Pushdown automata and constant height: decidability and bounds

Acta Informatica https://doi.org/10.1007/s00236-022-00434-0 ORIGINAL ARTICLE Pushdown automata and constant height: decidability and bounds Giovanni Pighizzini1 · Luca Prigioniero1 Received: 29 September 2020 / Accepted: 6 July 2022 © The Author(s) 2022 Abstract It cannot be decided whether a pushdown automaton accepts using a pushdown height, which does not depend on the input length, i.e., when it accepts using constant height. Furthermore, when a pushdown automaton accepts in constant height, the height can be arbitrarily large with respect to the size of the description of the machine, namely it does not exist any recursive function in the size of the description of the machine bounding the height of the pushdown. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the situation is different. First, acceptance in constant height is decidable. Moreover, in the case of acceptance in constant height, the height is at most exponential with respect to the size of the description of the pushdown automaton. We also prove a matching lower bound. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal. 1 Introduction The investigation of computational devices working with a limited amount of resources is a classical topic in automata theory. It is well known that by limiting the memory size of a device by some constant, the computational power of the resulting model cannot exceed that of finite automata. For instance, if we consider pushdown automata in which the maximum height of the pushdown is limited by some constant, the resulting devices, called constant-height pushdown automata, can recognize regular languages only. Despite their limited computational power, constant-height pushdown automata are interesting since they allow more succinct representations of regular languages than finite automata [8]. Further properties of these Extended version of a paper presented at the conference DCFS 2019-Descriptional Complexity of Formal Systems, Košice, Slovakia, July 17–19, 2019 [Lecture Notes in Computer Science, vol. 11612 Springer, 2019, pp. 260–271]. B Luca Prigioniero Giovanni Pighizzini 1 Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy 123 G. Pighizzini, L. Prigioniero devices have been recently considered. A double-exponential size increase when converting a nondeterministic constant-height pushdown automaton into an equivalent deterministic one, which cannot be avoided in the worst case, has been proven in [4]. A double-exponential size gap also holds for the conversion of deterministic and nondeterministic constant-height pushdown automata with a two-way input head into equivalent one-way devices [3]. Tight bounds for the size costs of Boolean operations on constant-height pushdown automata have been stated in [5]. A natural generative counterpart of constant-height pushdown automata are nonselfembedding context-free grammars, roughly context-free grammars without “true” recursion [7], which have been recently showed to be polynomially related in size to constant-height pushdown automata [10]. In this paper, we focus on pushdown automata with an unrestricted pushdown store, namely classical pushdown automata, that, however, are able to accept their inputs by making use only of a constant amount of pushdown store. More precisely, we say that a pushdown automaton M accepts in constant height h, for some given integer h ≥ 0, if, for each word in the language accepted by M, there exists at least one accepting computation in which the maximum height reached by the store is bounded by h. Notice that this does not prevent the existence of accepting or rejecting computations using an unbounded pushdown height. However, M can be converted into an equivalent constant-height pushdown automaton, which stops and rejects each time a computation tries to exceed the height limit h, and has a description whose size is a polynomial in both h and the size of the description of M. While studying these size relationships, we tried to understand how large can h be with respect to the size of the description of M. We discovered that h can be arbitrarily large. Indeed, in the first part of the paper we prove that there is no recursive function bounding the maximal height reached by the pushdown store in a pushdown automaton accepting in constant height, with respect to the size of its description. We also prove that it cannot be decided if a pushdown automaton accepts in constant height. We point out that this problem is different from the classical problem of deciding if a given context-free language is regular, which has been proven to be undecidable long time ago [2]. In fact, there exist pushdown automata that recognize regular languages using nonconstant height (an example is presented in the paper). Hence, while acceptance in constant height is sufficient for the regularity of the accepted language, it is not necessary. In the second part of the paper, we restrict the attention to the case of pushdown automata with a one-letter input alphabet, namely unary pushdown automata. By studying the structure of the computations of these devices, we are able to prove that, in contrast to the general case, it can be decided whether or not they accept in constant height. Furthermore, we also prove that if a unary pushdown automaton M accepts in height h, constant with respect to the input length, then h is bounded by an exponential function in the size of M. By presenting a suitable family of pushdown automata, we show that this bound cannot be reduced. In the final part of the paper, we consider pushdown automata that accept using height which is not constant in the input length. Our aim is to investigate how the pushdown height grows. In particular, we want to know if there exists a minimum growth of the pushdown height, with respect to the length of the input, when it is not constant. The answer to this question is already known, and it derives from results on Turing machines: the height of the store should grow at least as a double logarithmic function [1]. This lower bound cannot be increased, because a matching upper bound has been recently obtained in [6], where a witness language defined over an alphabet of 6 letters is presented. Using standard arguments, such language can be encoded on a binary alphabet, without changing the use of the pushdown store. Hence, in the case of an input alphabet with at least two letters, there are languages 123 Pushdown automata and constant height... accepted by using a pushdown height which is double logarithmic with respect to the input length. When we restrict to a unary alphabet, the situation is different. In fact, as a consequence of the constructions presented in the second part of the paper, we are able to prove that i (...truncated)