This paper compares several translation representations for a synchronous context-free grammar parse including CFGs/hypergraphs, finite-state automata (FSA), and pushdown automata (PDA). The representation choice is shown to determine the form and complexity of target LM intersection and shortest-path algorithms that follow. Intersection, shortest path, FSA expansion and RTN replacement algorithms are presented for PDAs. Chinese-toEnglish translation experiments using HiFST and HiPDT, FSA and PDA-based decoders, are presented using admissible (or exact) search, possible for HiFST with compact SCFG rulesets and HiPDT with compact LMs. For large rulesets with large LMs, we introduce a two-pass search strategy which we then analyze in terms of search errors and translation performance.