Inferring Different Types of Lindenmayer Systems Using Artificial Intelligence

Lindenmayer systems (L-systems) are a formal grammar system which consist of a set of rewriting rules. Each rewriting rule is comprised of a symbol to replace (predecessor), a replacement string (successor), and an optional condition that is necessary for replacement. Starting with an initial string, every symbol in the string is replaced in parallel in accordance with the conditions on the rewriting rules, to produce a new string. The replacement process iterates as needed to produce a sequence of strings. There are different types of L-systems, which allow for different types of conditions, and methods of selecting the rules to apply. Some symbols of the alphabet can be interpreted as instructions for simulation software towards process modelling, where each string describes another step of the simulated process. Typically, creating an L-system for a specific process is done by experts by making meticulous measurements and using a priori knowledge about the process. It would be desirable to have a method to automatically learn the L-systems (the simulation program) from data, such as from a temporal sequence of images. This thesis presents a suite of tools, collectively called the Plant Model Inference Tools or PMIT (despite the name, the tools are domain agnostic), for inferring different types of L-systems using only a sequence of strings describing the process over some initial time period. Variants of PMIT are created for deterministic context-free L-systems, stochastic L-systems, and parametric L-systems. They are each evaluated using existing known deterministic and parametric L-systems from the literature, and procedurally generated stochastic L-systems. Accuracy can be detected in various ways, such as checking whether the inferred L-system is equal to the original one. PMIT is able to correctly infer deterministic L-systems with up to 31 symbols in the alphabet compared to the previous state-of-the-art algorithm's limit of 2 symbols. Stochastic L-systems allow symbols in the alphabet to have multiple rewriting rules each with an associated probability of being selected. Evaluating stochastic L-system inference with 960 procedurally generated L-systems with multiple sequences of strings as input found the following: 1) when 3 input sequences are used, the inferred successors always matched the original successors for systems with up to 9 rewriting rules, 2) when 6 sequences of strings are used, the difference between the associated probabilities of the inferred and the original L-system is approximately 1%. Parametric L-systems allow symbols to have multiple rewriting rules with parameters that get passed during rewriting. Rule selection is based on an associated Boolean condition over the parameters that gets evaluated to choose the rule to be applied. Inference is done in two steps. In the first step, the successors are inferred, and in the second step, appropriate Boolean conditions are found. Parametric L-system inference was evaluated on 20 known parametric L-systems. For 18 of the 20 L-systems where all successors were non-empty, the successors were correctly identified, but the time taken was up to 26 days on a single core CPU for the largest L-system. The second step, inferring the Boolean conditions, was successful for all 20 systems in the test set. No previous algorithm from the literature had implemented stochastic or parametric L-system inference. Inferring L-systems of greater complexity algorithmically can save considerable time and effort versus constructing them manually; however, perhaps more importantly rather than relying on existing knowledge, inferring a simulation of a process from data can help reveal the underlying scientific principles of the process

Critical Exponents and Stabilizers of Infinite Words

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free binary words and all $(\frac 73)$-power-free binary words, but not over all $(\frac 73)^+$-power-free binary words; -- the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all $(\frac 73)$-power-free binary words; -- if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; -- the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC

Overlap-Free Words and Generalizations

The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. This thesis will consider several different generalizations of Thue's work. In particular we shall study the properties of infinite words avoiding various types of repetitions. In Chapter 1 we introduce the theory of combinatorics on words. We present the basic definitions and give an historical survey of the area. In Chapter 2 we consider the work of Thue in more detail. We present various well-known properties of the Thue-Morse word and give some generalizations. We examine Fife's characterization of the infinite overlap-free words and give a simpler proof of this result. We also present some applications to transcendental number theory, generalizing a classical result of Mahler. In Chapter 3 we generalize a result of Seebold by showing that the only infinite 7/3-power-free binary words that can be obtained by iterating a morphism are the Thue-Morse word and its complement. In Chapter 4 we continue our study of overlap-free and 7/3-power-free words. We discuss the squares that can appear as subwords of these words. We also show that it is possible to construct infinite 7/3-power-free binary words containing infinitely many overlaps. In Chapter 5 we consider certain questions of language theory. In particular, we examine the context-freeness of the set of words containing overlaps. We show that over a three-letter alphabet, this set is not context-free, and over a two-letter alphabet, we show that this set cannot be unambiguously context-free. In Chapter 6 we construct infinite words over a four-letter alphabet that avoid squares in any arithmetic progression of odd difference. Our constructions are based on properties of the paperfolding words. We use these infinite words to construct non-repetitive tilings of the integer lattice. In Chapter 7 we consider approximate squares rather than squares. We give constructions of infinite words that avoid such approximate squares. In Chapter 8 we conclude the work and present some open problems

Inklusion von Patternsprachen und verwandte Probleme

A pattern is a word that consists of variables and terminal symbols. The pattern language that is generated by a pattern A is the set of all terminal words that can be obtained from A by uniform replacement of variables with terminal words. For example, the pattern A = a x y a x (where x and y are variables, and the letter a is a terminal symbol) generates the set of all words that have some word a x both as prefix and suffix (where these two occurrences of a x do not overlap). Due to their simple definition, pattern languages have various connections to a wide range of other areas in theoretical computer science and mathematics. Among these areas are combinatorics on words, logic, and the theory of free semigroups. On the other hand, many of the canonical questions in formal language theory are surprisingly difficult. The present thesis discusses various aspects of the inclusion problem of pattern languages. It can be divide in two parts. The first one examines the decidability of pattern languages with a limited number of variables and fixed terminal alphabets. In addition to this, the minimizability of regular expressions with repetition operators is studied. The second part deals with descriptive patterns, the smallest generalizations of arbitrary languages through pattern languages ("smallest" with respect to the inclusion relation). Main questions are the existence and the discoverability of descriptive patterns for arbitrary languages.Ein Pattern ist ein Wort aus Variablen und Terminalsymbolen. Die von einem Pattern A erzeugte Patternsprache ist die Menge aller Terminalwörter, die durch eine uniforme Ersetzung der Variablen in A durch Terminalwörter erzeugt werden können. So beschreibt das Pattern A = a x y a x (wobei x und y Variablen sind und a ein Terminal ist) die Menge aller Wörter, die ein Wort der Form a x sowohl als Präfix, als auch als Suffix haben (ohne dass sich diese beiden Vorkommen von a x überlappen). Wegen ihrer einfachen Definition besitzen Patternsprachen eine Vielzahl von Verbindungen zu verschiedenen anderen Gebieten der theoretischen Informatik und Mathematik, unter anderem zur Wortkombinatorik, Logik und der Theorie freier Halbgruppen. Andererseits führen viele der üblichen sprachtheoretischen Fragestellungen bei Patternsprachen zu kombinatorischen Problemen von überraschender Schwierigkeit. Die vorliegende Dissertation widmet sich verschiedenen Aspekten des Inklusionsproblems von Patternsprachen und kann in zwei Teile unterteilt werden. Der erste Teil untersucht die Entscheidbarkeit des Inklusionsproblems für Sprachen, die von Pattern mit beschränkter Variablenzahl über Terminalalphabeten von beschränkter Größe erzeugt werden. Darüber hinaus werden verschiedene Aspekte der Minimierbarkeit von regulären Ausdrücken mit Rückreferenzen betrachtet. Der zweite Teil der Dissertation handelt von deskriptiven Pattern; d.h. denjenigen Pattern, die die (hinsichtlich der Inklusion) kleinsten Verallgemeinerungen einer gegebenen Sprache erzeugen. Hauptfragen sind hierbei die Existenz und die Auffindbarkeit deskriptiver Pattern für beliebige Sprachen