Number of Holes in Unavoidable Sets of Partial Words II

We are concerned with the complexity of deciding the avoidability of sets of partial words over an arbitrary alphabet. Towards this, we investigate the minimum size of unavoidable sets of partial words with a fixed number of holes. Additionally, we analyze the complexity of variations on the decision problem when placing restrictions on the number of holes and length of the words

Number of Holes in Unavoidable Sets of Partial Words II *

Abstract We are concerned with the complexity of deciding the avoidability of sets of partial words over an arbitrary alphabet. Towards this, we investigate the minimum size of unavoidable sets of partial words with a fixed number of holes. Additionally, we analyze the complexity of variations on the decision problem when placing restrictions on the number of holes and length of the words

Number of Holes in Unavoidable Sets of Partial Words I

Partial words are sequences over a finite alphabet that may contain some undefined positions called holes. We consider unavoidable sets of partial words of equal length. We compute the minimum number of holes in sets of size three over a binary alphabet (summed over all partial words in the sets). We also construct all sets that achieve this minimum. This is a step towards the difficult problem of fully characterizing all unavoidable sets of partial words of size three

Unavoidable Sets of Partial Words

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of ?do not know? symbols or ?holes,? appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a k-letter alphabet

Testing avoidability on sets of partial words is hard

We prove that the problem of deciding whether a finite set of partial words is unavoidable is NP-hard for any alphabet of size larger than or equal to two, which is in contrast with the well-known feasability results for unavoidability of a set of full words. We raise some related questions on avoidability of sets of partial words

Random subshifts of finite type

Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $\omega$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $\alpha$. Let $X_{\omega}$ be the (random) SFT built from the set $\omega$. For each $0\leq \alpha \leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_{\omega}$ is empty, as well as the limiting distribution of entropy for $X_{\omega}$. For $\alpha$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_{\omega}$ contains a unique irreducible component of positive entropy converges exponentially to 1. These results are obtained by studying certain sequences of random directed graphs. This version of "random SFT" differs significantly from a previous notion by the same name, which has appeared in the context of random dynamical systems and bundled dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP636 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hiding variables when decomposing specifications into GR(1) contracts

We propose a method for eliminating variables from component specifications during the decomposition of GR(1) properties into contracts. The variables that can be eliminated are identified by parameterizing the communication architecture to investigate the dependence of realizability on the availability of information. We prove that the selected variables can be hidden from other components, while still expressing the resulting specification as a game with full information with respect to the remaining variables. The values of other variables need not be known all the time, so we hide them for part of the time, thus reducing the amount of information that needs to be communicated between components. We improve on our previous results on algorithmic decomposition of GR(1) properties, and prove existence of decompositions in the full information case. We use semantic methods of computation based on binary decision diagrams. To recover the constructed specifications so that humans can read them, we implement exact symbolic minimal covering over the lattice of integer orthotopes, thus deriving minimal formulae in disjunctive normal form over integer variable intervals