The Size of One-Way Cellular Automata

International audienceWe investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are $\textsf{NLOGSPACE}$-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given

Adding Logical Operators to Tree Pattern Queries on Graph-Structured Data

As data are increasingly modeled as graphs for expressing complex relationships, the tree pattern query on graph-structured data becomes an important type of queries in real-world applications. Most practical query languages, such as XQuery and SPARQL, support logical expressions using logical-AND/OR/NOT operators to define structural constraints of tree patterns. In this paper, (1) we propose generalized tree pattern queries (GTPQs) over graph-structured data, which fully support propositional logic of structural constraints. (2) We make a thorough study of fundamental problems including satisfiability, containment and minimization, and analyze the computational complexity and the decision procedures of these problems. (3) We propose a compact graph representation of intermediate results and a pruning approach to reduce the size of intermediate results and the number of join operations -- two factors that often impair the efficiency of traditional algorithms for evaluating tree pattern queries. (4) We present an efficient algorithm for evaluating GTPQs using 3-hop as the underlying reachability index. (5) Experiments on both real-life and synthetic data sets demonstrate the effectiveness and efficiency of our algorithm, from several times to orders of magnitude faster than state-of-the-art algorithms in terms of evaluation time, even for traditional tree pattern queries with only conjunctive operations.Comment: 16 page

Limits of Diagonalization and the Polynomial Hierarchy

Determining the computational complexity of problems is a large area of study. It seeks to separate these problems into ones with efficient solutions, and those with inefficient solutions. Of course, the strata is much more fine-grain than this. Of special interest are two classes of problems: P and NP. These have been of much interest to complexity theorists for quite some time, because both contain many instances of important real-world problems, and finding efficient solutions for those in NP would be beneficial for computing applications. Yet with all this attention, there are still important unanswered questions about the two classes. It is known that P ⊆ NP, however it is still unknown whether P = NP or if P ⊂ NP. Before we discuss why this problem is so crucial to complexity theory, an overview of P, NP, and coNP is necessary. The class P is a model of the notion of efficiently solvable , and thus contains all languages (problems) that are decidable in deterministic polynomial time. This means that any language in P has a deterministic Turing Machine (algorithm) that will either accept or reject any input in n^k steps, where n is the length of the input string, and k is a constant. The class NP contains all languages that are decidable in nondeterministic polynomial time. A nondeterministic Turing Machine is one that is allowed to guess the correct path of computation, and seems to be able to reach an accept or reject state faster than if it was forced to run deterministically. It is unknown whether NP is closed under complementation because of this nondeterminism. It is quite easy to show a class of deterministically-solvable languages (such as P) is closed under complementation: we simply reverse the accept and reject states. This method is not viable for a nondeterministic machine, since switching the accept and reject states will result in machine that computes a completely different language. Thus the class coNP is defined as containing the complement of every language in NP. In the rest of this paper we will present structural definitions of P and NP as well as present example languages from each. These structural definitions will give insight into the arrangement of the polynomial hierarchy, which is discussed in section 3. A diagonalization proof is presented in section 4, and an explanation of the general usage of diagonalization follows. In section 5, universal languages are defined and an important result from Kozen is given. In the final section, the limits of diagonalization as they pertain to P and NP are outlined, as well as the same limits for relativized classes

Optimal Planning Modulo Theories

Planning for real-world applications requires algorithms and tools with the ability to handle the complexity such scenarios entail. However, meeting the needs of such applications poses substantial challenges, both representational and algorithmic. On the one hand, expressive languages are needed to build faithful models. On the other hand, efficient solving techniques that can support these languages need to be devised. A response to this challenge is underway, and the past few years witnessed a community effort towards more expressive languages, including decidable fragments of first-order theories. In this work we focus on planning with arithmetic theories and propose Optimal Planning Modulo Theories, a framework that attempts to provide efficient means of dealing with such problems. Leveraging generic Optimization Modulo Theories (OMT) solvers, we first present domain-specific encodings for optimal planning in complex logistic domains. We then present a more general, domain- independent formulation that allows to extend OMT planning to a broader class of well-studied numeric problems in planning. To the best of our knowledge, this is the first time OMT procedures are employed in domain-independent planning

Descriptive complexity of real computation and probabilistic independence logic

We introduce a novel variant of BSS machines called Separate Branching BSS machines (S-BSS in short) and develop a Fagin-type logical characterisation for languages decidable in non-deterministic polynomial time by S-BSS machines. We show that NP on S-BSS machines is strictly included in NP on BSS machines and that every NP language on S-BSS machines is a countable union of closed sets in the usual topology of R^n. Moreover, we establish that on Boolean inputs NP on S-BSS machines without real constants characterises a natural fragment of the complexity class existsR (a class of problems polynomial time reducible to the true existential theory of the reals) and hence lies between NP and PSPACE. Finally we apply our results to determine the data complexity of probabilistic independence logic.Peer reviewe

A System-Level Simulation Model for a Protocol Processor

As the development of integrated circuit technology continues to follow Moore’s law the complexity of circuits increases exponentially. Traditional hardware description languages such as VHDL and Verilog are no longer powerful enough to cope with this level of complexity and do not provide facilities for hardware/software codesign. Languages such as SystemC are intended to solve these problems by combining the powerful expression of high level programming languages and hardware oriented facilities of hardware description languages. To fully replace older languages in the desing flow of digital systems SystemC should also be synthesizable. The devices required by modern high speed networks often share the same tight constraints for e.g. size, power consumption and price with embedded systems but have also very demanding real time and quality of service requirements that are difficult to satisfy with general purpose processors. Dedicated hardware blocks of an application specific instruction set processor are one way to combine fast processing speed, energy efficiency, flexibility and relatively low time-to-market. Common features can be identified in the network processing domain making it possible to develop specialized but configurable processor architectures. One such architecture is the TACO which is based on transport triggered architecture. The architecture offers a high degree of parallelism and modularity and greatly simplified instruction decoding. For this M.Sc.(Tech) thesis, a simulation environment for the TACO architecture was developed with SystemC 2.2 using an old version written with SystemC 1.0 as a starting point. The environment enables rapid design space exploration by providing facilities for hw/sw codesign and simulation and an extendable library of automatically configured reusable hardware blocks. Other topics that are covered are the differences between SystemC 1.0 and 2.2 from the viewpoint of hardware modeling, and compilation of a SystemC model into synthesizable VHDL with Celoxica Agility SystemC Compiler. A simulation model for a processor for TCP/IP packet validation was designed and tested as a test case for the environment.Siirretty Doriast

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

Bounded Languages Meet Cellular Automata with Sparse Communication

Cellular automata are one-dimensional arrays of interconnected interacting finite automata. We investigate one of the weakest classes, the real-time one-way cellular automata, and impose an additional restriction on their inter-cell communication by bounding the number of allowed uses of the links between cells. Moreover, we consider the devices as acceptors for bounded languages in order to explore the borderline at which non-trivial decidability problems of cellular automata classes become decidable. It is shown that even devices with drastically reduced communication, that is, each two neighboring cells may communicate only constantly often, accept bounded languages that are not semilinear. If the number of communications is at least logarithmic in the length of the input, several problems are undecidable. The same result is obtained for classes where the total number of communications during a computation is linearly bounded