− In this paper, following the way opened by a previous paper deposited on arXiv, we give an upper bound to the number of states for a hyperbolic cellular automaton in the pentagrid. Indeed, we prove that there is a hyperbolic cellular automaton which is rotation invariant and whose halting problem is undecidable and which has 9 states.

In this paper we investigate the role of collective commitments in groups of agents involved in Cooperative Problem Solving (CPS). Collective intentions and collective commitmens are formalized in the logical framework designed by Rao and Georgee in 22]. The novelty of our approach is to base a collective commitment on a social plan, and deening it in terms of collective intentions and pairwise...

In this note we introduce a notion of a generically (strongly generically) NP-complete problem and show that the randomized bounded version of the halting problem is strongly generically NP-complete.

The problem of service availability is analyzed using the model developed in Part I. It is shown that the general service availability problem is undecidable, i.e. there is no single algorithm to determine whether a given service in a given system is available at a given time. In restricted cases, it is shown that the problem is decidable, but is NP-complete. The problem of service availability...

Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological To-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated ...

It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth of x, and can solve the Halting problem for all programs with length shorter than the m-depth of x. It is also shown that there are strings for which the al...

Thanks to the theory of graphons and random graphs, Feynman are new analytic tools for study infinities in (strongly coupled) gauge field theories. We formulate Halting problem graphon processes build a computation dealing with solutions combinatorial Dyson–Schwinger equations context Turing machines Manin’s renormalization Hopf algebra.

The fundamental proposal in this article is that logical formulas of the form (f ↔ ¬f) are not contradictions, and that formulas of the form (t ↔ t) are not tautologies. Such formulas, wherever they appear in mathematics, are instead reason to conclude that f and t have a third truth value, different from true and false. These formulas are circular definitions of f and t. We can interpret the i...

We consider a chip-firing game on finite directed graphs and give an answer to a question posed by Bjorner, Lovasz, and Shor in 1991: given an initial configuration of chips, when does it stabilize? The approach they took to address this halting problem involves computing a period vector p with the property that toppling the vertices according to p results in the original configuration, and the...