6 The word problem for automaton groups
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Jan Philipp Wächter
Abstract
In this chapter, we consider decision problems for automaton groups. These groups are defined using finite-state, letter-to-letter transducers, which are usually simply called automata in this context. In such an automaton, every state induces a function mapping input to output words. If the automaton is invertible, the closure of these functions under composition is a group, which is called an automaton group. Alternatively automaton groups can be viewed as certain subgroups of the automorphism group of an infinite regular tree. Our main focus is the word problem, which asks whether a given word over the generators-in the case of automaton groups, the states-represents the neutral element. We will discuss the complexity of this problem in various variants (ordinary vs. compressed; uniform vs. nonuniform) and also for various subclasses of automaton groups obtained via Sidki’s activity hierarchy. We introduce all necessary concepts from automaton theory including Turing machines and complexity theory. In particular, we give a complete proof chain to show that the uniform word problem for automaton groups is PSPACE-complete without requiring in-depth knowledge from theoretical computer science.
Abstract
In this chapter, we consider decision problems for automaton groups. These groups are defined using finite-state, letter-to-letter transducers, which are usually simply called automata in this context. In such an automaton, every state induces a function mapping input to output words. If the automaton is invertible, the closure of these functions under composition is a group, which is called an automaton group. Alternatively automaton groups can be viewed as certain subgroups of the automorphism group of an infinite regular tree. Our main focus is the word problem, which asks whether a given word over the generators-in the case of automaton groups, the states-represents the neutral element. We will discuss the complexity of this problem in various variants (ordinary vs. compressed; uniform vs. nonuniform) and also for various subclasses of automaton groups obtained via Sidki’s activity hierarchy. We introduce all necessary concepts from automaton theory including Turing machines and complexity theory. In particular, we give a complete proof chain to show that the uniform word problem for automaton groups is PSPACE-complete without requiring in-depth knowledge from theoretical computer science.
Chapters in this book
- Frontmatter I
- Contents V
- List of Contributing Authors VII
- 1 Cellular automata over shifts and subshifts: Garden of Eden theorems and surjunctivity 1
- 2 Languages, groups, and equations 63
- 3 Stallings’ automata 95
- 4 Groups with multiple context-free word problems 161
- 5 Parallel complexity in group theory 183
- 6 The word problem for automaton groups 265
- Index 397
Chapters in this book
- Frontmatter I
- Contents V
- List of Contributing Authors VII
- 1 Cellular automata over shifts and subshifts: Garden of Eden theorems and surjunctivity 1
- 2 Languages, groups, and equations 63
- 3 Stallings’ automata 95
- 4 Groups with multiple context-free word problems 161
- 5 Parallel complexity in group theory 183
- 6 The word problem for automaton groups 265
- Index 397
