시간과 공간을 압축하는 튜링 기계 시뮬레이션의 전자역학적 법칙
📝 Abstract
Standard simulations of Turing machines suggest a linear relationship between the temporal duration t of a run and the amount of information that must be stored by known simulations to certify, verify, or regenerate the configuration at time t. For deterministic multitape Turing machines over a fixed finite alphabet, this apparent linear dependence is not intrinsic: any length-t run can be simulated using O( √ t) work-tape cells via a Height Compression Theorem for succinct computation trees together with an Algebraic Replay Engine. In this paper we recast that construction in geometric and information-theoretic language. We interpret the execution trace as a spacetime DAG of local update events and exhibit a family of recursively defined holographic boundary summaries such that, along the square-root-space simulation, the total description length of all boundary data stored at any time is O( √ t). Using Kolmogorov complexity, we prove that every internal configuration has constant conditional description complexity given the appropriate boundary summary and time index, establishing that the spacetime bulk carries no additional algorithmic information beyond its boundary. We express this as a one-dimensional computational area law : there exists a simulation in which the information capacity of the active “holographic screen” needed to generate a spacetime region of volume proportional to t is bounded by O( √ t). In this precise sense, deterministic computation on a one-dimensional work tape admits a holographic representation, with the bulk history algebraically determined by data residing on a lower-dimensional boundary screen.
💡 Analysis
Standard simulations of Turing machines suggest a linear relationship between the temporal duration t of a run and the amount of information that must be stored by known simulations to certify, verify, or regenerate the configuration at time t. For deterministic multitape Turing machines over a fixed finite alphabet, this apparent linear dependence is not intrinsic: any length-t run can be simulated using O( √ t) work-tape cells via a Height Compression Theorem for succinct computation trees together with an Algebraic Replay Engine. In this paper we recast that construction in geometric and information-theoretic language. We interpret the execution trace as a spacetime DAG of local update events and exhibit a family of recursively defined holographic boundary summaries such that, along the square-root-space simulation, the total description length of all boundary data stored at any time is O( √ t). Using Kolmogorov complexity, we prove that every internal configuration has constant conditional description complexity given the appropriate boundary summary and time index, establishing that the spacetime bulk carries no additional algorithmic information beyond its boundary. We express this as a one-dimensional computational area law : there exists a simulation in which the information capacity of the active “holographic screen” needed to generate a spacetime region of volume proportional to t is bounded by O( √ t). In this precise sense, deterministic computation on a one-dimensional work tape admits a holographic representation, with the bulk history algebraically determined by data residing on a lower-dimensional boundary screen.
📄 Content
The tradeoff between deterministic time and space on Turing machines is a foundational theme in complexity theory, with a long history of upper and lower bounds relating these resources [1,2,3]. Together with classical simulations such as Savitch’s theorem relating nondeterministic and deterministic space [4], these results establish that many natural problems admit nontrivial transformations between time and space bounds. More recently, for deterministic multitape Turing machines it has been shown that a time-t computation can be simulated in space O √ t log t [5], sharpening the classical O(t/ log t) bound.
In a companion work (henceforth Height Compression) we prove a stronger square-root space simulation theorem
for deterministic multitape Turing machines, where space is measured in tape cells over a fixed finite alphabet [6]. The proof proceeds via three ingredients: a Height Compression Theorem for succinct computation trees, an Algebraic Replay Engine that regenerates local configurations from short summaries, and a rolling boundary discipline for traversing the compressed tree. Informally, the companion paper shows that the entire computation history of a deterministic run can be regenerated from a carefully chosen sequence of low-dimensional boundary summaries, using only O( √ t) active tape cells at any moment. The present paper develops a geometric and information-theoretic interpretation of this phenomenon. We treat the execution of a deterministic machine not only as a linear sequence of configurations, but as a finite directed acyclic graph (DAG) that we regard as a discrete spacetime object, equipped with a combinatorial notion of locality induced by the machine’s transition rules. We then reinterpret the technical machinery of height compression in geometric language:
• We repackage the interval summaries of height compression as holographic boundary states that encode all information flowing into and out of a spacetime sub-region.
• We reinterpret the height-compressed computation tree as a static causal tree of spacetime volumes, on which linear time appears as a particular depth-first traversal.
• We define an active holographic screen consisting of the boundary states and local replay window maintained by the simulator, and we show that its size satisfies a one-dimensional area law: the maximum screen size over the run is O( √ t).
On top of these formal correspondences, we prove a precise information-theoretic statement: the bulk of the spacetime object has O(1) conditional Kolmogorov complexity relative to its boundary summaries. That is, once the machine and the boundary data of a block are fixed, any internal configuration can be produced by a fixed, constant-complexity procedure. In this sense, the interior of a deterministic computation history is an information-theoretic vacuum: all nontrivial information resides in its boundary data, and the bulk is an algebraically determined evaluation trace.
Finally, we formulate conjectural extensions of the area law to higher-dimensional memory architectures, and we discuss analogies with holography in quantum gravity and with area laws for entanglement in many-body systems [8,9,10,11,12]. We distinguish carefully between rigorous theorems (which are purely combinatorial and information-theoretic) and speculative analogies.
We briefly recall the core constructions from the height-compression technique, specializing to the aspects needed for the present work. For background on standard models of computation and complexity-theoretic notation, see, e.g., [3]. For Kolmogorov complexity and encoding conventions, we follow [7] and make our choices explicit in Appendix A.
We fix a deterministic multitape Turing machine M = (Q, Σ, Γ, δ, q 0 , q acc , q rej ), with finite state set Q, input alphabet Σ, work alphabet Γ ⊇ Σ, and transition function
for some fixed number of tapes k ≥ 1. We assume the standard Lipschitz locality of head motion: in one time step, each head moves by at most one cell.
A run of M of length t is a sequence of configurations
where each C τ encodes the tape contents, head positions, and control state at time τ . The spacetime diagram of the run is the finite directed acyclic graph
where V consists of all local degrees of freedom (e.g., tape cells with time labels and the control state) and E consists of directed edges representing the causal dependencies induced by δ between consecutive configurations. For standard one-dimensional tapes we have V ⊆ Z 1+1 ; more generally one may consider higher-dimensional tape lattices. We will colloquially refer to M raw as a “spacetime manifold”, but for our purposes it is simply a finite DAG equipped with the adjacency relation defined by δ.
We will use the following notion of block locality, which is the one realized by the heightcompression construction in [6].
Definition 1 (Block-respecting run). Let b ∈ N and let c int ≥ 1 be a fixed constant. A length-t run of M is block-
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