Generalized Results on Monoids as Memory

We show that some results from the theory of group automata and monoid automata still hold for more general classes of monoids and models. Extending previous work for finite automata over commutative groups, we demonstrate a context-free language that can not be recognized by any rational monoid automaton over a finitely generated permutable monoid. We show that the class of languages recognized by rational monoid automata over finitely generated completely simple or completely 0-simple permutable monoids is a semi-linear full trio. Furthermore, we investigate valence pushdown automata, and prove that they are only as powerful as (finite) valence automata. We observe that certain results proven for monoid automata can be easily lifted to the case of context-free valence grammars.Comment: In Proceedings AFL 2017, arXiv:1708.0622

Fixed interval scheduling problem with minimal idle time with an application to music arrangement problem

The Operational Fixed Interval Scheduling Problem aims to find an assignment of jobs to machines that maximizes the total weight of the completed jobs. We introduce a new variant of the problem where we consider the additional goal of minimizing the idle time, the total duration during which the machines are idle. The problem is expressed using quadratic unconstrained binary optimization (QUBO) formulation, taking into account soft and hard constraints required to ensure that the number of jobs running at a time point is desirably equal to the number of machines. Our choice of QUBO representation is motivated by the increasing popularity of new computational architectures such as neuromorphic processors, coherent Ising machines, and quantum and quantum-inspired digital annealers for which QUBO is a natural input. An optimization problem that can be solved using the presented QUBO formulation is the music reduction problem, the process of reducing a given music piece for a smaller number of instruments. We use two music compositions to test the QUBO formulation and compare the performance of simulated, quantum, and hybrid annealing algorithms.Comment: 15 pages, 3 figure

State-efficient QFA Algorithm for Quantum Computers

The study of quantum finite automata (QFA's) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as $\mathtt{MOD}_{\rm p} = \{a^{j}~ |~ j \equiv 0 \mod p\}$, where $p$ is a prime number. In this paper, we present a modified MCQFA algorithm for the language $\mathtt{MOD}_{\rm p}$, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using less basis gates compared to the implementation of the original algorithm given in the literature