Forbidding and enforcing systems (fe-systems) provide a new way of defining classes of structures based on boundary conditions. Forbidding and enforcing systems on formal languages were inspired by molecular reactions and DNA computing. Initially, they were used to define new classes of languages (fe-families) based on forbidden subwords and enforced words. This paper considers a metric on languages and proves that the metric space obtained is homeomorphic to the Cantor space. This work studies Chomsky classes of families as subspaces and shows they are neither closed nor open. The paper investigates the fe-families as subspaces and proves the necessary and sufficient conditions for the fe-families to be open. Consequently, this proves that fe-systems define classes of languages different than Chomsky hierarchy. This work shows a characterization of continuous functions through fe-systems and includes results about homomorphic images of fe-families. This paper introduces a new notion of connecting graphs and a new way to study classes of graphs. Forbidding-enforcing systems on graphs define classes of graphs based on forbidden subgraphs and enforced subgraphs. Using fe-systems, the paper characterizes known classes of graphs, such as paths, cycles, trees, complete graphs and k-regular graphs. Several normal forms for forbidding and enforced sets are stated and proved. This work introduces the notion of forbidding and enforcing to posets where fe-systems are used to define families of subsets of a given poset, which in some sense generalizes language fe-systems. Poset fe-systems are, also, used to define a single subset of elements satisfying the forbidding and enforcing constraints. The latter generalizes graph fe-systems to an extent, but defines new classes of structures based on weak enforcing. Some properties of poset fe-systems are investigated. A series of normal forms for forbidding and enforcing sets is presented. This work ends with examples illustrating the computational potential of fe-systems. The process of cutting DNA by an enzyme and ligating is modeled in the setting of language fe-systems. The potential for use of fe-systems in information processing is illustrated by defining the solutions to the k-colorability problem.

This chapter sets out a new proposal for a coherent interdisciplinary prehistory of the Andes, based firstly on a long overdue reexamination of the relationships between the various regional ‘dialects’ within the Quechua language family; and secondly on the search for a far more satisfactory correlation with the archaeological record.Our founding principle is that language expansions do not ‘just happen’. Rather, they happen only for those very same reasons of socio-cultural change that archaeology seeks to describe through its own, independent data. Here is the true link between our disciplines, so we discard outdated, facile equations of ‘language equals culture equals genes’, in favour of the real correlation: that language families necessarily reflect past expansive processes, whose traces should also be clear in the material culture record. This principle is one that we can make use of to identify and assess correspondences between archaeological and linguistic patterns, on three levels: chronology, geography, and above all, causation. Or in other words: when, where and why did particular language expansions occur?In the Andes, in principle this entails that we should look to the Horizons, not the Intermediate Periods, as offering the most natural explanations for the major Quechua and Aymara dispersals. With the Incas too late to account for the time-depth of either family, the most plausible candidate for the first major expansion of Quechua turns out in our view to be the Wari Middle Horizon, with the Chavín Early Horizon more tentatively suggested as behind the earlier spread of the Aymara family. This effectively both upturns the traditional Torero hypothesis, and bears clear implications for the long debate in archaeology as to the nature, duration and extent of ‘Horizons’. / Este artículo propone una nueva visión de la prehistoria andina, que busca tejer un conjunto mas coherente entre las varias disciplinas que intentan entender el pasado precolombino. Se fundamenta, en primer lugar, en una reexaminación, pendiente ya desde décadas, de la clasificación tradicional de las relaciones entre los diversos "dialectos" regionales al interior de la familia lingüística quechua; y, en segundo lugar, en la búsqueda de una correlación mucho más satisfactoria con el registro arqueológico.El nuevo enfoque que aquí proponemos se enraíza en el principio fundamental que si algunas lenguas mayores han logrado dispersarse de manera espectacular, esto no pudo haber ocurrido sin ningún motivo. Más bien, tales expansiones lingüísticas se deben a las mismas razones —es decir, los mismos cambios socioculturales— que la arqueología también busca describir por medio de sus propios datos independientes. Allí radica el auténtico vínculo entre nuestras disciplinas, de manera que podemos descartar las ecuaciones simplistas y obsoletas del estilo "lengua=cultura=genes", en favor de la correlación verdadera: las familias de lenguas reflejan procesos expansivos pasados, cuyos indicios deberían quedar claros también en el registro de la cultura material. Este principio se aprovecha para identificar y evaluar las correspondencias entre los patrones arqueológicos y lingüísticos, y así en tres niveles: la cronología, la geografía y, sobre todo, la causalidad. En otras palabras: ¿cuando, dónde y porqué se difundieron determinadas lenguas?En los Andes esto implica que en principio debemos ver a los horizontes, y no a los periodos intermedios, como los que ofrecen las explicaciones más naturales para las dispersiones mayores del quechua y el aimara. Ya que el Imperio incaico remonta a una época demasiado tardía las explicaciones de la profundidad temporal de cada familia, es más bien el Horizonte Medio Wari el que se vuelve el candidato más verosímil para haber vehiculizado la primera gran expansión del quechua, según nuestro parecer. Asimismo, aunque de manera más tentativa, se sugiere que el Horizonte Temprano Chavín pudo haber impulsado la dispersión más temprana de la familia aimara. Esto, en efecto, trastoca la hipótesis tradicional de Torero, además de conllevar claras implicancias para el largo debate arqueológico acerca de la naturaleza, duración y extensión de los "horizontes".