Andreas Malcher

Complexity of multi-head finite automata: Origins and directions

Multi-head finite automata were introduced and first investigated by Rabin and Scott in 1964 and Rosenberg in 1966. Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. Additionally the conversions between different types of multi-head finite automata induce in most cases size bounds that cannot be bounded by any recursive function, so-called non-recursive trade-offs. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature.

Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature.

Fast reversible language recognition using cellular automata

We investigate cellular automata as acceptors for formal languages. In particular, we consider real-time devices which are reversible on the core of computation, i.e., from initial configuration to the configuration given by the time complexity. This property is called real-time reversibility. We study whether for a given real-time CA working on finite configurations with fixed boundary conditions there exists a reverse real-time CA with the same neighborhood. It is shown that real-time reversibility is undecidable, which contrasts the general case, where reversibility is decidable for one-dimensional devices. Moreover, we prove the undecidability of emptiness, finiteness, infiniteness, inclusion, equivalence, regularity, and context-freedom. First steps towards the exploration of the computational capacity are done and closure under Boolean operations is shown.

Real-time reversible iterative arrays

Iterative arrays are one-dimensional arrays of interconnected interacting finite automata. The cell at the origin is equipped with a one-way read-only input tape. We investigate iterative arrays as acceptors for formal languages. In particular, we consider real-time devices which are reversible on the core of computation, i.e., from initial configuration to the configuration given by the time complexity. This property is called real-time reversibility. It is shown that real-time reversible iterative arrays can simulate restricted variants of stacks and queues. It turns out that real-time reversible iterative arrays are strictly weaker than real-time reversible cellular automata. On the other hand, a non-semilinear language is accepted. We show that real-time reversibility itself is not even semidecidable, which extends the undecidability for cellular automata and contrasts with the general case, where reversibility is decidable for one-dimensional devices. Moreover, we prove the non-semidecidability of several other properties. Several closure properties are also derived.

Finite turns and the regular closure of linear context-free languages

Turn bounded pushdown automata with different conditions for beginning a new turn are investigated. Their relationships with closures of the linear context-free languages under regular operations are studied. For example, automata with an unbounded number of turns that have to empty their pushdown store up to the initial symbol in order to start a new turn are characterized by the regular closure of the linear languages. Automata that additionally have to re-enter the initial state are (almost) characterized by the Kleene star closure of the linear languages. For both a bounded and an unbounded number of turns, requiring to empty the pushdown store is a strictly stronger condition than requiring to re-enter the initial state. Several new language families are obtained which form a double-stranded hierarchy. Closure properties of these families under AFL operations are derived. The regular closure of the linear languages share the strong closure properties of the context-free languages, i.e., the family is a full AFL. Interestingly, three natural new language families are not closed under intersection with regular languages and inverse homomorphism. Finally, an algorithm is presented parsing languages from the new families in quadratic time.

One-Way Reversible Multi-head Finite Automata

One-way multi-head finite automata are considered towards their ability to perform reversible computations. It is shown that, for every number k ≥ 1 of heads, there are problems which can be solved by one-way k-head finite automata, but not by any one-way reversible k-head finite automaton. Additionally, a proper head hierarchy is obtained for one-way reversible multi-head finite automata. Finally, decidability problems are considered. It turns out that one-way reversible finite automata with two heads are still a powerful model, since almost all commonly studied problems are not even semidecidable.

Descriptional complexity of two-way pushdown automata with restricted head reversals

Two-way nondeterministic pushdown automata (2PDA) are classical nondeterministic pushdown automata (PDA) enhanced with two-way motion of the input head. In this paper, the subclass of 2PDA accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects. In particular, the effect of reducing the number of pushdown reversals to a constant number is of interest. It turns out that this reduction leads to an exponential blow-up in case of nondeterministic devices, and to a doubly-exponential blow-up in case of deterministic devices. If the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped, the resulting trade-offs are no longer bounded by recursive functions, and so-called non-recursive trade-offs are shown.

Cellular Automata with Sparse Communication

We investigate cellular automata whose internal inter-cell communication is bounded. The communication is quantitatively measured by the number of uses of the links between cells. It is shown that even the weakest non-trivial device in question, that is, one-way cellular automata where each two neighboring cells may communicate constantly often only, accept rather complicated languages. We investigate the computational capacity of the devices in question and prove an infinite strict hierarchy depending on the bound on the total number of communications during a computation. Despite their sparse communication even for the weakest devices, by reduction of Hilberts tenth problem undecidability of several problems is derived. Finally, the question whether a given real-time one-way cellular automaton belongs to the weakest class is shown to be undecidable. This result can be adapted to answer an open question posed in [16].

Regulated nondeterminism in pushdown automata

A generalization of pushdown automata towards regulated nondeterminism is studied. The nondeterminism is governed in such a way that the decision, whether or not a nondeterministic rule is applied, depends on the whole content of the stack. More precisely, the content of the stack is considered as a word over the stack alphabet, and the pushdown automaton is allowed to act nondeterministically, if this word belongs to some given set R of control words. Otherwise its behavior is deterministic. It turns out that non-context-free languages can be accepted if R is a context-free and non-regular language. On the other hand, if the control sets R are regular languages, then the resulting devices are not more powerful than nondeterministic pushdown automata. This raises the natural question of the relations between the structure and complexity of regular sets R on one hand and the computational capacity of the corresponding R-PDA on the other hand. The main result of the paper shows that an infinite proper hierarchy of regular control sets leads to an infinite proper hierarchy of the corresponding language classes. Additionally, closure properties and decision problems of these language classes are investigated.

Descriptional complexity of bounded context-free languages

Finite-turn pushdown automata (PDA) are investigated concerning their descriptional complexity. It is known that they accept exactly the class of ultralinear context-free languages. Furthermore, the increase in size when converting arbitrary PDAs accepting ultralinear languages to finite-turn PDAs cannot be bounded by any recursive function. The latter phenomenon is known as non-recursive trade-off. In this paper, finite-turn PDAs accepting letter-bounded languages are considered. It turns out that in this case the non-recursive trade-off is reduced to a recursive trade-off, more precisely, to an exponential trade-off. A conversion algorithm is presented and the optimality of the construction is shown by proving tight lower bounds. Furthermore, the question of reducing the number of turns of a given finite-turn PDA is studied. Again, a conversion algorithm is provided which shows that in this case the trade-off is at most polynomial.