Tinput-Driven Pushdown, Counter, and Stack Automata

Abstract

In input-driven automata the input alphabet is divided into distinct classes and different actions on the storage medium are solely governed by the input symbols. For example, in inputdriven pushdown automata (IDPDA) there are three distinct classes of input symbols determining the action of pushing, popping, or doing nothing on the pushdown store. Here, input-driven automata are extended in such a way that the input is preprocessed by a deterministic sequential transducer. IDPDAs extended in this way are called tinput-driven pushdown automata (TDPDA) and it turns out that TDPDAs are more powerful than IDPDAs but still not as powerful as real-time deterministic pushdown automata. Nevertheless, even this stronger model has still good closure and decidability properties. In detail, it is shown that TDPDAs are closed under the Boolean operations union, intersection, and complementation. Furthermore, decidability procedures for the inclusion problem as well as for the questions of whether a given automaton is a TDPDA or an IDPDA are developed. Additionally, representation theorems for the context-free languages using IDPDAs and TDPDAs are established. Two other classes investigated are on the one hand TDPDAs restricted to tinput-driven counter automata and on the other hand TDPDAs generalized to tinput-driven stack automata. In both cases, it is possible to preserve the good closure and decidability properties of TDPDAs, namely, the closure under the Boolean operations as well as the decidability of the inclusion problem.

Digging input-driven pushdown automata

“…So, the inclusion of Theorem 3.1 is strict. In [13], input-driven automata are extended in such a way that the input is preprocessed by a deterministic injective and length-preserving finite state transducer. The corresponding devices are called tinput-driven pushdown automata (TDPDA).…”

“…We use L = { a n b n+m a m | n, m ≥ 1 } as a witness language. It is known [13] that L does not belong to L (TDPDA). Since L (IDPDA) ⊂ L (TDPDA), L cannot be accepted by any IDPDA.…”

“…Proof. It is shown in [13] that the inclusion problem is not semidecidable for two IDPDAs with not necessarily compatible signatures. Moreover, it can be assumed that the pushdown store of the two IDPDAs is in fact used as a counter only.…”

“…Moreover, it can be assumed that the pushdown store of the two IDPDAs is in fact used as a counter only. The basic idea of the proof in [13] is to construct two IDPDAs M and M such that the intersection L(M ) ∩ L(M ) represents the suitably encoded version of the valid computations of a counter machine. Then, the inclusion problem can be related with the emptiness problem of a counter machine which is known to be not semidecidable.…”

“…Of particular interest is the impact of the new behavior of DIDPDAs on their power. So, among others the capacities are compared with the capacities of ordinary IDPDAs and so-called tinput-driven pushdown automata which are basically IDPDAs whose input may be preprocessed by length-preserving finite state transducers [13]. It turns out that the family of languages accepted by DIDPDAs is incomparable with the other devices.…”