This book constitutes the research papers presented at the 4th International Workshop, RP 2010 held in Brno, Czech Republic, August 28-29, 2010 and was co-located with Joint MFCS and CSL 2010 (35th International Symposiums on Mathematical Foundations of Computer Science and 19th EACSL Annual Conferences on Computer Science Logic). The revised 9 full papers and the 4 invited talks of this workshop reflect reachability problems that appear in algebraic structures, computational models, hybrid systems and verification. Reachability is a fundamental problem in the context of many models and abstractions which are describing various computational processes. Topics of interest include reachability problems in infinite state systems, rewriting systems, dynamical and hybrid systems, reachability problems in logic and verification, reachability analysis in different computational models, counter, timed, cellular, communicating automata, Petri-Nets, computational aspects of algebraic structures (semigroups, groups and rings), frontiers between decidable and undecidable reachability problems, predictability in iterative maps and new computational paradigms.