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We introduce p-equivalence by asymptotic probabilities, which is a weakalmost-equivalence based on zero-one laws in finite model theory. In thispaper, we consider the computational complexities of p-equivalence problems forregular languages and provide the following details. First, we give anrobustness of p-equivalence and a logical characterization for p-equivalence.The characterization is useful to generate some algorithms for p-equivalenceproblems by coupling with standard results from descriptive complexity. Second,we give the computational complexities for the p-equivalence problems by thelogical characterization. The computational complexities are the same as forthe (fully) equivalence problems. Finally, we apply the proofs forp-equivalence to some generalized equivalences.

The Almost Equivalence by Asymptotic Probabilities for Regular Languages and Its Computational Complexities