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(Le ture Notes for Álgebra de Pro essos) Mestrado em Matemáti a Computa ional Luís Soares Barbosa lsb di.uminho.pt Departamento de Informáti a Universidade do Minho Março 2003 1 Observation & Constru tion 1. Fun tions. One of the most elementary models of a omputational pro ess is that of a fun tion f : I −→ O whi h spe i es a transformation rule between two stru tures I and O. The behaviour of a fun tion is aptured by the output it produ es, whi h is ompletely determined by the supplied input. In a (metaphori al) sense, this may be dubbed as the `engineer's view' of reality: here is a re ipe (a tool, a te hnology) to build gnus from gnats. Often, however, reality is not so simple. For example, one may know how to produ e `gnus' from `gnats' but not in all ases. This is expressed by observing the output of f in a more re ned ontext: O is repla ed by O + 1 and f is said to be a partial fun tion. In other situations one may re ognise that there is some environmental (or ontext) information about `gnats' that, for some reason, should be hidden from input. It may be the ase that su h information is too extensive to be supplied to f by its user, or that it is shared by other fun tions as well. It might also be the ase that building gnus would eventually modify the environment, thus in uen ing latter produ tion of more `gnus'. For U a denotation of su h ontext information, the signature of f be omes