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Abstract History preserving bisimilarity (hp-bisimilarity) and hereditary history preserving bisimilarity (hhp-bisimilarity) are behavioural equivalences taking into account causal relationships between events of concurrent systems. Their prominent feature is being preserved under action refinement, an operation important for the top-down design of concurrent systems. We show that--unlike hp-bisimilarity--checking hhpbisimilarity for finite labelled asynchronous transition systems is not decidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent we introduce an intermediate problem of checking domino bisimilarity for origin constrained tiling systems, whose undecidability is interesting in its own right. We also argue that the undecidability of hhp-bisimilarity holds for finite labelled 1-safe Petri nets. 1 Introduction The notion of behavioural equivalence that has attracted most attention in con-currency theory is bisimilarity, originally introduced by Park [20] and Milner [15]; concurrent programs are considered to have the same meaning if they are bisim-ilar. The prominent role of bisimilarity is due to many pleasant properties it enjoys; we mention a few of them here. A process of checking whether two transition systems are bisimilar can beseen as a two player game which is in fact an Ehrenfeucht-Fra"iss'e type of game

We consider dictionary data structures based on expander graphs. We show that any one probe scheme with the properties of the previous data structure from [OP02] is indeed space optimal. We then construct four different dictionary data structures for various models of parallel external memory. All of them allows lookups using a single parallel probe. In the following n denotes the number of keys in the dictionary, and u the universe of possible keys. ∆opt denotes the space in bits required to store the n keys and their satellite data without any type of compression and d = O(log(u/n)). • A static dictionary data structure with error correcting codes using O(∆opt) bits of space, and one requiring O(ndlog d + ∆opt) bits of space without using error correcting codes. • A dynamic dictionary data structure for the parallel disk head model using O(ndlog n + ∆opt) bits of space, where updates take O(1) I/O’s amortized. • A dynamic dictionary data structure for the parallel disk model, with