Euclidean automata

In this note we introduce Euclidean automata (EA), a simple generalization of Finite State Automata (FSA). EA operate not on symbols from a finite alphabet as usual, but rather on vectors from a parameter space P , typically R. The main motivation for EA comes from classification problems involving a forced choice between a finite number (in the most important case, only two) alternatives. Since we want classifications to be stable under small perturbation of inputs, ideally the set of points in P classified to a given value should be open, yet it is evident that we cannot partition R or C into finitely many disjoint open sets. Approximate solutions thus must give up non-overlapping, e.g. by permitting probabilistic or fuzzy outcomes, or exhaustiveness, e.g. by leaving ‘gray areas’ near decision boundaries where the system produces no output. EA, as we shall see, sacrifice non-overlapping but maintain sharp, deterministic decision boundaries. Using EA we offer the beginnings of an analysis of being in a conflicted state, some situation where we know that we should do A, since it is ‘the right thing to do’ yet we have a strong compulsion to do some B (including doing nothing) instead. To anchor the discussion we will use several specific examples, such as refraining from or taking some drug such as tobacco, alcohol, or heroin, that is generally agreed to have pleasant short-term but harmful long-term effects; slipping into some recreational activity while there is still work to do; keeping or not keeping some promise; etc. The problem is complex, arguably it is the single most complex problem we have to face in everyday life. Therefore, some simplification will be necessary, and we will state the main problem in a way that already abstracts away from certain aspects that would take us far from our goal of analyzing internal conflict. First, we are not interested in defending the specific moral premisses used in the analysis of drug addiction, laziness, and similar examples of conflict: our focus is with the conflicted state itself, not the individual components. Second, we will pay only limited attention of the issue of how we know that A is the right thing rather than some alternative A′ or even B – we are interested in the situation when we already know that A is right and B is wrong. Third, real life conflicts are rarely between two, laboratory pure components: often there are multiple factors,