Attribute grammars are an extension of context-free grammars devised by Knuth as a formalism for specifying the semantics of a context-free language along with the syntax of the language. The syntactic phase of the translation process has been extensively studied and many techniques are available for automatically generating efficient parsers for context-free grammars. Attribute grammars offer the prospect of similarly automating the implementation of the semantic phase. In this paper we present a general method of constructing, for any non-circular attribute grammar, a deterministic translator which will perform the semantic evaluation of each syntax tree of the grammar in time linear with the size of the tree. Each tree is traversed in a manner particularly suited to the shape of the tree, yielding a near optimal evaluation order for that tree. Basically, the translator consists of a finite set of "Local Control Automata", one for each production; these are ordinary finite-state acyclic automata augmented with some special features, which are used to regulate the evaluation process of each syntax tree. With each node in the tree there will be associated the Local Control Automaton of the production applying at the node. At any given time during the translation process all Local Control Automata are inactive, except for the one associated with the currently processed node, which is responsible for directing the next steps taken by the translator until control is finally passed to a neighbour node, reactivating its Local Control Automaton. The Local Control Automata of neighbour nodes communicate with each other.The construction of the translator is custom tailored to each individual attribute grammar. The dependencies among the attributes occurring in the semantic rules are analysed to produce a near-optimal evaluation strategy for that grammar. This strategy ensures that during the evaluation process, each time the translator enters some subtree of the syntax tree, at least one new attribute evaluation will occur at each node visited. It is this property which distinguishes the method presented here from previously known methods of generating translators for unrestricted attribute grammars, and which causes the translators to be near-optimal.