Niels Christensen

This invited guest blog post focuses on Leonid Levin's Optimal Search Theorem. The theorem is one of few general results proving that it is not in vain to strive for the fastest solution to a computational task. In contrast to the "negative" result known as Blum's Speedup Theorem, Levin shows that a lot of tasks does have an optimal program. Or, if you prefer that perspective, it points to a severe problem in the concept of big-O-optimality.

The linear speedup theorem states, informally, that constants do not matter: It is essentially always possible to find a program solving any decision problem a factor of 2 faster. This result is a classical theorem in computing, but also one of the most debated. The main ingredient of the typical proof of the linear speedup theorem is tape compression, where a fast machine is constructed with tape alphabet or number of tapes far greater than that of the original machine. In this paper, we prove… Vis mere

The linear speedup theorem states, informally, that constants do not matter: It is essentially always possible to find a program solving any decision problem a factor of 2 faster. This result is a classical theorem in computing, but also one of the most debated. The main ingredient of the typical proof of the linear speedup theorem is tape compression, where a fast machine is constructed with tape alphabet or number of tapes far greater than that of the original machine. In this paper, we prove that limiting Turing machines to a fixed alphabet and a fixed number of tapes rules out linear speedup. Specifically, we describe a language that can be recognized in linear time (e. g., 1.51n), and provide a proof, based on Kolmogorov complexity, that the computation cannot be sped up (e. g., below 1.49n). Without the tape and alphabet limitation, the linear speedup theorem does hold and yields machines of time complexity of the form (1+ε)n for arbitrarily small ε > 0.

Earlier results negating linear speedup in alternative models of computation have often been based on the existence of very efficient universal machines. In the vernacular of programming language theory: These models have very efficient self-interpreters. As the second contribution of this paper, we define a class, PICSTI, of computation models that exactly captures this property, and we disprove the Linear Speedup Theorem for every model in this class, thus generalizing all similar, model-specific proofs. Vis mindre

In one year, 98% of the atoms in your body will have left you. Yet your memories will remain with you. It is the structures and their interactions that makes our memories remain. The same principle applies to digital storage.

This paper shows how the design of a digital repository can be quantitatively related to its longevity. I define a programmatic, probabilistic model of hardware failures and repair operations in a digital repository. The mean time to failure of this model is then… Vis mere

In one year, 98% of the atoms in your body will have left you. Yet your memories will remain with you. It is the structures and their interactions that makes our memories remain. The same principle applies to digital storage.

This paper shows how the design of a digital repository can be quantitatively related to its longevity. I define a programmatic, probabilistic model of hardware failures and repair operations in a digital repository. The mean time to failure of this model is then computed in a number of experiments based on simulation

Keywords: web archiving, bit preservation, mean time to failure, simulation of probabilistic models. Vis mindre