Deterministic Autopoietic Automata

In 2001, van Leeuwen and Wiedermann [1] have defined evolving interactive systems, in particular sequences of interactive finite automata with global states, to model infinite computations on an ever changing machine or system of machines. Modern computation does not just happen on an individual machine for a fixed time, but it goes on forever over an unbounded number of software and hardware changes. Evolving automata have also been called lineage of automata [3]. For more background information, see [4, 6]. All results of this paper are related to the paper by Wiedermann [5] studying autopoietic automata, a special kind of offspring-producing evolving machines. The offspring relation defines trees of autopoietic automata. Attention is often focused on a lineage of autopoietic automata, a path in a tree of autopoietic automata. Finite autopoietic automata as defined in [5] are finite automata augmented with the following special features. They have two input options, two output options and two modes. The two modes are the reproducing mode (defined by a subset R of states) and the transducer mode (defined by the complementary set Q−R of the states. In the reproducing mode the automaton uses a finite read-only input tape and a one-way output tape. The finite automaton operates like a Turing machine. It is a finite automaton though, because the input tape is of fixed length and there are no additional work tapes. In the transducer mode, the automaton does not change the tapes, but reads from an infinite input stream of which it can access one symbol of Σ at a time from an input buffer, and it writes one symbol of Σ at a time into an output buffer, producing an output stream. During the whole operation, the input tape of the automaton A actually contains the code of A, a straightforward description of the transition relation δ . The code is a sequence of 5-tuples in arbitrary order. Each 5-tuple consists of an observed symbol (on the tape or in the input buffer, depending on the mode), a current state, a new symbol (to be written onto the tape or into the output buffer, depending on