Superposition or critical pair computation is one of the key operations in the Knuth-Bendix completion procedure and its extensions. We propose a practical technique which can save computation of some critical pairs where the most general unifiers used to generate these critical pairs are less general than the most general unifiers used to generate other joinable critical pairs. Consequently, there is no need to superpose identical subterms at different positions in a rule more than once and there is also no need to superpose symmetric subterms in a rule more than once. The combination of this technique with other critical pair criteria proposed in the literature is also discussed. The technique has been integrated in the completion procedures for ordinary term rewriting systems as well as term rewriting systems with associative-commutative operators implemented in RRL, Rewrite Rule Laboratory. Performance of the completion procedures with and without this technique is compared on a number of examples.
This chapter discussed the concept of the black hole. Black holes were definedand differentiated into three groups; mini-black holes, intermediate black holes andsupermassive black holes. Black hole effects include extraordinary gravitational pull andthe emission of radiation. Matter is also crushed into zero volume, and the spaghettificationphenomenon means that black holes stretch things to death. Black holes are fearsomeobjects, in part because they are dangerous from a distance and also because you canaccidentally get relatively close to them. The future destruction of the Earth by a black holewas vividly described. There are about two dozen black holes in the Milky Way andmillions in the universe. The dangers posed by the accretion disk were explained anddescribed.
Unlike most chapters, whose content is mutually exclusive to that containedelsewhere in the same book, there was intentional redundancy between some of thematerial in this chapter and previous chapters. The reason is that both satellites andspace debris (each the subject of a previous chapter), were deemed individual andindependent risks to those residing on the Earth and spacefarers. But for the purposes ofthe present chapter, both are also important factors in space saturation. In this chapterspace saturation was defined and exemplified. We learned about the linear nature ofspace saturation. The constant reality of cosmic collisions was explained, exemplifiedand quantified. Collisions involving space satellites, asteroids, comets, planets,centaurs, galaxies and other space bodies were described. The past, present and futureincidence of collisions was estimated.
The purpose of this chapter was to attempt to quantify as closely as possible therisk posed to spacefarers and to the planet Earth by space dangers. The empirical record ofspace objects striking the Earth was examined, specifically with respect to NEOs, asteroids,meteors and comets. Potential threats from supernovas, black holes, space debris andgamma rays were also quantified. An overall aggregate risk assessment was provided.