On the Existence of Universal Finite or Pushdown Automata

It is well known that there exist universal Turing machines (UTM). Such a UTM simulates any special Turing machine (TM) M in a certain way. There are several ways of simulation. One is that a UTM U simulating a TM M with input w halts if and only if M halts on input w. Another possibility is that any computation step of M is simulated by U using some number of steps which are be restricted by some complexity function. In very small UTM’s this can be exponential. Almost all UTM’s constructed so far are deterministic, simulating deterministic TM’s. In [4] it has been shown that there exist UTM’s simulating all special TM’s with complexity constraints. These complexity constraints, for space or time, are from a subclass of all primitive recursive functions over one variable. The UTM’s have the same complexity constraints. In both cases, general TM’s and those with complexity constraint, the specific TM M and its input w ∈ Σ(M)∗, where Σ(M) is the alphabet of M, have to be encoded. Such an encoding, and also the decoding, can be achieved by deterministic finite state transducers (DFST), which means that encoding and decoding is bijective. The input for a UTM U , to simulate M with input w, can then have the form cm(M)#ci(w) where cm, ci are the encoding functions for M, w, respectively. If one intends to construct universal machines for weaker automata classes it should be kept in mind that encoding and decoding for such automata should not exceed the power of deterministic versions of those machines. Otherwise too much power and information could be hidden in the encoding. In [3] it has been shown, under this condition, that there don’t exist universal 1-way finite automata (FA), neither deterministic (DFA) nor nondeterministic (NFA) ones. The proof uses arguments on the number of states of such automata. So the question arises whether there exist universal universal pushdown automata (UPDA), and if so if encoding and decoding can be achieved by DFST’s, or if deterministic pushdown transducers (DPDT) are necessary. For general TM’s DFST’s suffice for encoding and decoding (e.g. [5]).