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+\section{Preliminary definitions}
+\label{sec:predefs}
+
+A number of types will turn out to be useful in general. We define:
+
+\lstinputlisting[firstline=25,lastline=28]{While/Common.dcl}
+
+The \CI{Error} type distinguishes between different kinds of errors, that can
+be further identified with a descriptive string. We could add more information
+to these types, like the position (file, line and column) of the input token
+that causes the error. The implementation thereof is straightforward and will
+for the sake of brevity be omitted here.
+
+We can model the partial state function as a function that yields an \CI{Either
+Error Int}. Whenever a variable is not defined, a \CI{Left (Runtime
+"Undefined")} will be returned.  Otherwise, a \CI{Right n} is returned. The
+initial state, which maps any variable to a \CI{Left} result, is implemented as
+an instance of the \CI{zero} class.
+
+\lstinputlisting[firstline=9,lastline=10]{While/WhileCommon.dcl}
+\lstinputlisting[firstline=12,lastline=13]{While/WhileCommon.icl}
+
+In order to build up our interpreter modularly, we wouldn't want to fix the
+type of the AST just yet. Rather, we want it to be possible to implement
+different kinds of abstract syntax trees: for example, it may be interesting to
+implement a While program as a list of statements (making composition
+implicit), while the straightforward AST would use a compositional constructor.
+To allow for different ASTs, we make the \CI{run} function, that will interpret
+a While program in some state, overloaded:
+
+\lstinputlisting[firstline=35,lastline=35]{While/WhileCommon.dcl}
+
+Similarly, we want to be able to use the same terminology of \emph{evaluation}
+for both arithmetic and boolean expressions. This can be done using overloading
+as well:
+
+\lstinputlisting[firstline=37,lastline=39]{While/WhileCommon.dcl}
+
+The \CI{eval} function may yield an error, for example when an undefined
+variable is used. It makes sense to separate evaluation and interpretation,
+because evaluation does not depend on the type for statements. Any AST using
+\CI{AExpr} may now use this \CI{eval} function (while it is also possible to
+create an AST that uses a different representation of arithmetic expressions).
+
+Although all these functions are presented in an overloaded fashion, we will
+only consider one implementation, the AST from \autoref{sec:ast} below. Other
+models do not introduce additional problems that would be interesting for the
+purposes of this paper.