Hereditary History Preserving Simulation is Undecidable

We show undecidability of hereditary history preserving simulation for finite asynchronous transition systems by a reduction from the halting problem of deterministic Turing machines. To make the proof more transparent we introduce an intermediate problem of deciding the winner in domino snake games. First we reduce the halting problem of deterministic Turing machines to domino snake games. Then we show how to model a domino snake game by a hereditary history simulation game on a pair of finite asynchronous transition systems. 1 Domino snake games A tiling system D = (D,V,H) consists of a set D of dominoes, and two relations: V ⊆ D2, called vertical compatibility, and H ⊆ D2, called horizontal compatibility. Intuitively, one can think of dominoes as unit squares with coloured sides (and with an orientation, i.e., the dominoes cannot be rotated.) In this metaphor the meaning of the vertical and horizontal compatibility relations can be described as follows: for a pair of dominoes d, d′ ∈ D, we have • (d, d′) ∈ V , if the top side of d has the same color as the bottom side of d′, • (d, d′) ∈ H, if the right-hand side of d has the same color as the left-hand side of d′. ∗Address: BRICS, Department of Computer Science, University of Aarhus, Ny Munkegade, Building 540, 8000 Aarhus C, Denmark. Emails: {mju,mn}@brics.dk. †Basic Research in Computer Science, Centre of the Danish National Research Foundation.