A deterministic finite automaton (DFA) is composite if its language can be decomposed into an intersection of languages of smaller DFAs. Otherwise, A is prime. This notion of primality was introduced by Kupferman and Mosheiff in 2013, and while they proved that we can decide whether a DFA is composite, the precise complexity of this problem is still open, with a doubly-exponential gap between the upper and lower bounds. In this work, we focus on permutation DFAs, i.e., those for which the transition monoid is a group. We provide an NP algorithm to decide whether a permutation DFA is composite, and show that the difficulty of this problem comes from the number of non-accepting states of the instance: we give a fixed-parameter tractable algorithm with the number of rejecting states as the parameter. Moreover, we investigate the class of commutative permutation DFAs. Their structural properties allow us to decide compositionality in NLOGSPACE, and even in LOGSPACE if the alphabet size is fixed. Despite this low complexity, we show that complex behaviors still arise in this class: we provide a family of composite DFAs each requiring polynomially many factors with respect to its size. We also consider the variant of the problem that asks whether a DFA is k-factor composite, that is, decomposable into k smaller DFAs, for some given integer k. We show that, for commutative permutation DFAs, restricting the number of factors makes the decision computationally harder, and yields a problem with tight bounds: it is NP-complete. Finally, we show that in general, this problem is in PSPACE, and it is in LOGSPACE for DFAs with a singleton alphabet.
The plastic deformation micro-mechanisms of extruded pure Zn deformed in tension along the extrusion direction were investigated by means of in situ scanning electron microscopy (SEM) integrated with electron back-scatter diffraction (EBSD). Plastic deformation began with the activation of < a > basal slip in grains with the highest Schmid factor while, the incompatibility of deformation between neighbour grains was accommodated by grain boundary sliding. The geometrically necessary dislocation density increased sharply from 1.53 x 10(13) m(-)(2) to 9.03 x 10(13) m(-)(2) when applied strain reached 6.7%, and this increase coincides with the strong initial strain hardening region. The incompatibility of deformation between neighbour grains was accommodated by grain boundary sliding at strains above 3.3%, which somehow limited the strain hardening rate. Evidence of < c + a > pyramidal II slip was also found through slip trace analysis from the early stages of deformation, i.e. 1.6% strain, but it was always limited to a small fraction of suitably oriented grains. Moreover, transmission electron microscopy (TEM) observations showed that many < c + a > pyramidal dislocations were dissociated into the basal plane and became sessile. {[Math Processing Error]}<[Math Processing Error]> compression twins were nucleated at 3.3% strain and the fraction of grains undergoing twinning as well as the area fraction of twins increased proportionally to the applied strain. Twinning was favoured by the fiber texture and the twin variant with the highest Schmid factor was primarily activated in each grain. The contribution of twinning to the total strain was limited (around 11% when the applied strain was 16.7%). The strain hardening rate decreased sharply beyond 6.7% and the hardening contribution of basal slip was balanced by grain boundary sliding and compression twinning. Finally, a high fraction of sub-grain boundaries that trigger recrystallization at larger strains was found at 16.7%. These observations reveal the sequence and interaction of plastic deformation mechanisms in Zn, which may help design novel Zn alloys with improved mechanical properties.
Lattice materials are widely used in many fields requiring lightweight structures with tunable properties. Designing an optimal architecture remains a challenging task due to a vast parameter space and resources required to numerically evaluate the performance of candidate structures. This study presents a workflow the optimization of 3D lattices constructed by a combination of different cell topologies. It combines genetic algorithm, mechanical simulations, and a 3D convolutional neural network. The workflow was tested two objectives: optimization for the maximum average specific elastic modulus between X, Y, and Z directions and optimization for the maximum in only one direction. The optimized lattices demonstrated up to 91.5% increase in specific elastic modulus compared to the pure single-type composition. The two tested objectives converged to distinct patterns: the single-direction case led to the formation of column-like features, while average between three directions favored more diverse designs. The proposed approach can be adapted the optimization of other physical properties or material types.
Blockchains face a scalability challenge due to the intrinsic throughput limitations of consensus protocols and the limitation in block sizes due to decentralization. An alternative to improve the number of transactions per second is to use Layer 2 (L2) rollups. L2s perform most computations offchain using blockchains (L1) minimally under-the-hood to guarantee correctness. A sequencer receives offchain L2 transaction requests, batches them, and commits compressed or hashed batches to L1. Hashing offers much better compression but requires a data availability committee (DAC) to translate hashes back into their corresponding batches. Current L2s consist of a centralized sequencer which receives and serializes all transactions and an optional DAC. Centralized sequencers can undesirably influence L2s evolution. We propose in this paper a fully decentralized implementation of a service that combines (1) a sequencer that posts hashes to the L1 blockchain and (2) the data availability committee that reverses the hashes. We call the resulting service a (decentralized) arranger. Our decentralized arranger is based on Set Byzantine Consensus (SBC), a service where participants can propose sets of values and consensus is reached on a subset of the union of the values proposed. We extend SBC for our fully decentralized arranger. Our main contributions are (1) a formal definition of arrangers; (2) two implementations, one with a centralized sequencer and another with a fully decentralized algorithm, with their proof of correctness; and (3) empirical evidence that our solution scales by implementing all building blocks necessary to implement a correct server.
The Gibbs free energies of competing phases in the Al-Mg-Si-Ag alloy system were determined by combining first-principles calculations, the cluster expansion method, and Monte Carlo simulations, with experimental validation. Based on these results, a thermodynamic database of the Al-Mg-Si-Ag system was developed, enabling accurate determination of phase stability and phase distributions. The analysis of short-range order parameters showed that the solute cluster was dominated by Mg-Si co-clusters in Al-Mg-Si alloys, while Mg-Ag co-clusters formed prior to the Mg-Si-Ag clusters when the Al-Mg-Si-Ag alloy is quenched from high temperature. Transmission electron microsocopy confirmed the existence of such clusters. Ag addition further modifies the crystal structure of Guinier-Preston zones, yielding Al1-zAgz, Al1-x-zMgxAgz and MgxAgz variants, whereas the structure of metastable beta '' and stable beta remian unchanged. Phase diagram calculations reveal that that the fcc+beta '/beta '(Ag) region progressively transforms into fcc+beta with increasing temperature, indicating stabilization of beta ' by Ag. These atomic-scale insights highlight the role of Ag in modifying early-stage clustering and precipitation. Hardness measurements demonstrate that a minor Ag addition enhances the peak hardness by similar to 20%, consistent with the predicted role of early Mg-Si-Ag clustering in promoting beta '' nucleation. These findings establish a direct link between atomic-scale thermodynamics and macroscopic hardening, offering guidance for designing high-performance Al-Mg-Si-Ag alloys.
