Causal Depth as an Invariant of Irreducible Computation: Reassessing Logical Depth, Causal Order, and Physical Time

Babu George

doi:10.20944/preprints202607.1938.v1

Submitted:

25 July 2026

Posted:

27 July 2026

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Abstract

Arguments that computation underlies physical time frequently identify the work embodied in a complex state with Bennett's logical depth, and identify that quantity in turn with relativistic proper time. This paper examines the identification and finds it untenable in the form usually presented, for reasons that nevertheless indicate a defensible reformulation. Three difficulties are set out. First, computational irreducibility in its informal form is not a theorem; its rigorous surrogate is the P-completeness of prediction problems, which is a statement about parallel time rather than about sequential duration. Second, logical depth is invariant only up to a machine-dependent time factor and a shift in its significance parameter, and it is model-relative in a way that becomes acute once quantum computation is admitted. Third, and most consequentially, logical depth is a minimum taken over producing programs whereas proper time is a maximum taken over timelike curves, so the two quantities are extremal in opposite directions. We propose causal depth, defined as the height of the causal dependency graph of a process, as the quantity that occupies the position the argument requires. Causal depth is invariant under reordering of causally independent events, is extremal in the same direction as proper time, and corresponds to the longest-chain construction through which causal set theory recovers proper time from discrete causal order. The resulting correspondence is stated as a conjecture with explicit refutation conditions. We argue that the substantive open question concerns the gap between the causal depth of a state and the height of the histories that actually produce it.

Keywords:

computational sirreducibility

;

logical depth

;

p-completeness

;

circuit depth

;

causal sets

;

quantum speed limits

;

discrete spacetime

Subject:

Computer Science and Mathematics - Computational Mathematics

I. Introduction

A recurring proposal in the foundations of computation holds that the temporal structure of physical processes is best understood as computational rather than metrical. On this view, duration is a derived quantity, since it depends on the observer's frame in relativity, on the substrate in engineering practice, and on available energy in quantum mechanics, whereas the sequence of causally ordered events that a process must traverse is intrinsic to the process itself. The proposal is attractive, and versions of it appear in discrete approaches to quantum gravity, in algorithmic information theory, and in the literature surrounding cellular automata.

The proposal is imprecise in most of its published forms at the point where its central identification is made. Three claims are often combined: that certain computations admit no shortcut; that the accumulated computational work embodied in a state is measured by Bennett's logical depth [1]; and that this quantity is invariant across substrates in the way that proper time is invariant across observers. The three claims have different evidential standing. The first is a conjecture whose rigorous surrogate is narrower than the informal statement. The second is a definition whose invariance properties are weaker than is sometimes assumed. The third does not hold in the usual formulation.

This paper separates the claims, identifies the limitation of each, and proposes a reformulation. The central observation is structural. Logical depth is defined as a minimum over programs that produce a given object, while proper time in general relativity is a maximum over timelike curves joining two events. Identifying the two therefore equates an infimum with a supremum. The mismatch is not a matter of convention; it indicates that the quantities describe different objects, one a property of an endpoint state and the other a property of a trajectory.

We argue that the quantity with the correct formal shape is the height of the causal dependency graph of a computation, which is the standard notion of circuit depth. This quantity, which we call causal depth in order to emphasise its interpretation, is invariant under every reordering of causally independent events, and is a longest-path quantity. It therefore matches the extremal direction of proper time, and it corresponds directly to the longest-chain construction by which causal set theory recovers proper time from discrete causal order [2,3]. The paper's contribution is conceptual clarification and reformulation rather than a new theorem; we make this explicit in Section IX.

Section II fixes notation. Section III examines what is provable about computational irreducibility. Section IV sets out the limits of logical depth as an invariant. Section V defines causal depth and states its elementary properties. Section VI develops the correspondence with discrete spacetime geometry and notes the constraints that Lorentz invariance imposes on network-based models. Section VII shows that quantum bounds constrain the rate of state transitions without constraining their number. Section VIII states the correspondence conjecture together with the conditions under which it would be refuted. Sections IX and X discuss limitations and conclude.

II. Preliminaries and Notation

Let U denote a fixed universal Turing machine and let K(x) denote the Kolmogorov complexity of a finite string x, that is, the length of a shortest program p with U(p) = x. The invariance theorem guarantees that K is independent of the choice of U up to an additive constant [4]. Kolmogorov complexity does not distinguish organised structure from randomness: a uniformly random string is incompressible and therefore maximally complex under K, while exhibiting no internal organisation.

Bennett introduced logical depth to capture the missing distinction [1]. Informally, the depth of an object is the running time of a near-minimal program that produces it, where the tolerance for departure from minimal program length is controlled by a significance parameter. An object is deep when no short program produces it quickly. Bennett established a slow growth law: deep objects cannot be produced quickly from shallow ones by any deterministic process, nor with substantial probability by a probabilistic one.

For the parallel notions we follow standard usage. NC denotes the class of problems decidable by uniform Boolean circuit families of polylogarithmic depth and polynomial size, equivalently by parallel machines running in polylogarithmic time with polynomially many processors. A problem is P-complete when it lies in P and every problem in P reduces to it under logarithmic-space reductions. P-complete problems belong to NC if and only if NC = P, an equality that is not expected to hold [5].

For a computation we write E for its set of elementary events and ≺ for the partial order on E generated by data dependencies, so that e ≺ f holds when the output of e is an input to f. The Hasse diagram of (E, ≺) is a directed acyclic graph, which we call the causal graph of the computation. A chain in (E, ≺) is a totally ordered subset, and the height of the computation is the number of elements in a longest chain.

III. The Formal Content of Computational Irreducibility

A. The Informal Notion

Wolfram's notion of computational irreducibility asserts that for certain systems the state at step t cannot be determined substantially faster than by performing the t steps [6]. Read literally, this is a claim about the nonexistence of an algorithm with a specified resource bound. Claims of that form are, in general, beyond what current complexity theory can establish for specific natural systems. There is no proof that any particular elementary cellular automaton is irreducible in this unrestricted sense, and no corresponding proof for any specific natural system. The literature occasionally treats irreducibility as a theorem about state space, but that status has not been established.

B. P-Completeness as the Rigorous Surrogate

What has been proven is narrower and considerably sharper. Consider the prediction problem for a cellular automaton: given an initial configuration, a cell index, and a step count t written in unary, decide the state of that cell at step t. Neary and Woods established that this problem is P-complete for the elementary rule 110 [7], building on Cook's demonstration that rule 110 is universal [8]. P-completeness of prediction has also been established for majority-vote automata and zero-temperature Ising dynamics [9], for the automaton known as life without death [10], and for sandpile models [11].

The resulting implication is conditional on NC ≠ P. Under that assumption, no parallel machine with polynomially many processors can determine the state of such a system in polylogarithmic time. The relevant obstruction is therefore not elapsed wall-clock duration, but the inability to compress the dependency structure into polylogarithmic depth. The formal content of irreducibility is consequently a claim about causal or circuit depth rather than a claim about sequential time. The consequences of this reformulation are examined in the following sections.

C. Three Scope Conditions

First, P-completeness is a worst-case statement quantified over inputs, whereas irreducibility as informally intended is a claim about typical trajectories from simple initial conditions. The distance between worst-case and average-case behaviour is not a technicality, and the two coincide for no general reason.

Second, irreducibility is relational rather than intrinsic. A predictor with computational resources strictly exceeding those of the predicted system can shortcut it by simulating it faster. Irreducibility constrains only observers whose computational capacity does not exceed that of the system observed. Any necessity claimed in this context is therefore conditional on the party to whom the necessity applies.

Third, irreducibility is a property of a system together with a choice of observable. Israeli and Goldenfeld constructed exact local coarse-grainings for elementary cellular automata across all of Wolfram's classes, including rule 110, under which large-scale behaviour becomes predictable even though the microscopic dynamics is universal [12,13]. It follows that a system may be irreducible with respect to fine-grained questions and reducible with respect to coarse-grained ones. Claims that a physical process forbids shortcuts must therefore specify the observables at issue, and the practical reach of irreducibility depends on how many observables of interest survive coarse-graining.

IV. Logical Depth and the Limits of Its Invariance

Logical depth is well suited to the purpose for which Bennett introduced it, namely separating organised complexity from randomness and from trivial regularity. Its use as the invariant of a physical theory faces four related difficulties.

A. Uncomputability

Depth inherits uncomputability from Kolmogorov complexity. This does not by itself disqualify a foundational quantity, but it does mean that depth cannot serve as a directly measurable unit, and that any empirical programme built on it must proceed through computable surrogates.

B. Machine Relativity

The invariance theorem protects Kolmogorov complexity because universal simulation costs only a fixed-length prefix, which is additive in program length. Time enjoys no comparable protection, because simulation overhead is multiplicative in running time. Bennett accordingly established machine independence for depth only in a weaker sense, modulo a shift in the significance parameter and a machine-dependent factor in time [1]. Depth is therefore not a number but an equivalence class under reparametrisation. It may still be called invariant, but the invariance has the character of a complexity class rather than that of a scalar such as proper time.

C. Model Relativity

The difficulty becomes acute once the class of admissible producing machines is widened. Depth as ordinarily defined quantifies over programs for a universal classical machine. If a state can be prepared by a polynomial-size quantum circuit while requiring superpolynomial classical time, its classical and quantum depths differ, and they coincide only if BQP = P. Shor's algorithm for integer factorisation is the standard illustration that no such equality is expected [14]. Claims that the number of steps required to reach a state is independent of whether the state is computed on a quantum machine or simulated classically are therefore unproven and inconsistent with prevailing complexity-theoretic expectations. A count of this kind is relative to a computational model, and the physically relevant model is the one instantiated by the underlying dynamics.

D. The Directional Mismatch

The most serious difficulty is structural. Logical depth is a minimum over the programs that produce an object. Proper time in general relativity is a maximum: among timelike curves joining two events, the geodesic is the one of greatest elapsed proper time, which is the content of the twin paradox. An identification of depth with proper time therefore pairs an infimum with a supremum.

The mismatch reflects a difference in the objects described. Depth is a property of an endpoint state, recording the least costly route by which an admissible process could have produced it. Proper time is a property of a trajectory, recording what one particular history actually cost. Treating them as the same quantity conflates a state property with a path property, and it leads to the further error of describing the shortest path through a causal structure as a geodesic. Section VI shows that the discrete analogue of the geodesic is the longest chain, not the shortest path.

V. Causal Depth

The quantity with the required formal properties is already available in standard computational terminology, although its interpretation in relation to physical time requires separate analysis.

Definition 1 (causal depth of a process).

Let a process be given by its set of elementary events E together with the dependency order≺. The causal depth of the process, written d(E,≺), is the number of elements in a longest chain of (E,≺).

Definition 2 (causal depth of a state).

Let P be a class of admissible processes and let s₀ be a designated initial condition. The causal depth of a state x relative to P, written Dₚ(x), is the minimum of d(E,≺) over all processes in P that carry s₀ to x.

Definition 1 is the height of the dependency graph and coincides with circuit depth when the process is a Boolean circuit. Definition 2 introduces a minimisation over admissible processes. The distinction between the two definitions is central to the argument: Definition 1 concerns a trajectory, whereas Definition 2 concerns an endpoint.

Proposition 1.

For every process p in P that produces x from s₀, d(p) ≥ Dₚ(x).

The proposition follows directly from Definition 2. Its relevance for the present argument is that every admissible history producing x has causal depth at least Dₚ(x), so the state quantity functions as a lower bound on the depth of any producing trajectory.

Proposition 2.

d(E,≺) is invariant under every relabelling of E and every scheduling of its events that preserves≺.

Proposition 2 also follows from the definition of height in a partially ordered set. Relabelling events does not alter the order relation, and a schedule that preserves ≺ is a linear extension of the same partial order. The set of chains is therefore unchanged, as is the maximum chain length. Consequently, observers or descriptions that differ only in the ordering assigned to causally unrelated events assign the same causal depth to the process. This invariance is limited in scope: causal depth is invariant across schedules of a fixed computation, while remaining model-relative across different computations of the same function. The reparametrisation problem identified in Section IV is therefore constrained rather than removed.

For rewriting systems, the invariance in Proposition 2 requires that the causal graph itself be independent of the order in which rewrites are applied. This is the property that the Wolfram model calls causal invariance and that the rewriting literature calls confluence or the Church-Rosser property [15,16]. Confluence is a property that some rule sets have and others lack; it is an assumption on the dynamics rather than a consequence of discreteness. Newman's lemma derives confluence from local confluence only for terminating systems [16], and a cosmological rewriting system does not terminate, so confluence in that setting must be established by other means or postulated outright.

VI. Correspondence with Discrete Spacetime Geometry

A. Causal Order and Counting

The physical component of the argument has an established formulation in the causal set literature. Malament proved that in past and future distinguishing spacetimes the causal structure determines the topological, differential, and conformal structure of the manifold [17]. Causal order thus fixes the geometry up to a conformal factor. The causal set programme supplies the remaining factor by counting, taking spacetime volume to be proportional to the number of discrete elements [2,3]. The programme is often summarised by the formula that order together with number yields geometry.

B. Proper Time as Longest Chain

Within that framework, the discrete analogue of the proper time separating two causally related elements is the length of the longest chain joining them, a proposal due to Myrheim [18] and substantiated for Minkowski space by Brightwell and Gregory, who showed that for a Poisson sprinkling the expected longest-chain length between two elements scales with the proper time separating their images [19].

This result fixes the direction of the extremal principle and confirms the diagnosis of Section IV-D. In a discrete causal structure the geodesic is the longest chain, not the shortest path. A short path through the order does not represent a rapid route between two events; it represents the discrete image of a worldline that has taken a spatial detour and accumulated little proper time. The twin who ages least is the one whose worldline contains the fewest elements. Causal depth, being a longest-chain quantity, therefore has the same extremal character as proper time, which logical depth does not.

C. Constraints on Network Models

Two constraints on discrete frameworks are relevant here, since arguments of this kind frequently rely on hypergraph rewriting models whose relation to relativity remains under active debate.

The first concerns Lorentz invariance, which is not automatic and is difficult to obtain. A regular lattice, or a network with bounded local valence, generically selects a preferred frame. The causal set answer is that elements must be distributed by a Poisson process, whose statistics are Lorentz invariant even though no single realisation is. Bombelli, Henson, and Sorkin proved that no equivariant measurable map exists from sprinklings into Minkowski space to spacetime directions, and drew the further consequence that no finite-valence graph can be associated with a sprinkling consistently with Lorentz invariance [20]. Hossenfelder has established a related no-go result for Poincare-invariant networks [21]. Models built on hypergraphs of bounded valence therefore face a specific documented obstacle, and the claim that special relativity emerges from such models [22,23] depends on assumptions, including causal invariance, that have not been derived from the dynamics.

The second concerns the standing of the frameworks themselves. The causal set programme is an active research area with a substantial technical literature and a review that sets out both its results and its open problems [24]. The hypergraph rewriting programme has not to date produced quantitative predictions that distinguish it from alternatives. The arguments of the present paper depend on the causal set results and not on the hypergraph programme; this distinction clarifies the dependency structure of the argument.

VII. Rate and Count in Physical Bounds

Quantum mechanics constrains the rate at which a system passes through distinguishable states. The Margolus-Levitin theorem bounds the minimum time required for a system with mean energy E above its ground state to evolve into an orthogonal state by πħ / 2E [25]. The Mandelstam-Tamm relation supplies a complementary bound in terms of the energy spread [26], and Levitin and Toffoli showed that the unified bound obtained from the two is tight [27]. Lloyd combined these results into a limit of order 2E / πħ elementary logical operations per second for any physical system [28].

The structure of these constraints supports the distinction drawn here. The bounds restrict the rate of state transitions and do not determine how many transitions a given task requires. Energy increases the number of transitions available per unit time; it does not reduce the length of a dependency chain. If a state has large causal depth relative to the physical model, additional energy can shorten the duration of each round but does not by itself reduce the number of rounds. The separation between rate and count is therefore the relevant physical counterpart of the computational distinction.

VIII. A Correspondence Conjecture

What remains is the relation between the computational and physical components, which should be stated as a conjecture with explicit conditions rather than as an identity.

Conjecture 1 (correspondence).

Let P be the class of processes realisable under the physical dynamics, and let x be a state of that dynamics. Then the elapsed proper time along any history producing x, measured in units of the discreteness scale, is bounded below by a quantity proportional to Dₚ(x).

The lower bound follows from Proposition 1 together with the identification of proper time with longest-chain length discussed in Section VI-B, provided that the causal graph of the physical process and the causal set of the emergent spacetime coincide under the correspondence posited by the causal set programme. This proviso carries the substantive content of the conjecture.

The stronger identification, in which elapsed proper time is taken to equal causal depth rather than merely to dominate it, requires an additional optimality assumption. It would require actual histories to approximate the least-depth admissible histories leading to their present configurations. No established result supports this assumption. Physical dynamics may be thermodynamically dissipative, and the dependency structure of a macroscopic history may contain redundancies that are absent from a least-depth construction. The causal depth of a state therefore supplies a lower bound on the depth of its histories without measuring the depth of any particular history. Proper time measures an actual trajectory, whereas causal depth measures an unavoidable dependency bound. The quantities coincide only under an additional optimality hypothesis; absent such a hypothesis, the relevant research problem is the estimation of the gap between them.

In this form the position has explicit refutation conditions. It would be refuted if a physically realisable process were exhibited that produced states of high causal depth in few causal rounds. It would be substantially weakened if the coarse-grained reducibility demonstrated by Israeli and Goldenfeld [12,13] proved to be generic rather than exceptional, since in that case the observables of physical interest would largely evade depth and the framework, while formally correct, would carry little explanatory weight. It would be refuted outright by a proof that NC = P, which would remove the parallel-time obstruction entirely. Each of these is a well-posed question in an existing research literature.

IX. Limitations

The contribution of this paper is a reformulation rather than a theorem. Propositions 1 and 2 are direct consequences of the definitions, and causal depth coincides formally with circuit depth. The proposed contribution is the identification of circuit depth, rather than logical depth, as the quantity with the formal structure needed for comparison with relativistic proper time. This identification is supported by the shared longest-chain extremal structure, which logical depth lacks.

Several further limitations should be recorded. Definition 2 quantifies over a class P of admissible processes whose specification is left open, and the interest of the resulting quantity depends entirely on how P is fixed; a physically motivated specification would require commitments about the fundamental dynamics that we do not make here. The correspondence in Section VIII inherits every assumption of the causal set programme, including the discreteness postulate and the volume-number correspondence. No computable surrogate for causal depth is offered, and the practical estimation of the quantity for any state of physical interest remains open. Finally, the conditional character of Section III should be kept in view: the parallel-time obstruction rests on the expectation that NC is properly contained in P, which is unproven.

X. Conclusions

Duration is a derived quantity and computational speed is a property of hardware. Neither provides the relevant invariant for the present account. The proposed replacement is an order-theoretic structure: the partial order of causal dependencies, together with a counting measure on that order that is defined up to reparametrisation and relative to a model of computation.

Within that structure the appropriate measure is the height of the dependency graph rather than the running time of a minimal program. Causal depth is invariant under reordering of causally independent events, is extremal in the same direction as proper time, and corresponds to the construction by which causal set theory recovers metrical time from discrete order. Bennett's logical depth remains the correct tool for the problem it was designed to address, which is the separation of organised complexity from randomness, but it is not the invariant that a computational account of physical time requires.

The gap between what a state's production could not have avoided and what its actual history cost is a substantive target for further work. On this account, that gap is the quantity a computational theory of physical time would need to estimate or explain, and it is currently unmeasured.

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