arXiv subjects

Kirill Osipov

Publications and source records attributed to Kirill Osipov.

4 recordsLinked to original sources

arXiv2026

Step Recursion: Mixed Stride Spectra, Path Factorization, and Synchronization Geometry

Step recursion allows recursive computation to move through a canonical hierarchy in jumps, or strides. Suppose a function algebra is allowed to use several primitive stride lengths $L$. Composition immediately produces sums of these lengths, but it is not clear whether arbitrary nesting of mixed recursions can create any genuinely new canonical stride. We prove that it cannot: at every fixed canonical row $n\ge2$ and lower basis $m<n$, the canonical strides definable from $L$ are exactly the additive monoid $\langle L\rangle$ generated by $L$. The proof gives more information than membership alone. Every sufficiently high dependency path computing a canonical descent of stride $p$ carries total label weight exactly $p$, and the possible path signatures are precisely the additive factorizations of $p$ by the primitive stride labels. Thus the computation remembers both which strides are definable and how each one can be assembled. After saturation, inclusion between mixed-stride classes is exactly inclusion between their additive stride monoids, giving a concrete lattice description and finite certificates for inclusion. We then study several synchronized recursion clocks. Joint descent maps can encode arbitrary additive submonoids of $\mathbb N^r$, already giving continuum order complexity for $r=2$. Ordinary scalar observation, however, forgets correlations between the clocks and retains only the independent coordinate strides. This identifies precisely where synchronization information is preserved and where it collapses.

cs.CC↗

arXiv2026

Step Recursion: Exact Depth Does Not Determine Algebraic Expressiveness

Step recursion is a form of bounded recursion in which each recursive call moves from an input y to a prescribed predecessor $ρ_g(y)$. Its depth $D_g(y)$ is the exact number of such moves needed to reach zero. A natural question is whether knowing this depth for every input determines the expressive power of the resulting function algebra. We prove that it does not. We first construct two simple generators with exactly the same depth map but different step-recursion algebras. One gives ordinary binary halving, $b(x)=2x+1$; the other is $p(0)=1$, $p(x)=x+2^{λ(x)}$ for $x>0$, where $λ$ is binary length. Although $D_p=D_b$ pointwise, the predecessor $ρ_p$ cannot be defined from any fixed-stride binary-halving descent at basis zero. Thus two recursion schemes may take exactly the same number of steps on every input and still have different expressive power. The phenomenon is much larger than this example. Whenever infinitely many depth levels allow more than one predecessor arrangement, a single exact depth profile supports $2^{\aleph_0}$ distinct step-recursion algebras over every countable basis containing zero and the projections. In the computable setting the corresponding effective family has exactly $\aleph_0$ distinct algebras. Hence exact recursion depth is an informative resource measure, but it is not a complete invariant: the geometry of predecessor choices inside each depth level carries additional algebraic information.

cs.LO↗

arXiv2026

Step Recursion: A Three-Parameter Refinement of the Grzegorczyk Hierarchy

We ask whether asymptotic recursion depth determines the expressive strength of a bounded recursive algebra, and prove that it does not. We replace ordinary predecessor recursion by generalized-inverse descent along a fixed iterate $g_n^{[l]}$ and obtain classes $H^m_{n,l}$, where $m$ measures initial-function strength, $n$ the growth row, and $l$ the traversal stride. For all rows $n,n'\ge2$ we prove an exact inclusion criterion. At a fixed row $n\ge2$ three regimes occur: below the critical basis ($m<n$), equal-row inclusion is exactly reverse divisibility $l'\mid l$; at $m=n$ every stride collapses to one class; and from $m=n+1$ this class is ordinary bounded recursion $E^m$. Hence pairwise $Θ$-equivalent descent depths can induce infinite descending chains, infinite antichains, and copies of every finite partial order. The separation is therefore controlled by traversal alignment rather than by growth rate or recursion depth alone. The proof combines exact-depth simulation, trace sparsity, and selected dependency chains. The exceptional doubling row has the same reverse-divisibility order at basis zero, but all strides collapse from basis one onward; from basis three it equals ordinary bounded recursion, while at basis two $H^2_{1,l}\subsetneq FP$. At basis zero the doubling-row classes are proper subclasses of deterministic functional logspace, so the same dual-divisibility order already occurs inside $FL$.

cs.LO↗

arXiv2026

Step Recursion: Resource Profiles and Descent Quotients

We develop a resource representation for step recursion in which mutable-state width and recursion descent are explicit and independent parameters. A width bound $u$ controls the size of the encoded machine state, while an effective descent $ρ$ determines the available recursion depth $δ_ρ(u)$. For generalized-inverse descents, we derive the depth directly from generator growth and characterize the increasing sequences that can occur as generator orbits. We then connect this depth--width geometry to standard finite-branching computation. Every deterministic bounded-state dynamics is realizable by a single ordinary bounded step recursion over a fixed finite numerical basis. Using deterministic, existential, universal, or alternating aggregation on the same local dynamics yields the corresponding machine semantics. After closure under the width reparameterizations needed to absorb fixed local cost, the resulting language classes are exactly the machine time--space classes on profiles $(δ_ρ(u),u)$. Finally, profile domination quotients effective descents by admissible width reparameterization. Some depth curves collapse, yet polynomial widths support an explicit infinite strict hierarchy between the canonical polynomial- and exponential-depth profiles. Thus descent remains a nonredundant resource coordinate after polynomial width reparameterization; standard complexity classes are calibration points.

cs.CC↗