We settle the exact complexity of the reachability problem in (stateless) vector addition systems (VAS) in fixed low dimension. In dimensions 2-4 it has only been known to be sandwiched between NP and PSPACE. We prove PSPACE-hardness of the reachability problem for symmetric vector addition systems in dimension 3 (3-VAS), a restricted fragment of general 3-VAS. Combined with previously established PSPACE upper bounds, our result settles the complexity of the problem to be PSPACE-complete in 3-VAS and 4-VAS, as well as in their symmetric fragments.
Sharpness and complexity are two central factors in the generalization analysis of deep neural networks. Existing quantitative evaluations of generalization measures have largely focused on individual scalar measures, leaving the joint explanatory power of sharpness and complexity largely unexplored. This work studies how far sharpness and complexity can jointly explain generalization. We use linear regression and introduce a Pareto-based analysis to quantitatively evaluate the joint explanatory power of these two factors. Beyond the existing parameter-level definitions, we further propose realizations of sharpness and complexity that are closer to function space and less dependent on raw parameter representations. We find that function-oriented definitions of these two quantities expand the explanatory scope of the two-factor view beyond what is achieved by existing parameter-level metrics. Overall, our results support the sharpness-complexity perspective as an informative lens for understanding generalization across diverse settings. At the same time, the remaining failures indicate that whether this two-factor view can serve as a complete theory of generalization remains open.
Finding the shortest program that generates a sequence is uncomputable, and for six decades that fact has been mistaken for a wall around finding any generating program. It is not a wall but a price, and this paper measures it. For every algorithm that learns about a candidate program only through its score, a class spanning Levin search, evolutionary methods, simulated annealing, and the cross-entropy method, we define the coupling width of a search problem and prove an unconditional worst-case lower bound, exponential in that width with base one less than the domain size. From it follows a conservation law: structural knowledge injected into a search trades one for one against the search it removes, and their sum can never fall below the length of the program sought. Levin's 1973 upper bound and the lower bound proved here are the two ends of one conserved quantity, closing on each other as the instruction set grows. The only escape is to read a candidate's structure rather than its score, and its price, which we prove for generic targets, is incompleteness. A deterministic engine built on this theory recovers a generating program, certified by compressing its data and predicting an unseen continuation, for 2,383 of 3,914 sequences across four independent populations, including 244 of the 256 elementary cellular automata, with measured discovery cost rising along program length more than an order of magnitude inside the score-oracle worst case.