By C. Ausnit, et al

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It follows that [ p] = [q] , which implies that T p = q. Ground-completeness of a model of some theory is an interesting property because it implies that the model does not validate any equations of closed terms that are not also derivable from the theory. The initial algebra of an equational theory is ground-complete for that theory by definition. However, 28 Preliminaries ground-completeness is not necessarily restricted to the initial algebra of an equational theory; other models of the theory may also satisfy this property.

An important observation is that, since the subject of study is interacting systems, not just the language generated by an automaton is important, but also the states traversed during a run or execution of the automaton. The term ‘transition system’ is the term most often used in reactive-systems modeling. Thus, also this book uses that term. The semantic domain serves as the basis for the remainder of the book. The meaning of the various equational theories for reasoning about reactive systems developed in the remaining chapters is defined in terms of the semantic domain, in the way explained in the previous chapter.

In this book, it is always assumed that there are as many variables as needed. Therefore, often, the set of variables is omitted from the notation T ( , V ), yielding the notation T ( ) for all the terms over signature . As a shorthand, terms over some signature are also referred to as -terms. A term that does not contain variables is called a closed or ground term. The set of all closed terms over signature is denoted C( ). To emphasize that arbitrary terms in T ( ) may contain variables, they are often referred to as open terms.

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