We discuss the relationship between discrete-time processes (chains) and
one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin)
systems, possibly with a grammar (exclusion rule). We establish conditions for
a stochastic process to define a Gibbs measure and vice versa. Our conditions
generalize well known equivalence results between ergodic Markov chains and
fields, as well as the known Gibbsian character of processes with exponential
continuity rate. Our arguments are purely probabilistic; they are based on the
study of regular systems of conditional probabilities (specifications).
Furthermore, we discuss the equivalence of uniqueness criteria for chains and
fields and we establish bounds for the continuity rates of the respective
systems of finite-volume conditional probabilities. As an auxiliary result we
prove a (re)construction theorem for specifications starting from single-site
conditioning, which applies in a more general setting (general spin space,
specifications not necessarily Gibbsian).