The Complexity of Reasoning with FODD and GFODD

Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a knowledge representation that is useful in mechanizing decision theoretic planning in relational domains. GFODDs generalize function-free first order logic and include numerical values and numerical generalizations of existential and universal quantification. Previous work presented heuristic inference algorithms for GFODDs and implemented these heuristics in systems for decision theoretic planning. In this paper, we study the complexity of the computational problems addressed by such implementations. In particular, we study the evaluation problem, the satisfiability problem, and the equivalence problem for GFODDs under the assumption that the size of the intended model is given with the problem, a restriction that guarantees decidability. Our results provide a complete characterization placing these problems within the polynomial hierarchy. The same characterization applies to the corresponding restriction of problems in first order logic, giving an interesting new avenue for efficient inference when the number of objects is bounded. Our results show that for $\Sigma_k$ formulas, and for corresponding GFODDs, evaluation and satisfiability are $\Sigma_k^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete. For $\Pi_k$ formulas evaluation is $\Pi_k^p$ complete, satisfiability is one level higher and is $\Sigma_{k+1}^p$ complete, and equivalence is $\Pi_{k+1}^p$ complete.Comment: A short version of this paper appears in AAAI 2014. Version 2 includes a reorganization and some expanded proof

Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of the same nam

Discovery of high-entropy ceramics via machine learning

AbstractAlthough high-entropy materials are attracting considerable interest due to a combination of useful properties and promising applications, predicting their formation remains a hindrance for rational discovery of new systems. Experimental approaches are based on physical intuition and/or expensive trial and error strategies. Most computational methods rely on the availability of sufficient experimental data and computational power. Machine learning (ML) applied to materials science can accelerate development and reduce costs. In this study, we propose an ML method, leveraging thermodynamic and compositional attributes of a given material for predicting the synthesizability (i.e., entropy-forming ability) of disordered metal carbides. The relative importance of the thermodynamic and compositional features for the predictions are then explored. The approach’s suitability is demonstrated by comparing values calculated with density functional theory to ML predictions. Finally, the model is employed to predict the entropy-forming ability of 70 new compositions; several predictions are validated by additional density functional theory calculations and experimental synthesis, corroborating the effectiveness in exploring vast compositional spaces in a high-throughput manner. Importantly, seven compositions are selected specifically, because they contain all three of the Group VI elements (Cr, Mo, and W), which do not form room temperature-stable rock-salt monocarbides. Incorporating the Group VI elements into the rock-salt structure provides further opportunity for tuning the electronic structure and potentially material performance

Precipitation strengthened high strength, high conductivity Cu-Cr-Nb alloys produced by chill block melt spinning

A series of Cu-based alloys containing 2 to 10 a/o Cr and 1 to 5 a/o Nb were produced by chill block melt spinning (CBMS). The melt spun ribbons were consolidated and hot rolled to sheet to produce a supersaturated Cu-Cr-Nb solid solution from which the high melting point intermetallic compound Cr2Nb could be precipitated to strengthen the Cu matrix. The results show that the materials possess electrical conductivities in excess of 90 percent that of pure Cu at 200 C and above. The strengths of the Cu-Cr-Nb alloys were much greater than Cu, Cu-0.6 Cr, NARloy-A, and NARloy-Z in the as-melt spun condition. The strengths of the consolidated materials were less than Cu-Cr and Cu-Cr-Zr below 500 C and 600 C respectively, but were significantly better above these temperatures. The strengths of the consolidated materials were greater than NARloy-Z, at all temperatures. The GLIDCOP possessed similar strength levels up to 750 C when the strength of the Cu-Cr-Nb alloys begins to degrade. The long term stability of the Cu-Cr-Nb alloys was measured by the microhardness of aged samples and the growth of precipitates. The microhardness measurements indicate that the alloys overage rapidly, but do not suffer much loss in strength between 10 and 100 hours which confirms the results of the electrical resistivity measurements taken during the aging of the alloys at 500 C. The loss in strength from peak strength levels is significant, but the strength remains exceptionally good. Transmission electron microscopy (TEM) of the as-melt spun samples revealed that Cr2Nb precipitates formed in the liquid Cu during the chill block melt spinning, indicating a very strong driving force for the formation of the precipitates. The TEM of the aged and consolidated materials indicates that the precipitates coarsen considerably, but remain in the submicron range