A new perspective of paramodulation complexity by solving massive 8 puzzles

A sliding puzzle is a combination puzzle where a player slide pieces along certain routes on a board to reach a certain end-configuration. In this paper, we propose a novel measurement of complexity of massive sliding puzzles with paramodulation which is an inference method of automated reasoning. It turned out that by counting the number of clauses yielded with paramodulation, we can evaluate the difficulty of each puzzle. In experiment, we have generated 100 * 8 puzzles which passed the solvability checking by countering inversions. By doing this, we can distinguish the complexity of 8 puzzles with the number of generated with paramodulation. For example, board [2,3,6,1,7,8,5,4, hole] is the easiest with score 3008 and board [6,5,8,7,4,3,2,1, hole] is the most difficult with score 48653. Besides, we have succeeded to obverse several layers of complexity (the number of clauses generated) in 100 puzzles. We can conclude that proposal method can provide a new perspective of paramodulation complexity concerning sliding block puzzles.

A Curious New Result of Resolution Strategies in Negation-Limited Inverters Problem

Generally, negation-limited inverters problem is known as a puzzle of constructing an inverter with AND gates and OR gates and a few inverters. In this paper, we introduce a curious new result about the effectiveness of two powerful ATP (Automated Theorem Proving) strategies on tackling negation limited inverter problem. Two resolution strategies are UR (Unit Resulting) resolution and hyper-resolution. In experiment, we come two kinds of automated circuit construction: 3 input/output inverters and 4 input/output BCD Counter Circuit. Both circuits are constructed with a few limited inverters. Curiously, it has been turned out that UR resolution is drastically faster than hyper-resolution in the measurement of the size of SOS (Set of Support). Besides, we discuss the syntactic and semantic criteria which might causes considerable difference of computation cost between UR resolution and hyper-resolution.

A constrained recursion algorithm for batch normalization of tree-sturctured LSTM

Tree-structured LSTM is promising way to consider long-distance interaction over hierarchies. However, there have been few research efforts on the hyperparameter tuning of the construction and traversal of tree-structured LSTM. To name a few, hyperparamters such as the interval of state initialization, the number of batches for normalization have been left unexplored specifically in applying batch normalization for reducing training cost and parallelization. In this paper, we propose a novel recursive algorithm for traversing batch normalized tree-structured LSTM. In proposal method, we impose the constraint on the recursion algorithm for the depth-first search of binary tree representation of LSTM for which batch normalization is applied. With our constrained recursion, we can control the hyperparameter in the traversal of several tree-structured LSTMs which is generated in the process of batch normalization. The tree traversal is divided into two steps. At first stage, the width-first search over models is applied for discover the start point of the latest tree-structured LSTM block. Then, the depth-first search is run to traverse tree-structured LSTM. Proposed method enables us to explore the optimized selection of hyperparameters of recursive neural network implementation by changing the constraints of our recursion algorithm. In experiment, we measure and plot the validation loss and computing time with changing the length of internal of state initialization of tree-structured LSTM. It has been turned out that proposal method is effective for hyperparameter tuning such as the number of batches and length of interval of state initialization of tree-structured LSTM.