COMPUTATIONS AND DECIDABILITY OF ITERATIVE ARRAYS WITH RESTRICTED COMMUNICATION

Iterative arrays (IAs) are linear arrays of interconnected interacting finite state machines, where one distinguished one is equipped with a one-way read-only input tape. We investigate IAs operating in real time whose inter-cell communication is bounded by a constant number of bits not depending on the number of states. Their capabilities are considered in terms of syntactical pattern recognition. It is known [17] that such devices can recognize rather complicated sets of unary patterns with a minimum amount of communication, namely one-bit communication. Some examples are the sets {a2n | n ≥ 1}, {an2 | n ≥ 1}, and {ap | p is prime}. Here, we consider non-unary patterns and it turns out that the non-unary case is quite different. We present several real-time constructions for certain non-unary syntactical patterns. For example, the sets {anbn | n ≥ 1}, {anbncn | n ≥ 1}, {an(bn)m | n, m ≥ 1}, and {anbamb(ba)n·m | n, m ≥ 1} are recognized in real time by IAs. Moreover, it is shown that real-time one-bit IAs can, in some sense, add and multiply integer numbers. Furthermore, decidability questions of communication restricted IAs are dealt with. Due to the constructions provided, undecidability results can be derived. It turns out that emptiness is still not even semidecidable for one-bit IAs despite their restricted communication. Moreover, also the questions of finiteness, infiniteness, inclusion, and equivalence are non-semidecidable.