On Communication Complexity in Evolution-Communication P Systems

Looking for a theory of communication complexity for P systems, we consider here so-called evolution-communication (EC for short) P systems, where objects evolve by multiset rewriting rules without target commands and pass through membranes by means of symport/antiport rules. (Actually, in most cases below we use only symport rules.) We first propose a way to measure the communication costs by means of “quanta of energy” (produced by evolution rules and) consumed by communication rules. EC P systems with such costs are proved to be Turing complete in all three cases with respect to the relation between evolution and communication operations: priority of communication, mixing the rules without priority for any type, priority of evolution (with the cost of communication increasing in this ordering in the universality proofs). More appropriate measures of communication complexity are then defined, as dynamical parameters, counting the communication steps or the number (and the weight) of communication rules used during a computation. Such parameters can be used in three ways: as properties of P systems (considering the families of sets of numbers generated by systems with a given communication complexity), as conditions to be imposed on computations (accepting only those computations with a communication complexity bounded by a given threshold), and as standard complexity measures (defining the class of problems which can be solved by P systems with a bounded complexity). Because we ignore the evolution steps, in all three cases it makes sense to consider hierarchies starting with finite complexity thresholds. We only give some preliminary results about these hierarchies (for instance, proving that already their lower levels contain complex – e.g., non-semilinear – sets), and we leave open many problems and research issues.Junta de Andalucía P08 – TIC 0420

Complexity Hierarchies Beyond Elementary

We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep updating the catalogue of problems from Section 6 in future revision

Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level $\Sigma_1$ of the dot-depth hierarchy are decidable in $NL$ (previously only the decidability was known). The same remains true if predicates mod $d$ for fixed $d$ are allowed. - If modular predicates for arbitrary $d$ are allowed, then the classes of the Boolean hierarchy over level $\Sigma_1$ are decidable. - For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level $\Sigma_2$ of the Straubing-Th\'erien hierarchy are decidable in $NL$. This is the first decidability result for this hierarchy. - The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for $NL$. - The membership problems for quasi-aperiodic languages and for $d$-quasi-aperiodic languages are logspace many-one complete for $PSPACE$

Parallel turing machines with one-head control units and cellular automata

Parallel Turing machines (PTM) can be viewed as a generalization of cellular automata (CA) where an additional measure called processor complexity can be defined which indicates the ``amount of parallelism\u27\u27 used. In this paper PTM are investigated with respect to their power as recognizers of formal languages. A combinatorial approach as well as diagonalization are used to obtain hierarchies of complexity classes for PTM and CA. In some cases it is possible to keep the space complexity of PTM fixed. Thus for the first time it is possible to find hierarchies of complexity classes (though not CA classes) which are completely contained in the class of languages recognizable by CA with space complexity n and in polynomial time. A possible collapse of the time hierarchy for these CA would therefore also imply some unexpected properties of PTM

Complexity Bounds for Ordinal-Based Termination

`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program running times. Our main tool for this are length function theorems, which provide complexity bounds on the use of well quasi orders. We illustrate how to prove such theorems in the simple yet until now untreated case of ordinals. We show how to apply this new theorem to derive complexity bounds on programs when they are proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability Problems (RP 2014, 22-24 September 2014, Oxford

Towards an Intelligent Database System Founded on the SP Theory of Computing and Cognition

The SP theory of computing and cognition, described in previous publications, is an attractive model for intelligent databases because it provides a simple but versatile format for different kinds of knowledge, it has capabilities in artificial intelligence, and it can also function like established database models when that is required. This paper describes how the SP model can emulate other models used in database applications and compares the SP model with those other models. The artificial intelligence capabilities of the SP model are reviewed and its relationship with other artificial intelligence systems is described. Also considered are ways in which current prototypes may be translated into an 'industrial strength' working system

Hierarchies of Inefficient Kernelizability

The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the "kernelization" of Leaf Out Branching(k) into a disjunction over n instances of size poly(k). Observing that Turing kernels are preserved by polynomial parametric transformations, we define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of ordinary parameterized complexity, by the PPT-closure of problems that seem likely to be fundamentally hard for efficient Turing kernelization. We find that several previously considered problems are complete for our fundamental hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log n