On store languages of language acceptors

Abstract

It is well known that the "store language" of every pushdown automaton -the set of store configurations (state and stack contents) that can appear as an intermediate step in accepting computations -is a regular language. Here many models of language acceptors with various store structures are examined, along with a study of their store languages. For each model, an attempt is made to find the simplest model that accepts their store languages. Some connections between store languages of one-way and two-way machines are demonstrated, as with connections between nondeterministic and deterministic machines. A nice application of these store language results is also presented, showing a general technique for proving families accepted by many deterministic models are closed under right quotient with regular languages, resolving some open questions (and significantly simplifying proofs for others that are known) in the literature. Lower bounds on the space complexity of Turing machines for having non-regular store languages are obtained.Hence, the following is immediate:Corollary 41. There is a middle log log n space-bounded 2DTM M such that S(M ) is not regular. Turning now to one-way machines:Proposition 42. If M is a middle s(n) space-bounded 1NTM and s(n) grows slower than log n, then S(M ) is regular.Proof. The proof is the same as the proof of Proposition 38 using Proposition 37, part 2, and noting that the M ′ constructed in that proof would also be one-way if M is one-way.Corollary 43. If M is a strongly s(n) space-bounded 1NTM and s(n) grows slower than log n, then S s (M ) is regular.

On store languages and applications

“…This is done as many of the models are familiar to those in the area, and the detail given is enough to understand how they operate. If the reader desires further details, the formal definitions can be found in our recent paper [5].…”

“…The precise definition of the store language S(M ), where M is from some machine model M, depends on the definition of the model. In [5], the store languages of many different models are defined in a general fashion by separating the definition of "store types" from machines using these types. A machine of any type is denoted by a tuple M = (Q, Σ, Γ, δ, q 0 , F ), where Q is the finite state set, Σ is the input alphabet, Γ is the store alphabet, δ is the finite transition function, q 0 ∈ Q is the initial state, and F ⊆ Q is the final state set.…”

“…Upon returning to the top of the stack, it can again push and pop. A reversal-bounded stack automaton, NRBSA, has a bound on the number of changes between pushing and popping, but also the number of changes of direction when reading from the inside of the stack [5]. Here, we also consider a nondeterministic Turing machine to have a one-way read-only input tape and a bi-infinite read/write worktape.…”

“…And, in [5], the store languages of other one-way machine models were shown to be regular as well, including r-flip pushdown automata (pushdown automata with the ability to "flip" their pushdown up to r times [6]), reversal-bounded queue automata (there is a bound on the number of switches between enqueuing and dequeueing), and nondeterministic Turing machines with a one-way read-only input tape, and a reversalbounded worktape (a bound on the number of changes of directions of the read/write head). The paper [5] also demonstrated some general connections of store languages between two-way and one-way machine models. Furthermore, it was shown that in any one-way machine model (defined properly) with only regular store languages, then the languages accepted by the deterministic machines in this class are closed under right quotient with regular languages.…”

On counting functions and slenderness of languages

“…Proof. Part 1 follows from the fact that the languages accepted by reversal-bounded DTM's are closed under right-quotient with regular languages [29]. For Part 2, clearly, if L is accepted by an unambiguous reversal-bounded NTM M , we can construct an unambiguous reversal-bounded NTM M accepting L{x} −1 .…”