# Interpretations

A map changes input structures to output structures. Another way to think about it is that, under a map, an output structure is some reflection of the input structure.

**Interpretations** specify maps by defining output structures in terms of the input structures (Engelfriet and Hoogeboom, 2001).

A **scheme** is thus a series of definitions of (output) properties.
A **boolean monadic recursive scheme** is thus a scheme in which each property in the scheme is **boolean** (i.e., returns a boolean value) and **monadic** (i.e. unary), and the definitions are **recursive** (i.e., the properties being defined can be used in their own definitions).

In phonological terms, output formulas assert the conditions under which a segment is \(+\) for a given feature in the output structure.

Thus, a BMRS for word-final devoicing is as follows:

\[ \begin{array}[t]{ll} [\mathrm{son}]_o(x) := [\mathrm{son}]_i(x) \\ [\mathrm{voi}]_o(x) := \mathtt{if}~\mathrm{D\#}_i(x)~\mathtt{then}~\bot~\mathrm{else}~[\mathrm{voi}]_i(x)\\ [\mathrm{cor}]_o(x) := [\mathrm{cor}]_i(x) \\ \end{array} \]

This obtains the truth value table above.