Improving the efficiency of AC matching and unification

This report consists of three independent parts that each study important steps in the matching and unification process for AC theories. In the first part we consider the problem of AC matching where there is just a single (variadic) AC symbol and no free function symbols in the pattern and subject. We show that even this restricted problem is NP-complete. We give some search methods and empirical results. In the second part we consider the full AC matching problem where there is no restriction on AC and free functions symbols allowed in the pattern and subject. Our approach is to build a hierarchy of bipartite graph matching problems which encodes all the possible solutions of subproblems. Certain sets of solutions to the graph problems are then used to construct simplified AC systems which are solved by a constrained search. In the final part we focus on one of the computationally intensive steps in current AC unification algorithms : the extraction of potential unifiers from a diophantine basis. We show that certain sub-problems are NP-complete and derive a new search algorithm which is shown to be at worst equivalent to the best published algorithm and which is potentially much better

Embedding and Weight Distribution for Quantum Annealing

Before being able to calculate on the D-Wave machine, its very restricted structure requires the embedding of the original problem graph onto the Chimera hardware graph. A precalculated embedding of a complete graph enables to map all problems with the same number of nodes or less straightforwardly. The problem of finding the largest complete graph minor and its embedding scheme in a Chimera graph with broken qubits can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general it is fixed parameter tractable in the number of broken qubits. By dropping specific matches the problem can be simplified. Some preliminary results comparing this heuristic approach to exact optimization are shown. After the structural embedding the actual embedded Ising model needs to be constructed from the original problem coefficient values, such that the minima of both are equivalent. That means in the solution of the embedded Ising model the values for each single qubit embedding should be synchronized. The resulting constraints can be derived to a graph property related to expansion, which is efficient to solve in the embedding framework. First results show an improvement over standard methods with respect to coefficient ratio