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This paper demonstrates a duality between the non-robustness of polynomial time dimension and the existence of one-way functions. Polynomial-time dimension (denoted $\mathrm{cdim}_\mathrm{P}$) quantifies the density of information of infinite sequences using polynomial time betting algorithms called $s$-gales. An alternate quantification of the notion of polynomial time density of information is using polynomial-time Kolmogorov complexity rate (denoted $K_{poly}$). Hitchcock and Vinodchandran (CCC 2004) showed that $\mathrm{cdim}_\mathrm{P}$ is always greater than or equal to $K_{poly}$. We first show that if one-way functions exist then there exists a polynomial-time samplable distribution with respect to which $\mathrm{cdim}_\mathrm{P}$ and $K_{poly}$ are separated by a uniform gap with probability $1$. Conversely, we show that if there exists such a polynomial-time samplable distribution, then (infinitely-often) one-way functions exist. Using our main results, we solve a long standing open problem posed by Hitchcock and Vinodchandran (CCC 2004) and Stull under the assumption that one-way functions exist. We demonstrate that if one-way functions exist, then there are individual sequences $X$ whose poly-time dimension strictly exceeds $K_{poly}(X)$, that is $\mathrm{cdim}_\mathrm{P}(X) > K_{poly}(X)$. The corresponding unbounded notions, namely, the constructive dimension and the asymptotic lower rate of unbounded Kolmogorov complexity are equal for every sequence. Analogous notions are equal even at polynomial space and finite-state level. In view of these results, it is reasonable to conjecture that the polynomial-time quantities are identical for every sequence and set of sequences. However, under a plausible assumption which underlies modern cryptography - namely the existence of one-way functions, we refute the conjecture thereby giving a negative answer to the open question posed by Hitchcock, Vinodchandran and Stull. Further, we show that the gap between these quantities can be made as large as possible (i.e. close to 1). We also establish similar bounds for strong poly-time dimension versus asymptotic upper Kolmogorov complexity rates. Our proof uses several new constructions and arguments involving probabilistic tools such as the Borel-Cantelli Lemma, the Kolmogorov inequality for martingales and the theorem on universal extrapolation by Ilango, Ren, and Santhanam. This work shows that the question of non-robustness of polynomial-time information density notions, which is prima facie different, is intimately related to questions which are of current interest in cryptography and meta-complexity.

This paper demonstrates a duality between the non-robustness of polynomial time dimension and the existence of one-way functions. Polynomial-time dimension (denoted $\mathrm{cdim}_\mathrm{P}$) quantifies the density of information of infinite sequences using polynomial time betting algorithms called $s$-gales. An alternate quantification of the notion of polynomial time density of information is using polynomial-time Kolmogorov complexity rate (denoted $K_{poly}$). Hitchcock and Vinodchandran (CCC 2004) showed that $\mathrm{cdim}_\mathrm{P}$ is always greater than or equal to $K_{poly}$. We first show that if one-way functions exist then there exists a polynomial-time samplable distribution with respect to which $\mathrm{cdim}_\mathrm{P}$ and $K_{poly}$ are separated by a uniform gap with probability $1$. Conversely, we show that if there exists such a polynomial-time samplable distribution, then (infinitely-often) one-way functions exist. Using our main results, we solve a long standing open problem posed by Hitchcock and Vinodchandran (CCC 2004) and Stull under the assumption that one-way functions exist. We demonstrate that if one-way functions exist, then there are individual sequences $X$ whose poly-time dimension strictly exceeds $K_{poly}(X)$, that is $\mathrm{cdim}_\mathrm{P}(X) > K_{poly}(X)$. The corresponding unbounded notions, namely, the constructive dimension and the asymptotic lower rate of unbounded Kolmogorov complexity are equal for every sequence. Analogous notions are equal even at polynomial space and finite-state level. In view of these results, it is reasonable to conjecture that the polynomial-time quantities are identical for every sequence and set of sequences. However, under a plausible assumption which underlies modern cryptography - namely the existence of one-way functions, we refute the conjecture thereby giving a negative answer to the open question posed by Hitchcock, Vinodchandran and Stull. Further, we show that the gap between these quantities can be made as large as possible (i.e. close to 1). We also establish similar bounds for strong poly-time dimension versus asymptotic upper Kolmogorov complexity rates. Our proof uses several new constructions and arguments involving probabilistic tools such as the Borel-Cantelli Lemma, the Kolmogorov inequality for martingales and the theorem on universal extrapolation by Ilango, Ren, and Santhanam. This work shows that the question of non-robustness of polynomial-time information density notions, which is prima facie different, is intimately related to questions which are of current interest in cryptography and meta-complexity.

This paper demonstrates a duality between the non-robustness of polynomial time dimension and the existence of one-way functions. Polynomial-time dimension (denoted $\mathrm{cdim}_\mathrm{P}$) quantifies the density of information of infinite sequences using polynomial time betting algorithms called $s$-gales. An alternate quantification of the notion of polynomial time density of information is using polynomial-time Kolmogorov complexity rate (denoted $K_{poly}$). Hitchcock and Vinodchandran (CCC 2004) showed that $\mathrm{cdim}_\mathrm{P}$ is always greater than or equal to $K_{poly}$. We first show that if one-way functions exist then there exists a polynomial-time samplable distribution with respect to which $\mathrm{cdim}_\mathrm{P}$ and $K_{poly}$ are separated by a uniform gap with probability $1$. Conversely, we show that if there exists such a polynomial-time samplable distribution, then (infinitely-often) one-way functions exist. Using our main results, we solve a long standing open problem posed by Hitchcock and Vinodchandran (CCC 2004) and Stull under the assumption that one-way functions exist. We demonstrate that if one-way functions exist, then there are individual sequences $X$ whose poly-time dimension strictly exceeds $K_{poly}(X)$, that is $\mathrm{cdim}_\mathrm{P}(X) > K_{poly}(X)$. The corresponding unbounded notions, namely, the constructive dimension and the asymptotic lower rate of unbounded Kolmogorov complexity are equal for every sequence. Analogous notions are equal even at polynomial space and finite-state level. In view of these results, it is reasonable to conjecture that the polynomial-time quantities are identical for every sequence and set of sequences. However, under a plausible assumption which underlies modern cryptography - namely the existence of one-way functions, we refute the conjecture thereby giving a negative answer to the open question posed by Hitchcock, Vinodchandran and Stull. Further, we show that the gap between these quantities can be made as large as possible (i.e. close to 1). We also establish similar bounds for strong poly-time dimension versus asymptotic upper Kolmogorov complexity rates. Our proof uses several new constructions and arguments involving probabilistic tools such as the Borel-Cantelli Lemma, the Kolmogorov inequality for martingales and the theorem on universal extrapolation by Ilango, Ren, and Santhanam. This work shows that the question of non-robustness of polynomial-time information density notions, which is prima facie different, is intimately related to questions which are of current interest in cryptography and meta-complexity.