Having a good length function on a group is the main ingredient in a recent combined memory/length approach to solving random equations in that group. Currently, the groups of greatest interest in this respect are Artin’s braid groups. We demonstrate, by a series of experiments, that a length function based on the Birman-Ko-Lee presentation of the braid group is substantially better than the co...

The globalized knowledge society generates virtual enterprises that are usually set up and managed on the web, and the new trend is to make the relevant technologies available on intelligent portable devices. The existence of trust is a mandatory condition to make such enterprises successful. Trust has many facets ranging from very theoretical ones to fully heuristic features. One point is that...

The problem of Π1−separating the hierarchy of bounded arithmetic has been studied in the paper. It is shown that the notion of Herbrand Consistency, in its full generality, cannot Π1−separate the theory I∆0 + ∧ j Ωj from I∆0; though it can Π1−separate I∆0 + Exp from I∆0. This extends a result of L. A. Ko lodziejczyk (2006), by showing the unprovability of the Herbrand Consistency of I∆0 in the ...

Human society is increasingly dependent on artifacts. The progress of artificial intelligence accelerates this tendency. In spite of strong concern about heavy dependence on artifacts, it appears an inevitable consequence of the knowledge society. In this chapter, I am seeking a better way of living with advanced artifacts to realize an artifact-mediated society where people are supported by hu...