A Classification of Trapezoidal Words

Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Learnability and Overgeneration in Computational Syntax

This paper addresses the hypothesis that unnatural patterns generated by grammar formalisms can be eliminated on the grounds that they are unlearnable. I consider three examples of formal languages thought to represent dependencies unattested in natural language syntax, and show that all three can be learned by grammar induction algorithms following the Distributional Learning paradigm of Clark and Eyraud (2007). While learnable language classes are restrictive by necessity (Gold, 1967), these facts suggest that learnability alone may be insufficient for addressing concerns of overgeneration in syntax

A Survey on the Local Divisor Technique

Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of localizable language class as a new abstract concept which unifies some of the proofs for the results above

Blockchain-Based Digitalization of Logistics Processes—Innovation, Applications, Best Practices

Blockchain technology is becoming one of the most powerful future technologies in supporting logistics processes and applications. It has the potential to destroy and reorganize traditional logistics structures. Both researchers and practitioners all over the world continuously report on novel blockchain-based projects, possibilities, and innovative solutions with better logistic service levels and lower costs. The idea of this Special Issue is to provide an overview of the status quo in research and possibilities to effectively implement blockchain-based solutions in business practice. This Special Issue reprint contained well-prepared research reports regarding recent advances in blockchain technology around logistics processes to provide insights into realized maturity

Decidability of membership problems for flat rational subsets of GL(2,Q)\mathrm{GL}(2,\mathbb{Q}) and singular matrices

This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove various new decidability results for $2\times 2$ matrices over $\mathbb{Q}$. For that, we introduce the concept of flat rational sets: if $M$ is a monoid and $N$ is a submonoid, then ``flat rational sets of $M$ over $N$'' are finite unions of the form $L_0g_1L_1 \cdots g_t L_t$ where all $L_i$'s are rational subsets of $N$ and $g_i\in M$. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of $GL(2,\mathbb{Q})$ over $GL(2,\mathbb{Z})$ is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further inside $GL(2,\mathbb{Q})$. We also show a dichotomy for nontrivial group extension of $GL(2,\mathbb{Z})$ in $GL(2,\mathbb{Q})$: if $G$ is a f.g. group such that $GL(2,\mathbb{Z}) < G \leq GL(2,\mathbb{Q})$, then either $G\cong GL(2,\mathbb{Z})\times Z^k$, for some $k\geq 1$, or $G$ contains an extension of the Baumslag-Solitar group $BS(1,q)$, with $q\geq 2$, of infinite index. In the first case of the dichotomy the membership problem for $G$ is decidable but the equality problem for rational subsets of $G$ is undecidable. In the second case, decidability of the membership problem for rational subsets in $G$ is open. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable (in doubly exponential time) for flat rational subsets of $Q^{2 \times 2}$ over the submonoid that is generated by the matrices from $Z^{2 \times 2}$ with determinants in $\{-1,0,1\}$