We present constant-time testing algorithms for generalized shogi (Japanese chess), chess, and xiangqi (Chinese chess). These problems are known or believed to be EXPTIME-complete. A testing algorithm (or a tester) for a property accepts an input if it has the property, and rejects it with high probability if it is far from having the property (e.g., at least 2/3) by reading only a constant part of the input. A property is said to be testable if a tester exists. Given any position on a ⌊√n⌋×⌊√n⌋ board with O(n) pieces, the generalized shogi, chess, and xiangqi problem are problems determining the property that “the player who moves first has a winning strategy.” We propose that this property is testable for shogi, chess, and xiangqi. The shogi tester and xiangqi tester have a one-sided-error, but surprisingly, the chess tester has no-error. Over the last decade, many problems have been revealed to be testable, but most of such problems belong to NP. This is the first result on the constant-time testability of EXPTIME-complete problems.

In this paper, we study lower bounds on the query complexity of testing algorithms for various problems. Given an oracle that returns information of an input object, a testing algorithm distinguishes the case that the object has a given property P from the case that it has a large distance to having P with probability at least . The query complexity of an algorithm is measured by the number of accesses to the oracle. We introduce two reductions that preserve the query complexity. One is derived from the gap-preserving local reduction and the other is from the L-reduction. By using the former reduction, we show linear lower bounds on the query complexity for testing basic NP-complete properties, i.e., 3-edge-colorability, directed Hamiltonian path/cycle, undirected Hamiltonian path/cycle, 3-dimensional matching and NP-complete generalized satisfiability problems. Also, using the second reduction, we show a linear lower bound on the query complexity of approximating the size of the maximum 3-dimensional matching.