Computing with cells: membrane systems - some complexity issues.

Membrane computing is a branch of natural computing which abstracts computing models from the structure and the functioning of the living cell. The main ingredients of membrane systems, called P systems, are (i) the membrane structure, which consists of a hierarchical arrangements of membranes which delimit compartments where (ii) multisets of symbols, called objects, evolve according to (iii) sets of rules which are localised and associated with compartments. By using the rules in a nondeterministic/deterministic maximally parallel manner, transitions between the system configurations can be obtained. A sequence of transitions is a computation of how the system is evolving. Various ways of controlling the transfer of objects from one membrane to another and applying the rules, as well as possibilities to dissolve, divide or create membranes have been studied. Membrane systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems once future biotechnology gives way to a practical bio-realization. In this paper we survey some interesting and fundamental complexity issues such as universality vs. nonuniversality, determinism vs. nondeterminism, membrane and alphabet size hierarchies, characterizations of context-sensitive languages and other language classes and various notions of parallelism

Linearly bounded infinite graphs

Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs

On determinism versus nondeterminism for restarting automata

AbstractA restarting automaton processes a given word by executing a sequence of local simplifications until a simple word is obtained that the automaton then accepts. Such a computation is expressed as a sequence of cycles. A nondeterministic restarting automaton M is called correctness preserving, if, for each cycle u⊢Mcv, the string v belongs to the characteristic language LC(M) of M, if the string u does. Our first result states that for each type of restarting automaton X∈{R,RW,RWW,RL,RLW,RLWW}, if M is a nondeterministic X-automaton that is correctness preserving, then there exists a deterministic X-automaton M1 such that the characteristic languages LC(M1) and LC(M) coincide. When a restarting automaton M executes a cycle that transforms a string from the language LC(M) into a string not belonging to LC(M), then this can be interpreted as an error of M. By counting the number of cycles it may take M to detect this error, we obtain a measure for the influence that errors have on computations. Accordingly, this measure is called error detection distance. It turns out, however, that an X-automaton with bounded error detection distance is equivalent to a correctness preserving X-automaton, and therewith to a deterministic X-automaton. This means that nondeterminism increases the expressive power of X-automata only in combination with an unbounded error detection distance

Distributed Graph Automata and Verification of Distributed Algorithms

Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants of our automata for which the emptiness problem is decidable. As an application, we suggest how suitable graph automata might be useful in formal verification of distributed algorithms, using Floyd-Hoare logic.Comment: 26 pages, 6 figures, includes a condensed version of the author's Master's thesis arXiv:1404.6503. (This version of the article (v2) is identical to the previous one (v1), except for minor changes in phrasing.

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