The unification problem for term rewriting systems (TRSs) is the problem of deciding, for a TRS R and two terms s and t, whether s and t are unifiable modulo R. We have shown that the problem is decidable for confluent simple TRSs. Here, a simple TRS means one where the right-hand side of every rewrite rule is a ground term or a variable. In this paper, we extend this result and show that the unification problem for confluent semi-constructor TRSs is decidable. Here, a semi-constructor TRS means one where all defined symbols appearing in the right-hand side of each rewrite rule occur only in its ground subterms.

To examine the computational complexity of cryptographic primitives such as the discrete logarithm problem, the factoring problem and the Diffie-Hellman problem, we define a new problem called square-root exponent, which is a problem to compute a value whose discrete logarithm is a square root of the discrete logarithm of a given value. We analyze reduction between the discrete logarithm problem modulo a prime and the factoring problem through the square-root exponent. We also examine reductions among the computational version and the decisional version of the square-root exponent and the Diffie-Hellman problem and show that the gap between the computational square-root exponent and the decisional square-root exponent partially overlaps with the gap between the computational Diffie-Hellman and the decisional Diffie-Hellman under some condition.

A pattern is a finite string of constant symbols and variable symbols. The language of a pattern is the set of all strings obtained by substituting any nonnull constant string for each variable symbol in the pattern. The class of pattern languages was introduced by Angluin in 1979 as a concrete class which is inferable from positive data. In this paper, we consider the decision problem whether for given two patterns there is a containment relation between their languages, which was posed by Angluin and its decidability remains open. We give some sufficient conditions to make this problem decidable. We also introduce the notions of generalizations and minimal generalizations common to a set of patterns. We characterize the above open problem using the minimal generalization.

Efficient probabilistic decision trees are required in various application areas such as character recognition. This paper presents a polynomial-time approximate algorithm for designing a probabilistic decision tree. The obtained tree is near-optimal for the cost, defined as the weighted sum of the expected test execution time and expected loss. The algorithm is advantageous over other reported heuristics from the viewpoint that the goodness of the solution is theoretically guaranteed. That is, the relative deviation of the obtained tree cost from the exact optimum is not more than a positive constant ε, which can be set arbitrarily small. When the given loss function is Hamming metric, the time efficiency is further improved by using the information theoretical lower bound on the tree cost. The time efficiency of the algorithm and the accuracy of the solutions were evaluated through computational experiments. The results show that the computing time increases very slowly with an increase in problem size and the relative error of the obtained solution is much less than the upper bound ε for most problems.