A word is square-free if it does not contain non-empty factors of the form XX.In 1906 Thue proved that there exist arbitrarily long square-free words over 3-letter alphabet. We consider a new type of square-free words. A square-free word is extremal if it cannot be extended to a new square-free word by inserting a single letter on arbitrary position. We prove that there exist infinitely many extremal words over 3-letter alphabet. Some parts of our construction relies on computer verifications. We also pose some related open problems. arXiv:1910.06226v1 [math.CO]
The paper deals with the concrete planning problem (CPP) -a stage of the Web Service Composition (WSC) in the PlanICS framework. The complexity of the problem is discussed. A novel SMT-based approach to CPP is defined and its performance is compared to the standard Genetic Algorithm (GA) and the OpenOpt numerical toolset planner in the framework of the PlanICS system. The discussion of all the approaches is supported by extensive experimental results.
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