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1. Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

   Lu, Zhenjian; Oliveira, Igor C.; Zimand, Marius (January 2022, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022))

   Mikołaj Bojańczyk and Emanuela Merelli and David P. Woodruff (Ed.)

   The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then K(x) ≤ log(1/δ) + O(1). In a recent work, Lu and Oliveira [31] established an unconditional time-bounded version of this result, by showing that if x can be efficiently sampled with probability δ then rKt(x) = O(log(1/δ)) + O(log n), where rKt denotes the randomized analogue of Levin’s Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a O(log(1/δ)) bound instead of the information-theoretic optimal log(1/δ). Motivated by this discrepancy, we investigate optimal coding theorems in the time-bounded setting. Our main contributions can be summarised as follows. • Efficient coding theorem for rKt with a factor of 2. Addressing a question from [31], we show that if x can be efficiently sampled with probability at least δ then rKt(x) ≤ (2 + o(1)) · log(1/δ) +O(log n). As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given x, the code of the sampler, and δ, it outputs, with probability ≥ 0.99, a probabilistic representation of x that certifies this rKt complexity bound. • Optimality under a cryptographic assumption. Under a hypothesis about the security of cryptographic pseudorandom generators, we show that no efficient coding theorem can achieve a bound of the form rKt(x) ≤ (2 − o(1)) · log(1/δ) + poly(log n). Under a weaker assumption, we exhibit a gap between efficient coding theorems and existential coding theorems with near-optimal parameters. • Optimal coding theorem for pKt and unconditional Antunes-Fortnow. We consider pKt complexity [17], a variant of rKt where the randomness is public and the time bound is fixed. We observe the existence of an optimal coding theorem for pKt, and employ this result to establish an unconditional version of a theorem of Antunes and Fortnow [5] which characterizes the worst-case running times of languages that are in average polynomial-time over all P-samplable distributions. 

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2. Search-To-Decision Reductions for Kolmogorov Complexity

   https://doi.org/10.4230/lipics.ccc.2024.34  

   Mazor, Noam; Pass, Rafael (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   {"Abstract":["A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-t program generating a given string x is at most s.\r\nIn this work, we consider the more "robust" version of the time-bounded Kolmogorov complexity problem, referred to as the GapMINKT problem, where given a size bound s and a running time bound t, the goal is to determine whether there exists a poly(t,|x|)-time program of length s+O(log |x|) that generates x. We present the first non-trivial search-to-decision reduction R for the GapMINKT problem; R has a running-time bound of 2^{ε n} for any ε > 0 and additionally only queries its oracle on "thresholds" s of size s+O(log |x|). As such, we get that any algorithm with running-time (resp. circuit size) 2^{α s} poly(|x|,t,s) for solving GapMINKT (given an instance (x,t,s), yields an algorithm for finding a witness with running-time (resp. circuit size) 2^{(α+ε) s} poly(|x|,t,s).\r\nOur second result is a polynomial-time search-to-decision reduction for the time-bounded Kolmogorov complexity problem in the average-case regime. Such a reduction was recently shown by Liu and Pass (FOCS'20), heavily relying on cryptographic techniques. Our reduction is more direct and additionally has the advantage of being length-preserving, and as such also applies in the exponential time/size regime.\r\nA central component in both of these results is the use of Kolmogorov and Levin’s Symmetry of Information Theorem."]} 

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3. The traveling salesman theorem for Jordan curves

   Bishop, Christopher J. (September 2023, Advances in mathematics)

   This gives an improvment of Peter Jones's traveling salesman theorem that holds for Jordan curves, but not for general sets. His theorem implies that the length of a Jordan arc is bounded by (1+delta)diameter + C(delta) beta-sum, and this paper shows this can be replaced by chord + O(beta-sum), where O(diameter) is replaced by the distance between the endpoints of the curve. This is true in all finite dimensions (with a dimension dependent constant). A corollary of our self-contained argument proves the ususal TST in all dimensions (a result of Okikiolu). An appendix proves a folklore result that several different formulation of Jones's theorem are all equivalent. 

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4. Analog Category and Complexity

   Knudsen, Ben; Weinberger, Shmuel (June 2024, SIAM journal on applied algebra and geometry)

   We study probabilistic variants of the Lusternik–Schnirelmann category and topological complexity, which bound the classical invariants from below. We present a number of computations illustrating both wide agreement and wide disagreement with the classical notions. In the aspherical case, where our invariants are group invariants, we establish a counterpart of the Eilenberg– Ganea theorem in the torsion-free case, as well as a contrasting universal upper bound in the finite case. 

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5. Gardner formula for Ising perceptron models at small densities

   Bolthausen, E.; Nakajima, S; Sun, N; Xu, C (January 2022, Conference on Learning Theory)

   We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth–Mezard 1989, see also Gardner–Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero. 

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