Incremental construction of minimal acyclic finite-state automata

In this paper, we describe a new method for constructing minimal, deterministic, acyclic finite-state automata from a set of strings. Traditional methods consist of two phases: the first to construct a trie, the second one to minimize it. Our approach is to construct a minimal automaton in a single phase by adding new strings one by one and minimizing the resulting automaton on-the-fly. We present a general algorithm as well as a specialization that relies upon the lexicographical ordering of the input strings.Comment: 14 pages, 7 figure

Out of Nowhere: Spacetime from causality: causal set theory

This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian W\"uthrich and under contract with Oxford University Press. (More information at www.beyondspacetime.net.) This chapter introduces causal set theory and identifies and articulates a 'problem of space' in this theory.Comment: 29 pages, 5 figure

Comparability in the graph monoid

Let $\Gamma$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $\Gamma$-action instead. Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^\Gamma$ with a $\Gamma$-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of $M_E^\Gamma$ is comparable, periodic, graphs such that every nonzero element of $M_E^\Gamma$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^\Gamma$ is periodic, and graphs such that no element of $M_E^\Gamma$ is aperiodic. The Graded Classification Conjecture can be formulated to state that $M_E^\Gamma$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^\Gamma.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.Comment: This version contains some modifications based on the input of a referee for the New York Journal of Mathematic