On the Size Complexity of Non-Returning Context-Free PC Grammar Systems

Improving the previously known best bound, we show that any recursively enumerable language can be generated with a non-returning parallel communicating (PC) grammar system having six context-free components. We also present a non-returning universal PC grammar system generating unary languages, that is, a system where not only the number of components, but also the number of productions and the number of nonterminals are limited by certain constants, and these size parameters do not depend on the generated language

Matter and Anti-Matter in Membrane Systems

The concept of a matter object being annihilated when meeting its corresponding anti-matter object is investigated in the context of membrane systems, i.e., of (distributed) multiset rewriting systems applying rules in the maximally parallel way. Computational completeness can be obtained with using only non-cooperative rules besides these matter/anti-matter annihilation rules if these annihilation rules have priority over the other rules. Without this priority condition, in addition catalytic rules with one single catalyst are needed to get computational completeness. Even deterministic systems are obtained in the accepting case. Universal P systems with a rather small number of rules – 57 for computing systems, 59 for generating and 52 for accepting systems – can be constructed when using non-cooperative rules together with matter/anti-matter annihilation rules having weak priority. Allowing anti-matter objects as input and/or output, we even get a computationally complete computing model for computations on integer numbers. Interpreting sequences of symbols taken in from and/or sent out to the environment as strings, we get a model for computations on strings, which can even be interpreted as representations of elements of a group based on a computable finite presentation

The Collatz Process Embeds a Base Conversion Algorithm

The Collatz process is defined on natural numbers by iterating the map T(x)=T0(x)=x/2 when x∈N is even and T(x)=T1(x)=(3x+1)/2 when x is odd. In an effort to understand its dynamics, and since Generalised Collatz Maps are known to simulate Turing Machines [Conway, 1972], it seems natural to ask what kinds of algorithmic behaviours it embeds. We define a quasi-cellular automaton that exactly simulates the Collatz process on the square grid: on input x∈N , written horizontally in base 2, successive rows give the Collatz sequence of x in base 2. We show that vertical columns simultaneously iterate the map in base 3. This leads to our main result: the Collatz process embeds an algorithm that converts any natural number from base 3 to base 2. We also find that the evolution of our automaton computes the parity of the number of 1s in any ternary input. It follows that predicting about half of the bits of the iterates Ti(x) , for i=O(logx) , is in the complexity class NC 1 but outside AC 0 . These results show that the Collatz process is capable of some simple, but non-trivial, computation in bases 2 and 3, suggesting an algorithmic approach to thinking about prediction and existence of cycles in the Collatz process