<p>Strictly Local (SL) languages have proven useful for the modeling of local phonotactic dependencies and functional analogues that can describe local phonological processes were recently defined by exploiting a convenient property: namely that a word’s suffix (up to a particular finite length) determines its grammatical continuations. Strictly Piecewise (SP) languages are much like the SL languages and have been used to model non-local phonotactic dependencies, the difference between SL and SP languages being the choice of strict or general precedence. A functional extension of the SP languages would be useful for describing non-local phonological processes, but the SP languages do not exhibit a residual-defining property directly parallel to that which formed the basis of SL functions. The Boolean closure of the SP languages, known as the Piecewise Testable (PT) languages, do exhibit such a property, and we show how a class of SP functions can be defined by first defining a class of PT functions and then imposing specific restrictions onto them. We then characterize the PT functions using finite-state transducers, and consider an alternative for the SP functions since the size of a PT transducer is exponential in the size of the alphabet. By using a transducer-like object that operates by checking its memory store against sets of logical formulae, an exponential reduction in the size of the overall machine is possible, though it comes at the cost of an increase in run-time polynomially proportional to the size of the alphabet. The increased runtime is directly comparable to that of the factored acceptors for SP languages which inspired our alternative to simple finite-state transducers. This characterization also highlights a distinction between PT and SP functions that mirrors how the PT languages are the Boolean closure of the SP languages.</p>