Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming

Here we study the NP-complete $K$-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random $K$-SAT, with $\alpha=M/N$ as control parameter, can be solved quickly for small enough values of $\alpha$. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically hardest problems are located only close to this phase transition. Here we study $K$-SAT ($K=3,4$) and the related optimization problem MAX-SAT by a linear programming approach, which is widely used for practical problems and allows for polynomial run time. In contrast to branch and bound it operates outside the space of feasible configurations. On the other hand, finding a solution within polynomial time is not guaranteed. We investigated several variants like including artificial objective functions, so called cutting-plane approaches, and a mapping to the NP-complete vertex-cover problem. We observed several easy-hard transitions, from where the problems are typically solvable (in polynomial time) using the given algorithms, respectively, to where they are not solvable in polynomial time. For the related vertex-cover problem on random graphs these easy-hard transitions can be identified with structural properties of the graphs, like percolation transitions. For the present random $K$-SAT problem we have investigated numerous structural properties also exhibiting clear transitions, but they appear not be correlated to the here observed easy-hard transitions. This renders the behaviour of random $K$-SAT more complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure

On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces

We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions considered. For example, the logic with the interior-connectedness predicate (and without contact) is undecidable over polygons or regular closed sets in the Euclidean plane, NP-complete over regular closed sets in three-dimensional Euclidean space, and ExpTime-complete over polyhedra in three-dimensional Euclidean space.Comment: Accepted for publication in the IJCAI 2011 proceeding

Models in science

Models are of central importance in many scientific contexts. The centrality of models such as the billiard ball model of a gas, the Bohr model of the atom, the MIT bag model of the nucleon, the Gaussian-chain model of a polymer, the Lorenz model of the atmosphere, the Lotka-Volterra model of predator-prey interaction, the double helix model of DNA, agent-based and evolutionary models in the social sciences, or general equilibrium models of markets in their respective domains are cases in point. Scientists spend a great deal of time building, testing, comparing and revising models, and much journal space is dedicated to introducing, applying and interpreting these valuable tools. In short, models are one of the principle instruments of modern science. Philosophers are acknowledging the importance of models with increasing attention and are probing the assorted roles that models play in scientific practice. The result has been an incredible proliferation of model-types in the philosophical literature. Probing models, phenomenological models, computational models, developmental models, explanatory models, impoverished models, testing models, idealized models, theoretical models, scale models, heuristic models, caricature models, didactic models, fantasy models, toy models, imaginary models, mathematical models, substitute models, iconic models, formal models, analogue models and instrumental models are but some of the notions that are used to categorize models. While at first glance this abundance is overwhelming, it can quickly be brought under control by recognizing that these notions pertain to different problems that arise in connection with models. For example, models raise questions in semantics (what is the representational function that models perform?), ontology (what kind of things are models?), epistemology (how do we learn with models?), and, of course, in philosophy of science (how do models relate to theory?; what are the implications of a model based approach to science for the debates over scientific realism, reductionism, explanation and laws of nature?). • 1. Semantics: Models and Representation o 1. Semantics: Models and Representation Models can perform two fundamentally different representational functions. On the one hand, a model can be a representation of a selected part of the world (the 'target system'). Depending on the nature of the target, such models are either models of phenomena or models of data. On the other hand, a model can represent a theory in the sense that it interprets the laws and axioms of that theory. These two notions are not mutually exclusive as scientific models can be representations in both senses at the same time. Representational models I: models of phenomena Many scientific models represent a phenomenon, where 'phenomenon' is used as an umbrella term covering all relatively stable and general features of the world that are interesting from a scientific point of view. Empiricists like A first step towards a discussion of the issue of scientific representation is to realize that there is no such thing as the problem of scientific representation. Rather, there are different but related problems. It is not yet clear what specific set of questions a theory of representation has to come to terms with, but whatever list of questions one might put on the agenda of a theory of scientific representation, there are two problems that will occupy center stage in the discussion 2 Somewhat surprisingly, until recently this question has not attracted much attention in twentieth century philosophy of science, despite the fact that the corresponding problems in the philosophy of mind and in aesthetics have been discussed extensively for decades (there is a substantial body of literature dealing with the question of what it means for a mental state to represent a certain state of affairs; and the question of how a configuration of flat marks on a canvass can depict something beyond this canvass has puzzled aestheticians for a long time). However, some recent publications address this and other closely related problems (Bailer-Jones 2003 The second problem is concerned with representational styles. It is a commonplace that one can represent the same subject matter in different ways. This pluralism does not seem to be a prerogative of the fine arts as the representations used in the sciences are not all of one kind either. Weizsäcker's liquid drop model represents the nucleus of an atom in a manner very different from the shell model, and a scale model of the wing of an air plane represents the wing in a way that is different from how a mathematical model of its shape does. What representational styles are there in the sciences? Although this question is not explicitly addressed in the literature on the so-called semantic view of theories, some answers seem to emerge from its understanding of models. One version of the semantic view, one that builds on a mathematical notion of models (see Sec. 2), posits that a model and its target have to be isomorphic Further notions that can be understood as addressing the issue of representational styles have been introduced in the literature on models. Among them, scale models, idealized models, analogical models and phenomenological models play an important role. These categories are not mutually exclusive; for instance, some scale models would also qualify as idealized models and it is not clear where exactly to draw the line between idealized and analogue models. Scale models. Some models are basically down-sized or enlarged copies of their target systems Idealized models. An idealization is a deliberate simplification of something complicated with the objective of making it more tractable. Frictionless planes, point masses, infinite velocities, isolated systems, omniscient agents, or markets in perfect equilibrium are but some well-know examples. Philosophical debates over idealization have focused on two general kinds of idealizations: so-called Aristotelian and Galilean idealizations. Aristotelian idealization amounts to 'stripping away', in our imagination, all properties from a concrete object that we believe are not relevant to the problem at hand. This allows us to focus on a limited set of properties in isolation. An example is a classical mechanics model of the planetary system, describing the planets as objects only having shape and mass, disregarding all other properties. Other labels for this kind of idealization include 'abstraction' (Cartwright 1989, Ch. 5), 'negligibility assumptions' Galilean idealizations are ones that involve deliberate distortions. Physicists build models consisting of point masses moving on frictionless planes, economists assume that agents are omniscient, biologists study isolated populations, and so on. It was characteristic of Galileo's approach to science to use simplifications of this sort whenever a situation was too complicated to tackle. For this reason it is common to refer to this sort of idealizations as 'Galilean idealizations' (McMullin 1985); another common label is 'distorted models'. Galilean idealizations are beset with riddles. What does a model involving distortions of this kind tell us about reality? How can we test its accuracy? In reply to these questions Laymon (1991) has put forward a theory which understands idealizations as ideal limits: imagine a series of experimental refinements of the actual situation which approach the postulated limit and then require that the closer the properties of a system come to the ideal limit, the closer its behavior has to come to the behavior of the ideal limit (monotonicity). But these conditions need not always hold and it is not clear how to understand situations in which no ideal limit exists. We can, at least in principle, produce a series of table tops that are ever more slippery but we cannot possibly produce a series of systems in which Planck's constant approaches zero. This raises the question of whether one can always make an idealized model more realistic by de-idealizing it. We will come back to this issue in section 5.1. Galilean and Aristotelian idealizations are not mutually exclusive. On the contrary, they often come together. Consider again the mechanical model of the planetary system: the model only takes into account a narrow set of properties and distorts these, for instance by describing planets as ideal spheres with a rotation-symmetric mass distribution. Models that involve substantial Galilean as well as Aristotelian idealizations are sometimes referred to as 'caricatures At this point we would like to mention a notion that seems to be closely related to idealization, namely approximation. Although the terms are sometimes used interchangeably, there seems to be a substantial difference between the two. Approximations are introduced in a mathematical context. One mathematical item is an approximation of another one if it is close to it in some relevant sense. What this item is may vary. Sometimes we want to approximate one curve with another one. This happens when we expand a function into a power series and only keep the first two or three terms. In other situations we approximate an equation by another one by letting a control parameter tend towards zero Analogical models. Standard examples of analogical models include the hydraulic model of an economic system, the billiard ball model of a gas, the computer model of the mind or the liquid drop model of the nucleus. At the most basic level, two things are analogous if there are certain relevant similarities between them. Hesse (1963) distinguishes different types of analogies according to the kinds of similarity relations in which two objects enter. A simple type of analogy is one that is based on shared properties. There is an analogy between the earth and the moon based on the fact that both are large, solid, opaque, spherical bodies, receiving heat and light from the sun, revolving around their axes, and gravitating towards other bodies. But sameness of properties is not a necessary condition. An analogy between two objects can also be based on relevant similarities between their properties. In this more liberal sense we can say that there is an analogy between sound and light because echoes are similar to reflections, loudness to brightness, pitch to color, detectability by the ear to detectability by the eye, and so on. Analogies can also be based on the sameness or resemblance of relations between parts of two systems rather than on their monadic properties. It is this sense that some politicians assert that the relation of a father to his children is analogous to the relation of the state to the citizens. The analogies mentioned so far have been what Hesse calls 'material analogies'. We obtain a more formal notion of analogy when we abstract from the concrete features the systems possess and only focus on their formal set-up. What the analogue model then shares with its target is not a set of features, but the same pattern of abstract relationships (i.e. the same structure, where structure is understood in the formal sense). This notion of analogy is closely related to what Hesse calls 'formal analogy'. Two items are related by formal analogy if they are both interpretations of the same formal calculus. For instance, there is a formal analogy between a swinging pendulum and an oscillating electric circuit because they are both described by the same mathematical equation. 5 A further distinction due to Hesse is the one between positive, negative and neutral analogies. The positive analogy between two items consists in the properties or relations they share (both gas molecules and billiard balls have mass), the negative analogy in the ones they do not share (billiard balls are colored, gas molecules are not). The neutral analogy comprises the properties of which it is not known yet whether they belong to the positive or the negative analogy (do gas molecules obeying Newton's laws of collision exhibit an approach to equilibrium?). Neutral analogies play an important role in scientific research because they give rise to questions and suggest new hypotheses. In this vein, various authors have emphasized the heuristic role that analogies play in theory construction and in creative thought Phenomenological models. Phenomenological models have been defined in different, though related, ways. A traditional definition takes them to be models that only represent observable properties of their targets and refrain from postulating hidden mechanisms and the like. Another approach, due to Concluding remarks. Each of these notions is still somewhat vague, suffering from internal problems, and much work needs to be done to tighten them. But more pressing than these is the question of how the different notions relate to each other. Are analogies fundamentally different from idealizations, or do they occupy different areas on a continuous scale? How do icons differ from idealizations and analogies? At the present stage we do not know how to answer these questions. What we need is a systematic account of the different ways in which models can relate to reality and of how these ways compare to each other. Representational models II: models of data Another kind of representational models are so-called 'models of data ' (Suppes 1962). A model of data is a corrected, rectified, regimented, and in many instances idealized version of the data we gain from immediate observation, the so-called raw data. Characteristically, one first eliminates errors (e.g. removes points from the record that are due to faulty observation) and then present the data in a 'neat' way, for instance by drawing a smooth curve through a set of points. These two steps are commonly referred to as 'data reduction' and 'curve fitting'. When we investigate the trajectory of a certain planet, for instance, we first eliminate points that are fallacious from the observation records and then fit a smooth curve to the remaining ones. Models of data play a crucial role in confirming theories because it is the model of data and not the often messy and complex raw data that we compare to a theoretical prediction. 6 The construction of a data model can be extremely complicated. It requires sophisticated statistical techniques and raises serious methodological as well as philosophical questions. How do we decide which points on the record need to be removed? And given a clean set of data, what curve do we fit to it? The first question has been dealt with mainly within the context of the philosophy of experiment (see for instance Models of theory In modern logic, a model is a structure that makes all sentences of a theory true, where a theory is taken to be a (usually deductively closed) set of sentences in a formal language (see Bell and Machover 1977 or Hodges 1997 for details). The structure is a 'model' in the sense that it is what the theory represents. As a simple example consider Euclidean geometry, which consists of axioms-e.g. 'any two points can be joined by a straight line'-and the theorems that can be derived from these axioms. Any structure of which all these statements are true is a model of Euclidean geometry. A structure S = <U, O, R> is a composite entity consisting of (i) a non-empty set U of individuals called the domain (or universe) of S, (ii) an indexed set O (i.e. an ordered list) of operations on U (which may be empty), and (iii) a non-empty indexed set R of relations on U. It is important to note that nothing about what the objects are matters for the definition of a structure-they are mere dummies. Similarly, operations and functions are specified purely extensionally; that is, n-place relations are defined as classes of ntuples, and functions taking n arguments are defined as classes of (n+1)-tuples. If all sentences of a theory are true when its symbols are interpreted as referring to either objects, relations, or functions of a structure S, then S is a model of this theory. Many models in science carry over from logic the idea of being the interpretation of an abstract calculus. This is particularly pertinent in physics, where general laws-such as Newton's equation of motion-lie at the heart of a theory. These laws are applied to a particular system-e.g. a pendulum-by choosing a special force function, making assumptions about the mass distribution of the pendulum etc. The resulting model then is an interpretation (or realization) of the general law. Ontology: What Are Models? There is a variety of things that are commonly referred to as models: physical objects, fictional objects, set-theoretic structures, descriptions, equations, or combinations of some of these. However, these categories are neither mutually exclusive nor jointly 7 exhaustive. Where one draws the line between, say, fictional objects and set-theoretical structures may well depend on one's metaphysical convictions, and some models may fall into yet another class of things. What models are is, of course, an interesting question in its own right, but, as briefly indicated in the last section, it has also important implications for semantics and, as we will see below, for epistemology. Physical objects Some models are straightforward physical objects. These are commonly referred to as 'material models'. The class of material models comprises anything that is a physical entity and that serves as a scientific representation of something else. Among the members of this class we find stock examples like wooden models of bridges, planes, or ships, analogue models like electric circuit models of neural systems or pipe models of an economy, or Watson and Crick's model of DNA. But also more cutting edge cases, especially from the life sciences, where certain organisms are studied as stand-ins for others, belong to this category. Material models do not give rise to any ontological difficulties over and above the wellknown quibbles in connection with objects, which metaphysicians deal with (e.g. the nature of properties, the identity of objects, parts and wholes, and so on). Fictional objects Many models are not material models. The Bohr model of the atom, a frictionless pendulum, or isolated populations, for instance, are in the scientist's mind rather than in the laboratory and they do not have to be physically realized and experimented upon to perform their representational function. It seems natural to view them as fictional entities. This position can be traced back to the German neo-Kantian Vaihinger (1911), who emphasized the importance of fictions for scientific reasoning. Giere has recently advocated the view that models are abstract entities (1988, 81). It is not entirely clear what Giere means by 'abstract entities', but his discussion of mechanical models seems to suggest that he uses the term to designate fictional entities. This view squares well with scientific practice, where scientists often talk about models as if they were objects, as well as with philosophical views that see the manipulation of models as an essential part of the process of scientific investigation (Morgan 1999). It is natural to assume that one can manipulate something only if it exists. Furthermore, models often have more properties than we explicitly attribute to them when we construct them, which is why they are interesting vehicles of research. A view that regards models as objects can easily explain this without further ado: when we introduce a model we use an identifying description, but the object itself is not exhaustively characterized by this description. Research then simply amounts to finding out more about the object thus identified. The drawback of this suggestion is that fictional entities are notoriously beset with ontological riddles. This has led many philosophers to argue that there are no such things as fictional entities and that apparent ontological commitments to them must be renounced. The most influential of these deflationary accounts goes back to Set-theoretic structures An influential point of view takes models to be set-theoretic structures. This position can be traced back to Descriptions A time-honored position has it that what scientists display in scientific papers and textbooks when they present a model are more or less stylized descriptions of the relevant target systems (Achinstein 1968 Equations Another group of things that is habitually referred to as 'models', in particular in economics, is equations (which are then also termed 'mathematical models'). The BlackScholes model of the stock market or the Mundell-Fleming model of an open economy are cases in point. The problem with this suggestion is that equations are syntactic items and as such they face objections similar to the ones put forward against descriptions. First, one can describe the same situation using different co-ordinates and as a result obtain different equations; but we do not seem to obtain a different model. Second, the model has properties different from the equation. An oscillator is three-dimensional but the equation describing its motion is not. Equally, an equation may be inhomogeneous but the system it describes is not. Gerrymandered ontologies The proposals discussed so far have tacitly assumed that a model belongs to one particular class of objects. But this assumption is not necessary. It might be the case that models are a mixture of elements belonging to different ontological categories. In this vein Morga

A GLOBALIZAÇÃO E O ESTADO-NAÇÃO: RUMO À PÓS-MODERNIDADE E AO ESTADO COSMOPOLITA?

O presente artigo analisa a crise enfrentada pelo Estado-nação diante da novarealidade mundial trazida sobretudo pela globalização. Para isso, faz-se uma retomada dos elementos estruturais do Estado-nação e, num segundo momento, contrapõe-se tal estrutura aos fenômenos atuais resultantes da mundialização, perpassando, inclusive, idéias fundamentais sobre o cosmopolitismo, sobretudo mediante a análise do pensamento habermasiano, e sobre o comunitarismo. Em seguida, traz-se à discussão os problemas apontados por Michael Hardt e Antonio Negri em sua obra ‘O Império’, na qual retratam bem o lado negro e prejudicial do desaparecimento das fronteiras entre os países, para, ao final, reforçar que, embora inevitável e causador de graves problemas, o processo de globalização deverá, ao menos, pressupor a defesa da democracia

S6K1 and 4E-BP1 Are Independent Regulated and Control Cellular Growth in Bladder Cancer

Aberrant activation and mutation status of proteins in the phosphatidylinositol-3-kinase (PI3K)/Akt/mammalian target of rapamycin (mTOR) and the mitogen activated protein kinase (MAPK) signaling pathways have been linked to tumorigenesis in various tumors including urothelial carcinoma (UC). However, anti-tumor therapy with small molecule inhibitors against mTOR turned out to be less successful than expected. We characterized the molecular mechanism of this pathway in urothelial carcinoma by interfering with different molecular components using small chemical inhibitors and siRNA technology and analyzed effects on the molecular activation status, cell growth, proliferation and apoptosis. In a majority of tested cell lines constitutive activation of the PI3K was observed. Manipulation of mTOR or Akt expression or activity only regulated phosphorylation of S6K1 but not 4E-BP1. Instead, we provide evidence for an alternative mTOR independent but PI3K dependent regulation of 4E-BP1. Only the simultaneous inhibition of both S6K1 and 4E-BP1 suppressed cell growth efficiently. Crosstalk between PI3K and the MAPK signaling pathway is mediated via PI3K and indirect by S6K1 activity. Inhibition of MEK1/2 results in activation of Akt but not mTOR/S6K1 or 4E-BP1. Our data suggest that 4E-BP1 is a potential new target molecule and stratification marker for anti cancer therapy in UC and support the consideration of a multi-targeting approach against PI3K, mTORC1/2 and MAPK

Helium nanodroplet isolation ro-vibrational spectroscopy: methods and recent results

In this article, recent developments in HElium NanoDroplet Isolation (HENDI) spectroscopy are reviewed, with an emphasis on the infrared region of the spectrum. Topics discussed include experimental details, comparison of radiation sources, symmetry issues of the helium solvation structure, sources of line broadening, changes in spectroscopic constants upon solvation, and applications including formation of novel chemical structures.Comment: 24 pages, 8 figures, 3 tables; to be published in the Journal of Chemical Physic