Push-Down Automata with Gap-Order Constraints

We consider push-down automata with data (PDAD) that operate on variables ranging over the set of natural numbers. The conditions on variables are defined via gap-order constraint. Gap-order constraints allow to compare variables for equality, or to check that the gap between the values of two variables exceeds a given natural number. The messages inside the stack are equipped with values that are natural numbers reflecting their “values”. When a message is pushed to the stack, its value may be defined by a variable in the program. When a message is popped, its value may be copied to a variable. Thus, we obtain a system that is infinite in two dimensions, namely we have a stack that may contain an unbounded number of messages each of which is equipped with a natural number. We present an algorithm for solving the control state reachability problem for PDAD based on two steps. We first provide a translation to the corresponding problem for context-free grammars with data (CFGD). Then, we use ideas from the framework of well quasi-orderings in order to obtain an algorithm for solving the reachability problem for CFGDs.