We construct a collection of new topological Ramsey spaces of trees. It is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. We give an example of its application by proving a partition theorem for profinite graphs.

Studies to neutrosophic graphs happens to be not only innovative and interesting, but gives a new dimension to graph theory. The classic coloring of edge problem happens to give various results. Neutrosophic tree will certainly find lots of applications in data mining when certain levels of indeterminacy is involved in the problem. Several open problems are suggested.

Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author’s recent result for the triangle-free Henson graph, we prove (...) that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey’s Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker. (shrink)

This paper presents an analysis of Seneca's 58th letter to Lucilius and Porphyry's Isagoge, which were the origin of the tree diagrams that became popular in philosophy and logic from the early Middle Ages onwards. These diagrams visualise the extent to which a concept can be understood as a category, genus, species or individual and what the method of dihairesis (division) means. The paper explores the dissimilarities between Seneca's and Porphyry's tree structures, scrutinising them through the perspective of modern (...) class='Hi'>graph theory. Although traditional trees can be interpreted in a modern way, their terms and definitions defy modern standardisation. These distinctions are not only useful for differentiating between Seneca's and Porphyry's trees, but also for comprehending the evolution of this philosophical tradition. Thus the paper highlights the continued significance of these tree diagrams in analysing concepts and categories in philosophy and structuring objects and information in modern fields such as artificial intelligence, computer science, bioinformatics, graph theory, and many more. (shrink)

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] (...) and culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica26(2) (2006) 183–205], the Ramsey theory of H3 had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin.2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math.185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph G, there is a finite number T(G) such that for any coloring of all copies of G in H3 into finitely many colors, there is a subgraph of H3 which is again universal homogeneous triangle-free in which the coloring takes no more than T(G) colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code H3 and the development of their Ramsey theory. (shrink)

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook and Rabern, Rabern, and Macauley. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It (...) is shown that this notion of paradoxicality coincides with that of Kripke. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs. (shrink)

In this article the structure of the models of decidable monadic theories of planar graphs is investigated. It is shown that if the monadic theory of a class K of planar graphs is decidable, then the tree-width in the sense of Robertson and Seymour of the elements of K is universally bounded and there is a class T of trees such that the monadic theory of K is interpretable in the monadic theory of T.

Contemporary generative grammar assumes that syntactic structure is best described in terms of sets, and that locality conditions, as well as cross-linguistic variation, is determined at the level of designated functional heads. Syntactic operations (merge, MERGE, etc.) build a structure by deriving sets from lexical atoms and recursively (and monotonically) yielding sets of sets. Additional restrictions over the format of structural descriptions limit the number of elements involved in each operation to two at each derivational step, a head and a (...) non-head. In this paper, we will explore an alternative direction for minimalist inquiry based on previous work, e.g., Frank (2002, 2006), albeit under novel assumptions. We propose a view of syntactic structure as a specification of relations in graphs, which correspond to the extended projection of lexical heads; these are _elementary trees_ in Tree Adjoining Grammars. We present empirical motivation for a lexicalised approach to structure building, where the units of the grammar are elementary trees. Our proposal will be based on cross-linguistic evidence; we will consider the structure of elementary trees in Spanish, English and German. We will also explore the consequences of assuming that nodes in elementary trees are addresses for purposes of tree composition operations, substitution and adjunction. (shrink)

This article combines Social Choice Theory with Discrete Optimization. We assume that individuals have preferences over edges of a graph that need to be aggregated. The goal is to find a socially “best” spanning tree in the graph. As ranking all spanning trees is becoming infeasible even for small numbers of vertices and/or edges of a graph, our interest lies in finding algorithms that determine a socially “best” spanning tree in a simple manner. This problem (...) is closely related to the minimum (or maximum) spanning tree problem in Discrete Optimization. Our main result shows that for the various underlying ranking rules on the set of spanning trees discussed in this article, the sets of “best” spanning trees coincide. Moreover, a greedy algorithm based on a transitive group ranking on the set of edges will always provide such a “best” spanning tree. (shrink)

In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it is (...) not known if these implications reverse, BGS also showed that those theorems imply finite choice (in some cases, with additional induction assumptions). This motivated us to study the proof-theoretic strength of finite choice. Using a variant of Steel forcing with tagged trees, we show that finite choice is not provable from the $\Delta ^1_1$ -comprehension scheme (even over $\omega $ -models). We also show that finite choice is a consequence of the arithmetic Bolzano–Weierstrass theorem (introduced by Friedman and studied by Conidis), assuming $\Sigma ^1_1$ -induction. Our results were used by BGS to show that several theorems in graph theory cannot be proved using $\Delta ^1_1$ -comprehension. Our results also strengthen results of Conidis. (shrink)

König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Acta Litterarum ac Scientiarum, Szeged, 3, 121–30] includes a graph theoretic formulation: an infinite, locally finite and connected graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite tree with a (...) root has a infinite branch. This tree lemma can be traced to [Beth, E. W. 1955. ‘Semantic entailment and formal derivability’, Mededelingen der Kon. Ned. Akad. v. Wet., new series 18, 13, 309–42]. It is argued that Beth independently discovered the tree lemma in the early 1950s and that it was later recognized among logicians that Beth's result was a consequence of the infinity lemma. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree. (shrink)

We introduce values for rooted-tree and sink-tree digraph games axiomatically and provide their explicit formula representation. These values may be considered as natural extensions of the lower equivalent and upper equivalent solutions for line-graph games studied in van den Brink et al. (Econ Theory 33:349–349, 2007). We study the distribution of Harsanyi dividends. We show that the problem of sharing a river with a delta or with multiple sources among different agents located at different levels along the riverbed (...) can be embedded into the framework of a rooted-tree or sink-tree digraph game correspondingly. (shrink)

In every undirected graph or, more generally, in every undirected hypergraph of bounded rank, one can specify an orientation of the edges or hyperedges by monadic second-order formulas using quantifications on sets of edges or hyperedges. The proof uses an extension to hypergraphs of the classical notion of a depth-first spanning tree. Applications are given to the characterization of the classes of graphs and hypergraphs having decidable monadic theories.

We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $ -regular trees, $\Delta \in \omega $. We show that such a problem admits a computable solution on every highly computable $\Delta $ -regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $ -regular forest. We also show that if such a problem admits a computable solution on every (...) computable maximum degree $\Delta $ forest then it admits a continuous solution on every maximum degree $\Delta $ Borel graph with appropriate topological hypotheses, though the converse does not hold. (shrink)

The long-awaited revision of the industry standard on phylogenetics Since the publication of the first edition of this landmark volume more than twenty-five years ago, phylogenetic systematics has taken its place as the dominant paradigm of systematic biology. It has profoundly influenced the way scientists study evolution, and has seen many theoretical and technical advances as the field has continued to grow. It goes almost without saying that the next twenty-five years of phylogenetic research will prove as fascinating as the (...) first, with many exciting developments yet to come. This new edition of Phylogenetics captures the very essence of this rapidly evolving discipline. Written for the practicing systematist and phylogeneticist, it addresses both the philosophical and technical issues of the field, as well as surveys general practices in taxonomy. Major sections of the book deal with the nature of species and higher taxa, homology and characters, trees and tree graphs, and biogeography—the purpose being to develop biologically relevant species, character, tree, and biogeographic concepts that can be applied fruitfully to phylogenetics. The book then turns its focus to phylogenetic trees, including an in-depth guide to tree-building algorithms. Additional coverage includes: Parsimony and parsimony analysis Parametric phylogenetics including maximum likelihood and Bayesian approaches Phylogenetic classification Critiques of evolutionary taxonomy, phenetics, and transformed cladistics Specimen selection, field collecting, and curating Systematic publication and the rules of nomenclature Providing a thorough synthesis of the field, this important update to Phylogenetics is essential for students and researchers in the areas of evolutionary biology, molecular evolution, genetics and evolutionary genetics, paleontology, physical anthropology, and zoology. (shrink)

Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal \, any winning strategy has complexity at least \}\). Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are (...) explored. The Turing computability of a binary relation used to classify cop-win graphs is studied, and the computational difficulty of determining the winner for locally finite computable graphs is discussed. (shrink)

For [Formula: see text], the coarse similarity class of [Formula: see text], denoted by [Formula: see text], is the set of all [Formula: see text] such that the symmetric difference of [Formula: see text] and [Formula: see text] has asymptotic density [Formula: see text]. There is a natural metric [Formula: see text] on the space [Formula: see text] of coarse similarity classes defined by letting [Formula: see text] be the upper density of the symmetric difference of [Formula: see text] and (...) [Formula: see text]. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form [Formula: see text] where [Formula: see text] is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of [Formula: see text]. We then define a distance between Turing degrees based on Hausdorff distance in the metric space [Formula: see text]. We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly [Formula: see text], [Formula: see text], and [Formula: see text], and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree [Formula: see text] to be attractive if the class of all degrees at distance [Formula: see text] from [Formula: see text] has measure [Formula: see text], and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually [Formula: see text]-random, as well as a perfect set whose elements are mutually [Formula: see text]-generic. Finally, we study the completeness of [Formula: see text] from the perspectives of computability theory and reverse mathematics. (shrink)

We construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with Σ b 1 -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.

Let $\mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of (...) forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi $ the corresponding forbidden induced subgraphs. This machinery fails on finite graphs as shown by our results: – There is a class $\mathscr {C}$ of finite graphs that is definable in first-order logic and closed under induced subgraphs but has no finite set of forbidden induced subgraphs. – Even if we only consider classes $\mathscr {C}$ of finite graphs that can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from a first-order sentence $\varphi $ that defines $\mathscr {C}$ and the size of the characterization cannot be bounded by $f(|\varphi |)$ for any computable function f. Besides their importance in graph theory, the above results also significantly strengthen similar known theorems for arbitrary structures. (shrink)

Previous work employing graph theory and nonlinear analysis has found increased spatial and temporal disorder, respectively, of functional brain connectivity in schizophrenia. We present a new method combining graph theory and nonlinear techniques that measures the temporal disorder of functional brain connections. Multichannel electroencephalographic data were windowed and functional networks were reconstructed using the minimum spanning trees of correlation matrices. Using a method based on Shannon entropy, we found elevated connection entropy in gamma activity of (...) patients with schizophrenia; however, gamma connection entropy remained elevated in patients with schizophrenia even after a reduction in symptoms due to treatment with antipsychotics. Our results are consistent with several possibilities: aberrant functional connectivity is epiphenomenal to schizophrenia, aberrant functional connectivity is a central feature but antipsychotics reduce symptoms by an independent mechanism, or connection entropy is not an appropriately sensitive measure of brain abnormalities in schizophrenia. (shrink)

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees.................................................................................................... 31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 2.4. Humean (...) Events............................................................................................43 2.5. Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (shrink)

We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the least (...) number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs. (shrink)

Consideration of the amazing organized intricacy of human cortical anatomy entails a deeper appreciation of nature that is fully consistent with a mature religious spirit. A brain seems at first glance to be a mere lump of grayish claylike stuff, but facts of basic neuroanatomy compel us to consider that this particular kind of stuff may really contain all the richly tangible and richly ghostly inner essences of emotion, thought, and behavior. Humans are the “college graduates” of evolution. The human (...) cortex is 3,400 times the volume of, yet only slightly thicker (about 3 millimeters) than, that of the mouse. This remarkable sheet is as thin as a graduation‐day “mortarboard” cap, but its 2,600 square centimeter area is four times as large (about 20 × 20 inches if a square; both metric and English units used deliberately). Zooming in, there are about 50 billion cortical neurons; though named after “pyramids,” they are more like tiny “magic trees,” with branches and roots so long and fine that there are 1 or 2 miles of these electrically scintillating fibers within each cubic millimeter of cortex. Cortical neurons communicate intimately: viewed from above, beneath a single square millimeter 100,000 nerve cells intertwine; each such neuron makes 5,000 or more connections with others. These and many additional amazing facts about brain tissue, together with some conjectures about dense connectedness in the mathematics of graph theory, help to bear out the groundwork prepared by such pioneers as Ralph Wendell Burhoe that the spirit and knowledge of science might rejoin that of religion. If it takes enchanted matter to contain consciousness, this is a kind of enchantment that science may well be able to penetrate for eventual thoroughgoing understanding. Inevitable by‐products will be greater reverence for nature and greater awe at the mystery of nature's origin. (shrink)

We study human performance in two classical NP‐hard optimization problems: Set Cover and Maximum Coverage. We suggest that Set Cover and Max Coverage are related to means selection problems that arise in human problem‐solving and in pursuing multiple goals: The relationship between goals and means is expressed as a bipartite graph where edges between means and goals indicate which means can be used to achieve which goals. While these problems are believed to be computationally intractable in general, they become (...) more tractable when the structure of the network resembles a tree. Thus, our main prediction is that people should perform better with goal systems that are more tree‐like. We report three behavioral experiments which confirm this prediction. Our results suggest that combinatorial parameters that are instrumental to algorithm design can also be useful for understanding when and why people struggle to choose between multiple means to achieve multiple goals. (shrink)

The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations 〈A i 〉, 〈B〉, 〈A i B〉, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation (...) 〈A 1 A 2〉 associated to different choices for B. The same analysis applies to the Hardy model and to the “perfect correlations” discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality. (shrink)

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148–155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83–127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207–219, 1987) are two natural ways of generalizing “context-free” grammar formalisms for string and (...) tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191–229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15–29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars. (shrink)

PREFACEProf. Göran Sundholm of Leiden University inspired the group of Logic at Lille and Valparaíso to start a fundamental review of the dialogical conception of logic by linking it to constructive type logic. One of Sundholm's insights was that inference can be seen as involving an implicit interlocutor. This led to several investigations aimed at exploring the consequences of joining winning strategies to the proof-theoretical conception of meaning. The leading idea is, roughly, that while introduction rules lay down the conditions (...) under which a winning strategy for the Proponent may be built, the elimination rules lay down those elements of the Opponent's assertions that the Proponent has the right to use for building winning strategy. It is the pragmatic and ethical features of obligations and rights that naturally lead to the dialogical interpretation of natural deduction.During the Visiting Professorship of Prof. Sundholm at Lille in (2012) the group of Lille started delving into the ways of implementing Per Martin-Löf's Constructive Type Theory with the dialogical perspective. In particular, Aarne Ranta's (1988) paper, the first publication on the subject, was read and discussed during Sundholm’s seminar. The discussions strongly suggested that the game-theoretical conception of quantifiers as deploying interdependent moves provide a natural link between CTT and dialogical logic. This idea triggered several publications by the group of Lille in collaboration with Nicolas Clerbout and Juan Redmond at the University of Valparaíso, including the publication of the book (2015) by Clerbout / Rahman that provides a systematic development of this way of linking CTT and the dialogical conception of logic. However the Clerbout / Rahman book was written from the CTT perspective on dialogical logic, rather than the other way round. The present book should provide the perspective from the other side of the dialogue between the Dialogical Framework and Constructive Type Theory. The main idea of our present study is that Sundholm's (1997) notion of epistemic assumption is closely linked to the Copy-Cat Rule or Socratic Rule that distinguishes the dialogical framework from other game-theoretical approaches. One way to read the present book is as a further development of Sundholm’s extension of Austin's (1946, p. 171) remark on acts of assertion to inference. Indeed, Sundholm (2013, p. 17) gave the following forceful formulation: When I say therefore, I give others my authority for asserting the conclusion, given theirs for asserting the premisses.Per Martin-Löf, in recent lectures, have utilized the dialogical perspective on epistemic assumptions to get out of a certain circle that threatens the explanation of the notions of inference and demonstration. A demonstration may be explained as a chain of (immediate) inferences starting from no premisses. That an inference J1... Jn—————Jis valid means that the conclusion J can be made evident on the assumption that J1, …, Jn are known. The notion of epistemic assumption thus enters in the explanation of valid inference. We cannot, however, in this explanation understand 'known' in the sense of demonstrated, for then we are explaining the notion of inference in terms of demonstration, whereas demonstration has been explained in terms of inference. Martin-Löf suggests that we here understand 'known' in the sense of asserted, so that epistemic assumptions are judgements others have made, judgements for which others have taken the responsibility; that the inference is valid then means that, given that others have taken responsbility for the premisses, I can take responsibility for the conclusion:The circularity problem is this: if you define a demonstration to be a chain of immediate inferences, then you are defining demonstration in terms of inference. Now we are considering an immediate inference and we are trying to give a proper explanation of that; but, if that begins by saying: Assume that J1, …, Jn have been demonstrated – then you are clearly in trouble, because you are about to explain demonstration in terms of the notion of immediate inference, hence when you are giving an account of the notion of immediate inference, the notion of demonstration is not yet at your disposal. So, to say: Assume that J1, …, Jn have already been demonstrated, makes you accusable of trying to explain things in a circle. The solution to this circularity problem, it seems to me now, comes naturally out of this dialogical analysis. […]The solution is that the premisses here should not be assumed to be known in the qualified sense, that is, to be demonstrated, but we should simply assume that they have been asserted, which is to say that others have taken responsibility for them, and then the question for me is whether I can take responsibility for the conclusion. So, the assumption is merely that they have been asserted, not that they have been demonstrated. That seems to me to be the appropriate definition of epistemic assumption in Sundholm's sense. One of the main tenets of the present study is that the further move of relating the Socratic Rule rule with judgemental equality provides both a simpler and more direct way to implement the constructive type theoretical approach within the dialogical framework. Such a reconsideration of the Socratic Rule, roughly speaking, amounts to the following. 1.A move from P that brings-forward a play-object in order to defend an elementary proposition A can be challenged by O. 2.The answer to such a challenge, involves P bringing forward a definitional equality that expresses the fact that the play-object chosen by P copies the one O has chosen while positing A. For short, the equality expresses at the object-languaje level that the defence of P relies on the authority given to O to be able to produce the play-objects she brings forward. More generally, according to this view, a definitional equality established by P and brought forward while defending the proposition A, expresses the equality between a play-object (introduced by O) and the instruction for building a play-object deployed by O while affirming A. So it can be read as a computation rule that indicates how to compute O's instructions. Let us recall that from the strategic point of view, O's moves correspond to elimination rules of demonstrations. Thus, the dialogical rules that prescribe how to introduce a definitional equality – correspond – at the strategic level – to the definitional equality rules for CTT as applied to the selector-functions involved in the elimination rules. So, what we are doing here is extending the dialogical interpretation of Sundholm's epistemic assumption to the rules that set up the definitional equality of a type.The present work is structured in the following way:•In the Introduction we state the main tenets that guide our book•Chapter II contains a brief overview of Constructive Type Theory penned by Ansten Klev with exercises and their solutions. •Chapter III offers a novel presentation of the dialogical conception of logic. In fact, the first two sections of the chapter provide an overview of standard dialogical logic – including solved exercises, slightly adapted to the content of the present volume. The rest of the sections develop a new dialogical approach to Constructive Type Theory. •Chapters IV and V contain the main body of our study, namely the relation between the Socratic Rule, epistemic assumptions, and equality. The chapter provides the reader with thoroughly worked out examples with comments. •After the Final Remarks the book containsa.An appendix containing a an overview of all the rules for the new formulation of dialogical approach to constructive type theory developed in the book; b.a brief outline of Per Martin-Löf's informal presentation of the demonstration of the axiom of choice.c.two examples of a tree-shaped graph of an extensive strategy. Acknowledgments: Many thanks to Mark van Atten (Paris1), Giuliano Bacigalupo (Zürich), Charles Zacharie Bowao (Univ. Ma Ngouabi de Brazzaville, Congo), Christian Berner (Lille3), Michel Crubellier (Lille3), Pierre Cardascia (Lille3), Oumar Dia (Dakar), Marcel Nguimbi ((Univ. Marien Ngouabi de Brazzaville, Congo), Adjoua Bernadette Dango (Lille3 / Alassane Ouattara de Bouaké, Côte d'Ivoire), Steephen Rossy Eckoubili (Lille3), Johan Georg Granström (Zürich), Gerhard Heinzmann (Nancy2), Muhammad Iqbal (Lille3), Matthieu Fontaine (Lille3), Radmila Jovanovic (Belgrad), Hanna Karpenko (Lille3), Laurent Keiff (Lille3), Sébastien Magnier (Saint Dennis, Réunion), Clément Lion (Lille3), Gildas Nzokou (Libreville), Fachrur Rozie (Lille3), Helge Rückert (Mannheim), Mohammad Shafiei (Paris1), and Hassan Tahiri (Lisbon), for rich interchanges, suggestions on the main claims of the present study and proof-reading of specific sections of the text. Special thanks to Gildas Nzokou (Libreville), Steephen Rossy Eckoubili (Lille3) and Clément Lion (Lille3), with whom a Pre-Graduate Textbook in French on dialogical logic and CTT is in the process of being worked out – during the visiting-professorship in 2016 of the former. Eckoubili is the author of the sections on exercises for dialogical logic, Nzokou cared of those involving the development of CTT-demonstrations out of winning-strategies. Special thanks also to Ansten Klev (Prague) for helping with technical details and conceptual issues on CTT, particularly so as author of the chapter introducing CTT. (shrink)

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. (...) Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (shrink)

Goldman (1971) analyzed interrelations between act-statements by inducing a structure by means of the relationship by, e.g.: "He turned on the light by flipping the switch." Generally, the structure is represented by act-diagrams, e.g. act-trees. In the present article the mathematical theory of directed graphs (digraphs), specifically the concepts of partially or strictly ordered sets, graph-theoretical trees, semi-lattices etc. are shown to be applicable and conducive to the formal and a more general description of networks of (...) act statements generated by a (relative) basic actionstatement and by the relation by. The well-known problem of identity of acts described by corresponding statements connected by by is differentiated by introducing the graph-theoretical equivalence relation of belonging to the same-graph {graph-sameness or graph-identity) admitting of a more refined classification and logical description of the interdependence of actions, acttypes, act-properties etc. (shrink)

Compared to existing classical approaches to semiotics which are dyadic (signifier/signified, F. de Saussure) and triadic (symbol/concept/object, Ch. S. Peirce), this theory can be characterized as tetradic ([sign/semion]//[object/noema]) and is the result of either doubling the dyadic approach along the semiotic/ordinary dimension or splitting the ‘concept’ of the triadic one into two (semiotic/ordinary). Other important features of this approach are (a) the distinction made between concepts (only functional pairs of extent and intent) and categories (as representations of expressions) and (...) (b) the indication of the need for providing the mathematical passage from the duality between two sets (where one is a singleton) within systems of sets to category-theoretical monoids within systems of categories while waiting for the solution of this problem in the field of logic. Last but not least, human language expressions are the most representative physical instances of semiotic objects. Moreover, as computational experiments which are possible with linguistic objects present a high degree of systematicity (of oppositions), in general, it is relatively easy to elucidate their dependence on the concepts underlying signs. This new semiotic theory or rather this new research program emerged as the fruit of experimentation and reflection on the application of data science tools elaborated within the frameworks of Rough Set Theory (RST), Formal Context Analysis (FCA) and, though only theoretically, Distributed Information Logic (DIL). The semiotic objects (s-objects) of this theory can be described in tabular datasets. Nevertheless, at this stage of formalisation of the theory, lattices (not trees) can be used as working representation structures for characterizing the components of concept systems and graphs for categories of each layer. (shrink)

This new volume on logic follows a recognizable format that deals in turn with the topics of mathematical logic, moving from concepts, via definitions and inferences, to theories and axioms. However, this fresh work offers a key innovation in its ‘pyramidal’ graph system for the logical formalization of all these items. The author has developed this new methodology on the basis of original research, traditional logical instruments such as Porphyrian trees, and modern concepts of classification, in which pyramids (...) are the central organizing concept. The pyramidal schema enables both the content of concepts and the relations between the concept positions in the pyramid to be read off from the graph. Logical connectors are analyzed in terms of the direction in which they connect within the pyramid. Additionally, the author shows that logical connectors are of fundamentally different types: only one sort generates propositions with truth values, while the other yields conceptual expressions or complex concepts. On this basis, strong arguments are developed against adopting the non-discriminating connector definitions implicit in Wittgensteinian truth-value tables. Special consideration is given to mathematical connectors so as to illuminate the formation of concepts in the natural sciences. To show what the pyramidal method can contribute to science, a pyramid of the number concepts prevalent in mathematics is constructed. The book also counters the logical dogma of ‘false’ contradictory propositions and sheds new light on the logical characteristics of probable propositions, as well as on syllogistic and other inferences. (shrink)

continent. 2.1 (2012): 22–28. Jeroen Mettes burst onto the Dutch poetry scene twice. First, in 2005, when he became a strong presence on the nascent Dutch poetry blogosphere overnight as he embarked on his critical project Dichtersalfabet (Poet’s Alphabet). And again in 2011, when to great critical acclaim (and some bafflement) his complete writings were published – almost five years after his far too early death. 2005 was the year in which Dutch poetry blogging exploded. That year saw the foundation (...) of the influential, polemical, and populistically inclined weblog De Contrabas (The Double Bass), which became a strong force for internet poetry in Dutch in the years to follow. In the summer of that year, a lively debate raged in the aftermath of Bas Belleman’s article “ Doet poëzie er nu eindelijk toe ?” (“Does poetry finally matter now?”), on a blog specifically devoted to this question. Up to that point, the poetical debate in the Netherlands had largely been confined to literary reviews (which were often subsidized), having become mostly marginalized in more mainstream media, where poetry could be covered by only a small number of so-called authorities. As a result, literary debate had acquired a rather placid quality. Though a variety of camps with different aesthetics could be discerned, most poetical positions shared a general acceptance of poetry as a form of art somewhat apart from fundamental political concerns. Late modernists would pursue subtlety and density of reference. Others would insist poetry was best understood as a form of entertainment that should ideally be accessible and work well on the stage. Still others would insist that poetry is mostly a play with forms. Linguistically disruptive strategies were valued highly by some, but mostly for their aesthetic effect. Values of disinterested playfulness reigned supreme everywhere. Any idea that poetry could be a field in which one confronts politics and the world was decidedly marginal. This led to a climate in which most attempts at polemics were DOA, often based on far too superficial positioning and analysis. The greatest polemical debates were revolving around the question of whether poetry should be difficult or easy, with both camps defining their ideas of difficulty and accessibility in ways that were so utterly shallow as to make the entire point moot. Debates were performed, rather than engaged with. It was a postmodern hell of underarticulated poetics. Half-consciously, people were yearning for new forms of criticism that could put the oomph back into poetry. Weblogs provided for ways to explore debate directly outside of the clotted older channels of the reviews and the newspapers. Belleman’s essay and the resulting online activity had shown that there was a widespread eagerness to take poetry more seriously as a social art form. It was in this environment that Mettes started his remarkable project Dichtersalfabet . At that moment, Mettes was active mostly in academic circles, having become noted at Leiden University as a particularly gifted student of literary theory. Within the Netherlands, the field of literary theory has a very odd relationship to literature as it is practiced in the country. Academic theory tends to have a mostly international view and engage with international debates of cultural criticism, literary theory, and philosophy, with academics often publishing in English and attending conferences around the world. Literature itself however is much more concerned with domestic traditions. Consequently, in the Netherlands, there exists a language gap between academic theoretical practice (as it is studied in the literary theory departments) and literary practice (which, academically, gets studied in specialized departments of Dutch literature). The Dichtersalfabet can be seen as Mettes’s attempt to close this gap. It is also an attempt to bridge the divide between theory and practice, in which he could apply his theoretical knowledge in a very unorthodox and unacademic critical mode that moreover could reach far beyond the domain of conventional criticism. Mettes’s goal was to trace a diagonal through Dutch poetic culture, to “strangle” what he perceived to be its dominant oppressive traditions of agreeable irrelevance, in order to see whatever might be able to survive his critical assaults. But he could only do so by means of a very serious engagement with poetry itself. To this end, he would go systematically through the poetry bookshelf of the Verwijs bookshop (part of a mainstream chain of booksellers) in The Hague, buying one publication per blog item, starting from A and working his way through the alphabet, reading whatever he might encounter that way in the restaurant of the HEMA store (another big commercial chain in the country). He would subsequently write down his reading experiences, refraining however from trying to write a nuanced book review. Rather, he would write about anything that caught his attention and sparked his critical interest. This way of working would yield vast, at times somewhat rambling, dense, lively, and generally brilliant essays, in which he held no punches. He never hesitated to pull out his entire arsenal of concepts from the international theory traditions, while never degenerating into mere academic exercise and pointless intertextualities. The attempt was rather to live the poetry that he read, and to engage it with the full range of political, academic, cultural, and personal references that he had at his disposal—all that composed the individual named Jeroen Mettes as a reader. Often what he wrote would not be according to the standards of what we usually think of as a critical review of a book of poetry. Sometimes he would even be a little sloppy in his judgments of poets or representations of the books he read, for example by basing an entire essay on the blurb of a book rather than its poetry content. But what he did was always brilliant writing nonetheless—virtuoso riffs on poetic fragments randomly found within capitalist society, exposing an incisive and insistent poetical sensibility. Mettes read poetry for political reasons, to see whether poetry could offer him a way to deal with a political world he detested. The right-wing horrors of the Bush years, the Iraq war, and the turn of Dutch public opinion towards ever more conservative, narrow-minded, and xenophobic views alongside a complete failure of the political left to present any credible alternative, were weighing heavily on the times in which Mettes reported on his reading. Poetry was to measure this world, diagram it, to lay bare its inconsistencies and faults, to indicate where lines of flight might be found. Amid the ruins of a world wrecked by imperialist policies, corporate capitalism, and doctrinal neoliberalism it would have to show the possibility of a new community. And it was, through its rhythmical workings, to release the reading subject from his confinement to ideologically conditioned individuality and lead him into the immanent paradise of reading. The stakes were high. Much higher than anything Dutch poetry had seen for many years. Mettes’s blog was widely read from the start. His posts sparked lively debates. Some of these subsequently led to the publication of extensive essays on a few key poets in some literary journals, particularly Parmentier and the Flemish journal yang , for which Mettes would become a member of the editorial board, a few months after starting the Dichtersalfabet . This could have been the start of a brilliant career, but this was not to be. The initial manic energy that fueled the blog gradually subsided. The Alfabet was updated less and less regularly. Mettes sometimes just disappeared for many weeks, then suddenly returning with a brilliant essay. Until, on September 21, 2006, he posted his final blogpost, consisting of no text whatsoever. That night I learned from his mentor at Leiden University that he had committed suicide. Mettes and I had had some fruitful exchanges on poetry, rhythm, music, and form, mostly on the blogs, but also by email. Three weeks before his death was the last time I heard from him: a very sudden, uncharacteristically curt note saying “My old new sentence epic.” Attached to that message I found a DOC-file of a work so major that I felt intimidated. This was N30 , a text he had been working on for over five years. After his death, it took me a long time before I dared to read it in its entirety. In the meantime, the work of preparing the manuscript for publication was entrusted by his relatives to his colleagues at yang magazine. It took them a few years to brush up the text and to edit the Dichtersalfabet -blog (which, apart from the Alphabet project itself, incorporated many other fragments of political, polemical, and theoretical writing) into book form along with the essays. The result of this labor was finally published in 2011 as a two-book set, and Mettes burst onto the Dutch poetry scene for the second time. The work was widely reviewed, on blogs, in journals, magazines, and newspapers. Many critics who had not followed the blogs in 2005 showed themselves surprised, baffled even, by the intensity of Mettes’s critical writing. But for those who had read the blog, the main surprise was in the poetry. During Mettes’s lifetime, some of his poems had already been published in Parmentier . Although these were strong texts by themselves, in no way did they prepare readers for N30 . Nothing like it had been written in Dutch before. Instead, N30 explicitly follows the American tradition of Language Writing, directly referencing Ron Silliman and his concept of The New Sentence. However, it would seem that much of the poetical thinking around his use of this technique puts him closer to a writer such as Bruce Andrews. For Mettes, using non sequiturs as a unit of poetic construction was not only a way of reinventing formal textual construction, but it was another way of finding the fault lines in the social fabric. From the perspective of the Language tradition, one may put N30 somewhere between Silliman and Andrews. N30 shares an autobiographical element with Silliman’s New Sentence projects, and as in Andrews, there is a concern for mapping out social totality within text—what Mettes refers to as a “textual world civil war.” Again this shows a formal textual strategy for allowing the person “Jeroen Mettes” to be absorbed by the world, which here appears as a whirlwind of demotic and demonic chatter, full of violence, humor, intensity, beauty, disgust, sex, commerce, and strife. Influenced as it may by American precursors, Mettes’s tone and form end up quite different from his American counterparts, consistently referencing a world that is Dutch, all too Dutch, taking on the oppressive orderliness of Dutch society with its endemic penchant for consensus by introducing chaos into its daily life and laying bare its implicit aggressions. The work’s 31 chapters each have a different feel and rhythmical outline, but none of them follow a predetermined pattern. Rather, Mettes would consistently edit and reedit the text, randomly rewriting parts of it, as he explains in his poetical creed Politieke Poëzie (Political Poetry). N30 – referring to the 1999 antiglobalist protest in Seattle – was to be the first text of a trilogy. The work itself was written “in the mode of the present.” A second text was to be written in the mode of the future, and a third one, in the mode of the past, was going to be an epic poem about the Paris Commune, and to form an alternative poetic constitution for the European project. I still deeply regret that Jeroen Mettes never got to complete those projects, just as I would be very keen on knowing what he might have had to say about more recent political developments. Instead, in 2006, he remained stuck in the horrors of the present, that ended up consuming him completely. He left Dutch literature with some of its most piercing criticism and its most profoundly moving, exciting and powerful poetry. Excerpts from N30 Translated by Vincent W.J. van Gerven Oei from Jeroen Mettes. "N30." In N30+ . Amsterdam: De wereldbibliotheek, 2011. Published with permission of Uitgeverij Wereldbibliotheek, Amsterdam. Chapter 1 1999. A day is a space too. And another man, who had chained himself, had his ribs crushed, and a motor has driven over somebody’s legs. Dutch health care system spends ±145 million guilders per year on worriers. A spiderweb vibrates as I pass by. Randstad renovating. She slaps her bag against her ass: “Hurry up!” OPINION IS TRUE FRIENDSHIP Your skin. It doesn’t express anything. “But the use of the sword, that’s what I learned, and you’ll need nothing more for the moment.” Just try to interrogate a guy like that. Gullit in Sierra Leone. Codes silently lying all around. But that’s simply what belongs to “that it’s just allowed”: that sigh of “world” (a word expressing that the trees are now standing along the water like black men with white bags in their hair); that’s nothing else right? And you see how everything has to move, and first of all what cannot do so. Without Elysium and without savings, barbarians lashing out, horny for an enemy, staring across the water, staring into the air—staring to get out of it. “You’ve never showed me more than the mall,” she said. All those “dreams” in the end—and now? It was lying on the stairway, so I picked it up and took it upstairs. Chapter 3 “You know what?” Telecommunication. For love… I don’t really like that cheap cultural pessimism, but… The holy city is on pilgrimage in the earthly bodies of the faithful until the time of the heavenly kingdom has come. The end of an exhausting autumn day behind the computer, my eyes filled with tears of fatigue. KITCHEN / INSTALLATION / SPECIALIST. Network integration. In the sun, stretched out on a sheet. (…) I don’t believe what I’m reading, because I want to believe something else. An illusion? Suits me. There’s a variety of shapes and tastes… “So what?” you may think. 102 dalmatians can’t be wrong. But I want more, dear… A feel good movie. I’m smashing the burned body. So what? We continue to save the European civilization. What’s there to win? Plato with poets = Stalin without gulag? Ball against the crossbar. No wonder. She comes straight to her point. She’s standing in the kitchen eating an apple. (…) The godless Napoleon had used her as a stable and wanted to have her taken down. “Our” Rutger Hauer. Ready or not here I come. Psst… are you also wearing a string? Nobody understands our desire. Cliffs breaking the waves and shattering the sunset. I used to be a real romantic (as a poet). A typical fantasy used to be the one in which I brutally raped mother and daughter Seaver from the sitcom Growing Pains . Nevertheless you only contain bad words. Eyelashes. Automatic or manual? That your skin always in the afternoon. Integration. The air is empty. Too bad! Hand in hand on their lonely way. Alaska! Chapter 12 May 5, 2001 [10:00-10:30] A dust cloud on a hill. Globe. Indian (British) (tie) / pope. Damascus. Rape. We’re carrying the ayatollah’s portrait through the streets. At the moment the girl is mostly suede jacket with white ribbons on her sleeves. A small explosion flares up/impact. Camouflage. Close up. We’re analyzing the situation. He’s dead right? Dead dead. Dead. Everything without, these, and only with the body. Indices signal death. Dollar bills are printed in factories. Holes. Light patch. Globe in a box. Microphone. What’s the situation? Grey impact on a green hill (field?). The water is blue. He has no lips. Interns on the background with skirts that are too long. This is an example of a sonnet. An Islamic woman pushes against the door of an electronics shop. Arrows (percentages (prices)). Is this what awaits the American? Touch screen interface. The word, an island, can only be a sign in that situation. We pull up a chair, join in on the fun. On the shelves only books about computers. One glance in the distance is enough to lighten up a luna park in the distance. She’s really desperate, especially when she laughs. Click. Ah. Next. And now it’s raining, but that’s ok. Yellow stains sliding over the south. Shallow caves light: clothes, boots, electrical equipment. 45. 22:10. Nothing gives you the right to eat more than people starving to death. The Hague. Slam dunk. Traffic light. Two H’s, one L (standing for the L (little prick)). We’re happy to say something. Clouds, small suns, temperatures, cities. The truth is never an excuse. Yellow. Yellow. Green. Yellow. Yellow. Yellow. Yellow. Green. Green. Yellow. Will you email me? Skeleton: “No.” Ex-nerds in brand-new and brightly red sport cars. $$$. I love. Shihab. Hooves in the sand. Skinny senior with over-sized sunglasses; old jockey (cap, trophy) smiling in slow motion. And there I am again, flashback, crying with my head in between my hands. Sometimes I’ve got the feeling that cannibals. Eyes: blue. Cancer. Why would I wait until tomorrow? Golden beams protruding from the lifted/lit earth. May 5, 2001 [11:30-12:45] You’ll remember this for the rest of your life. Graphs, diagrams. Bu$ine$$. Blue shirt, white collar, no neck (porn star). A name lights up. I’m hysteric. Will you join us? Letters falling in their words. Fingers set up a tent and start to dance. Young entrepreneurs from poor neighborhoods (read: black) guided by Microsoft. Kinda makes me happy, that sort of kitsch. A sense of exhaustion/impotence to see anything but the present. (…) Wouldn’t you like to? Orange explosion in an industrial zone. YOU’RE DOING THIS FOR AMNESTY INTERNATIONAL. Would you. A familiar face. Clouds and blossom. Sunflowers. Supermodels. Mountainous area in a rectangle: shades of brown, from dark to beige, more green toward the south. Tents and next to them (it’s all a blur) people. Plane. Stadium. Geometrical block of people. No, I ain’t crying. I don’t speak no more. I just want. Quote + photo. (Positive:) screaming crowd. Three-piece suit, seen from the back, before entering the arena. On the back: “Daddy abused me.” Oh, bummer. State of emergency has been declared and everyone has to cooperate. She’s cut her wrists. What we do know (…) is that there’s never been a unique word, an imperative name, nor will there ever be. [Click here for work that suits you.] Barefooted children are watching it (coherent pieces revelation of what’s lying below). Who knows how she’s changed during those two years. “Everything used to be better” + sigh. And here we are. An empty field of parquet. A city lying behind it. Explosion. Blue. A rain drop falling in my coffee. May 5, 2001 [14:30-15:30] A young Arafat on video speaking with raised finger. “I’m calling from my convertible.” Names on walls, victims, numbers… Tourists. Yellow. Yellow. “Your own child! Really, what kind of human are you?” I don’t want to hear it no more. A woman jumping out of the water in a yellow bikini against a background of fireworks and the Cheops pyramid. Thy sorrow shall become good fortune, thy complaints laudation. All planets will float and wander. Wo die Welt zum Bild wird, kommt das System (…) zur Herrschaft. It is something, but is it? May 5, 2001 [18:30-19:00] Iris. Leaves. NASDAQ. Open / and white and. For the one who’s doing nothing, just waiting. (…) NO DEFEAT is made entirely up of defeat -- since / the world it opens is always a place / formerly / unsuspected. October 2002. “Jeroen, I’m leaving for the cemetery, byeeee.” The rise of the middle class. My entire oeuvre is an ode to the. My entire head is a fight against the. God always demands what you cannot sacrifice. You may take that the easy way, but… “The state hasn’t made us, but we make the state” (Hitler). A stork exits the elevator. Skeletons of. Moscow. Helsinki. Palermo. Paris. Chapter 30 Like your paradises: nothing. United Desire, as only remaining superpower. And even though the sea is now calmer and the wind is blowing pleasantly in my face… Heart! Who determines whether a tradition is “alive”? The yellow leaf or the white branch? Mars. This sentence is a typical example. Most Dutch people are happy. No consolation. When I see a girl sitting at a table with a book, a notepad, a pen, a bottle of mineral water, her hand writing in the light—then I consider that one thing. “Presents,” “poetry,” “classics.” We are what we cannot make from ourselves. “Left”: mendicant orders, missionaries. Saint-Just: “A republic is founded on the destruction of its enemies.” She crosses the street with a banana peel between her fingers. (…) We chose our own wardens, torturers, it was us who called all this insanity upon ourselves, we created this nightmare… But “no”? Girl (just like a beach ball) talking rapid Spanish (Portuguese?) in a mobile phone. Do I have a chance now that her boyfriend is getting bold? CLIO, horny bitch. What else do you want? An old woman, between the doors of the C1000, is suddenly unable to go on; her husband stretches out his hand, speaking a few encouraging words. Selection from. Der Führer schenkt den Jüden ein Stadt. How can it reach us if we haven’t been already reached somehow? It doesn’t “speak.” No problem. Each word she uses is a small miracle, as if she doesn’t belong to it, to language, but wanders around with a pocket light looking for the exit; she’s never desperate (maybe a little nervous), lighting up heavy words from the inside. But indeed, we’re free. But the predicate is not an attribute, but an event, and the subject is not a subject, but a shell. That’s why also samurai, knights, and warriors raised the blossom as emblem: they knew how to die. Locked up in a baby carriage with a McDonald’s balloon. Blue helicopter, the blue sky. Whether you want to refer? The point is. How / Motherfucker can I sing a sad song / When I remember Zion? You’ll feel so miserable and worthless that you think: “If only I were dead!,” or: “Just put an end to it!” “So you’re an economist?” Her card—two little birds building a nest, her handwriting shaking—is still on the mantelpiece. Guevara: “No, a communist.” A straw fire, such was our life: rapidly it flared up, rapidly it passed. I’m fleeing, coming from nowhere. (…) Eazy-E drinking coffee with the American president. If I’d scream, would that be an event? Drown it: the cleaner it will rise up from the depths. No! The night, so fast… As if there’s something opened up in that face. Come on, we may not curse life. He shows me his methadone: “If you drink that all at once, you’ll die instantly.” The last one dictates how we should behave to deserve happiness. One shine / above the earth. “I want to go to Bosnia,” I said bluntly. I don’t even know the name of the current mayor. Let’s despise our success! “There is no future; this is the future. Hope is a weakness that we've overcome. We have found happiness!” Sun. Sushi. Volvo. I feel like a bomb about to explode at any moment. Makes a difference for the reconstruction right? The decor moves forward. Daughter of Nereus, you nymphs of the sea, and you Thetis, you should have kept his tired head above the waves! Alas! This sentence has been written wearing a green cap. I receive my orders from the future. A frog jumps into it. Her husband has turned the Intifada, which he follows daily on CCN, into his hobby, “to forget that he doesn’t have his driver’s license yet." Suddenly the sun slides over the crosswalk. Her (his?) foot is playing with the slipper under the table. Is this how I’m writing this book now? I’m not a fellow man. I hate you and I want to hurt you. These are my people. Their screaming doesn’t rise above the constantly wailing sirens which we've learned to ignore. My whole body became warm and suddenly started to tremble. Unfortunate is he who is standing on the threshold of the most beautiful time, but awaits a better one. Arafat’s “removal” is contrary to American interests. Jeep drives into boy. What you can do alone, you should do alone. A food gift from the people of the United States of America. Two seagulls. [...]. (shrink)

Goldman (1971) analyzed interrelations between act-statements by inducing a structure by means of the relationship by, e.g.: "He turned on the light by flipping the switch." Generally, the structure is represented by act-diagrams, e.g. act-trees. In the present article the mathematical theory of directed graphs (digraphs), specifically the concepts of partially or strictly ordered sets, graph-theoretical trees, semi-lattices etc. are shown to be applicable and conducive to the formal and a more general description of networks of (...) act statements generated by a (relative) basic actionstatement and by the relation by. The well-known problem of identity of acts described by corresponding statements connected by by is differentiated by introducing the graph-theoretical equivalence relation of belonging to the same-graph {graph-sameness or graph-identity) admitting of a more refined classification and logical description of the interdependence of actions, acttypes, act-properties etc. (shrink)