Computational Necessity Hierarchy

• Computational Necessity Hierarchy is a stratified framework that specifies which computational behaviors and resource demands are inevitable under given assumptions.
• It organizes systems—from natural processes to quantum algorithms—by criteria such as provability, geometric regularity, and resource limitations.
• The framework shows necessity emerging from competition, logical constraints, and structural depth, guiding design in proof theory, optimization, and learning.

Computational necessity hierarchy denotes a family of stratified frameworks for specifying which computational behaviors, resources, or structural features are unavoidable under stated assumptions. In different research settings, the hierarchy orders natural processes by attraction toward simplicity or universality, total computable functions by what a formal system can prove about their time complexity, analytical problems by the number of nested limits required for solution, optimization models by their placement in the polynomial hierarchy or by the regularity needed to prevent pathological value functions, cellular automata by emulation power, quantum algorithms by the state structure necessary for speedup, and distributed or augmented learners by the resources required to avoid regret or hallucination. Taken together, these works suggest not a single canonical formalism but a recurrent explanatory pattern: necessity is imposed by competition, provability, geometric regularity, dynamical structure, computational budget, or oracle access.

1. Core meanings of hierarchy and necessity

Across the literature, a computational necessity hierarchy can be a trichotomy, an infinite ascending chain, a preorder, an integer-valued index, or a hardness placement in a standard complexity hierarchy. What unifies these uses is that each framework asks which computational levels are stable or reachable, and which are hidden, disfavored, or impossible without stronger assumptions.

Setting	Hierarchical object	Necessity criterion
Natural computation	simple, intermediate, universal processes	persistence under competition and cognition
Time complexity and logic	$F, F^{(1)}, F^{(2)}, \dots$	provability of totality and time-class membership
Scientific computing	$\mathrm{SCI}=0,1,2,\dots$	minimum number of nested limits
Bilevel optimization	value-function classes and $\Sigma_2^p$-hardness	lower-level regularity
Cellular automata	emulation preorder	ability to emulate other automata
Quantum algorithms	state families with $p,q$	macroscopic superposition for speedup
Federated and augmented learning	compute budgets, oracle access	collaboration or augmentation needed for performance

A common misconception is that “hierarchy” always means a total order. Several of these frameworks explicitly reject that interpretation. The emulation relation for elementary cellular automata is a preorder rather than a total order, the natural-computation picture in Joosten is an explanatory pressure rather than a mathematically proven total order of degrees, and the quantum results state necessary rather than sufficient conditions. In this sense, necessity is often weaker than classification by exact equivalence classes, but stronger than an informal claim of difficulty (Joosten, 2012, Ben-Artzi et al., 2015, Hudcová et al., 2021, Bolte et al., 2024, Li et al., 2024, Shi et al., 10 Aug 2025).

2. Natural computation: simplicity, universality, and cognitive filtering

A foundational formulation appears in Joosten’s account of why nature seems to exhibit many universal processes and few intermediate ones. Against the backdrop of Wolfram’s Principle of Computational Equivalence, the paper defines universality by simulation with easy coding and decoding,

$\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$

and, for its first section, treats a system as complex when it can be easily perceived as a universal computational process. This notion is paired with

$$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$

the Church–Turing Thesis,

$$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$

and the Generalized Natural Selection principle,

$$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$

The resulting hierarchy distinguishes three regimes: simple processes, complex but non-universal or intermediate processes, and universal computation. The proposed mechanism is explicitly resource-based. If natural processes are computational, then they compete for “the three main resources space, matter, and time.” A process that can simulate or subsume another can use it rather than be displaced by it; universal processes are the limiting case because they can simulate any competitor. The paper therefore states

$$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$

and, for natural processes only,

$\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$

This is presented as correspondence rather than strict equivalence.

The paper is equally careful to deny that the hierarchy is established by theorem alone. It acknowledges that theoretical computer science contains “explicit undecidable intermediate degrees” and that “the structure of such degrees is actually known to be very rich,” but treats the known examples as highly artificial and therefore unlikely to appear naturally. The second explanatory layer is cognitive. Complexity is redefined relative to resource bounds as whatever “comprehending or describing it exceeds or supersedes all available resources (time, space, description size),” and diagonalization is used as a model of framework extension. The proof sketch defining

$\mathrm{SCI}=0,1,2,\dots$0

illustrates how one passes to a stronger setting, and the paper suggests that human cognition may not possess a similarly granular mechanism for detecting intermediate levels. A plausible implication is that the empirical scarcity of intermediate phenomena may be jointly produced by natural selection among computations and by cognitive under-sampling of the middle range.

The empirical case study on small Turing machines reinforces that picture without proving it. In $\mathrm{SCI}=0,1,2,\dots$1-space, $\mathrm{SCI}=0,1,2,\dots$2-space, and related small-machine settings, most machines either halt quickly or never halt, halting distributions show “pronounced phase-transitions,” and richer-looking machines often compute simple functions. Rule 2205 always outputs one black cell despite a localized diagram; rule 1351 computes the identity function after exponential time while exhibiting recursive structure. The paper also reports, for all Turing machines in $\mathrm{SCI}=0,1,2,\dots$3-space, the correspondence $\mathrm{SCI}=0,1,2,\dots$4 iff $\mathrm{SCI}=0,1,2,\dots$5 computes in linear time and $\mathrm{SCI}=0,1,2,\dots$6 iff $\mathrm{SCI}=0,1,2,\dots$7 computes in exponential time, linking a geometrical fractal dimension to time complexity. The overall hierarchy is thus not a proof that the middle is empty, but an argument that simplicity and universality are favored attractors while intermediate complexity is rare, fragile, artificial, or cognitively obscured (Joosten, 2012).

3. Provability gaps and hidden time-complexity classes

A different computational necessity hierarchy arises in proof theory and complexity. The “hidden machine” framework begins with a sufficiently expressive, consistent formal axiomatic system $\mathrm{SCI}=0,1,2,\dots$8 that interprets ZFC and can express Kleene’s predicate $\mathrm{SCI}=0,1,2,\dots$9, provability predicates $\Sigma_2^p$0, and running-time predicates $\Sigma_2^p$1. Totality is formalized as

$\Sigma_2^p$2

Using diagonalization relative to provability in $\Sigma_2^p$3, the paper constructs a total computable function $\Sigma_2^p$4 that outruns every function whose totality is provable in $\Sigma_2^p$5. Its machine $\Sigma_2^p$6 enumerates indices $\Sigma_2^p$7 with short proofs of totality in $\Sigma_2^p$8 and returns

$\Sigma_2^p$9

The critical point is that $p,q$0 is computable and intuitively total, but $p,q$1 cannot prove that $p,q$2 is total.

The paper then transports this phenomenon from totality to time complexity. For a definable set $p,q$3 of time bounds, membership in a deterministic time class is expressed by

$p,q$4

with the formal witness condition

$p,q$5

The first incompleteness theorem states that there exists a total computable function $p,q$6 such that, if $p,q$7 is consistent,

$p,q$8

and also

$p,q$9

Thus a function may have some deterministic time bound in the intended model while membership in any such class remains unprovable within the system.

The hierarchy becomes explicit by iteration. Setting $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$0, $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$1, and

$\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$2

the construction defines $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$3 diagonally relative to the stronger theory $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$4. The result is an infinite denumerable hierarchy

$\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$5

paired with

$\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$6

such that each $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$7 is total and computable but not provably recognizable by the corresponding system as belonging to any time class. The second central theorem states that for each stage $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$8, if $\mathcal{C}^{-1}(\Pi(C(x)))=\Theta(x),$9 is any set of time-bounds whose fastest-growing member is asymptotically dominated by the runtime of $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$0, then there exists a larger set $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$1 such that $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$2 but $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$3.

Here necessity is neither ecological nor cognitive. It is proof-theoretic. “Hiddenness” has two layers: totality hiddenness and complexity hiddenness. A machine exists, is legitimate, and has objective complexity behavior, yet the formal system cannot prove the statement that expresses its membership in a time complexity class. The paper’s simulation estimate,

$$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$4

shows that higher classes can still capture the runtime of the simulation, even when lower-class membership remains unprovable. The hierarchy is therefore a hierarchy of formal accessibility rather than of computability alone (Abrahão et al., 2020).

4. Nested limits, optimization hardness, and regularity as a structural constraint

In numerical analysis and optimization, computational necessity is often expressed not as proof-theoretic hiddenness but as a lower bound on algorithmic structure. The Solvability Complexity Index (SCI) counts the minimum number of nested limits required to compute a problem. For a computational problem $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$5, a tower of algorithms of height $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$6 is a family

$$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$7

such that

$$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$8

This yields a hierarchy in which $$\mathbf{PCE}: \text{Almost all processes that are not obviously simple can be viewed as computations of equivalent and maximal sophistication,}$$9 means finite-time solvability, $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$0 means one limit is enough, $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$1 means two nested limits are needed, and so on. The framework classifies bounded-operator spectrum, Schrödinger spectra, polynomial root finding, inverse problems, optimization, and decision problems. For the full class of bounded operators on $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$2, for example,

$$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$3

so three limits are necessary in the general case. For self-adjoint or normal operators the problem drops to SCI $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$4, and some bounded-potential Schrödinger settings are even in $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$5. Necessity here is the structural depth of computation: some problems cannot be solved by any lower tower, regardless of ingenuity.

Bilevel programming supplies a complementary hierarchy in classical complexity and representation theory. Unconstrained $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$6 smooth bilevel programming is shown to be “as hard as general extended-real-valued lower semicontinuous minimization,” and any proper lower semicontinuous function with closed domain is representable as the value function of an optimistic smooth bilevel problem with $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$7. In the polynomial setting, any extended-real-valued semi-algebraic function is the value function of a polynomial bilevel problem whose lower-level problem is unconstrained, while for bounded-box lower levels the optimistic and pessimistic value-function classes become $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$8 and $$\mathbf{CT}: \text{Everything that is algorithmically computable is computable by a Turing Machine,}$$9. On the complexity side, the decision version of polynomial bilevel programming is $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$0-hard: given degree-at-most-five polynomials $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$1 and $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$2, a bounded box $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$3, and a constant $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$4, deciding whether the optimistic bilevel optimum is strictly smaller than $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$5 is one level above NP in the polynomial hierarchy. The paper’s conclusion is explicit: some form of regularity or qualification condition at the lower level is computationally necessary if one wants tractable theory or algorithms.

The moment-SOS hierarchy makes the same point in a different language. At a fixed level $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$6, the relaxation is an SDP with matrix size

$$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$7

which is polynomial in $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$8 when $$\mathbf{GNS}: \text{In nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication.}$$9 is fixed. Yet polynomial-size SDP does not imply polynomial-time computability in the Turing model, because feasible solutions and SOS certificates can have exponential or doubly-exponential bit-complexity. The negative examples of O’Donnell and of Raghavendra–Weitz show exactly that. The paper then supplies sufficient conditions under which fixed-level computation is genuinely polynomial-time: explicit boundedness

$$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$0

polynomial bit-complexity of the moments $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$1, and lower bounds of order $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$2 on the smallest nonzero eigenvalues of the moment and localizing matrices. A purely geometric sufficient condition is that the feasible set contain a ball $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$3 with

$$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$4

A plausible synthesis is that these three literatures describe the same meta-principle at different levels of abstraction: absent enough regularity, bounded geometry, or additional limiting depth, expressive computational formalisms rise into higher hardness classes or become inaccessible to lower-grade algorithms (Ben-Artzi et al., 2015, Bolte et al., 2024, Gribling et al., 2023).

5. Dynamical systems and quantum algorithms

In discrete dynamics, the hierarchy can be defined by emulation rather than by explicit resource bounds. For elementary cellular automata (ECA), the basic relation asks how many other automata a given rule can emulate. If $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$5 and $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$6, then $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$7 when $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$8 is a subautomaton of the block dynamics $$\mathbf{PCE}\Longrightarrow \mathbf{CT}+\mathbf{GNS},$$9. Equivalently, there exists an injective encoding $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$0 such that

$\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$1

for every local neighborhood triple $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$2. The induced relation $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$3 is reflexive and transitive, hence a preorder. It preserves computational power: if $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$4 is Turing complete and $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$5, then $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$6 is also Turing complete. The computed emulation graph for supercell sizes $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$7 to $\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$8 shows a dense partial hierarchy, not a linear ranking. Rules 0, 204, 170, 240, 51, and 128 function as frequently emulated canonical targets, whereas rules 30, 45, 86, and 89 cannot emulate any ECA non-trivially within the tested range. This leads to a definition of computational chaos: a 1D nearest-neighbor CA is computationally chaotic if it cannot emulate any 1D nearest-neighbor CA with more than one state non-trivially.

Quantum computation introduces yet another necessity criterion: certain many-body state structures are necessary for large quantum speedup. The relevant observables are additive operators

$\mathbf{PCE'}\approx \mathbf{CT}+\mathbf{GNS}.$9

and the index $\mathrm{SCI}=0,1,2,\dots$00 for pure-state families is defined through

$\mathrm{SCI}=0,1,2,\dots$01

with $\mathrm{SCI}=0,1,2,\dots$02. A family has a superposition of macroscopically distinct states if $\mathrm{SCI}=0,1,2,\dots$03. For mixed-state families, the analogous index $\mathrm{SCI}=0,1,2,\dots$04 is defined via

$\mathrm{SCI}=0,1,2,\dots$05

and again satisfies $\mathrm{SCI}=0,1,2,\dots$06, with macroscopic superposition at $\mathrm{SCI}=0,1,2,\dots$07. The paper proves that $\mathrm{SCI}=0,1,2,\dots$08 iff $\mathrm{SCI}=0,1,2,\dots$09 for pure states, and introduces the efficiently computable index $\mathrm{SCI}=0,1,2,\dots$10, derived from the variance-covariance matrix, with $\mathrm{SCI}=0,1,2,\dots$11.

The conjectural hierarchy here is explicitly one of necessity, not sufficiency. If a quantum computer achieves exponential speedup, then states with $\mathrm{SCI}=0,1,2,\dots$12 on a subsystem of size $\mathrm{SCI}=0,1,2,\dots$13 must appear during the computation for some infinite set of hard instances. The paper further extends the claim to Grover’s algorithm, showing that for $\mathrm{SCI}=0,1,2,\dots$14, for $\mathrm{SCI}=0,1,2,\dots$15, and for $\mathrm{SCI}=0,1,2,\dots$16 with $\mathrm{SCI}=0,1,2,\dots$17, relevant intermediate states satisfy $\mathrm{SCI}=0,1,2,\dots$18. The final state may or may not have $\mathrm{SCI}=0,1,2,\dots$19, depending on the structure of the solution set, but the necessity claim concerns states during the computation. This places macroscopic superposition above weaker entanglement-based conditions in a hierarchy of necessary quantum state structures for computational advantage (Hudcová et al., 2021, Shimizu et al., 2010).

6. Resource-bounded learning, hallucination boundaries, and escape mechanisms

In distributed online learning, necessity can be triggered by limited local computation rather than by logical impossibility. For online model selection with decentralized data over $\mathrm{SCI}=0,1,2,\dots$20 clients and $\mathrm{SCI}=0,1,2,\dots$21 candidate hypothesis spaces, the central question is when collaboration is actually required. The answer is sharply two-case. Without additional computational constraints, collaboration is unnecessary: noncooperative strategies can match the optimal regret rate, and the lower bound scales as

$\mathrm{SCI}=0,1,2,\dots$22

Under per-client time complexity $\mathrm{SCI}=0,1,2,\dots$23, however, collaboration becomes necessary. The paper formalizes this by letting each client process only $\mathrm{SCI}=0,1,2,\dots$24 hypotheses per round. When $\mathrm{SCI}=0,1,2,\dots$25, both collaborative and noncooperative methods achieve the same order of regret. When $\mathrm{SCI}=0,1,2,\dots$26, any noncooperative algorithm suffers a lower bound of order

$\mathrm{SCI}=0,1,2,\dots$27

while the proposed federated method improves on that regime through a combination of federated online mirror descent, an improved Bernstein inequality for martingales, and decoupling model selection from prediction. This hierarchy does not assert that collaboration is universally beneficial; it asserts that collaboration becomes computationally necessary only below a threshold local budget.

A more radical notion of necessity appears in recent work on LLMs modeled as probabilistic Turing machines. There the hierarchy consists of three boundaries at which hallucination becomes inevitable: a diagonalization boundary, an uncomputability boundary, and an information-theoretic boundary. A probabilistic LLM is defined as

$\mathrm{SCI}=0,1,2,\dots$28

For relational truth $\mathrm{SCI}=0,1,2,\dots$29, straying hallucination is

$\mathrm{SCI}=0,1,2,\dots$30

and for probabilistic truth, distortion hallucination is

$\mathrm{SCI}=0,1,2,\dots$31

At the diagonalization boundary, every enumerable sequence of probabilistic LLMs can be defeated by a computable truth function that excludes each model’s most probable answer on its diagonal input. At the uncomputability boundary, if truth is defined by the halting oracle, any standard model must exhibit significant distortion hallucination on infinitely many inputs. At the information-theoretic boundary, the learner pump lemma states that finite information capacity $\mathrm{SCI}=0,1,2,\dots$32 implies a threshold beyond which sufficiently complex truths force hallucination; the key expression is

$\mathrm{SCI}=0,1,2,\dots$33

This framework also proposes escape mechanisms. Retrieval-augmented generation is formalized as oracle access, producing an “absolute escape” through computational jumps: there exists a ground-truth function $\mathrm{SCI}=0,1,2,\dots$34 on which every standard model hallucinates but an oracle-augmented model $\mathrm{SCI}=0,1,2,\dots$35 has zero hallucination. Continual learning is formalized as an “internalized oracle” via a sequence of models $\mathrm{SCI}=0,1,2,\dots$36, with the claim that the effective capacity changes roughly as

$\mathrm{SCI}=0,1,2,\dots$37

That escape is explicitly limited: it addresses static information-theoretic failure, not diagonalization or uncomputability in general. The paper’s concluding design principle, Computational Class Alignment, states that deployment in high-stakes settings requires the intrinsic complexity of the assigned task to lie strictly within the computational class of the agent or its augmented system. Read together with the federated model-selection result, this suggests a broad contemporary reformulation of computational necessity: a capability may be unnecessary in one regime and indispensable in another, depending on whether the operative boundary is local compute, formal proof, external knowledge, or computational class itself (Li et al., 2024, Shi et al., 10 Aug 2025).