Deciding equivalence of deterministic one-counter automata in polynomial time with applications to learning

A polynomial time algorithm is presented for deciding the equivalence of deterministic one-counter automata (DOCAs), solving an open problem posed by Valiant and Paterson. Unlike the previously known exponential time algoithm, which used a non-deterministic simulation to reduce the problem to PDA emptiness decidability, this new procedure constructs a joint description of the equivalence classes of configurations of the DOCAs being compared. The primary tool used by the algorithm is the notion of linguistic height. If we equate a string $u$ with the DOCA configuration reached on input $u$, then the linguistic height of the configuration $u$ with respect to strings $\alpha$, $\beta$, $\gamma$ is the least positive integer $k$ such that exactly one of $u$ $\alpha\beta\sp{k}\gamma$ and $u$ $\alpha\gamma$ is an accepting configuration. The algorithm exploits the fact that linguistic heights of arbitrary configurations can be extrapolated from the linguistic heights of configurations with polynomial bounded counter values. When the linguistic height of $u$ is defined, it can be determined by sequences of test inputs of the form $u$ $\alpha\beta\sp{k}\gamma$. Thus, the equivalence algorithm suggests a way of querying a "black box" DOCA in order to deduce the structure of its equivalence classes of configurations. A learning algorithm for DFA's, devised by Angluin, provides a framework for a DOCA learning algorithm--the learner constructs a sequence of DOCA approximations, using query strings of the form $\pi\rho\sp{j}\sigma\alpha\beta\sp{k}\gamma$ for various words $\pi$, $\rho$, $\sigma$ and $\alpha$, $\beta$, $\gamma$. By structuring the queries in this way, patterns of access to DOCA configurations can be determined; linguistic heights verify the action of the counter. In support of this strategy a new machine model, the linear pattern automaton, is defined and shown to be equivalent to the DOCA model.