The complexity of the equivalence problem for counter machines, semilinear sets, and simple programs

It is shown that the class of relations (functions) definable by Presburger formulas is exactly the class of relations (functions) computable by finite-reversal multicounter machines. An upper bound of 2<supscrpt>c(N/logN)</supscrpt><supscrpt>4</supscrpt> on the deterministic time complexity of the equivalence problem for such machines is established. In fact, it is proved that the inequivalence problem is NP-complete. These results are used to derive some upper bounds on the complexity of the equivalence problem for semilinear sets and simple programs. For example, it is shown that the equivalence problem for semilinear sets (these sets are exactly the Presburger relations) is decidable in deterministic time 2<supscrpt>2</supscrpt><supscrpt>cN</supscrpt><supscrpt>2</supscrpt>. A class of programs which realize exactly the relations (functions) definable by Presburger formulas is shown to have an NP-complete inequivalence problem. Hence, its equivalence problem is decidable in deterministic time 2<supscrpt>p(N)</supscrpt>. This bound is a four-level exponential improvement over a previously known result.

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