Computing Partial Recursive Functions by Transition P Systems

In this paper a variant of transition P systems with external output designed to compute partial functions on natural numbers is presented. These P systems are stable under composition, iteration and unbounded minimization (μ–recursion) of functions. We prove that every partial recursive function can be computed by such P systems, from which the computational completeness of this model can be deduced.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Reactive Turing Machines

We propose reactive Turing machines (RTMs), extending classical Turing machines with a process-theoretical notion of interaction, and use it to define a notion of executable transition system. We show that every computable transition system with a bounded branching degree is simulated modulo divergence-preserving branching bisimilarity by an RTM, and that every effective transition system is simulated modulo the variant of branching bisimilarity that does not require divergence preservation. We conclude from these results that the parallel composition of (communicating) RTMs can be simulated by a single RTM. We prove that there exist universal RTMs modulo branching bisimilarity, but these essentially employ divergence to be able to simulate an RTM of arbitrary branching degree. We also prove that modulo divergence-preserving branching bisimilarity there are RTMs that are universal up to their own branching degree. Finally, we establish a correspondence between executability and finite definability in a simple process calculus

Computing Partial Recursive Functions by Virus Machines

Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted Virus Machines are unbounded Virus Machines, in the sense that no restriction on the number of hosts, the number of instructions and the number of viruses contained in any host along any computation is placed on them. The computational completeness of these machines has been obtained by simulating register machines. In this paper, Virus Machines as function computing devices are considered. Then, the universality of non-restricted virus machines is proved by showing that they can compute all partial recursive functions.Ministerio de Economía y Competitividad TIN2012- 3743

Forward Analysis for WSTS, Part III: Karp-Miller Trees

This paper is a sequel of "Forward Analysis for WSTS, Part I: Completions" [STACS 2009, LZI Intl. Proc. in Informatics 3, 433-444] and "Forward Analysis for WSTS, Part II: Complete WSTS" [Logical Methods in Computer Science 8(3), 2012]. In these two papers, we provided a framework to conduct forward reachability analyses of WSTS, using finite representations of downward-closed sets. We further develop this framework to obtain a generic Karp-Miller algorithm for the new class of very-WSTS. This allows us to show that coverability sets of very-WSTS can be computed as their finite ideal decompositions. Under natural effectiveness assumptions, we also show that LTL model checking for very-WSTS is decidable. The termination of our procedure rests on a new notion of acceleration levels, which we study. We characterize those domains that allow for only finitely many accelerations, based on ordinal ranks

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Zipf's law and L. Levin's probability distributions

Zipf's law in its basic incarnation is an empirical probability distribution governing the frequency of usage of words in a language. As Terence Tao recently remarked, it still lacks a convincing and satisfactory mathematical explanation. In this paper I suggest that at least in certain situations, Zipf's law can be explained as a special case of the a priori distribution introduced and studied by L. Levin. The Zipf ranking corresponding to diminishing probability appears then as the ordering determined by the growing Kolmogorov complexity. One argument justifying this assertion is the appeal to a recent interpretation by Yu. Manin and M. Marcolli of asymptotic bounds for error--correcting codes in terms of phase transition. In the respective partition function, Kolmogorov complexity of a code plays the role of its energy. This version contains minor corrections and additions.Comment: 19 page

Bridging the Gap between Enumerative and Symbolic Model Checkers

We present a method to perform symbolic state space generation for languages with existing enumerative state generators. The method is largely independent from the chosen modelling language. We validated this on three different types of languages and tools: state-based languages (PROMELA), action-based process algebras (muCRL, mCRL2), and discrete abstractions of ODEs (Maple).\ud Only little information about the combinatorial structure of the\ud underlying model checking problem need to be provided. The key enabling data structure is the "PINS" dependency matrix. Moreover, it can be provided gradually (more precise information yield better results).\ud \ud Second, in addition to symbolic reachability, the same PINS matrix contains enough information to enable new optimizations in state space generation (transition caching), again independent from the chosen modelling language. We have also based existing optimizations, like (recursive) state collapsing, on top of PINS and hint at how to support partial order reduction techniques.\ud \ud Third, PINS allows interfacing of existing state generators to, e.g., distributed reachability tools. Thus, besides the stated novelties, the method we propose also significantly reduces the complexity of building modular yet still efficient model checking tools.\ud \ud Our experiments show that we can match or even outperform existing tools by reusing their own state generators, which we have linked into an implementation of our ideas