Maurice Margenstern

Frontier between decidability and undecidability: a survey

Abstract After recalling the definition of decidability and universality, we first give a survey of results on the as exact as possible border betweeen a decidable problem and the corresponding undecidablity question in various models of discrete computation: diophantine equations, word problem, Post systems, molecular computations, register machines, neural networks, cellular automata, tiling the plane and Turing machines with planar tape. We then go on with results more specific to classical Turing machines, with a survey of results and a sketchy account on technics. We conclude by an illustration on simulating the 3x+1 problem.

Cellular Automata in Hyperbolic Spaces

The chapter presents a bit more than fifteen years of research on cellular automata in hyperbolic spaces. After a short historical section, we remind the reader what is needed to know from hyperbolic geometry. Then we sum up the results which where obtained during the considered period. We focus on results about universal cellular automata, giving the main ideas which were used in the quest for universal hyperbolic cellular automata with a number of states as small as possible.

Context-free insertion-deletion systems

We consider a class of insertion-deletion systems which have not been investigated so far, those without any context controlling the insertion-deletion operations. Rather unexpectedly, we found that context-free insertion-deletion systems characterize the recursively enumerable languages. Moreover, this assertion is valid for systems with only one axiom, and also using inserted and deleted strings of a small length. As direct consequences of the main result we found that set-conditional insertion-deletion systems with two axioms generate any recursively enumerable language (this solves an open problem), as well as that membrane systems with one membrane having context-free insertion-deleletion rules without conditional use of them generate all recursively enumerable languages (this improves an earlier result). Some open problems are also formulated.

NP problems are tractable in the space of cellular automata in the hyperbolic plane

In this paper, we define cellular automata on a grid of the hyperbolic plane that is based on the tessellation obtained from the regular pentagon with right angles. Owing to the properties of that grid, we show that 3-SAT can be solved in polynomial time in that peculiar setting; then we extend that result for any NP problem. On this ground, several directions are indicated.

Accepting hybrid networks of evolutionary processors

We consider time complexity classes defined on accepting hybrid networks of evolutionary processors (AHNEP) similarly to the classical time complexity classes defined on the standard computing model of Turing machine. By definition, AHNEPs are deterministic. We prove that the classical complexity class NP equals the set of languages accepted by AHNEPs in polynomial time.

A universal cellular automaton in the hyperbolic plane

The paper gives the construction of a universal CA with 22 states in the regular rectangular pentagonal grid of the hyperbolic plane. The CA implements a railway circuit which simulates a register machine and which improves a bit already known railway simulations of a Turing machine.

Universality of Reversible Hexagonal Cellular Automata

We define a kind of cellular automaton called a hexagonal partitioned cellular automaton (HPCA), and study logical universality of a reversible HPCA. We give a specific 64-state reversible HPCA H 1 , and show that a Fredkin gate can be embedded in this cellular space. Since a Fredkin gate is known to be a universal logic element, logical universality of H 1 is concluded. Although the number of states of H 1 is greater than those of the previous models of reversible CAs having universality, the size of the configuration realizing a Fredkin gate is greatly reduced, and its local transition function is still simple. Comparison with the previous models, and open problems related to these model are also discussed.

Generalized communicating P systems

This paper considers a generalization of various communication models based on the P system paradigm where two objects synchronously move across components. More precisely, the model uses blocks of four cells such that pairs of objects from two input cells travel together to target output cells. It is shown that the model introduced, based on interactions between blocks, is complete, being able to generate all recursively enumerable sets of natural numbers. It is also proven that completeness is achievable by using a minimal interaction between blocks, i.e. every pair of cells is the input or output for at most one block. It is also shown that the concepts introduced in this paper to define the model may be simulated by more particular communication primitives, including symport, antiport and uniport rules. This enables us to automatically translate a system using interaction rules in any of minimal symport, minimal antiport or conditional uniport P systems.

A Universal Cellular Automaton on the Ternary Heptagrid

In this paper, we construct the first weakly universal cellular automaton on the ternary heptagrid. It requires six states only. It provides a universal automaton with less states than in the case of the pentagrid where the best result is nine states, a result also recently established by the authors.

P colonies with a bounded number of cells and programs

We continue the investigation of P colonies, a class of abstract computing devices composed of very simple agents (computational tools), acting and evolving in a shared environment. We show that if P colonies are initialized by placing a number of copies of a certain object in the environment, then they can generate any recursively enumerable set of numbers with a bounded number of cells, each cell containing a bounded number of programs (of bounded length), for constant bounds.