An in-between "implicit" and "explicit" complexity: Automata

Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model

Bounded Counter Languages

We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters

Automata with modulo counters and nondeterministic counter bounds

We introduce and investigate Nondeterministically Bounded Modulo Counter Automata (NBMCA), which are two-way one-head automata that comprise a constant number of modulo counters, where the counter bounds are nondeterministically guessed, and this is the only element of nondeterminism. NBMCA are tailored to recognising those languages that are characterised by the existence of a specific factorisation of their words, e. g., pattern languages. In this work, we subject NBMCA to a theoretically sound analysis

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

Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur

Tradeoffs for language recognition on alternating machines

AbstractThe alternating machine having a separate input tape with k two-way, read-only heads, and a certain number of internal configurations, AM(k), is considered as a parallel computing model. For the complexity measure TIME · SPACE · PARALLELISM (TSP), the optimal lower bounds Ω(n2) and Ω(n3/2) respectively are proved for the recognition of specific languages on AM(1) and AM(k) respectively. For the complexity measure REVERSALS · SPACE · PARALLELISM (RSP), the lower bound Ω(n1/2) is established for the recognition of a specific language on AM(k). This result implies a polynomial lower bound on PARALLEL TIME · HARDWARE of parallel RAM's.Lower bounds on the complexity measures TIME · SPACE and REVERSALS · SPACE of nondeterministic machines are direct consequences of the result introduced above.All lower bounds obtained are substantially improved in the case that SPACE⩾ nɛ for 0<ɛ<1. Several strongest lower bounds for two-way and one-way alternating (deterministic, nondeterministic) multihead finite automata are obtained as direct consequences of these results. The hierarchies for the complexity measures TSP, RSP, TS and RS can be immediately achieved too

Automata with Modulo Counters and Nondeterministic Counter Bounds

We introduce and investigate Nondeterministically Bounded Modulo Counter Automata (NBMCA), which are two-way multi-head automata that comprise a constant number of modulo counters, where the counter bounds are nondeterministically guessed, and this is the only element of nondeterminism. NBMCA are tailored to recognising those languages that are characterised by the existence of a specific factorisation of their words, e. g., pattern languages. In this work, we subject NBMCA to a theoretically sound analysis