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It is well-known that pushdown systems (PDS) effectively preserve regularity. This property implies the decidability of the reachability problem for PDS and has been applied to automatic program verification. The backward regularity preservation property was also shown for an extension of PDS by adding registers. This paper aims to show the forward regularity preservation property. First, we provide a concise definition of the register model called register pushdown systems (RPDS). Second, we show the forward regularity preservation property of RPDS by providing a saturation algorithm that constructs a register automaton (RA) recognizing post * P (L(A)) where A and P are a given RA and an RPDS, respectively, and post * P is the forward image of the mapping induced by P. We also give an example of applying the proposed algorithm to malware detection.

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Register context-free grammars (RCFG) is an extension of context-free grammars to handle data values in a restricted way. In RCFG, a certain number of data values in registers are associated with each nonterminal symbol and a production rule has the guard condition, which checks the equality between the content of a register and an input data value. This paper starts with RCFG and introduces register type, which is a finite representation of a relation among the contents of registers. By using register type, the paper provides a translation of RCFG to a normal form and ε-removal from a given RCFG. We then define a generalized RCFG (GRCFG) where an arbitrary binary relation can be specified in the guard condition. Since the membership and emptiness problems are shown to be undecidable in general, we extend register type for GRCFG and introduce two properties of GRCFG, simulation and progress, which guarantee the decidability of these problems. As a corollary, these problems are shown to be EXPTIME-complete for GRCFG with a total order over a dense set.