Finite Automata Models: Algorithm, Application, and Semigroup Study

Abstract

A finite automaton consists of states and state transitions labeled by letters of a finite alphabet. Every letter describes a transformation of the automaton's state space, generating a corresponding transformation semigroup. A stochastic version of automata has a probability distribution over the alphabet at every state, that allows assigning the likelihoods to generated sequences of letters. In this thesis, we explore two models: deterministic probabilistic finite automaton (DPFA) considered from the practical side, and the sandpile model considered from the theoretical side. For the first model, the presented study describes an algorithm for reconstructing DPFA from sequences of discrete observations using an n-gram merging method, including the practical implementation of the method, its experimental evaluation on case studies, and the application of this automata-based technique to the neurobiological data. For considered examples the performance is compared to the causal state splitting reconstruction (CSSR) technique: both methods achieve high quality in approximating the probability distribution over strings. Considering if transformation semigroups of reconstructed automata contain subgroups corresponding to those in examples, CSSR shows a good result for preserving cyclic permutation groups, but not the n-gram merging method, whose transformation semigroup is generally aperiodic. The application considered in this study uses electroencephalographic (EEG) microstate sequences and considers the question of distinguishing the participant groups (meditators and controls) and cognitive modes (mind-wandering, verbalization, visualization) by separating DPFA machines inferred from EEG data in a metric space. The separation between participant groups is achieved for many parameter settings with linear criterion (requiring non-overlapping clusters) and for a few instances with strict criterion (requiring dense distant clusters). Both criteria show great reliability when validated using permuted data. The separation of cognitive modes only demonstrated partial success with noticeably better performance within the group of controls and more instances of separation corresponding to the visualization condition. For the second, sandpile model, the presented study concentrates on the properties of their transformation semigroups for the standard Abelian sandpiles on circle graphs and the modified model, non-Abelian sandpiles on rooted trees. The exploration addresses the structure of recurrent configurations, the wreath product decomposition of semigroups, and the decomposition-based complexity measure. The identity and generator configurations of the recurrent group of Abelian sandpiles on circles are described, giving an explicit alternative way of understanding their well-known cyclic structure. The complexity of arbitrary finite Abelian semigroup is shown to be at most one. The embedding of the sandpile semigroup into the wreath product of flip-flop semigroups is constructed for non-Abelian sandpiles on rooted trees, implying its aperiodic complexity is the depth of the tree.