Pattern avoidance: themes and variations

AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems

On Periodically Iterated Morphisms

We investigate the computational power of periodically iterated morphisms, also known as D0L systems with periodic control, PD0L systems for short. These systems give rise to a class of one-sided infinite sequences, called PD0L words. We construct a PD0L word with exponential subword complexity, thereby answering a question raised by Lepisto (1993) on the existence of such words. We solve another open problem concerning the decidability of the first-order theories of PD0L words; we show it is already undecidable whether a certain letter occurs in a PD0L word. This stands in sharp contrast to the situation for D0L words (purely morphic words), which are known to have at most quadratic subword complexity, and for which the monadic theory is decidable. The main result of our paper, leading to these answers, is that every computable word w over an alphabet Sigma can be embedded in a PD0L word u over an extended alphabet Gamma in the following two ways: (i) such that every finite prefix of w is a subword of u, and (ii) such that w is obtained from u by erasing all letters from Gamma not in Sigma. The PD0L system generating such a word u is constructed by encoding a Fractran program that computes the word w; Fractran is a programming language as powerful as Turing Machines. As a consequence of (ii), if we allow the application of finite state transducers to PD0L words, we obtain the set of all computable words. Thus the set of PD0L words is not closed under finite state transduction, whereas the set of D0L words is. It moreover follows that equality of PD0L words (given by their PD0L system) is undecidable. Finally, we show that if erasing morphisms are admitted, then the question of productivity becomes undecidable, that is, the question whether a given PD0L system defines an infinite word

Cubefree words with many squares

We construct infinite cubefree binary words containing exponentially many distinct squares of length n . We also show that for every positive integer n , there is a cubefree binary square of length 2n.The first author is supported by an NSERC Discovery Grant. The second author is supported by an NSERC Postdoctoral Fellowship.http://dmtcs.episciences.org/48

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

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