and evaluated through the nonlinear interaction metric

$p$8

Introducing aspiration-based agents into imitator populations generally deteriorates cooperation for small or moderate mutation rates, often yielding $p$9. Mutation also lowers $1-p$0, and imitator-dominated populations are especially vulnerable because mutation breaks cooperative clusters maintained by imitation. The most notable exception occurs under high mutation and low aspiration: for $1-p$1 and $1-p$2, $1-p$3 becomes concave and $1-p$4 across broad ranges of $1-p$5; for $1-p$6 and also $1-p$7, the reported phase plots show $1-p$8 often exceeding $1-p$9 over substantial regions of $\Pr(\mathrm{DB})=q$0. In that regime, low-$\Pr(\mathrm{DB})=q$1 aspiration agents act as persistent cooperators and raise both $\Pr(\mathrm{DB})=q$2 and $\Pr(\mathrm{DB})=q$3 (Takesue, 2019).

A different decomposition is studied in the coexistence of imitation and Death–Birth updating on square lattices of degree $\Pr(\mathrm{DB})=q$4. There the same two rules can be mixed in two sharply different ways. In the annealed case, rule choice is time-wise: $\Pr(\mathrm{DB})=q$5 In the quenched case, each site is permanently assigned $\Pr(\mathrm{DB})=q$6 with $\Pr(\mathrm{DB})=q$7, so the population is spatially partitioned into two subdynamics. The model further attaches coevolving learning or teaching activity to players, with imitation probabilities scaled by $\Pr(\mathrm{DB})=q$8 or $\Pr(\mathrm{DB})=q$9 (Szolnoki et al., 2018).

The macroscopic effect depends primarily on the form of divisibility. Under annealed mixing, small DB weight $\Pr(\mathrm{imitation})=1-q$0 can already produce strong synergy, and with $\Pr(\mathrm{imitation})=1-q$1 available, full cooperation can be reached at relatively high $\Pr(\mathrm{imitation})=1-q$2; high-learning types are eliminated and only low-learning types survive in the coexistence phase. Under quenched mixing, by contrast, the paper reports no synergy between rules at the microscopic level: the system behaves as two coupled subsystems, cooperation appears only when DB sites percolate, and the actual values of learning or teaching activity become irrelevant. Divisibility across time is therefore dynamically generative in a way that divisibility across agents need not be (Szolnoki et al., 2018).

3. Path independence, ambiguous beliefs, and conditional updating

In imprecise-probability and ambiguity models, divisibility is usually defined as path independence of conditioning. For events $\Pr(\mathrm{imitation})=1-q$3 and $\Pr(\mathrm{imitation})=1-q$4, the rule is divisible if updating first on $\Pr(\mathrm{imitation})=1-q$5 and then on $\Pr(\mathrm{imitation})=1-q$6 yields the same posterior as updating once on $\Pr(\mathrm{imitation})=1-q$7: $\Pr(\mathrm{imitation})=1-q$8 This notion is especially explicit for lower probabilities, Choquet capacities, and set-valued posteriors (Gong et al., 2017).

Relative Maximum Likelihood (RML) updating starts from a convex, closed set of priors $\Pr(\mathrm{imitation})=1-q$9, identifies the maximum-likelihood subset

$U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$0

and contracts toward it by

$U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$1

before applying Bayes’ rule to each prior in $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$2. Full Bayesian updating is the special case $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$3, and Maximum Likelihood is $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$4. The rule is not divisible in general because direct update on a refined event $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$5 uses $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$6, whereas sequential update through $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$7 uses $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$8; unless these maximizing subsets align, the direct and sequential posterior sets differ. Full Bayesian updating is divisible, while Maximum Likelihood is divisible only under strong alignment conditions on the maximum-likelihood subsets (Cheng, 2019).

Conditional Maximum Likelihood (CML) updates ambiguous information differently. With unique prior $U_{B_2}(U_{B_1}[P])=U_{B_1\cap B_2}[P]$9 on states and ambiguous interpretations $M=\bigcap_{\alpha<\mu} X_\alpha$0, the posterior after observing $M=\bigcap_{\alpha<\mu} X_\alpha$1 is

$M=\bigcap_{\alpha<\mu} X_\alpha$2

For two independent signals with factorized interpretations $M=\bigcap_{\alpha<\mu} X_\alpha$3, the maximum over the Cartesian product $M=\bigcap_{\alpha<\mu} X_\alpha$4 factorizes into the product of maxima, so updating by $M=\bigcap_{\alpha<\mu} X_\alpha$5 then $M=\bigcap_{\alpha<\mu} X_\alpha$6 yields the same expression as one-shot updating by $M=\bigcap_{\alpha<\mu} X_\alpha$7. In that setting, the posterior is unaffected by the order in which independent signals arrive (Tang, 2020).

Extended Relative Maximum Likelihood transports the RML idea from sets of priors to convex capacities. For a convex capacity $M=\bigcap_{\alpha<\mu} X_\alpha$8 and event $M=\bigcap_{\alpha<\mu} X_\alpha$9, the conditional capacity is defined by

$R_j\equiv R_{j-1}+a_j w_j \pmod{k}$0

Here $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$1 gives the Dempster–Shafer rule and $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$2 gives the Fagin–Halpern rule. Proposition 1 in the note represents this update as a lower envelope over

$R_j\equiv R_{j-1}+a_j w_j \pmod{k}$3

where $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$4 is the core of $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$5 and $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$6 is its maximum-likelihood subset. The note uses this representation, together with Bayesian associativity on each prior and the comonotonicity of the core, to argue divisibility of DS, FH, and Extended RML under the standing convex-capacity assumptions (Cheng, 2021).

A different contrast appears in the study of updating Choquet capacities of order $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$7. There, Dempster’s rule and the Geometric rule are divisible for sequential deterministic evidence, because retaining focal sets through successive intersections is equivalent to conditioning once on the final intersection. The generalized Bayes rule is algebraically path-independent at the level of lower and upper ratios when one always refers back to the original $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$8, but the paper stresses that its extremizing prior $R_j\equiv R_{j-1}+a_j w_j \pmod{k}$9 depends on the realized evidence, so practical path dependence remains when posteriors are fed forward. Divisibility is therefore distinct from coherence and distinct from informativeness: the generalized Bayes rule is coherent and yields the widest intervals, but both generalized Bayes and the Geometric rule cannot update from vacuous priors; Dempster’s rule can sharpen vacuous intervals, yet can also induce sure loss, and Dempster’s and Geometric rules can mathematically contradict each other with respect to dilation and contraction (Gong et al., 2017).

4. Stratified modular semantics in hybrid knowledge bases

In hybrid knowledge representation, divisibility is realized as a stratum-by-stratum update semantics. A hybrid knowledge base has the form

$L\times L$0

where $L\times L$1 is a DL ontology and $L\times L$2 is a ground generalized logic program interpreted under Hybrid MKNF semantics. The key structural device is a splitting set $L\times L$3 of predicate symbols, which induces a bottom $L\times L$4, a top $L\times L$5, and a reduct $L\times L$6. The splitting theorem states that $L\times L$7 is an MKNF model of $L\times L$8 iff there exist $L\times L$9 and $k_i=8$0 such that $k_i=8$1 is an MKNF model of $k_i=8$2, $k_i=8$3 is an MKNF model of $k_i=8$4, and

$k_i=8$5

This is lifted to splitting sequences $k_i=8$6, yielding the representation

$k_i=8$7

for a solution $k_i=8$8 along the strata (Slota et al., 2011).

Dynamic updates are then defined only for updatable dynamic hybrid knowledge bases, namely those admitting an update-enabling splitting sequence. The technical point is that each layer becomes basic: either O-based and handled by classical minimal-change update, or P-based and handled by refined dynamic stable model semantics. The updated model is computed inductively. $k_i=8$9 is the dynamic MKNF model of the bottom layer; for each successor layer,

$p$00

is the dynamic MKNF model of the reduct formed from $p$01 and the intersection of earlier layers; at limits, $p$02 is the set of all Herbrand interpretations. The resulting semantics is modular, and the solution-independence proposition states that different update-enabling splitting sequences yield the same dynamic MKNF models (Slota et al., 2011).

This stratified divisibility is not merely a proof technique. It produces an operator that generalizes static Hybrid MKNF models, classical minimal-change update for pure ontological sequences, and dynamic stable models for pure rule sequences. The semantics satisfies primacy of new information,

$p$03

and its realism is illustrated by the cargo-imports example, where a nontrivial splitting sequence separates commodity and shipment DL axioms, importer rules, producers and countries together with low-risk EU commodity rules, and final inspection rules. Divisibility here is a condition for applicability: when an update-enabling splitting sequence does not exist, the modular procedure does not apply (Slota et al., 2011).

5. Scheduling, locality, and update realization in dynamical and logical systems

In cellular automata, divisible updating rules are update schemas that partition when lattice sites are updated. The study of order-3 bi-dimensional binary outer-totalistic automata distinguishes synchronous updating, asynchronous random single-site updating, and sequential row-wise updating. These schemes strongly affect local structures: asynchronous and sequential updates destroy characteristic Life structures and often produce labyrinth phases. Yet the global classification of rules by the control parameter

$p$04

remains qualitatively robust, with the first-order transition between high-density and absorbing regimes still organized around $p$05 under all three schemas. Divisibility of the schedule therefore changes morphology more than phase identity (Reia et al., 2015).

In distributed aggregative games with coupled equality constraints, divisibility is realized across agents through a decomposition into local primal control, dual coordination, and aggregator estimation. For double-integrator agents,

$p$06

while each agent also runs local multiplier dynamics and a fast dynamic-average-consensus estimator for $p$07. For multi-integrator agents, the controller adds a state feedback $p$08 chosen so that the closed-loop transfer function is strictly positive real. Under the paper’s convexity, strong monotonicity, Lipschitz, graph-connectivity, gain, and small-$p$09 assumptions, the resulting distributed strategy-updating rules drive the network exponentially to the unique variational generalized Nash equilibrium and a common multiplier $p$10 (Cai et al., 2021).

A related decomposition appears in continuous-time noncooperative games with discrete-time communication. There, each double-integrator agent maintains a local estimate vector $p$11 of all strategies and updates according to a continuous-time law combining damping, subgradient descent, and Laplacian consensus. The divisible feature is temporal: the same continuous-time rule can be implemented through periodic synchronous broadcasts or through dynamic event-triggered asynchronous broadcasts. Both schemes preserve asymptotic convergence to the Nash equilibrium; the event-triggered scheme is proved Zeno-free and, in the reported Cournot example, substantially reduces transmissions relative to periodic communication with $p$12 s (Cai et al., 2021).

In dynamic complexity, update rules are first-order formulas that maintain auxiliary relations under tuple insertions and deletions. The parity of a unary relation is maintainable with quantifier-free rules, but the parity-exists query

$p$13

is not maintainable in Prop. More sharply, for $p$14, the degree-restricted query $p$15 is in $p$16-ary Prop but not in $p$17-ary Prop, producing a strict arity hierarchy for Boolean graph queries under quantifier-free update rules. Fixed-degree variants are maintainable in DynFO, and the $p$18 version is maintainable in DynFO with binary auxiliary relations under a built-in linear order and BIT. This suggests another notion of divisibility: the ability to distribute a modular counting task across auxiliary relations of bounded arity and across update events on the input structure (Vortmeier et al., 2019).

6. Divisible combinatorial updates: partitions and permutation arithmetic

A structurally different use of divisibility concerns random set partitions whose block sizes must remain divisible by $p$19. For $p$20, ordinary restriction does not preserve divisibility, so a random deletion procedure is required. The deletion map $p$21 first performs a sequence of random displacements inside the block containing $p$22, encoded by transpositions

$p$23

and only then deletes the group $p$24. This randomized deletion is the projective counterpart of a divisible Chinese-restaurant-type seating rule in which $p$25 new arrivals undergo sequential displacement and are then seated as a single unit according to CRP weights. The resulting family $p$26 is exchangeable and consistent under deletion, and the associated Markov chains on $p$27-divisible partitions with at most $p$28 blocks are reversible, satisfying

$p$29

with stationary law $p$30 (Crane et al., 2011).

In arithmetic, divisibility becomes an incremental update rule for remainders when integers are written in factoradic form or viewed as tame permutations. Every nonnegative integer has

$p$31

and the factoradic digits are inversion counts

$p$32

For fixed modulus $p$33, only the first $p$34 digits matter because $p$35 for $p$36. With weights $p$37,

$p$38

This yields the incremental remainder recursion

$p$39

so divisibility by $p$40 is determined by a prefix-wise update of the remainder. For composite $p$41, the truncation can be sharpened to $p$42, where $p$43 is the smallest $p$44 such that $p$45. These rules mirror base-$p$46 digit tests but exploit factorial weights and permutation inversion structure (Oliver et al., 21 Feb 2025).

Across these literatures, divisible updating rules do not form a single doctrine. Some versions require exact associativity, some require only local decomposability, and some preserve invariants under randomized deletion, grouped insertion, or modular prefix computation. A plausible implication is that the key technical issue is not merely how an update is performed, but which decompositions preserve the semantics that the model treats as fundamental.