State Complexity of Linear Relations and Linear Subsequences of Automatic Sequences

Abstract

In this thesis, we study the state complexity of specific formal languages; for example, we study the number of states required in the minimal automaton reading the representation of two integers $i, j$ in parallel and accepting them if and only $i+c = j$ for some constant integer $c \geq 1$. We also study the state complexity of linear subsequences of automatic sequences; for example, we study the number of states required in the minimal automaton generating the linear subsequence $(h(i+c))_{i \geq 0}$ for some automatic sequences $(h(i))_{i \geq 0}$ and some constant integer $c \geq 1$. Moreover, we study the runtime complexity of generating automata for specific formal languages and linear subsequences of automatic sequences using a reasonable interpretation of B\"uchi arithmetic; for example we study the runtime complexity of creating an automaton reading the representation of two integers $i, j$ in parallel and accepting them if and only if $ni=j$ for some constant $n \geq 2$. We also state some open problems. The above topics are studied both for automata with input in base-$k$ representation for some integer $k \geq 2$ and for automata with input in Fibonacci representation. Most results are for automata reading input in most-significant-digit-first format and some results are stated for automata reading input in least-significant-digit-first format.