Finite-State Dimension and Lossy Decompressors

This paper examines information-theoretic questions regarding the difficulty of compressing data versus the difficulty of decompressing data and the role that information loss plays in this interaction. Finite-state compression and decompression are shown to be of equivalent difficulty, even when the decompressors are allowed to be lossy. Inspired by Kolmogorov complexity, this paper defines the optimal *decompression *ratio achievable on an infinite sequence by finite-state decompressors (that is, finite-state transducers outputting the sequence in question). It is shown that the optimal compression ratio achievable on a sequence S by any *information lossless* finite state compressor, known as the finite-state dimension of S, is equal to the optimal decompression ratio achievable on S by any finite-state decompressor. This result implies a new decompression characterization of finite-state dimension in terms of lossy finite-state transducers.Comment: We found that Theorem 3.11, which was basically the motive for this paper, was already proven by Sheinwald, Ziv, and Lempel in 1991 and 1995 paper

Replica Condensation and Tree Decay

We give an intuitive method--using local, cyclic replica symmetry--to isolate exponential tree decay in truncated (connected) correlations. We give an expansion and use the symmetry to show that all terms vanish, except those displaying {\em replica condensation}. The condensation property ensures exponential tree decay. We illustrate our method in a low-temperature Ising system, but expect that one can use a similar method in other random field and quantum field problems. While considering the illustration, we prove an elementary upper bound on the entropy of random lattice surfaces

Deep Faith: A Review of Early North African Christianity

One of the chief virtues of this book is how clearly and elegantly it is written. It is designed by an experienced teacher and an expert in his field. Posting about ­­­­­­­­the book Early North African Christianity from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation. https://inallthings.org/deep-faith-a-review-of-early-north-african-christianity

Thoughts on River Elegy, June 1988-June 2011

I remember watching the legendary six-part CCTV miniseries River Elegy in my dorm room at Peking University in June of 1988—the last June in the Chinese historical calendar that would not have its 4th day permanently stained red. At the time I could not understand much of the stentorian voice-over (I was only a couple of years into my lifelong struggle with Chinese), but during the week the show was broadcast it became clear that the documentary had hit academic circles like an atomic bomb. The series’ content—a sweeping, brutally painful critique of the deep structure of Chinese culture—was the topic of conversation among many of the Peking University grad students I was hanging out with. They had seen nothing like it. “At last,” they would say to me, “a TV show that tells the truth (shuo shihua).” It was common to see handwritten postings discussing the documentary on the outdoor bulletin boards at Peking University’ssanjiaodi, “triangle area,” and informal discussion sessions on the topics of the program were organized at Tsinghua other universities throughout China