Pushdown Automata, Graphs, Ends, Second-Order Logic, and Reachability Problems

We have discovered a very strong connection between certain areas of theoretical computer science—the theory of context-free languages and pushdown automata, tiling problems, cellular automata, and vector addition systems—and certain concepts from group theory, topology, and second-order logic. We use these concepts to investigate a rather wide class of graphs which we call context-free graphs. Using the results obtained and Rabin's theorem that the monadic second-order theory of the infinite binary tree is decidable, we are able to show that the monadic second-order theory of any context-free graph is decidable. Cellular automata and vector addition systems are usually considered as involving the grid of integer lattice points in n-dimensional space. We show that such systems make sense on a very general class of graphs and, in contrast to the classical case, all the relevant algorithmic problems concerning such systems are solvable on context-free graphs.

Authors

• David E. Muller
• Paul E. Schupp

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